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data/en.wikipedia.org/wiki/A_History_of_Vector_Analysis-0.md

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+---
+title: "A History of Vector Analysis"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/A_History_of_Vector_Analysis"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:13.416934+00:00"
+instance: "kb-cron"
+---
+
+A History of Vector Analysis (1967) is a book on the history of vector analysis by Michael J. Crowe, originally published by the University of Notre Dame Press.
+As a scholarly treatment of a reformation in technical communication, the text is a contribution to the history of science. In 2002, Crowe gave a talk summarizing the book, including an entertaining introduction in which he covered its publication history and related the award of a Jean Scott prize of $4000.  Crowe had entered the book in a competition for "a study on the history of complex and hypercomplex numbers" twenty-five years after his book was first published.
+
+
+== Summary of book ==
+The book has eight chapters: the first on the origins of vector analysis including Ancient Greek and 16th and 17th century influences; the second on the 19th century William Rowan Hamilton and quaternions; the third on other 19th and 18th century vectorial systems including equipollence of Giusto Bellavitis and the exterior algebra of Hermann Grassmann.
+Chapter four is on the general interest in the 19th century on vectorial systems including analysis of journal publications as well as sections on major figures and their views (e.g., Peter Guthrie Tait as an advocate of Quaternions and James Clerk Maxwell as a critic of Quaternions); the fifth chapter describes the development of the modern system of vector analysis by Josiah Willard Gibbs and Oliver Heaviside.
+In chapter six, "Struggle for existence",
+Michael J. Crowe delves into the zeitgeist that pruned down quaternion theory into vector analysis on three-dimensional space. He makes clear the ambition of this effort by considering five major texts as well as a couple dozen articles authored by participants in "The Great Vector Debate". These are the books:
+
+Elementary Treatise on Quaternions (1890) Peter Guthrie Tait
+Elements of Vector Analysis (1881,1884) Josiah Willard Gibbs
+Electromagnetic Theory (1893,1899,1912) Oliver Heaviside
+Utility of Quaternions in Physics (1893) Alexander McAulay
+Vector Analysis and Quaternions (1906) Alexander Macfarlane
+Twenty of the ancillary articles appeared in Nature; others were in Philosophical Magazine, London or Edinburgh Proceedings of the Royal Society, Physical Review, and Proceedings of the American Association for the Advancement of Science. The authors included Cargill Gilston Knott and a half-dozen other hands.
+The "struggle for existence" is a phrase from Charles Darwin’s Origin of Species and Crowe quotes Darwin:  "...young and rising naturalists,...will be able to view both sides of the question with impartiality." After 1901 with the Gibbs/Wilson/Yale publication Vector Analysis, the question was decided in favour of the vectorialists with separate dot and cross products. The pragmatic temper of the times set aside the four-dimensional source of vector algebra.
+Crowe's chapter seven is a survey of "Twelve major publications in Vector Analysis from 1894 to 1910". Of these twelve, seven are in German, two in Italian, one in Russian, and two in English. Whereas the previous chapter examined a debate in English, the final chapter notes the influence of Heinrich Hertz' results with radio and the rush of German research using vectors. Joseph George Coffin of MIT and Clark University published his Vector Analysis in 1909; it too leaned heavily into applications. Thus Crowe provides a context for Gibbs and Wilson’s famous textbook of 1901.
+The eighth chapter is the author's summary and conclusions. The book relies on references in chapter endnotes instead of a bibliography section.  Crowe also states that the Bibliography of the Quaternion Association, and its supplements to 1912, already listed all the primary literature for the study.
+
+
+== Summary of reviews ==
+There were significant reviews given near the time of original publication. Stanley Goldberg wrote "The polemics on both sides make very rich reading, especially when they are spiced with the sarcastic wit of a Heaviside, and the fervent, almost religious railing of a Tait." Morris Kline begins his 1969 review with "Since historical publications on modern developments are rare, this book is welcome." and ends with "the subtitle [,The Evolution of the Idea of a Vectorial System,] is a better description of the contents than the title proper."  Then William C. Waterhouse—picking up where Kline's review left off—writes in 1972 "Crowe's book on vector analysis seems a little anemic in comparison, perhaps because its title is misleading. ... [Crowe] does succeed in his goal of tracing the genealogy of the 3-space system, concluding that it was developed out of quaternions by physicists."
+Karin Reich wrote that Arnold Sommerfeld's name was missing from the book. As assistant to Felix Klein, Sommerfeld was assigned the project of unifying vector concepts and notations for Klein's encyclopedia.
+In 2003 Sandro Caparrini challenged Crowe’s conclusions by noting that "geometrical representations of forces and velocities by means of directed line segments...was already fairly well known by the middle of the eighteenth century" in his essay "Early Theories of Vectors". Caparrini cites several sources, in particular Gaetano Giorgini (1795 — 1874) and his appreciation in an 1830 article by Michel Chasles. Caparrini goes on to indicate that moments of forces and angular velocities were recognized as vectorial entities in the second half of the eighteenth century.
+
+
+== See also ==
+History of quaternions
+Hypercomplex number
+Vector space
+
+
+== Notes and references ==
+
+
+== External links ==
+A History of Vector Analysis from Goodreads

data/en.wikipedia.org/wiki/A_Mathematical_Theory_of_Communication-0.md

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 source: "https://en.wikipedia.org/wiki/A_Mathematical_Theory_of_Communication"
 category: "reference"
 tags: "science, encyclopedia"
-date_saved: "2026-05-05T08:36:35.522445+00:00"
+date_saved: "2026-05-05T08:45:40.516645+00:00"
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data/en.wikipedia.org/wiki/A_Metric_America-0.md

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+---
+title: "A Metric America"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/A_Metric_America"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:50.989387+00:00"
+instance: "kb-cron"
+---
+
+A Metric America: A Decision Whose Time Has Come was a 1971 book by the United States National Bureau of Standards (now the National Institute of Standards and Technology) printed by the Government Printing Office.
+In 1968, in the Metric Study Act (Pub. L. 90-472, August 9, 1968, 82 Stat. 693), Congress authorized a three-year study of systems of measurement in the U.S., with particular emphasis on the feasibility of adopting the SI. This detailed U.S. Metric Study was conducted by the Department of Commerce. A 45-member advisory panel consulted with and took testimony from hundreds of consumers, business organizations, labor groups, manufacturers, and state and local officials.
+A Metric America: "A Decision Whose Time Has Come" – For Real (NISTIR 4858) was a June 1992 follow-up to this book.
+
+
+== See also ==
+Metrication in the United States
+
+
+== External links ==
+A Metric America; A Decision Whose Time Has Come
+A Metric America: A Decision Whose Time Has Come - For Real (NISTIR 4858) at the Wayback Machine (archived November 12, 2011)

data/en.wikipedia.org/wiki/A_New_Era_of_Thought-0.md

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+---
+title: "A New Era of Thought"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/A_New_Era_of_Thought"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:55.623559+00:00"
+instance: "kb-cron"
+---
+
+A New Era of Thought is a non-fiction work written by Charles Howard Hinton, published in 1888 and reprinted in 1900 by Swan Sonnenschein & Co. Ltd., London. A New Era of Thought is about four-dimensional space and its implications on human thinking. The preface was written by Alicia Boole and H. John Falk. They also rewrote Part II which Hinton had sketched. The book has xvi and 230 pages.
+
+
+== Context ==
+A New Era of Thought is inspired by Plato's allegory of the cave and is influenced by the works of Immanuel Kant, Carl Friedrich Gauss and Nikolai Lobachevsky. It influenced the work of P.D. Ouspensky, particularly his book Tertium Organum where it is frequently quoted; Scientific American writer Martin Gardner, who mentioned this book in some of his articles; and Rudy Rucker's The Fourth Dimension.
+
+
+== Synopsis ==
+A New Era of Thought consists of two parts. The first part is a collection of philosophical and mathematical essays on the fourth dimension. These essays are somewhat disconnected. They teach the possibility of thinking four-dimensionally and about the religious and philosophical insights thus obtainable. In the second part Hinton develops a system of coloured cubes. These cubes serve as model to get a four-dimensional perception as a basis of four-dimensional thinking. This part describes how to visualize a tesseract by looking at several 3-D cross sections of it. The system of cubic models in A New Era of Thought is a forerunner of the cubic models in Hinton's book The Fourth Dimension.
+
+
+== Contents ==
+Preface
+Table of Contents
+Introductory Note to Part I
+Part I
+Introduction
+Chapter I.
+Scepticism and Science.
+Beginning of Knowledge.
+Chapter II.
+Apprehension of Nature.
+Intelligence.
+Study of Arrangement or Shape.
+Chapter III.
+The Elements of Knowledge.
+Chapter IV.
+Theory and Practice.
+Chapter V.
+Knowledge: Self-Elements.
+Chapter VI.
+Function of Mind.
+Space against Metaphysics.
+Self-Limitations and its Test.
+A Plane World.
+Chapter VII.
+Self Elements in our Consciousness.
+Chapter VIII.
+Relation of Lower and Higher Space.
+Theory of the Aether.
+Chapter IX.
+Another View of the Aether.
+Material and Aetherial Bodies.
+Chapter X.
+Higher Space and Higher Being.
+Perception and Inspiration.
+Chapter IX.
+Space the Scientific Basis of Altruism and Religion.
+Part II
+Chapter I.
+Three-space.
+Genesis of a Cube.
+Appearances of a Cube to a Plane-being.
+Chapter II.
+Further Appearances of a Cube to a Plane-being.
+Chapter III.
+Four-space.
+Genesis of a Tessaract; its Representation in Three-space.
+Chapter IV.
+Tessaract moving through Three-space.
+Models of the Sections.
+Chapter V.
+Representation of Three-space by Names and in a Plane.
+Chapter VI.
+The Means by which a Plane-being would Acquire a Conception of our Figures.
+Chapter VII.
+Four-space: its Representation in Three-space.
+Chapter VIII.
+Representation of Four-space by Name.
+Study of Tessaracts.
+Chapter IX.
+Further Study of Tessaracts.
+Chapter X.
+Cyclical Projections.
+Chapter XI.
+A Tessaractic Figure and its Projections.
+Appendices
+
+A. 100 Names used for Plane Space.
+B. 216 Names used for Cubic Space.
+C. 256 Names used for Tessaractic Space.
+D. List of Colours, Names and Symbols.
+E. A Theorem in Four-Space.
+F. Exercises on Shapes of Three Dimensions.
+G. Exercises on Shapes of Four Dimensions.
+H. Sections of the Tessaract.
+K. Drawings of the Cubic Sides and Sections of the Tessaract (Models 1–12) with Colours and Names.
+
+
+== Notes ==
+
+
+== External links ==
+Hinton's writings contains some abridged passages of the first part of A New Era of Thought, from ibiblio.
+A New Era of Thought (pdf) from Australian National Library
+A New Era of Thought from Google Books.

data/en.wikipedia.org/wiki/Extensions_of_First_Order_Logic-0.md

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+---
+title: "Extensions of First Order Logic"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Extensions_of_First_Order_Logic"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:39.252783+00:00"
+instance: "kb-cron"
+---
+
+Extensions of First Order Logic is a book on mathematical logic. It was written by María Manzano, and published in 1996 by the Cambridge University Press as volume 19 of their book series Cambridge Tracts in Theoretical Computer Science.
+
+
+== Topics ==
+The book concerns forms of logic that go beyond first-order logic, and in particular (following the work of Leon Henkin) the project of unifying them by translating all of these extensions into a specific form of logic, many-sorted logic. Beyond many-sorted logic, its topics include second-order logic (including its incompleteness and relation with Peano arithmetic), second-order arithmetic, type theory (in relational, functional, and equational forms), modal logic, and dynamic logic.
+It is organized into seven chapters. The first concerns second-order logic in its standard form, and it proves several foundational results for this logic. The second chapter introduces the sequent calculus, a method of making sound deductions in second-order logic, and its incompleteness. The third continues the topic of second-order logic, showing how to formulate Peano arithmetic in it, and using Gödel's first incompleteness theorem to provide a second proof of incompleteness of second-order logic. Chapter four formulates a non-standard semantics for second-order logic (from Henkin), in which quantification over relations is limited to only the definable relations. It defines this semantics in terms of "second-order frames" and "general structures", constructions that will be used to formulate second-order concepts within many-sorted logic. In the fifth chapter, the same concepts are used to give a non-standard semantics to type theory. After these chapters on other types of logic, the final two chapters introduce many-sorted logic, prove its soundness, completeness, and compactness, and describe how to translate the other forms of logic into it.
+
+
+== Audience and reception ==
+Although the book is intended as a textbook for advanced undergraduates or beginning graduate students, reviewer Mohamed Amer suggests that it does not have enough exercises to support a course in its subject, and that some of its proofs are lacking in detail. Reviewer Hans Jürgen Ohlbach suggests that it would be more usable as a reference than a textbook, and states that "it is certainly not suitable for undergraduates".
+Reviewer Yde Venema wonders how much of the logical power and useful properties of the various systems treated in this book have been lost in the translation to many-sorted logic, worries about the jump in computational complexity of automated theorem proving caused by the translation, complains about the book's clarity of exposition becoming lost in case analysis, and was disappointed at the lack of coverage of Montague grammar, fixed-point logic, and non-monotonic logic. Nevertheless, Venema recommends the book for courses introducing students to second-order and many-sorted logics, praising the book for its "overwhelming and catching enthusiasm". And reviewer B. Boričić calls it "nice and clearly written", "an appropriate introduction and reference", recommending it to researchers in several disciplines (mathematics, computer science, linguistics, and philosophy) where advanced forms of logic are important.
+
+
+== References ==
+
+
+== External links ==
+Extensions of First Order Logic at Cambridge University Press

data/en.wikipedia.org/wiki/Felix_Klein_and_Sophus_Lie-0.md

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+---
+title: "Felix Klein and Sophus Lie"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Felix_Klein_and_Sophus_Lie"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:40.437862+00:00"
+instance: "kb-cron"
+---
+
+Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century is a 1988 book by I. M. Yaglom, translated from the Russian into English by Sergei Sossinsky, on the history of the notion of symmetry and the mathematical works of Felix Klein and Sophus Lie besides other mathematicians, such as Camille Jordan, without focusing on biographical details, but on their ideas. It was published by Birkhäuser.
+
+
+== Editions ==
+The original Russian edition of the book was published in 1977 and translated into English by Sergei Sossinky and edited by Hardy Grant and Abe Shenitzer in 1988. In 2009, the book was republished by Ishi Press as Geometries, Groups and Algebras in the Nineteenth Century. The new edition, designed by Sam Sloan, has a foreword by Richard Bozulich.
+
+
+== Contents ==
+Felix Klein and Sophus Lie examines the evolution of mathematical ideas that converged in Klein and Lie's work. Yaglom based the work on his lectures to graduate-level students at Yaroslavl State University; Douglas Quadling described it as "a sharply-focused (though appropriately discursive) study" rather than "a catalogue of names and dates." The book begins with Camille Jordan, who discovers an unanswered letter from Galois among Cauchy's papers. This discovery leads Jordan to become the principal advocate of group theory. When Klein and Lie set off for Paris in 1870 to meet Jordan, their visit is cut short by the Franco-Prussian War, but it proves long enough to direct both mathematicians toward the applications of group theory that would define their careers.
+The book's central chapters trace three major developments in nineteenth-century geometry: projective geometry, non-Euclidean geometries, and multidimensional spaces. During this period, the scene was dominated by figures like Carl Friedrich Gauss, who attended a lecture by Bernhard Riemann on geometry at age 77, where the ideas presented were "so far ahead of their time that only Gauss could have understood them." Other prominent names include August Ferdinand Möbius, Jakob Steiner, János Bolyai, Nikolai Lobachevsky, Arthur Cayley, Hermann Grassmann, and William Rowan Hamilton.
+Alongside the main text runs an extensive apparatus of 312 footnotes that occupy nearly as much space as the main story. These notes provide additional mathematical detail, biographical information about minor contributors, and discussions of deeper philosophical issues.
+
+
+== Reception ==
+Douglas Quadling praised Felix Klein and Sophus Lie as "a work of considerable scholarship" that "tells a fascinating story in a style which consistently commands attention" and recommended it to mathematics teachers and advanced students. Quadling suggested that the book could serve as a natural sequel to E. T. Bell's popular Men of Mathematics.
+Ed Barbeau found the work both informative and frustrating. While acknowledging it as "a 'good read'" and "a useful source of succinct accounts of the principal developments of nineteenth century geometry," he was disappointed by the book's failure to maintain focus on its stated theme. Barbeau wrote that the thread of symmetry gets lost in "the wealth of material" and disappears for "large tracts of the book" only to be "quickly dissolved into a discussion of group theory" when it does appear.
+David Rowe delivered the harshest assessment, particularly regarding Yaglom's historical methodology. Rowe wrote that "many of the standard weaknesses found in historical studies undertaken by mathematicians," with interpretations that appear "based on a combination of folklore, conjecture, and superficial reading of popular (and sometimes notoriously unreliable) secondary work." Despite this, Rowe acknowledged the book's value "as a popular introduction to the historical role of symmetry in modern mathematics."
+The absence of an index was criticized by both Barbeau and Quadling.
+
+
+== Original version ==
+Феликс Клейн и Софус Ли. (1977).
+
+
+== See also ==
+Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert
+
+
+== References ==
+
+
+== External links ==
+Limited preview: Felix Klein and Sophus Lie via Internet Archive

data/en.wikipedia.org/wiki/Finding_Ellipses-0.md

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+---
+title: "Finding Ellipses"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Finding_Ellipses"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:42.735764+00:00"
+instance: "kb-cron"
+---
+
+Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, and the Numerical Range Know about Each Other is a mathematics book on "some surprising connections among complex analysis, geometry, and linear algebra", and on the connected ways that ellipses can arise from other subjects of study in all three of these fields. It was written by Ulrich Daepp, Pamela Gorkin, Andrew Shaffer, and Karl Voss, and published in 2019 by the American Mathematical Society and Mathematical Association of America as volume 34 of the Carus Mathematical Monographs, a series of books aimed at presenting technical topics in mathematics to a wide audience.
+
+
+== Topics ==
+Finding Ellipses studies a connection between Blaschke products, Poncelet's closure theorem, and the numerical range of matrices.
+A Blaschke product is a rational function that maps the unit disk in the complex plane to itself, and maps some given points within the disk to the origin. In the main case considered by the book, there are three distinct given points 
+  
+    
+      
+        0
+      
+    
+    {\displaystyle 0}
+  
+, 
+  
+    
+      
+        a
+      
+    
+    {\displaystyle a}
+  
+, and 
+  
+    
+      
+        b
+      
+    
+    {\displaystyle b}
+  
+, and their Blaschke product has the formula
+
+  
+    
+      
+        B
+        (
+        z
+        )
+        =
+        z
+        ⋅
+        
+          
+            
+              z
+              −
+              a
+            
+            
+              1
+              −
+              
+                
+                  
+                    a
+                    ¯
+                  
+                
+              
+              z
+            
+          
+        
+        ⋅
+        
+          
+            
+              z
+              −
+              b
+            
+            
+              1
+              −
+              
+                
+                  
+                    b
+                    ¯
+                  
+                
+              
+              z
+            
+          
+        
+        .
+      
+    
+    {\displaystyle B(z)=z\cdot {\frac {z-a}{1-{\bar {a}}z}}\cdot {\frac {z-b}{1-{\bar {b}}z}}.}
+  
+
+For this function, each point on the unit circle has three preimages, also on the unit circle. These triples of preimages form triangles inscribed in the unit circle, and (it turns out) they all circumscribe an ellipse with foci at 
+  
+    
+      
+        a
+      
+    
+    {\displaystyle a}
+  
+ and 
+  
+    
+      
+        b
+      
+    
+    {\displaystyle b}
+  
+. Thus, they form an infinite system of polygons inscribed in and circumscribing two conics, which is the kind of system that Poncelet's theorem describes. This theorem states that, whenever one polygon is inscribed in a conic and circumscribes another conic, it is part of an infinite family of polygons of the same type, one through each point of either conic. The family of triangles constructed from the Blaschke product is one of these infinite families of Poncelet's theorem.
+The third part of the connection surveyed by the book is the numerical range of a matrix, a region within which the eigenvalues of the matrix can be found. In the case of a 
+  
+    
+      
+        2
+        ×
+        2
+      
+    
+    {\displaystyle 2\times 2}
+  
+ complex matrix, the numerical range is an ellipse, by a result commonly called the elliptical range theorem, with the eigenvalues as its foci. For a certain matrix whose coefficients are derived from the two given points, and having these points on its diagonal, this ellipse is the one circumscribed by the triangles of Poncelet's theorem. More, the numerical range of any matrix is the intersection of the numerical ranges of its unitary dilations, which in this case are 
+  
+    
+      
+        3
+        ×
+        3
+      
+    
+    {\displaystyle 3\times 3}
+  
+ unitary matrices each having one of the triangles of Poncelet's theorem as its numerical range and the three vertices of the triangle as its eigenvalues.
+Finding Ellipses is arranged into three parts. The first part develops the mathematics of Blaschke products, Poncelet's closure theorem, and numerical ranges separately, before revealing the close connections between them. The second part of the book generalizes these ideas to higher-order Blaschke products, larger matrices, and Poncelet-like results for the corresponding numerical ranges, which generalize ellipses. These generalizations connect to more advanced topics in mathematics: "Lebesgue theory, Hardy spaces, functional analysis, operator theory and more". The third part consists of projects and exercises for students to develop this material beyond the exposition in the book. An online collection of web applets allow students to experiment with the constructions in the book.
+
+
+== Audience and reception ==
+Finding Ellipses is primarily aimed at advanced undergraduates in mathematics, although more as a jumping-off point for undergraduate research projects than as a textbook for courses. The first part of the book uses only standard undergraduate mathematics, but the second part is more demanding, and reviewer Bill Satzer writes that "even the best students might find themselves paging backward and forward in the book, feeling frustrated while trying to make connections". Despite that, Line Baribeau writes that it is "clear and engaging", and appealing in its use of modern topics. Yunus Zeytuncu is even more positive, calling it a "delight" that "realizes the dream" of bringing this combination of disciplines together into a neat package that is accessible to undergraduates.
+
+
+== References ==

data/en.wikipedia.org/wiki/Fondements_de_la_Géometrie_Algébrique-0.md

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+---
+title: "Fondements de la Géometrie Algébrique"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Fondements_de_la_Géometrie_Algébrique"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:44.112183+00:00"
+instance: "kb-cron"
+---
+
+Fondements de la Géometrie Algébrique (FGA) is a book that collected together seminar notes of Alexander Grothendieck. It is an 
+important source for his pioneering work on scheme theory, which laid foundations for algebraic geometry in its modern technical developments. 
+The title is a translation of the title of André Weil's book Foundations of Algebraic Geometry.
+It contained material on descent theory, and existence theorems including that for the Hilbert scheme. The Technique de descente et théorèmes d'existence en géometrie algébrique is one series of seminars within FGA.
+Like the bulk of Grothendieck's work of the IHÉS period, duplicated notes were circulated, but the publication was not as a conventional book.
+
+
+== Contents ==
+These are Séminaire Bourbaki notes, by number, from the years 1957 to 1962.
+
+Fondements de la géométrie algébrique. Commentaires [Séminaire Bourbaki, t. 14, 1961/62, Complément];
+Théorème de dualité pour les faisceaux algébriques cohérents [Séminaire Bourbaki, t. 9, 1956/57, no. 149]; (coherent duality)
+Géométrie formelle et géométrie algébrique [Séminaire Bourbaki, t. 11, 1958/59, no. 182]; (formal geometry)
+Technique de descente et théorèmes d'existence en géométrie algébrique. I-VI
+I. Généralités. Descente par morphismes fidèlement plats [Séminaire Bourbaki, t. 12, 1959/60, no. 190];
+II. Le théorème d'existence en théorie formelle des modules [Séminaire Bourbaki, t. 12, 1959/60, no. 195];
+III. Préschémas quotients [Séminaire Bourbaki, t. 13, 1960/61, no. 212];
+IV. Les schémas de Hilbert [Séminaire Bourbaki, t. 13, 1960/61, no. 221];
+V. Les schémas de Picard. Théorèmes d'existence [Séminaire Bourbaki, t. 14, 1961/62, no. 232];
+VI. Les schémas de Picard. Propriétés générales [Séminaire Bourbaki, t. 14, 1961/62, no. 236]
+
+
+== See also ==
+Éléments de géométrie algébrique
+Séminaire de Géométrie Algébrique du Bois Marie
+
+
+== References ==
+Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vistoli, Angelo (2005), Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3541-8, MR 2222646
+Fantechi, Barbara; Göttsche, Lothar (2005), "Local properties and Hilbert schemes of points", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 139–178, MR 2223408
+Grothendieck, Alexander (1962), Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957--1962.] (PDF), Paris: Secrétariat Mathématique, MR 0146040, archived from the original (PDF) on 2011-11-05, retrieved 2010-03-03
+Illusie, Luc (2005), "Grothendieck's existence theorem in formal geometry", Fundamental algebraic geometry (PDF), Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 179–233, MR 2223409, archived from the original (PDF) on 2006-12-08, retrieved 2006-09-27
+Kleiman, Steven L. (2005), "The Picard scheme", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 235–321, arXiv:math/0504020, Bibcode:2005math......4020K, MR 2223410
+Nitsure, Nitin (2005), "Construction of Hilbert and Quot schemes", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 105–137, arXiv:math/0504590, Bibcode:2005math......4590N, MR 2223407
+Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 1–104, arXiv:math/0412512, Bibcode:2004math.....12512V, MR 2223406
+SGA, EGA, FGA Archived 2022-07-04 at the Wayback Machine By Mateo Carmona

data/en.wikipedia.org/wiki/Formulario_mathematico-0.md

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 category: "reference"
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-date_saved: "2026-05-05T07:23:53.723865+00:00"
+date_saved: "2026-05-05T08:44:45.297151+00:00"
 instance: "kb-cron"
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data/en.wikipedia.org/wiki/Foundations_of_Algebraic_Geometry-0.md

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+---
+title: "Foundations of Algebraic Geometry"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Foundations_of_Algebraic_Geometry"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:46.432627+00:00"
+instance: "kb-cron"
+---
+
+Foundations of Algebraic Geometry is a book by André Weil (1946, 1962) that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory by defining the local intersection multiplicity of two subvarieties.
+Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over finite fields.
+Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective.  
+In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic geometry, all of which were superseded by Grothendieck's development of schemes.
+
+
+== See also ==
+Weil cohomology theory
+
+
+== References ==
+Raynaud, Michel (1999), "André Weil and the foundations of algebraic geometry" (PDF), Notices of the American Mathematical Society, 46 (8): 864–867, MR 1704257
+van der Waerden, Bartel Leendert (1971), "The foundation of algebraic geometry from Severi to André Weil", Archive for History of Exact Sciences, 7 (3): 171–180, doi:10.1007/BF00357215, MR 1554142, S2CID 189787203
+Weil, André (1947), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, vol. 29, Providence, R.I.: American Mathematical Society, MR 0023093 ISBN 9780821874622
+Weil, André (1962), Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications, vol. 29 (2 ed.), Providence, R.I.: American Mathematical Society, MR 0144898 ISBN 978-0-8218-1029-3
+Zariski, Oscar (1948), "Book Review: Foundations of algebraic geometry", Bulletin of the American Mathematical Society, 54 (7): 671–675, Bibcode:1948Sci...107...75W, doi:10.1090/S0002-9904-1948-09040-1, MR 1565074
+
+
+== External links ==
+Extracts from the preface of Foundations of Algebraic Geometry

data/en.wikipedia.org/wiki/Fundamenta_nova_theoriae_functionum_ellipticarum-0.md

@@ -0,0 +1,21 @@
+---
+title: "Fundamenta nova theoriae functionum ellipticarum"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Fundamenta_nova_theoriae_functionum_ellipticarum"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:49.966123+00:00"
+instance: "kb-cron"
+---
+
+Fundamenta nova theoriae functionum ellipticarum  (from Latin: New Foundations of the Theory of Elliptic Functions) is a treatise on elliptic functions by German mathematician Carl Gustav Jacob Jacobi. The book was first published in 1829, and has been reprinted in volume 1 of his collected works and on several later occasions. The book introduces Jacobi elliptic functions and the Jacobi triple product identity.
+
+
+== References ==
+Citations
+
+General
+Conway, John Horton (1980), "Monsters and moonshine", The Mathematical Intelligencer, 2 (4): 165–171, doi:10.1007/BF03028594, ISSN 0343-6993, MR 0600222, S2CID 121787388
+Cooke, Roger (2005), "Chapter 31 C. F. G. Jacobi, book on elliptic functions", in Grattan-Guinness, Ivor (ed.), Landmark writings in western mathematics 1640–1940, Elsevier B. V., Amsterdam, pp. 412–430, ISBN 978-0-444-50871-3, MR 2169816
+Jacobi, C. G. J. (1829), Fundamenta nova theoriae functionum ellipticarum (in Latin), Königsberg: Borntraeger, ISBN 978-1-108-05200-9, Reprinted by Cambridge University Press 2012 {{citation}}: ISBN / Date incompatibility (help)
+Jacobi, C. G. J. (1969) [1881], Gesammelte Werke, Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften, vol. I–VIII (2nd ed.), New York: Chelsea Publishing Co., MR 0260557, archived from the original on 2013-05-13, retrieved 2012-10-14

data/en.wikipedia.org/wiki/Games,_Puzzles,_and_Computation-0.md

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+---
+title: "Games, Puzzles, and Computation"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Games,_Puzzles,_and_Computation"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:51.099239+00:00"
+instance: "kb-cron"
+---
+
+Games, Puzzles, and Computation is a book on game complexity, written by Robert Hearn and Erik Demaine, and published in 2009 by A K Peters. It is revised from Hearn's doctoral dissertation, which was supervised by Demaine. The Basic Library List Committee of the Mathematical Association of America has recommended it for inclusion in undergraduate mathematics libraries.
+
+
+== Topics ==
+Games, Puzzles, and Computation concerns the computational complexity theory of solving logic puzzles and making optimal decisions in two-player and multi-player combinatorial games. Its focus is on games and puzzles that have seen real-world play, rather than ones that have been invented for a purely mathematical purpose. In this area it is common for puzzles and games such as sudoku, Rush Hour, reversi, and chess (in generalized forms with arbitrarily large boards) to be computationally difficult: sudoku is NP-complete, Rush Hour and reversi are PSPACE-complete, and chess is EXPTIME-complete. Beyond proving new results along these lines, the book aims to provide a unifying framework for proving such results, through the use of nondeterministic constraint logic, an abstract combinatorial problem that more closely resembles game play than the more classical problems previously used for completeness proofs.
+It is divided into three parts. The first part concerns constraint logic, which involving assigning orientations to the edges of an undirected graph so that each vertex has incoming edges with large-enough total weight. The second part of this book applies constraint logic in new proofs of hardness of various real-world games and puzzles,  by showing that, in each case, the vertices and edges of a constraint logic instance can be encoded by the moves and pieces of the game. Some of these hardness proofs simplify previously-known proofs; some ten of them are new, including the discovery that optimal play in certain multiplayer games can be an undecidable problem. A third part of the book provides a compendium of known hardness results in game complexity, updating a much shorter list of complete problems in game complexity from the 1979 book Computers and Intractability. An appendix provides a review of the methods from computational complexity theory needed in this study, for readers unfamiliar with this area.
+
+
+== Audience and reception ==
+Although primarily a research monograph and reference work for researchers in this area, reviewer Oswin Aichholzer recommends the book more generally to anyone interested in the mathematics of games and their complexity. Liljana Babinkostova writes that Games, Puzzles, and Computation is enjoyable reading, successful in its "purpose of building a bridge between games and the theory of computation".
+Leon Harkleroad is somewhat more critical, writing that the book feels padded in places, and Joseph O'Rourke complains that its organization, with many pages of abstract mathematics before reaching the real-world games, does not lend itself to cover-to-cover reading. However, both Harkleroad and O'Rourke agree that the book is well produced and thought-provoking.
+
+
+== References ==

data/en.wikipedia.org/wiki/Geometric_Exercises_in_Paper_Folding-0.md

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+---
+title: "Geometric Exercises in Paper Folding"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Geometric_Exercises_in_Paper_Folding"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:53.464312+00:00"
+instance: "kb-cron"
+---
+
+Geometric Exercises in Paper Folding is a book on the mathematics of paper folding. It was written by Indian mathematician T. Sundara Row, first published in India in 1893, and later republished in many other editions. Its topics include paper constructions for regular polygons, symmetry, and algebraic curves. According to the historian of mathematics Michael Friedman, it became "one of the main engines of the popularization of folding as a mathematical activity".
+
+
+== Publication history ==
+Geometric Exercises in Paper Folding was first published by Addison & Co. in Madras in 1893. The book became known in Europe through a remark of Felix Klein in his book Vorträge über ausgewählte Fragen der Elementargeometrie (1895) and its translation Famous Problems Of Elementary Geometry (1897). Based on the success of Geometric Exercises in Paper Folding in Germany, the Open Court Press of Chicago published it in the US, with updates by Wooster Woodruff Beman and David Eugene Smith. Although Open Court listed four editions of the book, published in 1901, 1905, 1917, and 1941, the content did not change between these editions. The fourth edition was also published in London by La Salle, and both presses reprinted the fourth edition in 1958.
+The contributions of Beman and Smith to the Open Court editions have been described as "translation and adaptation", despite the fact that the original 1893 edition was already in English. Beman and Smith also replaced many footnotes with references to their own work, replaced some of the diagrams by photographs, and removed some remarks specific to India. In 1966, Dover Publications of New York published a reprint of the 1905 edition, and other publishers of out-of-copyright works have also printed editions of the book.
+
+
+== Topics ==
+Geometric Exercises in Paper Folding shows how to construct various geometric figures using paper-folding in place of the classical Greek Straightedge and compass constructions.
+The book begins by constructing regular polygons beyond the classical constructible polygons of 3, 4, or 5 sides, or of any power of two times these numbers, and the construction by Carl Friedrich Gauss of the heptadecagon, it also provides a paper-folding construction of the regular nonagon, not possible with compass and straightedge. The nonagon construction involves angle trisection, but Rao is vague about how this can be performed using folding; an exact and rigorous method for folding-based trisection would have to wait until the work in the 1930s of Margherita Piazzola Beloch. The construction of the square also includes a discussion of the Pythagorean theorem. The book uses high-order regular polygons to provide a geometric calculation of pi.
+A discussion of the symmetries of the plane includes congruence, similarity, and collineations of the projective plane; this part of the book also covers some of the major theorems of projective geometry including Desargues's theorem, Pascal's theorem, and Poncelet's closure theorem.
+Later chapters of the book show how to construct algebraic curves including the conic sections, the conchoid, the cubical parabola, the witch of Agnesi, the cissoid of Diocles, and the Cassini ovals. The book also provides a gnomon-based proof of Nicomachus's theorem that the sum of the first 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+ cubes is the square of the sum of the first 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+ integers, and material on other arithmetic series, geometric series, and harmonic series.
+There are 285 exercises, and many illustrations, both in the form of diagrams and (in the updated editions) photographs.
+
+
+== Influences ==
+Tandalam Sundara Row was born in 1853, the son of a college principal, and earned a bachelor's degree at the Kumbakonam College in 1874, with second-place honours in mathematics. He became a tax collector in Tiruchirappalli, retiring in 1913, and pursued mathematics as an amateur. As well as Geometric Exercises in Paper Folding, he also wrote a second book, Elementary Solid Geometry, published in three parts from 1906 to 1909.
+One of the sources of inspiration for Geometric Exercises in Paper Folding was Kindergarten Gift No. VIII: Paper-folding. This was one of the Froebel gifts, a set of kindergarten activities designed in the early 19th century by Friedrich Fröbel. The book was also influenced by an earlier Indian geometry textbook, First Lessons in Geometry, by Bhimanakunte Hanumantha Rao (1855–1922). First Lessons drew inspiration from Fröbel's gifts in setting exercises based on paper-folding, and from the book Elementary Geometry: Congruent Figures by Olaus Henrici in using a definition of geometric congruence based on matching shapes to each other and well-suited for folding-based geometry.
+In turn, Geometric Exercises in Paper Folding inspired other works of mathematics. A chapter in Mathematische Unterhaltungen und Spiele [Mathematical Recreations and Games] by Wilhelm Ahrens (1901) concerns folding and is based on Rao's book, inspiring the inclusion of this material in several other books on recreational mathematics. Other mathematical publications have studied the curves that can be generated by the folding processes used in Geometric Exercises in Paper Folding. In 1934, Margherita Piazzola Beloch began her research on axiomatizing the mathematics of paper-folding, a line of work that would eventually lead to the Huzita–Hatori axioms in the late 20th century. Beloch was explicitly inspired by Rao's book, titling her first work in this area "Alcune applicazioni del metodo del ripiegamento della carta di Sundara Row" ["Several applications of the method of folding a paper of Sundara Row"].
+
+
+== Audience and reception ==
+The original intent of Geometric Exercises in Paper Folding was twofold: as an aid in geometry instruction,
+and as a work of recreational mathematics to inspire interest in geometry in a general audience. Edward Mann Langley, reviewing the 1901 edition, suggested that its content went well beyond what should be covered in a standard geometry course. And in their own textbook on geometry using paper-folding exercises, The First Book of Geometry (1905), Grace Chisholm Young and William Henry Young heavily criticized Geometric Exercises in Paper Folding, writing that it is "too difficult for a child, and too infantile for a grown person". However, reviewing the 1966 Dover edition, mathematics educator Pamela Liebeck called it "remarkably relevant" to the discovery learning techniques for geometry instruction of the time, and in 2016 computational origami expert Tetsuo Ida, introducing an attempt to formalize the mathematics of the book, wrote "After 123 years, the significance of the book remains."
+
+
+== References ==
+
+
+== External links ==
+Madras edition and Open Court edition of Geometric Exercises in Paper Folding on the Internet Archive

data/en.wikipedia.org/wiki/Geometric_Folding_Algorithms-0.md

@@ -0,0 +1,45 @@
+---
+title: "Geometric Folding Algorithms"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Geometric_Folding_Algorithms"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:54.562391+00:00"
+instance: "kb-cron"
+---
+
+Geometric Folding Algorithms: Linkages, Origami, Polyhedra is a monograph on the mathematics and computational geometry of mechanical linkages, paper folding, and polyhedral nets, by Erik Demaine and Joseph O'Rourke. It was published in 2007 by Cambridge University Press (ISBN 978-0-521-85757-4).
+A Japanese-language translation by Ryuhei Uehara was published in 2009 by the Modern Science Company (ISBN 978-4-7649-0377-7).
+
+
+== Audience ==
+Although aimed at computer science and mathematics students, much of the book is accessible to a broader audience of mathematically-sophisticated readers with some background in high-school level geometry.
+Mathematical origami expert Tom Hull has called it "a must-read for anyone interested in the field of computational origami".
+It is a monograph rather than a textbook, and in particular does not include sets of exercises.
+The Basic Library List Committee of the Mathematical Association of America has recommended this book for inclusion in undergraduate mathematics libraries.
+
+
+== Topics and organization ==
+The book is organized into three sections, on linkages, origami, and polyhedra.
+Topics in the section on linkages include
+the Peaucellier–Lipkin linkage for converting rotary motion into linear motion,
+Kempe's universality theorem that any algebraic curve can be traced out by a linkage,
+the existence of linkages for angle trisection,
+and the carpenter's rule problem on straightening two-dimensional polygonal chains.
+This part of the book also includes applications to motion planning for robotic arms, and to protein folding.
+The second section of the book concerns the mathematics of paper folding, and mathematical origami. It includes the NP-completeness of testing flat foldability,
+the problem of map folding (determining whether a pattern of mountain and valley folds forming a square grid can be folded flat),
+the work of Robert J. Lang using tree structures and circle packing to automate the design of origami folding patterns,
+the fold-and-cut theorem according to which any polygon can be constructed by folding a piece of paper and then making a single straight cut,
+origami-based angle trisection,
+rigid origami,
+and the work of David A. Huffman on curved folds.
+In the third section, on polyhedra, the topics include polyhedral nets and Dürer's conjecture on their existence for convex polyhedra, the sets of polyhedra that have a given polygon as their net, Steinitz's theorem characterizing the graphs of polyhedra, Cauchy's theorem that every polyhedron, considered as a linkage of flat polygons, is rigid, and Alexandrov's uniqueness theorem stating that the three-dimensional shape of a convex polyhedron is uniquely determined by the metric space of geodesics on its surface.
+The book concludes with a more speculative chapter on higher-dimensional generalizations of the problems it discusses.
+
+
+== References ==
+
+
+== External links ==
+Authors' web site for Geometric Folding Algorithms including contents, errata, and advances on open problems

data/en.wikipedia.org/wiki/Geometric_Origami-0.md

@@ -0,0 +1,69 @@
+---
+title: "Geometric Origami"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Geometric_Origami"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:55.725835+00:00"
+instance: "kb-cron"
+---
+
+Geometric Origami is a book on the mathematics of paper folding, focusing on the ability to simulate and extend classical straightedge and compass constructions using origami. It was written by Austrian mathematician Robert Geretschläger and published by Arbelos Publishing (Shipley, UK) in 2008. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
+
+
+== Topics ==
+The book is divided into two main parts. The first part is more theoretical. It outlines the Huzita–Hatori axioms for mathematical origami, and proves that they are capable of simulating any straightedge and compass construction. It goes on to show that, in this mathematical model, origami is strictly more powerful than straightedge and compass: with origami, it is possible to solve any cubic equation or quartic equation. In particular, origami methods can be used to trisect angles, and for doubling the cube, two problems that have been proven to have no exact solution using only straightedge and compass.
+The second part of the book focuses on folding instructions for constructing regular polygons using origami, and on finding the largest copy of a given regular polygon that can be constructed within a given square sheet of origami paper. With straightedge and compass, it is only possible to exactly construct regular 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+-gons for which 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+ is a product of a power of two with distinct Fermat primes (powers of two plus one): this allows 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+ to be  3, 5, 6, 8, 10, 12, etc. These are called the constructible polygons. With a construction system that can trisect angles, such as mathematical origami, more numbers of sides are possible, using Pierpont primes in place of Fermat primes, including 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+-gons for 
+  
+    
+      
+        n
+      
+    
+    {\displaystyle n}
+  
+ equal to 7, 13, 14, 17, 19, etc. Geometric Origami provides explicit folding instructions for 15 different regular polygons, including those with 3, 5, 6, 7, 8, 9, 10, 12, 13, 17, and 19 sides. Additionally, it discusses approximate constructions for polygons that cannot be constructed exactly in this way.
+
+
+== Audience and reception ==
+This book is quite technical, aimed more at mathematicians than at amateur origami enthusiasts looking for folding instructions for origami artworks. However, it may be of interest to origami designers, looking for methods to incorporate folding patterns for regular polygons into their designs. Origamist David Raynor suggests that its methods could also be useful in constructing templates from which to cut out clean unfolded pieces of paper in the shape of the regular polygons that it discusses, for use in origami models that use these polygons as a starting shape instead of the traditional square paper.
+Geometric Origami may also be useful as teaching material for university-level geometry and abstract algebra, or for undergraduate research projects extending those subjects, although reviewer Mary Fortune cautions that "there is much preliminary material to be covered" before a student would be ready for such a project. Reviewer Georg Gunther summarizes the book as "a delightful addition to a wonderful corner of mathematics where art and geometry meet", recommending it as a reference for "anyone with a working knowledge of elementary geometry, algebra, and the geometry of complex numbers".
+
+
+== References ==

data/en.wikipedia.org/wiki/Geometric_and_Topological_Inference-0.md

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+---
+title: "Geometric and Topological Inference"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Geometric_and_Topological_Inference"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:52.312144+00:00"
+instance: "kb-cron"
+---
+
+Geometric and Topological Inference is a monograph in computational geometry, computational topology, geometry processing, and topological data analysis, on the problem of inferring properties of an unknown space from a finite point cloud of noisy samples from the space. It was written by Jean-Daniel Boissonnat, Frédéric Chazal, and Mariette Yvinec, and published in 2018 by the Cambridge University Press in their Cambridge Texts in Applied Mathematics book series. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
+
+
+== Topics ==
+The book is subdivided into four parts and 11 chapters. The first part covers basic tools from topology needed in the study, including simplicial complexes, Čech complexes and Vietoris–Rips complex, homotopy equivalence of topological spaces to their nerves, filtrations of complexes, and the data structures needed to represent these concepts efficiently in computer algorithms. A second introductory part concerns material of a more geometric nature, including Delaunay triangulations and Voronoi diagrams, convex polytopes, convex hulls and convex hull algorithms, lower envelopes, alpha shapes and alpha complexes, and witness complexes.
+With these preliminaries out of the way, the remaining two sections show how to use these tools for topological inference. The third section is on recovering the unknown space itself (or a topologically equivalent space, described using a complex) from sufficiently well-behaved samples. The fourth part shows how, with weaker assumptions about the samples, it is still possible to recover useful information about the space, such as its homology and persistent homology.
+
+
+== Audience and reception ==
+Although the book is primarily aimed at specialists in these topics, it can also be used to introduce the area to non-specialists, and provides exercises suitable for an advanced course. Reviewer Michael Berg evaluates it as an "excellent book" aimed at a hot topic, inference from large data sets, and both Berg and Mark Hunacek note that it brings a surprising level of real-world applicability to formerly-pure topics in mathematics.
+
+
+== References ==

data/en.wikipedia.org/wiki/Geometry_From_Africa-0.md

@@ -0,0 +1,32 @@
+---
+title: "Geometry From Africa"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Geometry_From_Africa"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:58.111842+00:00"
+instance: "kb-cron"
+---
+
+Geometry From Africa: Mathematical and Educational Explorations is a book in ethnomathematics by Paulus Gerdes. It analyzes the mathematics behind geometric designs and patterns from multiple African cultures, and suggests ways of connecting this analysis with the mathematics curriculum. It was published in 1999 by the Mathematical Association of America, in their Classroom Resource Materials book series.
+
+
+== Background ==
+The book's author, Paulus Gerdes (1952–2014), was a mathematician from the Netherlands who became a professor of mathematics at the Eduardo Mondlane University in Mozambique, rector of Maputo University, and chair of the African Mathematical Union Commission on the History of Mathematics in Africa. He was a prolific author, especially of works on the ethnomathematics of Africa. However, as many of his publications were written in Portuguese, German, and French, or published only in Mozambique, this book makes his work in ethnomathematics more accessible to English-speaking mathematicians.
+
+
+== Topics ==
+The book is heavily illustrated, and describes geometric patterns in the carvings, textiles, drawings and paintings of multiple African cultures. Although these are primarily decorative rather than mathematical, Gerdes adds his own mathematical analysis of the patterns, and suggests ways of incorporating this analysis into the mathematical curriculum.
+It is divided into four chapters. The first of these provides an overview of geometric patterns in many African cultures, including examples of textiles, knotwork, architecture, basketry, metalwork, ceramics, petroglyphs, facial tattoos, body painting, and hair styles. The second chapter presents examples of designs in which squares and right triangles can be formed from elements of the patterns, and suggests educational activities connecting these materials to the Pythagorean theorem and to the theory of Latin squares. For instance, basket-weavers in Mozambique form square knotted buttons out of folded ribbons, and the resulting pattern of oblique lines crossing the square suggests a standard dissection-based proof of the theorem. The third chapter uses African designs, particularly in basket-weaving, to illustrate themes of symmetry, polygons and polyhedra, area, volume, and the theory of fullerenes. In the final chapter, the only one to concentrate on a single African culture, the book discusses the sona sand-drawings of the Chokwe people, in which a single self-crossing curve surrounds and separates a grid of points. These drawings connect to the theory of Euler tours, fractals, arithmetic series, and polyominos.
+
+
+== Audience and reception ==
+The book is aimed at primary and secondary school mathematics teachers. Reviewer Karen Dee Michalowicz, a school teacher, writes that although the connections between culture and mathematics are sometimes contrived, "every mathematics educator would benefit" from the book.
+Ethnomathematician Marcia Ascher suggests that the book would have benefited from an index and a map of the cultures from which the material of the book was drawn. Nevertheless, reviewer Richard Kitchen evaluates this book as "the most complete volume available on the ethnomathematics of Africa". Reviewer Steve Abbott writes that the book "provides many opportunities for developing cross-curricular and multi-cultural approaches" to mathematics education, and that it is "an important book that deserves to be widely read".
+
+
+== References ==
+
+
+== External links ==
+Geometry From Africa on the Internet Archive

data/en.wikipedia.org/wiki/Geshu_bu-0.md

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+---
+title: "Geshu bu"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Geshu_bu"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:00.452687+00:00"
+instance: "kb-cron"
+---
+
+The Geshu bu (格術補), translated to English as the Supplement to Geometric Optics, Science Updates, or Optic Updates, is a book on optics written by the Chinese Guangdong gentry and intellectual Zou Boqi (1819–1869). The book was written in the late Qing Dynasty period in China. It covers the principles of lenses and is based on mathematical theory.
+
+
+== Background ==
+
+Zou Boqi wrote two significant works in optics: the Geshu bu and the "Notes on the Mechanism for Capturing Images" (Sheying zhi qi ji). The Geshu bu tackles the Chinese principles on optics and its related literature while the Sheyi zhi qi ji delves more on the results of Zou's experiments as well as principles on photography. The Geshu bu was published in 1874, after Zou Boqi's death. In the late nineteenth century, Western knowledge, especially in optics, began to influence China. Zou's book, Geshu bu, introduced basic concepts of optics and lens, building on earlier Jesuit translations. This work challenged the traditional Mohist views and laid the groundwork for geometrical optics in China.
+
+
+== Contents ==
+Based from one fascicle of the book, published in 1877, it aims to expand on ancient Chinese mathematics. The term geshu highlights thorough investigation as explained in the preface by the Chinese philosopher Chen Li, drawing from Western lens-making techniques.
+In the book, Zou discussed different mirror shapes and principles. He also explained convex lenses, focusing on how they concentrate sunlight to ignite a fire. The focal point determines how close or far an image appears.
+The text introduces trigonometric calculations for lens reflections, providing a mathematical basis for understanding convex mirrors as well as adding details on telescopes and microscopes. The book also corrected some mistakes made by J. Adam Schall von Bell and Zhen Fuguang, writers on optics in China.
+
+
+=== Reception ===
+Due to its unique approach of integrating mathematical calculations in lens-making principles, the book received high praise from Qing scholars. In 1886, Zhu Kebao wrote about optical methods in his biography of Zou Boqi. Zou's book helped circulated the knowledge of optics and lenses in the print market. Such circulation was reflected in a winning essay from the 1889 Polytechnic Institute (Gezhi shuyuan, 格致書院) contest, linking optics and the "mirror that lights a fire."
+
+
+== References ==
+
+
+== External links ==
+Geshu bu (1877 edition), available on Google Books

data/en.wikipedia.org/wiki/Glossary_of_Principia_Mathematica-0.md

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+---
+title: "Glossary of Principia Mathematica"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Glossary_of_Principia_Mathematica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:01.640169+00:00"
+instance: "kb-cron"
+---
+
+This is a list of the notation used in Alfred North Whitehead and Bertrand Russell's Principia Mathematica (1910–1913).
+The second (but not the first) edition of Volume I has a list of notation used at the end.
+
+
+== Glossary ==
+This is a glossary of some of the technical terms in Principia Mathematica that are no longer widely used or whose meaning has changed.
+
+apparent variable
+bound variable
+
+atomic proposition
+A proposition of the form R(x,y,...) where R is a relation.
+
+Barbara
+A mnemonic for a certain syllogism.
+
+class
+A subset of the members of some type
+
+codomain
+The codomain of a relation R is the class of y such that xRy for some x.
+
+compact
+A relation R is called compact if whenever xRz there is a y with xRy and yRz
+
+concordant
+A set of real numbers is called concordant if all nonzero members have the same sign
+
+connected
+connexity
+A relation R is called connected if for any 2 distinct members x, y either xRy or yRx.
+
+continuous
+A continuous series is a complete totally ordered set isomorphic to the reals. *275
+
+correlator
+bijection
+
+couple
+1.  A cardinal couple is a class with exactly two elements
+2.  An ordinal couple is an ordered pair (treated in PM as a special sort of relation)
+
+Dedekindian
+complete (relation) *214
+
+definiendum
+The symbol being defined
+
+definiens
+The meaning of something being defined
+
+derivative
+A derivative of a subclass of a series is the class of limits of non-empty subclasses
+
+description
+A definition of something as the unique object with a given property
+
+descriptive function
+A function taking values that need not be truth values, in other words what is not called just a function.
+
+diversity
+The inequality relation
+
+domain
+The domain of a relation R is the class of x such that xRy for some y.
+
+elementary proposition
+A proposition built from atomic propositions using "or" and "not", but with no bound variables
+
+Epimenides
+Epimenides was a legendary Cretan philosopher
+
+existent
+non-empty
+
+extensional function
+A function whose value does not change if one of its arguments is changed to something equivalent.
+
+field
+The field of a relation R is the union of its domain and codomain
+
+first-order
+A first-order proposition is allowed to have quantification over individuals but not over things of higher type.
+
+function
+This often means a propositional function, in other words a function taking values "true" or "false". If it takes other values it is called a "descriptive function". PM allows two functions to be different even if they take the same values on all arguments.
+
+general proposition
+A proposition containing quantifiers
+
+generalization
+Quantification over some variables
+
+homogeneous
+A relation is called homogeneous if all arguments have the same type.
+
+individual
+An element of the lowest type under consideration
+
+inductive
+Finite, in the sense that a cardinal is inductive if it can be obtained by repeatedly adding 1 to 0. *120
+
+intensional function
+A function that is not extensional.
+
+logical
+1.  The logical sum of two propositions is their logical disjunction 
+2.  The logical product of two propositions is their logical conjunction
+
+matrix
+A function with no bound variables. *12
+
+median
+A class is called median for a relation if some element of the class lies strictly between any two terms. *271
+
+member
+element (of a class)
+
+molecular proposition
+A proposition built from two or more atomic propositions using "or" and "not"; in other words an elementary proposition that is not atomic.
+
+null-class
+A class containing no members
+
+predicative
+A century of scholarly discussion has not reached a definite consensus on exactly what this means, and Principia Mathematica gives several different explanations of it that are not easy to reconcile. See the introduction and *12. *12 says that a predicative function is one with no apparent (bound) variables, in other words a matrix.
+
+primitive proposition
+A proposition assumed without proof
+
+progression
+A sequence (indexed by natural numbers)
+
+rational
+A rational series is an ordered set isomorphic to the rational numbers
+
+real variable
+free variable
+
+referent
+The term x in xRy
+
+reflexive
+infinite in the sense that the class is in one-to-one correspondence with a proper subset of itself (*124)
+
+relation
+A propositional function of some variables (usually two). This is similar to the current meaning of "relation".
+
+relative product
+The relative product of two relations is their composition
+
+relatum
+The term y in xRy
+
+scope
+The scope of an expression is the part of a proposition where the expression has some given meaning (chapter III)
+
+Scott
+Sir Walter Scott, author of Waverley.
+
+second-order
+A second order function is one that may have first-order arguments
+
+section
+A section of a total order is a subclass containing all predecessors of its members.
+
+segment
+A subclass of a totally ordered set consisting of all the predecessors of the members of some class
+
+selection
+A choice function: something that selects one element from each of a collection of classes.
+
+sequent
+A sequent of a class α in a totally ordered class is a minimal element of the class of terms coming after all members of α. (*206)
+
+serial relation
+A total order on a class
+
+significant
+well-defined or meaningful
+
+similar
+of the same cardinality
+
+stretch
+A convex subclass of an ordered class
+
+stroke
+The Sheffer stroke (only used in the second edition of PM)
+
+type
+As in type theory. All objects belong to one of a number of disjoint types.
+
+typically
+Relating to types; for example, "typically ambiguous" means "of ambiguous type".
+
+unit
+A unit class is one that contains exactly one element
+
+universal
+A universal class is one containing all members of some type
+
+vector
+1.  Essentially an injective function from a class to itself (for example, a vector in a vector space acting on an affine space)
+2.  A vector-family is a non-empty commuting family of injective functions from some class to itself (VIB)
+
+
+== Symbols introduced in Principia Mathematica, Volume I ==
+
+
+== Symbols introduced in Principia Mathematica, Volume II ==
+
+
+== Symbols introduced in Principia Mathematica, Volume III ==
+
+
+== See also ==
+Glossary of set theory
+
+
+== Notes ==
+
+
+== References ==
+Whitehead, Alfred North, and Bertrand Russell. Principia Mathematica, 3 vols, Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1), 1927 (Vols. 2, 3).
+
+
+== External links ==
+List of notation in Principia Mathematica at the end of Volume I
+"The Notation in Principia Mathematica" by Bernard Linsky.
+Principia Mathematica online (University of Michigan Historical Math Collection):
+Volume I
+Volume II
+Volume III
+Proposition ✸54.43 in a more modern notation (Metamath)

data/en.wikipedia.org/wiki/God_Created_the_Integers-0.md

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+---
+title: "God Created the Integers"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/God_Created_the_Integers"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:02.760295+00:00"
+instance: "kb-cron"
+---
+
+God Created the Integers: The Mathematical Breakthroughs That Changed History is a 2005 anthology, edited by Stephen Hawking, of "excerpts from thirty-one of the most important works in the history of mathematics."
+Each chapter of the work focuses on a different mathematician and begins with a biographical overview. Within each chapter, Hawking examines the mathematician's key discoveries, presents formal proofs of significant results, and explains their impact on the development of the mathematical field.
+The title of the book is a reference to a quotation attributed to mathematician Leopold Kronecker, who once wrote that "God made the integers; all else is the work of man."
+
+
+== Content ==
+The works are grouped by author and ordered chronologically. Each section is prefaced by notes on the mathematician's life and work. The anthology includes works by the following mathematicians:
+
+Selections from the works of Euler, Bolyai, Lobachevsky and Galois, which are included in the second edition of the book (published in 2007), were not included in the first edition.
+
+
+== Editions ==
+Hawking, Stephen (2005). God Created the Integers: The Mathematical Breakthroughs That Changed History. Running Press Book Publishers. pp. 1160 (Hardback). ISBN 0-7624-1922-9.
+Hawking, Stephen (2007). God Created the Integers: The Mathematical Breakthroughs That Changed History. Running Press Book Publishers. pp. 1358 (Paperback). ISBN 978-0-7624-3004-8.
+
+
+== References ==

data/en.wikipedia.org/wiki/Grundlagen_der_Mathematik-0.md

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+---
+title: "Grundlagen der Mathematik"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Grundlagen_der_Mathematik"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:06.338584+00:00"
+instance: "kb-cron"
+---
+
+Grundlagen der Mathematik (English: Foundations of Mathematics) is a two-volume work by David Hilbert and Paul Bernays.  Originally published in 1934 and 1939, it presents fundamental mathematical ideas and introduced second-order arithmetic.
+
+
+== Publication history ==
+1934/1939 (Vol. I, II) First German edition, Springer
+1944 Reprint of first edition by J. W. Edwards, Ann Arbor, Michigan.
+1968/1970 (Vol. I, II) Second revised German edition, Springer
+1979/1982 (Vol. I, II) Russian translation of 1968/1970, Nauka Publ., Moscow
+2001/2003 (Vol. I, II) French translation, L’Harmattan, Paris
+2011/2013 (Parts A and B of Vol. I, prefaces and sections 1-5) English translation of 1968 and 1934, bilingual with German facsimile on the left-hand sides.
+The Hilbert Bernays Project is producing an English translation.
+
+
+== See also ==
+Hilbert–Bernays paradox
+
+
+== References ==
+Sieg, Wilfried; Ravaglia, Mark (2005), "Chapter 77. David Hilbert and Paul Bernays, Grundlagen der Mathematik" (PDF), in Grattan-Guinness, Ivor (ed.), Landmark writings in western mathematics 1640–1940, Elsevier B. V., Amsterdam, pp. 981–99, doi:10.1016/B978-044450871-3/50158-3, ISBN 978-0-444-50871-3, MR 2169816, archived from the original (PDF) on 2011-05-14, retrieved 2011-02-01
+Hilbert, David; Bernays, Paul (1934), Grundlagen der Mathematik. I, Die Grundlehren der mathematischen Wissenschaften, vol. 40, Berlin, New York: Springer-Verlag, ISBN 978-3-540-04134-4, JFM 60.0017.02, MR 0237246, archived from the original on 2011-05-17 {{citation}}: ISBN / Date incompatibility (help)
+Hilbert, David; Bernays, Paul (1939), Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, vol. 50, Berlin, New York: Springer-Verlag, ISBN 978-3-540-05110-7, JFM 65.0021.02, MR 0272596, archived from the original on 2011-05-17 {{citation}}: ISBN / Date incompatibility (help)
+Hilbert, David; Bernays, Paul (1968) [1934], Grundlagen der Mathematik. I, Die Grundlehren der mathematischen Wissenschaften, vol. 40 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3642868955, MR 0237246
+Hilbert, D.; Bernays, Paul (1970), Grundlagen der Mathematik. II, Die Grundlehren der mathematischen Wissenschaften, vol. 50 (2 ed.), Berlin-New York: Springer-Verlag, ISBN 978-3642868979, MR 0272596
+Hilbert, David; Bernays, Paul (2011) [1934/68], Grundlagen der Mathematik I — Foundations of Mathematics I, Part A: Prefaces and §§ 1–2 (in German and English) (1st ed.), London: College Publications, ISBN 978-1-84890-033-2, MR 3027390
+Hilbert, David; Bernays, Paul (2013) [1934/68], Grundlagen der Mathematik I — Foundations of Mathematics I, Part B: §§ 3–5 and Deleted Part I of the 1st Edn. (in German and English) (1st ed.), London: College Publications, ISBN 978-1-84890-075-2
+
+
+== External links ==
+Hilbert Bernays Project which aims to produce an English translation of Grundlagen der Mathematik.

data/en.wikipedia.org/wiki/Grundzüge_der_Mengenlehre-0.md

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+---
+title: "Grundzüge der Mengenlehre"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Grundzüge_der_Mengenlehre"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:07.505560+00:00"
+instance: "kb-cron"
+---
+
+Grundzüge der Mengenlehre (English: Basics of Set Theory) is a book on set theory written by Felix Hausdorff.
+First published in April 1914, Grundzüge der Mengenlehre was the first comprehensive introduction to set theory. In addition to the systematic treatment of known results in set theory, the book also contains chapters on measure theory and topology, which were then still considered parts of set theory. Hausdorff presented and developed original material that later became the basis for those areas.
+In 1927, Hausdorff published an extensively revised second edition under the title Mengenlehre (English: Set Theory), omitting many topics from the first edition. In 1935, a third German edition was released, which in 1957 was translated into English by John R. Aumann et al. under the title Set Theory.
+
+
+== References ==
+Blumberg, Henry (1920), "Hausdorff's Grundzüge der Mengenlehre", Bulletin of the American Mathematical Society, 27 (3): 116–129, doi:10.1090/S0002-9904-1920-03378-1.
+Gehman, H. M. (1927), "Hausdorff's Revised Mengenlehre", Bull. Amer. Math. Soc., 33 (6): 778–781, doi:10.1090/S0002-9904-1927-04478-0
+Hausdorff, Felix (1914), Grundzüge der Mengenlehre, Leipzig: Veit, ISBN 978-0-8284-0061-9 {{citation}}: ISBN / Date incompatibility (help) Reprinted by Chelsea Publishing Company in 1944, 1949 and 1965 [1].
+Hausdorff, F. (1935) [1927], Mengenlehre (3 ed.), Berlin-Leipzig: de Gruyter Republished by Dover Publications, New York, N. Y., 1944
+Hausdorff, Felix (1962) [1957], Set theory (2 ed.), New York: Chelsea Publishing Company, ISBN 978-0821838358 {{citation}}: ISBN / Date incompatibility (help) Republished by AMS-Chelsea 2005.
+Scholz, Erhard (2005), Felix Hausdorff and the Hausdorff edition (PDF). Extended edition of a chapter in The Princeton Companion to Mathematics.

data/en.wikipedia.org/wiki/Harmonice_Mundi-0.md

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 category: "reference"
 tags: "science, encyclopedia"
-date_saved: "2026-05-05T08:33:39.488056+00:00"
+date_saved: "2026-05-05T08:45:08.718356+00:00"
 instance: "kb-cron"
 ---
 

data/en.wikipedia.org/wiki/Harmonice_Mundi-1.md

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 source: "https://en.wikipedia.org/wiki/Harmonice_Mundi"
 category: "reference"
 tags: "science, encyclopedia"
-date_saved: "2026-05-05T08:33:39.488056+00:00"
+date_saved: "2026-05-05T08:45:08.718356+00:00"
 instance: "kb-cron"
 ---
 

data/en.wikipedia.org/wiki/Higher_Topos_Theory-0.md

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+---
+title: "Higher Topos Theory"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Higher_Topos_Theory"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:11.067998+00:00"
+instance: "kb-cron"
+---
+
+Higher Topos Theory is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory. Since 2018, Lurie has been transferring the contents of Higher Topos Theory (along with new material) to Kerodon, an "online resource for homotopy-coherent mathematics" inspired by the Stacks Project.
+
+
+== Topics ==
+Higher Topos Theory covers two related topics: ∞-categories and ∞-topoi (which are a special case of the former). The first five of the book's seven chapters comprise a rigorous development of general ∞-category theory in the language of quasicategories, a special class of simplicial set which acts as a model for ∞-categories. The path of this development largely parallels classical category theory, with the notable exception of the ∞-categorical Grothendieck construction; this correspondence, which Lurie refers to as "straightening and unstraightening", gains considerable importance in his treatment.
+The last two chapters are devoted to ∞-topoi, Lurie's own invention and the ∞-categorical analogue of topoi in classical category theory. The material of these chapters is original, and is adapted from an earlier preprint of Lurie's. There are also appendices discussing background material on categories, model categories, and simplicial categories.
+
+
+== History ==
+Higher Topos Theory followed an earlier work by Lurie, On Infinity Topoi, uploaded to the arXiv in 2003. Algebraic topologist Peter May was critical of this preprint, emailing Lurie's then-advisor Mike Hopkins "to say that Lurie’s paper had some interesting ideas, but that it felt preliminary and needed more rigor." Lurie released a draft of Higher Topos Theory on the arXiv in 2006, and the book was finally published in 2009.
+Lurie released a second book on higher category theory, Higher Algebra, as a preprint on his website in 2017. This book assumes the content of Higher Topos Theory and uses it to study algebra in the ∞-categorical context.
+
+
+== External links ==
+http://ncatlab.org/nlab/show/Higher+Topos+Theory
+If I want to study Jacob Lurie's books "Higher Topoi Theory", "Derived AG", what prerequisites should I have?
+https://www.math.ias.edu/~lurie/
+https://kerodon.net/about
+
+
+== References ==

data/en.wikipedia.org/wiki/History_of_the_Theory_of_Numbers-0.md

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+---
+title: "History of the Theory of Numbers"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/History_of_the_Theory_of_Numbers"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:12.238942+00:00"
+instance: "kb-cron"
+---
+
+History of the Theory of Numbers is a three-volume work by Leonard Eugene Dickson  summarizing work in number theory up to about 1920. The style is unusual in that Dickson mostly just lists results by various authors, with little further discussion. The central topic of quadratic reciprocity and higher reciprocity laws is barely mentioned; this was apparently going to be the topic of a fourth volume that was never written (Fenster 1999).
+
+
+== Volumes ==
+Volume 1 - Divisibility and Primality - 486 pages
+Volume 2 - Diophantine Analysis - 803 pages
+Volume 3 - Quadratic and Higher Forms - 313 pages
+
+
+== References ==
+Carmichael, Robert D. (1919), "Recent Publications: Reviews: History of the Theory of Numbers. Volume I: Divisibility and Primality", The American Mathematical Monthly, 26 (9): 396–403, doi:10.2307/2971917, ISSN 0002-9890, JSTOR 2971917
+Carmichael, Robert D. (1921), "Recent Publications: Reviews: History of the Theory of Numbers: History of the Theory of Numbers", The American Mathematical Monthly, 28 (2): 72–78, doi:10.2307/2973042, ISSN 0002-9890, JSTOR 2973042
+Carmichael, Robert D. (1923), "Recent Publications: Reviews: History of the Theory of Numbers", The American Mathematical Monthly, 30 (5): 259–262, doi:10.2307/2299094, ISSN 0002-9890, JSTOR 2299094
+Dickson, Leonard Eugene (2005) [1919], History of the theory of numbers. Vol. I: Divisibility and primality., New York: Dover Publications, ISBN 978-0-486-44232-7, MR 0245499, Zbl 1214.11001
+Dickson, Leonard Eugene (2005) [1920], History of the theory of numbers. Vol. II: Diophantine analysis, New York: Dover Publications, ISBN 978-0-486-44233-4, MR 0245500, Zbl 1214.11002
+Dickson, Leonard Eugene (2005) [1923], History of the theory of numbers. Vol. III: Quadratic and higher forms, New York: Dover Publications, ISBN 978-0-486-44234-1, MR 0245501
+Fenster, Della D. (1999), "Leonard Dickson's History of the theory of numbers: an historical study with mathematical implications", Revue d'Histoire des Mathématiques. Journal for the History of Mathematics, 5 (2): 159–179, ISSN 1262-022X, MR 1793101
+Fenster, Della Dumbaugh (1999), "Why Dickson left quadratic reciprocity out of his History of the theory of numbers", The American Mathematical Monthly, 106 (7): 618–627, doi:10.2307/2589491, ISSN 0002-9890, JSTOR 2589491, MR 1720467
+Furtwängler, Ph. (1923), "Literaturberichte: History of the theory of numbers", Monatshefte für Mathematik und Physik, 33 (1): A6–A7, doi:10.1007/BF01705606, ISSN 0026-9255, S2CID 116618046
+Lehmer, D. H. (1919), "Book Review: History of the Theory of Numbers", Bulletin of the American Mathematical Society, 26 (3): 125–132, doi:10.1090/S0002-9904-1919-03280-7, ISSN 0002-9904
+Lehmer, D. H. (1920), "Dickson's history of the theory of numbers", Bulletin of the American Mathematical Society, 26 (6): 281–282, doi:10.1090/S0002-9904-1920-03305-7, ISSN 0002-9904
+Vandiver, H. S. (1924), "Book Review: History of the Theory of Numbers", Bulletin of the American Mathematical Society, 30 (1): 65–70, doi:10.1090/S0002-9904-1924-03852-X, ISSN 0002-9904
+
+
+== External links ==
+History of the Theory of Numbers - Volume 1 at the Internet Archive.
+History of the Theory of Numbers - Volume 2 at the Internet Archive.
+History of the Theory of Numbers - Volume 3 at the Internet Archive.

data/en.wikipedia.org/wiki/Horologium_Oscillatorium-0.md

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+---
+title: "Horologium Oscillatorium"
+chunk: 1/3
+source: "https://en.wikipedia.org/wiki/Horologium_Oscillatorium"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:14.621466+00:00"
+instance: "kb-cron"
+---
+
+Horologium Oscillatorium: Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae (English: The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks) is a book published by Dutch mathematician and physicist Christiaan Huygens in 1673 and his major work on pendula and horology. It is regarded as one of the three most important works on mechanics in the 17th century, the other two being Galileo’s Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) and Newton’s Philosophiæ Naturalis Principia Mathematica (1687).
+Much more than a mere description of clocks, Huygens's Horologium Oscillatorium is the first modern treatise in which a physical problem (the accelerated motion of a falling body) is idealized by a set of parameters then analyzed mathematically and constitutes one of the seminal works of applied mathematics. The book is also known for its strangely worded dedication to Louis XIV. The appearance of the book in 1673 was a political issue, since at that time the Dutch Republic was at war with France; Huygens was anxious to show his allegiance to his patron, which can be seen in the obsequious dedication to Louis XIV.
+
+== Overview ==
+
+The motivation behind Horologium Oscillatorium (1673) goes back to the idea of using a pendulum to keep time, which had already been proposed by people engaged in astronomical observations such as Galileo. Mechanical clocks at the time were instead regulated by balances that were often very unreliable. Moreover, without reliable clocks, there was no good way to measure longitude at sea, which was particularly problematic for a country dependent on sea trade like the Dutch Republic.
+Huygens interest in using a freely suspended pendulum to regulate clocks began in earnest in December 1656. He had a working model by the next year which he patented and then communicated to others such as Frans van Schooten and Claude Mylon. Although Huygens’s design, published in a short tract entitled Horologium (1658), was a combination of existing ideas, it nonetheless became widely popular and many pendulum clocks by Salomon Coster and his associates were built on it. Existing clock towers, such as those at Scheveningen and Utrecht, were also retrofitted following Huygens's design.
+Huygens continued his mathematical studies on free fall shortly after and, in 1659, obtained a series of remarkable results. At the same time, he was aware that the periods of simple pendula are not perfectly tautochronous, that is, they do not keep exact time but depend to some extent on their amplitude. Huygens was interested in finding a way to make the bob of a pendulum move reliably and independently of its amplitude. The breakthrough came later that same year when he discovered that the ability to keep perfect time can be achieved if the path of the pendulum bob is a cycloid. However, it was unclear what form to give the metal cheeks regulating the pendulum to lead the bob in a cycloidal path. His famous and surprising solution was that the cheeks must also have the form of a cycloid, on a scale determined by the length of the pendulum. These and other results led Huygens to develop his theory of evolutes and provided the incentive to write a much larger work, which became the Horologium Oscillatorium.
+After 1673, during his stay in the Academie des Sciences, Huygens studied harmonic oscillation more generally and continued his attempt at determining longitude at sea using his pendulum clocks, but his experiments carried on ships were not always successful.
+
+== Contents ==
+
+In the Preface, Huygens states:
+
+For it is not in the nature of a simple pendulum to provide equal and reliable measurements of time… But by a geometrical method we have found a different and previously unknown way to suspend the pendulum… [so that] the time of the swing can be chosen equal to some calculated value
+The book is divided into five interconnected parts. Parts I and V of the book contain descriptions of clock designs. The rest of the book is made of three, highly abstract, mathematical and mechanical parts dealing with pendular motion and a theory of curves. Except for Part IV, written in 1664, the entirety of the book was composed in a three-month period starting in October 1659.
+
+=== Part I: Description of the oscillating clock ===
+Huygens spends the first part of the book describing in detail his design for an oscillating pendulum clock. It includes descriptions of the endless chain, a lens-shaped bob to reduce air resistance, a small weight to adjust the pendulum swing, an escapement mechanism for connecting the pendulum to the gears, and two thin metal plates in the shape of cycloids mounted on either side to limit pendular motion. This part ends with a table to adjust for the inequality of the solar day, a description on how to draw a cycloid, and a discussion of the application of pendulum clocks for the determination of longitude at sea.
+
+=== Part II: Fall of weights and motion along a cycloid ===
+In the second part of the book, Huygens states three hypotheses on the motion of bodies, which can be seen as precursors to Newton's three laws of motion. They are essentially the law of inertia, the effect of gravity on uniform motion, and the law of composition of motion:

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+---
+title: "Horologium Oscillatorium"
+chunk: 2/3
+source: "https://en.wikipedia.org/wiki/Horologium_Oscillatorium"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:14.621466+00:00"
+instance: "kb-cron"
+---
+
+If there is no gravity, and the air offers no resistance to the motion of bodies, then any one of these bodies admits of a single motion to be continued with an equal velocity along a straight line.
+Now truly this motion becomes, under the action of gravity and for whatever the direction of the uniform motion, a motion composed from that constant motion that a body now has or had previously, together with the motion due gravity downwards.
+Also, either of these motions can be considered separately, neither one to be impeded by the other.
+He uses these three rules to re-derive geometrically Galileo's original study of falling bodies, including linear fall along inclined planes and fall along a curved path. He then studies constrained fall, culminating with a proof that a body falling along an inverted cycloid reaches the bottom in a fixed amount of time, regardless of the point on the path at which it begins to fall. This in effect shows the solution to the tautochrone problem as given by a cycloid curve. In modern notation:
+
+  
+    
+      
+        (
+        π
+        
+          /
+        
+        2
+        )
+        √
+        (
+        2
+        D
+        
+          /
+        
+        g
+        )
+      
+    
+    {\displaystyle (\pi /2)\surd (2D/g)}
+  
+
+The following propositions are covered in Part II:
+
+=== Part III: Size and evolution of the curve ===
+
+In the third part of the book, Huygens introduces the concept of an evolute as the curve that is "unrolled" (Latin: evolutus) to create a second curve known as the involute. He then uses evolutes to justify the cycloidal shape of the thin plates in Part I. Huygens originally discovered the isochronism of the cycloid using infinitesimal techniques but in his final publication he resorted to proportions and reductio ad absurdum, in the manner of Archimedes, to rectify curves such as the cycloid, the parabola, and other higher order curves.
+The following propositions are covered in Part III:
+
+=== Part IV: Center of oscillation or movement ===
+The fourth and longest part of the book contains the first successful theory of the center of oscillation, together with special methods for applying the theory, and the calculations of the centers of oscillation of several plane and solid figures. Huygens introduces physical parameters into his analysis while addressing the problem of the compound pendulum. 
+It starts with a number of definitions and proceeds to derive propositions using Torricelli's Principle: If some weights begin to move under the force of gravity, then it is not possible for the center of gravity of these weights to ascend to a greater height than that found at the beginning of the motion. Huygens called this principle "the chief axiom of mechanics" and used it like a conservation of kinetic energy principle, without recourse to forces or torques. In the process, he obtained solutions to dynamical problems such as the period of an oscillating pendulum as well as a compound pendulum, the center of oscillation and its interchangeability with the pivot point, and the concept of moment of inertia and the constant of gravitational acceleration. Huygens made use, implicitly, of the formula for free fall. In modern notation:
+
+  
+    
+      
+        d
+        =
+        1
+        
+          /
+        
+        2
+        g
+        
+          t
+          
+            2
+          
+        
+      
+    
+    {\displaystyle d=1/2gt^{2}}
+  
+
+The following propositions are covered in Part IV:
+
+=== Part V: Alternative design and centrifugal force ===
+The last part of the book returns to the design of a clock where the motion of the pendulum is circular, and the string unwinds from the evolute of a parabola. It ends with thirteen propositions regarding bodies in uniform circular motion, without proofs, and states the laws of centripetal force for uniform circular motion. These propositions were studied closely at the time, although their proofs were only published posthumously in the De Vi Centrifuga (1703).
+
+=== Summary ===
+Many of the propositions found in the Horologium Oscillatorium had little to do with clocks but rather point to the evolution of Huygens’s ideas. When an attempt to measure the gravitational constant using a pendulum failed to give consistent results, Huygens abandoned the experiment and instead idealized the problem into a mathematical study comparing free fall and fall along a circle. 
+Initially, he followed Galileo’s approach to the study of fall, only to leave it shortly after when it was clear the results could not be extended to curvilinear fall. Huygens then tackled the problem directly by using his own approach to infinitesimal analysis, a combination of analytic geometry, classical geometry, and contemporary infinitesimal techniques. Huygens chose not to publish the majority of his results using these techniques but instead adhered as much as possible to a strictly classical presentation, in the manner of Archimedes.
+
+== Legacy ==

data/en.wikipedia.org/wiki/Horologium_Oscillatorium-2.md

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+---
+title: "Horologium Oscillatorium"
+chunk: 3/3
+source: "https://en.wikipedia.org/wiki/Horologium_Oscillatorium"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:14.621466+00:00"
+instance: "kb-cron"
+---
+
+=== Reception ===
+Initial reviews of Huygens's Horologium Oscillatorium in major research journals at the time were generally positive. An anonymous review in Journal de Sçavans (1674) praised the author of the book for his invention of the pendulum clock "which brings the greatest honor to our century because it is of utmost importance... for astronomy and for navigation" while also noting the elegant, but difficult, mathematics needed to fully understand the book. Another review in the Giornale de' Letterati (1674) repeated many of the same points than the first one, with further elaboration on Huygens's trials at sea. The review in the Philosophical Transactions (1673) likewise praised the author for his invention but mentions other contributors to the clock design, such as William Neile, that in time would lead to a priority dispute.
+In addition to submitting his work for review, Huygens sent copies of his book to individuals throughout Europe, including statesmen such as Johan De Witt, and mathematicians such as Gilles de Roberval and Gregory of St. Vincent. Their appreciation of the text was due not exclusively on their ability to comprehend it fully but rather as a recognition of Huygens’s intellectual standing, or of his gratitude or fraternity that such gift implied. Thus, sending copies of the Horologium Oscillatorium worked in a manner similar to a gift of an actual clock, which Huygens had also sent to several people, including Louis XIV and the Grand Duke Ferdinand II.
+
+=== Mathematical style ===
+
+Huygens's mathematics in the Horologium Oscillatorium and elsewhere is best characterized as geometrical analysis of curves and of motions. It closely resembled classical Greek geometry in style, as Huygens preferred the works of classical authors, above all Archimedes. He was also proficient in the analytical geometry of Descartes and Fermat, and made use of it particularly in Parts III and IV of his book. With these and other infinitesimal tools, Huygens was quite capable of finding solutions to hard problems that today are solved using mathematical analysis, such as proving a uniqueness theorem for a class of differential equations, or extending approximation and inequalities techniques to the case of second order differentials.
+Huygens's manner of presentation (i.e., clearly stated axioms, followed by propositions) also made an impression among contemporary mathematicians, including Newton, who studied the propositions on centrifugal force very closely and later acknowledged the influence of Horologium Oscillatorium on his own major work. Nonetheless, the Archimedean and geometrical style of Huygens's mathematics soon fell into disuse with the advent of the calculus, making it more difficult for subsequent generations to appreciate his work.
+
+=== Appraisal ===
+Huygens’s most lasting contribution in the Horologium Oscillatorium is his thorough application of mathematics to explain pendulum clocks, which were the first reliable timekeepers fit for scientific use. His mastery of geometry and physics to design and analyze a precision instrument arguably anticipated the advent of mechanical engineering.
+Huygens's analyses of the cycloid in Parts II and III would later lead to the studies of many other such curves, including the caustic, the brachistochrone, the sail curve, and the catenary. Additionally, his exacting mathematical dissection of physical problems into a minimum of parameters provided an example for others (such as the Bernoullis) on work in applied mathematics that would be carry on in the following centuries, albeit in the language of the calculus.
+
+== Editions ==
+Huygens’s own manuscript of the book is missing, but he bequeathed his notebooks and correspondence to the Library of the University of Leiden, now in the Codices Hugeniorum. Much of the background material is in Oeuvres Complètes, vols. 17-18.
+Since its publication in France in 1673, Huygens’s work has been available in Latin and in the following modern languages:
+
+First publication. Horologium Oscillatorium, Sive De Motu Pendulorum Ad Horologia Aptato Demonstrationes Geometricae. Latin. Paris: F. Muguet, 1673. [14] + 161 + [1] pages.[1].
+Later edition by W.J. ’s Gravesande. In Christiani Hugenii Zulichemii Opera varia, 4 vols. Latin. Leiden: J. vander Aa, 1724, 15–192. [Repr. as Christiani Hugenii Zulichemii opera mechanica, geometrica, astronomica et miscellenea, 4 vols., Leiden: G. Potvliet et alia, 1751].
+Standard edition. In Oeuvres Complètes, vol. 18. French and Latin. The Hague: Martinus Nijhoff, 1934, 68–368.
+German translation. Die Pendeluhr (trans. A. Heckscher and A. von Oettingen), Leipzig: Engelmann, 1913 (Ostwalds Klassiker der exakten Wissenschaften, no. 192).
+Italian translation. L’orologio a pendolo (trans. C. Pighetti), Florence: Barbèra, 1963. [Also includes an Italian translation of Traité de la Lumière].
+French translation. L’Horloge oscillante (trans. J. Peyroux), Bordeaux: Bergeret, 1980. [Photorepr. Paris: Blanchard, 1980].
+English translation. Christiaan Huygens’ The Pendulum Clock, or Geometrical Demonstrations Concerning the Motion Of Pendula As Applied To Clocks (trans. R.J. Blackwell), Ames: Iowa State University Press, 1986.
+Dutch translation. Christiaan Huygens: Het Slingeruurwerk, een studie (transl. J. Aarts), Utrecht: Epsilon Uitgaven, 2015.
+
+== References ==

data/en.wikipedia.org/wiki/Hydrodynamica-0.md

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+---
+title: "Hydrodynamica"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Hydrodynamica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:15.780405+00:00"
+instance: "kb-cron"
+---
+
+Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii (Latin for Hydrodynamics, or commentaries on the forces and motions of fluids) is a book published by Daniel Bernoulli in 1738. The title of this book eventually christened the field of fluid mechanics as hydrodynamics.
+This book introduced the Bernoulli's principle, stating the first form of conservation of energy in fluid dynamics.
+
+
+== Description ==
+The book deals with fluid mechanics and is organized around preliminary versions of the conservation of energy, as received from Christiaan Huygens's formulation of vis viva (Latin for living forces). The book describes the theory of water flowing through a tube and of water flowing from a hole in a container. In doing so, Bernoulli explained the nature of hydrodynamic pressure and discovered the role of loss of vis viva in fluid flow, which would later be known as the Bernoulli principle. The book also discusses hydraulic machines and introduces the notion of work and efficiency of a machine.
+In the tenth chapter, Bernoulli discussed a primitive version of kinetic theory of gases. Assuming that heat increases the velocity of the gas particles, he first demonstrated that the pressure of air is proportional to kinetic energy of gas particles, thus making the temperature of gas proportional to this kinetic energy as well. In this chapter Bernoulli introduces a correction to the volume that appears in Boyle's law, anticipating the Van der Waals equation by more than a century. However most of Bernoulli's theories of this chapter were ignored historically.
+
+
+== Table of contents ==
+The book is divided in 13 sections:
+
+Which is the introduction, and contains various matters to be considered initially
+Which discusses the equilibrium of fluids at rest, both within themselves, as well as related to other causes
+Concerning the velocities of fluids flowing from some kind of vessel through an opening of any kind
+Concerned with the various times, which are desired in the efflux of the water
+Concerning the motion of water from vessels being filled constantly
+Concerning fluids not flowing out, or, moving within the walls of the vessels
+Concerning the motion of water through submerged vessels, where it is shown by examples, either how significantly useful the principle of the conservation of living forces shall be, or as in these cases in which a certain amount is agreed to be lost from these continually.
+Concerning the motion both of homogeneous as well as heterogeneous fluids through vessels of irregular construction divided up into several parts, where the individual phenomena of the trajectories of the fluids through a number of openings may be explained and a part of the motion may be absorbed continually from the theory of living forces; and with the general rules for the motions of the fluids defined everywhere
+Concerning the motion of fluids which are not ejected by their own weight but by certain other forces, and which concern hydraulic machines, especially where the highest degree of perfection of the same can be given, and how they can be perfected further both by the mechanics of solids as well as of fluids
+Concerning the properties and motions of elastic fluids, but especially those of air.
+Concerning fluids acting in a vortex, also those which may be contained in moving vessels. This is a relatively short chapter, in which Bernoulli tries to reconcile the vortex theory of planetary motion with Newton’s Law of gravitation, as well as presenting the theory of fluid vortices, and some interesting experiments involving fluids in accelerating frames of reference.
+Which presents the static properties of moving fluids, what I call static-hydraulics
+Concerning the reaction of fluids flowing out of vessels, and with the impulse of the same after they have flowed out, on planes which they meet.
+
+
+== Reception ==
+Leonhard Euler, friend of Daniel Bernoulli, sent his criticism as soon as the book was published. Bernoulli accepted some of the criticism but considered that Euler's work on fluids was too abstract and did not describe the real world.
+A rivalry priority dispute started between Daniel and his father Johann Bernoulli who had also written on the matter. Johann claimed priority on the Bernoulli's principle. Johann's book Hydraulica was published in 1743 but falsely dated 1732.
+
+
+== See also ==
+Analyse des infiniment petits pour l'intelligence des lignes courbes by Guillaume de l'Hôpital derived from Johann Bernoulli's work
+
+
+== References ==
+
+
+== Bibliography ==
+Mikhailov, G.K. (2005). "Hydrodynamica". In Grattan-Guinness, Ivor (ed.). Landmark Writings in Western Mathematics 1640–1940. Elsevier. pp. 131–42. ISBN 978-0-08-045744-4.
+Bernoulli, Daniel (1738). Hydrodynamica, sive de viribus et motibus fluidorum commentarii (in Latin, source ETH-Bibliothek Zürich, Rar 5503). sumptibus Johannis Reinholdi Dulseckeri; Typis Joh. Deckeri, typographi Basiliensis. doi:10.3931/e-rara-3911.

data/en.wikipedia.org/wiki/IJP_The_Book_of_Surfaces-0.md

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+---
+title: "IJP The Book of Surfaces"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/IJP_The_Book_of_Surfaces"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:16.947814+00:00"
+instance: "kb-cron"
+---
+
+IJP the book of surfaces is a book by George L. Legendre, with a foreword by Mohsen Mostafavi.
+
+
+== Overview ==
+IJP the Book of Surfaces was released in 2003 by the publishing arm of the London-based Architectural Association School of Architecture. The book features six essays on the notion of surface written from an architectural, philosophical, literary, mathematical, and computational angle, as well as several lighter asides ranging from cookery to poetry. These threads have been given a particular typographic and graphic design treatment meant to weave them together into a continuous narrative.
+
+
+== Background and literary references ==
+The book addresses some significant developments of the decade, such as the explosion of computational tools; the emergence of the 3D surface as an architectural signifier of the Digital Revolution;  the profession's fascination with the formal possibilities of surface cladding; and the rise of innovative manufacturing technologies. It can be compared to contemporary titles like Mohsen Mostafavi’s and David Leatherbarrow’s Surface Architecture, an essay on the phenomenology of architectural façades, and Ellen Lupton’s collection Skin: Surface, Substance + Design, which explores the working metaphor of artificial skin in Materials science, fashion and the visual arts. By comparison, Illa Berman notes that IJP the Book of Surfaces withdraws from external cultural currents and their contexts and emerges from within the formal and computational specificity of the surface itself.   As a piece of writing, it is indebted to the literary school Oulipo. Its treatment of one theme as a collection of vignettes written in different voices (linguistic, mathematical, computational, mock-literary, and pop-cultural) nods back to Raymond Queneau’s 1947 Exercises in Style, in which the same trivial event is told and re-told in different idioms.
+
+
+== Form and content ==
+In keeping with the literary/mathematical spirit of Oulipo, layout, typography, and pagination form an integral part of the book's thesis. The pagination taps the formal affinity between a publisher's book spread and a mathematician's surface, both of which draw on the concept of mathematical matrix.  Similar mathematical references apply to the title of the work, which combines 'i' and 'j', two symbols commonly used in matrix algebra, with the symbol 'p' (for point), introduced by the author in reference to Euclidean space.
+
+
+== Reception ==
+The book's argument and restrained use of computer graphics by the standards of the day (dominated then as now by computer-generated renderings) elicited a mixed reception. Historians and theorists noted its withdrawal from the wider cultural context, its consistent argument, graphic sobriety, and theoretical reach. Some readers noted its lack of engagement with other pressing issues of the day, such as sustainability and ecology, as well as its blank and solipsistic tone, occasionally questionable syntax, and pedestrian graphic design. The book's reception may have reflected disagreements in the architectural community at the time over the meaning of innovation and the finality of computational tools. After 2015, similar disagreements arose between the trade press and the self-identifying avant-garde movement of architectural parametricism. Legendre has since distanced himself from the broader cultural claims of the movement).
+
+
+== See also ==
+Design computing – Computing as applied to design
+Digital architecture – Architecture using digital technology
+
+
+== References ==

data/en.wikipedia.org/wiki/In_Pursuit_of_the_Traveling_Salesman-0.md

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+---
+title: "In Pursuit of the Traveling Salesman"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/In_Pursuit_of_the_Traveling_Salesman"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:18.117712+00:00"
+instance: "kb-cron"
+---
+
+In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation is a book on the travelling salesman problem, by William J. Cook, published in 2011 by the Princeton University Press, with a paperback reprint in 2014. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
+
+
+== Topics ==
+The travelling salesman problem asks to find the shortest cyclic tour of a collection of points, in the plane or in more abstract mathematical spaces.
+Because the problem is NP-hard, algorithms that take polynomial time are unlikely to be guaranteed to find its optimal solution; on the other hand a brute-force search of all permutations would always solve the problem exactly but would take far too long to be usable for all but the smallest problems. Threading a middle ground between these too-fast and too-slow running times, and developing a practical system that can find the exact solution of larger instances, raises difficult questions of algorithm engineering, which have sparked the development of "many of the concepts and techniques of combinatorial optimization".
+The introductory chapter of the book explores the limits of calculation on the problem, from 49-point problems solved by hand in the mid-1950s by George Dantzig, D. R. Fulkerson, and Selmer M. Johnson to a problem with 85,900 points solved optimally in 2006 by the Concorde TSP Solver, which Cook helped develop. The next chapters covers the early history of the problem and of related problems, including Leonhard Euler's work on the Seven Bridges of Königsberg, William Rowan Hamilton's Icosian game, and Julia Robinson first naming the problem in 1949. Another chapter describes real-world applications of the problem, ranging "from genome sequencing and designing computer processors to arranging music and hunting for planets". Reviewer Brian Hayes cites "the most charming revelation" of the book as being the fact that one of those real-world applications has been route planning for actual traveling salesmen in the early 20th century.
+Chapters four through seven, "core of the book", discuss methods for solving the problem, leading from heuristics and metaheuristics, linear programming relaxation, and cutting-plane methods, up to the branch and bound method that combines these techniques and is used by Concorde. The next two chapters also cover technical material, on the performance of computer implementations and on the Computational complexity theory of the problem.
+The remaining chapters are more human-centered, covering human and animal problem-solving strategies, and the incorporation of TSP solutions into the artworks of Julian Lethbridge, Robert A. Bosch, and others. A short final summary chapter suggests possible future directions, including the possibility of progress on the P versus NP problem.
+
+
+== Audience ==
+The book is intended for a non-specialist audience, avoids technical detail and is written "in an easy to understand style". It includes many historical asides, examples, applications, and biographical information and photographs of key players in the story, making it accessible to readers without a mathematical background.
+Although In Pursuit of the Traveling Salesman is not a textbook, reviewer Christopher Thompson suggests that some of its material on the use of linear programming and on applications of the problem "would be well-suited for classroom use", citing in particular the way it links multiple fields including numerical analysis, graph theory, algorithm design, logic, and statistics.
+Reviewer Stan Wagon writes that "any reader with an interest in combinatorial algorithms will find much of value in this book". Jan Karel Lenstra and David Shmoys write that "The writing is relaxed and entertaining; the presentation is excellent. We greatly enjoyed reading it." And reviewer Haris Aziz concludes "The book is highly recommended to any one with a mathematical curiosity and interest in the
+development of ideas.".
+
+
+== Related works ==
+More details of Cook's work with Concorde, suitable for more serious researchers on the problem and on related topics, can be found in an earlier book by Cook with David Applegate, Robert E. Bixby
+and Václav Chvátal, The Traveling Salesman Problem: A Computational Study (2007).
+Other books on the travelling salesman problem, also more technical than In Pursuit of the Traveling Salesman, include The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (by Lawler, Lenstra, Rinnooy Kan, and Shmoys, 1985) and The Traveling Salesman Problem and Its Variations (by Gutin and Punnen, 2006).
+
+
+== References ==
+
+
+== Further reading ==
+Ellenberg, Jordan (March 10, 2012), "The fuzzy path may be shortest (review of In Pursuit of the Traveling Salesman)", The Wall Street Journal
+McLemee, Scott (March 21, 2012), "Algorithm of a salesman (review of In Pursuit of the Traveling Salesman)", Inside Higher Education

data/en.wikipedia.org/wiki/Indra's_Pearls_(book)-0.md

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+---
+title: "Indra's Pearls (book)"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Indra's_Pearls_(book)"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:19.251800+00:00"
+instance: "kb-cron"
+---
+
+Indra's Pearls: The Vision of Felix Klein is a geometry book written by David Mumford, Caroline Series and David Wright, and published by Cambridge University Press in 2002 and 2015.
+The book explores the patterns created by iterating conformal maps of the complex plane called Möbius transformations, and their connections with symmetry and self-similarity. These patterns were glimpsed by German mathematician Felix Klein, but modern computer graphics allows them to be fully visualised and explored in detail.
+
+
+== Title ==
+The book's title refers to Indra's net, a metaphorical object described in the Buddhist text of the Flower Garland Sutra. Indra's net consists of an infinite array of gossamer strands and pearls. The frontispiece to Indra's Pearls quotes the following description:
+
+In the glistening surface of each pearl are reflected all the other pearls ... In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end.
+The allusion to Felix Klein's "vision" is a reference to Klein's early investigations of Schottky groups and hand-drawn plots of their limit sets. It also refers to Klein's wider vision of the connections between group theory, symmetry and geometry - see Erlangen program.
+
+
+== Contents ==
+The contents of Indra's Pearls are as follows:
+
+Chapter 1. The language of symmetry – an introduction to the mathematical concept of symmetry and its relation to geometric groups.
+Chapter 2. A delightful fiction – an introduction to complex numbers and mappings of the complex plane and the Riemann sphere.
+Chapter 3. Double spirals and Möbius maps – Möbius transformations and their classification.
+Chapter 4. The Schottky dance – pairs of Möbius maps which generate Schottky groups; plotting their limit sets using breadth-first searches.
+Chapter 5. Fractal dust and infinite words – Schottky limit sets regarded as fractals; computer generation of these fractals using depth-first searches and iterated function systems.
+Chapter 6. Indra's necklace – the continuous limit sets generated when pairs of generating circles touch.
+Chapter 7. The glowing gasket – the Schottky group whose limit set is the Apollonian gasket; links to the modular group.
+Chapter 8. Playing with parameters – parameterising Schottky groups with parabolic commutator using two complex parameters; using these parameters to explore the Teichmüller space of Schottky groups.
+Chapter 9. Accidents will happen – introducing Maskit's slice, parameterised by a single complex parameter; exploring the boundary between discrete and non-discrete groups.
+Chapter 10. Between the cracks – further exploration of the Maskit boundary between discrete and non-discrete groups in another slice of parameter space; identification and exploration of degenerate groups.
+Chapter 11. Crossing boundaries – ideas for further exploration, such as adding a third generator.
+Chapter 12. Epilogue – concluding overview of non-Euclidean geometry and Teichmüller theory.
+
+
+== Importance ==
+Indra's Pearls  is unusual because it aims to give the reader a sense of the development of a real-life mathematical investigation, rather than just a formal presentation of the final results. It covers a broad span of topics, showing interconnections among geometry, number theory, abstract algebra and computer graphics.  It shows how computers are used by contemporary mathematicians. It uses computer graphics, diagrams and cartoons to enhance its written explanations. In the authors' own words:
+
+Our dream is that this book will reveal to our readers that mathematics is not alien and remote but just a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty - Indra's Pearls p viii.
+
+
+== References ==
+Mumford, David; Series, Caroline; Wright, David (2002), Indra's pearls (Hardback ed.), Cambridge University Press, ISBN 978-0-521-35253-6, MR 1913879
+Mumford, David; Series, Caroline; Wright, David (2015), Indra's pearls (Paperback ed.), Cambridge University Press, ISBN 978-1-107-56474-9
+
+
+== External links ==
+Indra's Pearls Web Site

data/en.wikipedia.org/wiki/International_Encyclopedia_of_Statistical_Science-0.md

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+---
+title: "International Encyclopedia of Statistical Science"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/International_Encyclopedia_of_Statistical_Science"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:21.535662+00:00"
+instance: "kb-cron"
+---
+
+The International Encyclopedia of Statistical Science is a statistical sciences reference published by Springer. It has been described as one of the scientific projects with the largest number of involved countries ever, since it includes contributors coming from 105 countries and six continents. It contains the last papers written by Hirotugu Akaike, Nobel Laureate Sir Clive Granger, John Nelder and Erich Leo Lehmann.
+The team has been nominated for the 2026 Nobel Peace Prize by qualified nominators from Cambodia and Spain, with additional nominations anticipated .
+The first edition, in three volumes, was edited by Miodrag Lovrić and appeared in December 2010. It is published by Springer and it is available in print and online form.
+
+
+== See also ==
+Encyclopedia of Statistical Sciences
+
+
+== References ==
+
+
+== External links ==
+Publisher's webpage for this Encyclopedia

data/en.wikipedia.org/wiki/Introduction_to_Circle_Packing-0.md

@@ -0,0 +1,33 @@
+---
+title: "Introduction to Circle Packing"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Introduction_to_Circle_Packing"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:22.697059+00:00"
+instance: "kb-cron"
+---
+
+Introduction to Circle Packing: The Theory of Discrete Analytic Functions is a mathematical monograph concerning systems of tangent circles and the circle packing theorem. It was written by Kenneth Stephenson and published in 2005 by the Cambridge University Press.
+
+
+== Topics ==
+Circle packings, as studied in this book, are systems of circles that touch at tangent points but do not overlap, according to a combinatorial pattern of adjacencies specifying which pairs of circles should touch. The circle packing theorem states that a circle packing exists if and only if the pattern of adjacencies forms a planar graph; it was originally proved by Paul Koebe in the 1930s, and popularized by William Thurston, who rediscovered it in the 1970s and connected it with the theory of conformal maps and conformal geometry. As a topic, this should be distinguished from sphere packing, which considers higher dimensions (here, everything is two dimensional) and is more focused on packing density than on combinatorial patterns of tangency.
+The book is divided into four parts, in progressive levels of difficulty. The first part introduces the subject visually, encouraging the reader to think about packings not just as static objects but as dynamic systems of circles that change in predictable ways when the conditions under which they are formed (their patterns of adjacency) change. The second part concerns the proof of the circle packing theorem itself, and of the associated rigidity theorem: every maximal planar graph can be associated with a circle packing that is unique up to Möbius transformations of the plane. More generally the same result holds for any triangulated manifold, with a circle packing on a topologically equivalent Riemann surface that is unique up to conformal equivalence.
+The third part of the book concerns the degrees of freedom that arise when the pattern of adjacencies is not fully triangulated (it is a planar graph, but not a maximal planar graph). In this case, different extensions of this pattern to larger maximal planar graphs will lead to different packings, which can be mapped to each other by corresponding circles. The book explores the connection between these mappings, which it calls discrete analytic functions, and the analytic functions of classical mathematical analysis. The final part of the book concerns a conjecture of William Thurston, proved by Burton Rodin and Dennis Sullivan, that makes this analogy concrete: conformal mappings from any topological disk to a circle can be approximated by filling the disk by a hexagonal packing of unit circles, finding a circle packing that adds to that pattern of adjacencies a single outer circle, and constructing the resulting discrete analytic function. This part also includes applications to number theory and the visualization of brain structure.
+Stephenson has implemented algorithms for circle packing and used them to construct the many illustrations of the book, giving to much of this work the flavor of experimental mathematics, although it is also mathematically rigorous. Unsolved problems are listed throughout the book, which also includes nine appendices on related topics such as the ring lemma and Doyle spirals.
+
+
+== Audience and reception ==
+The book presents research-level mathematics, and is aimed at professional mathematicians interested in this and related topics. Reviewer Frédéric Mathéus describes the level of the material in the book as "both mathematically rigorous and accessible to the novice mathematician", presented in an approachable style that conveys the author's love of the material. However, although the preface to the book states that no background knowledge is necessary, and that the book can be read by non-mathematicians or used as an undergraduate textbook, reviewer Michele Intermont disagrees, noting that it has no exercises for students and writing that "non-mathematicians will be nothing other than frustrated with this book". Similarly, reviewer David Mumford finds the first seven chapters (part I and much of part II) to be at an undergraduate level, but writes that "as a whole, the book is suitable for graduate students in math".
+
+
+== Publication ==
+Stephenson, Kenneth (2005), Introduction to circle packing: the theory of discrete analytic functions, New York: Cambridge University Press, ISBN 9780521823562, OCLC 55878014
+
+
+== References ==
+
+
+== External links ==
+Ken Stephenson's CirclePack software

data/en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknowns-0.md

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+---
+title: "Jade Mirror of the Four Unknowns"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknowns"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:23.909555+00:00"
+instance: "kb-cron"
+---
+
+Jade Mirror of the Four Unknowns, Siyuan yujian (simplified Chinese: 四元玉鉴; traditional Chinese: 四元玉鑒), also referred to as Jade Mirror of the Four Origins, is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie.
+The book consists of an introduction and three books, with a total of 288 problems. The first four  problems in the introduction illustrate his method of the four unknowns. He showed how to convert a problem stated verbally into a system of polynomial equations (up to the 14th order), by using up to four unknowns: 天 Heaven, 地 Earth, 人 Man, 物 Matter, and then how to reduce the system to a single polynomial equation in one unknown by successive elimination of unknowns. He then solved the high-order equation  by Southern Song dynasty mathematician Qin Jiushao's "Ling long kai fang" method published in Shùshū Jiǔzhāng (“Mathematical Treatise in Nine Sections”) in 1247 (more than 570 years before English mathematician William Horner's method using synthetic division). To do this, he makes use of the Pascal triangle, which he labels as the diagram of an ancient method first discovered by Jia Xian before 1050.
+Zhu also solved square and cube roots problems by solving quadratic and cubic equations, and added to the understanding of series and progressions, classifying them according to the coefficients of the Pascal triangle. He also showed how to solve systems of linear equations by reducing the matrix of their coefficients to diagonal form.
+Jade Mirror of the Four Unknowns consists of four books, with 24 classes and 288 problems, in which 232 problems deal with Tian yuan shu, 36 problems deal with variable of two variables, 13 problems of three variables, and 7 problems of four variables.
+
+== Introduction ==
+
+The four quantities are x, y, z, w  can be presented with the following diagram
+
+x
+y 太w
+z
+The square of which is:
+
+=== The Unitary Nebuls ===
+This section deals with Tian yuan shu or problems of one unknown.
+
+Question: Given the product of huangfan and zhi ji equals to 24 paces, and the sum of vertical and hypotenuse equals to 9 paces, what is the value of the base?
+Answer: 3 paces
+Set up  unitary tian as the base (that is let the base be the unknown quantity x)
+Since the product of huangfang and zhi ji = 24
+in which 
+
+huangfan is defined as：
+  
+    
+      
+        (
+        a
+        +
+        b
+        −
+        c
+        )
+      
+    
+    {\displaystyle (a+b-c)}
+  
+
+zhi ji：
+  
+    
+      
+        a
+        b
+      
+    
+    {\displaystyle ab}
+  
+
+therefore  
+  
+    
+      
+        (
+        a
+        +
+        b
+        −
+        c
+        )
+        a
+        b
+        =
+        24
+      
+    
+    {\displaystyle (a+b-c)ab=24}
+  
+
+Further, the sum of vertical and hypotenuse is
+
+  
+    
+      
+        b
+        +
+        c
+        =
+        9
+      
+    
+    {\displaystyle b+c=9}
+  
+
+Set up the unknown unitary tian as the vertical
+
+  
+    
+      
+        x
+        =
+        a
+      
+    
+    {\displaystyle x=a}
+  
+
+Then use Pythagoras to isolate c and b: 
+  
+    
+      
+        
+          a
+          
+            2
+          
+        
+        +
+        
+          b
+          
+            2
+          
+        
+        =
+        
+          c
+          
+            2
+          
+        
+        ⟺
+        
+          c
+          
+            2
+          
+        
+        −
+        
+          b
+          
+            2
+          
+        
+        =
+        
+          a
+          
+            2
+          
+        
+        ⟺
+        (
+        c
+        −
+        b
+        )
+        (
+        c
+        +
+        b
+        )
+        =
+        
+          a
+          
+            2
+          
+        
+        ⟺
+        c
+        −
+        b
+        =
+        
+          
+            
+              a
+              
+                2
+              
+            
+            
+              c
+              +
+              b
+            
+          
+        
+        =
+        
+          
+            
+              x
+              
+                2
+              
+            
+            9
+          
+        
+      
+    
+    {\displaystyle a^{2}+b^{2}=c^{2}\Longleftrightarrow c^{2}-b^{2}=a^{2}\Longleftrightarrow (c-b)(c+b)=a^{2}\Longleftrightarrow c-b={\frac {a^{2}}{c+b}}={\frac {x^{2}}{9}}}
+  
+
+such that we obtain:
+
+  
+    
+      
+        2
+        c
+        =
+        (
+        c
+        +
+        b
+        )
+        +
+        (
+        c
+        −
+        b
+        )
+        =
+        9
+        +
+        
+          
+            
+              x
+              
+                2
+              
+            
+            9
+          
+        
+      
+    
+    {\displaystyle 2c=(c+b)+(c-b)=9+{\frac {x^{2}}{9}}}
+  
+
+  
+    
+      
+        2
+        b
+        =
+        (
+        c
+        +
+        b
+        )
+        −
+        (
+        c
+        −
+        b
+        )
+        =
+        9
+        −
+        
+          
+            
+              x
+              
+                2
+              
+            
+            9
+          
+        
+      
+    
+    {\displaystyle 2b=(c+b)-(c-b)=9-{\frac {x^{2}}{9}}}
+  
+
+Combining everything, we obtain the following equation:
+
+     （
+  
+    
+      
+        
+          x
+          
+            5
+          
+        
+        −
+        9
+        
+          x
+          
+            4
+          
+        
+        −
+        81
+        
+          x
+          
+            3
+          
+        
+        +
+        729
+        
+          x
+          
+            2
+          
+        
+        =
+        3888
+      
+    
+    {\displaystyle x^{5}-9x^{4}-81x^{3}+729x^{2}=3888}
+  
+）
+ 太
+
+Solve it and obtain x=3
+
+=== The Mystery of Two Natures ===
+太 Unitary
+
+equation:  
+  
+    
+      
+        −
+        2
+        
+          y
+          
+            2
+          
+        
+        −
+        x
+        
+          y
+          
+            2
+          
+        
+        +
+        2
+        x
+        y
+        +
+        2
+        
+          x
+          
+            2
+          
+        
+        y
+        +
+        
+          x
+          
+            3
+          
+        
+        =
+        0
+      
+    
+    {\displaystyle -2y^{2}-xy^{2}+2xy+2x^{2}y+x^{3}=0}
+  
+;
+from the given
+
+太
+
+equation:  
+  
+    
+      
+        2
+        
+          y
+          
+            2
+          
+        
+        −
+        x
+        
+          y
+          
+            2
+          
+        
+        +
+        2
+        x
+        y
+        +
+        
+          x
+          
+            3
+          
+        
+        =
+        0
+      
+    
+    {\displaystyle 2y^{2}-xy^{2}+2xy+x^{3}=0}
+  
+;
+we get:
+
+太
+
+  
+    
+      
+        8
+        x
+        +
+        4
+        
+          x
+          
+            2
+          
+        
+        =
+        0
+      
+    
+    {\displaystyle 8x+4x^{2}=0}
+  
+
+and
+
+太
+
+  
+    
+      
+        2
+        
+          x
+          
+            2
+          
+        
+        +
+        
+          x
+          
+            3
+          
+        
+        =
+        0
+      
+    
+    {\displaystyle 2x^{2}+x^{3}=0}
+  
+
+by method of elimination, we obtain a quadratic equation
+
+  
+    
+      
+        
+          x
+          
+            2
+          
+        
+        −
+        2
+        x
+        −
+        8
+        =
+        0
+      
+    
+    {\displaystyle x^{2}-2x-8=0}
+  
+
+solution:  
+  
+    
+      
+        x
+        =
+        4
+      
+    
+    {\displaystyle x=4}
+  
+.
+
+=== The Evolution of Three Talents ===
+Template for solution of problem of three unknowns
+Zhu Shijie explained the method of elimination in detail. His example has been quoted frequently in scientific literature.
+Set up three equations as follows
+
+太
+
+  
+    
+      
+        −
+        y
+        −
+        z
+        −
+        
+          y
+          
+            2
+          
+        
+        x
+        −
+        x
+        +
+        x
+        y
+        z
+        =
+        0
+      
+    
+    {\displaystyle -y-z-y^{2}x-x+xyz=0}
+  
+ .... I
+
+  
+    
+      
+        −
+        y
+        −
+        z
+        +
+        x
+        −
+        
+          x
+          
+            2
+          
+        
+        +
+        x
+        z
+        =
+        0
+      
+    
+    {\displaystyle -y-z+x-x^{2}+xz=0}
+  
+.....II
+太
+
+  
+    
+      
+        
+          y
+          
+            2
+          
+        
+        −
+        
+          z
+          
+            2
+          
+        
+        +
+        
+          x
+          
+            2
+          
+        
+        =
+        0
+        ;
+      
+    
+    {\displaystyle y^{2}-z^{2}+x^{2}=0;}
+  
+....III
+Elimination of unknown between II and  III
+by manipulation of exchange of variables
+We obtain
+
+      太
+
+  
+    
+      
+        −
+        x
+        −
+        2
+        
+          x
+          
+            2
+          
+        
+        +
+        y
+        +
+        
+          y
+          
+            2
+          
+        
+        +
+        x
+        y
+        −
+        x
+        
+          y
+          
+            2
+          
+        
+        +
+        
+          x
+          
+            2
+          
+        
+        y
+      
+    
+    {\displaystyle -x-2x^{2}+y+y^{2}+xy-xy^{2}+x^{2}y}
+  
+ ...IV
+and
+
+太
+
+  
+    
+      
+        −
+        2
+        x
+        −
+        2
+        
+          x
+          
+            2
+          
+        
+        +
+        2
+        y
+        −
+        2
+        
+          y
+          
+            2
+          
+        
+        +
+        
+          y
+          
+            3
+          
+        
+        +
+        4
+        x
+        y
+        −
+        2
+        x
+        
+          y
+          
+            2
+          
+        
+        +
+        x
+        
+          y
+          
+            2
+          
+        
+      
+    
+    {\displaystyle -2x-2x^{2}+2y-2y^{2}+y^{3}+4xy-2xy^{2}+xy^{2}}
+  
+.... V
+Elimination of unknown between  IV and V we obtain a 3rd order equation
+
+  
+    
+      
+        
+          x
+          
+            4
+          
+        
+        −
+        6
+        
+          x
+          
+            3
+          
+        
+        +
+        4
+        
+          x
+          
+            2
+          
+        
+        +
+        6
+        x
+        −
+        5
+        =
+        0
+      
+    
+    {\displaystyle x^{4}-6x^{3}+4x^{2}+6x-5=0}
+  
+
+Solve to this 3rd order equation to obtain 
+  
+    
+      
+        x
+        =
+        5
+      
+    
+    {\displaystyle x=5}
+  
+;
+Change back the variables
+We obtain the hypothenus =5 paces
+
+=== Simultaneous of the Four Elements ===
+This section deals with simultaneous equations of four unknowns.
+
+  
+    
+      
+        
+          
+            {
+            
+              
+                
+                  −
+                  2
+                  y
+                  +
+                  x
+                  +
+                  z
+                  =
+                  0
+                
+              
+              
+                
+                  −
+                  
+                    y
+                    
+                      2
+                    
+                  
+                  x
+                  +
+                  4
+                  y
+                  +
+                  2
+                  x
+                  −
+                  
+                    x
+                    
+                      2
+                    
+                  
+                  +
+                  4
+                  z
+                  +
+                  x
+                  z
+                  =
+                  0
+                
+              
+              
+                
+                  
+                    x
+                    
+                      2
+                    
+                  
+                  +
+                  
+                    y
+                    
+                      2
+                    
+                  
+                  −
+                  
+                    z
+                    
+                      2
+                    
+                  
+                  =
+                  0
+                
+              
+              
+                
+                  2
+                  y
+                  −
+                  w
+                  +
+                  2
+                  x
+                  =
+                  0
+                
+              
+            
+            
+          
+        
+      
+    
+    {\displaystyle {\begin{cases}-2y+x+z=0\\-y^{2}x+4y+2x-x^{2}+4z+xz=0\\x^{2}+y^{2}-z^{2}=0\\2y-w+2x=0\end{cases}}}
+  

data/en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknowns-1.md

@@ -0,0 +1,710 @@
+---
+title: "Jade Mirror of the Four Unknowns"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Jade_Mirror_of_the_Four_Unknowns"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:23.909555+00:00"
+instance: "kb-cron"
+---
+
+Successive elimination of unknowns to get
+
+   
+  
+    
+      
+        4
+        
+          x
+          
+            2
+          
+        
+        −
+        7
+        x
+        −
+        686
+        =
+        0
+      
+    
+    {\displaystyle 4x^{2}-7x-686=0}
+  
+
+Solve this and obtain 14 paces
+
+== Book I ==
+
+=== Problems of Right Angle Triangles and Rectangles ===
+There are 18 problems in this section.
+Problem 18
+Obtain a tenth order  polynomial equation:
+
+  
+    
+      
+        16
+        
+          x
+          
+            10
+          
+        
+        −
+        64
+        
+          x
+          
+            9
+          
+        
+        +
+        160
+        
+          x
+          
+            8
+          
+        
+        −
+        384
+        
+          x
+          
+            7
+          
+        
+        +
+        512
+        
+          x
+          
+            6
+          
+        
+        −
+        544
+        
+          x
+          
+            5
+          
+        
+        +
+        456
+        
+          x
+          
+            4
+          
+        
+        +
+        126
+        
+          x
+          
+            3
+          
+        
+        +
+        3
+        
+          x
+          
+            2
+          
+        
+        −
+        4
+        x
+        −
+        177162
+        =
+        0
+      
+    
+    {\displaystyle 16x^{10}-64x^{9}+160x^{8}-384x^{7}+512x^{6}-544x^{5}+456x^{4}+126x^{3}+3x^{2}-4x-177162=0}
+  
+
+The root of which is x = 3,  multiply by 4, getting 12. That is the final answer.
+
+=== Problems of Plane Figures ===
+There are 18 problems in this section
+
+=== Problems of Piece Goods ===
+There are 9 problems in this section
+
+=== Problems on Grain Storage ===
+There are 6 problems in this section
+
+=== Problems on Labour ===
+There are 7 problems in this section
+
+=== Problems of Equations for Fractional Roots ===
+There are 13 problems in this section
+
+== Book II ==
+
+=== Mixed Problems ===
+
+=== Containment of Circles and Squares ===
+
+=== Problems on Areas ===
+
+=== Surveying with Right Angle Triangles ===
+There are eight problems in this section
+
+Problem 1
+
+Question: There is a rectangular town of unknown dimension which has one gate on each side. There is a pagoda located at 240 paces from the south gate. A man walking 180 paces from the west gate can see the pagoda, he then walks towards the south-east corner for 240 paces and reaches the pagoda; what is the length and width of the rectangular town?
+Answer: 120 paces in length and width one li
+Let tian yuan unitary as half of the length, we obtain a 4th order equation
+
+  
+    
+      
+        
+          x
+          
+            4
+          
+        
+        +
+        480
+        
+          x
+          
+            3
+          
+        
+        −
+        270000
+        
+          x
+          
+            2
+          
+        
+        +
+        15552000
+        x
+        +
+        1866240000
+        =
+        0
+      
+    
+    {\displaystyle x^{4}+480x^{3}-270000x^{2}+15552000x+1866240000=0}
+  
+
+solve it and obtain x=240 paces, hence length =2x= 480 paces=1 li and 120 paces.
+Similarity, let tian yuan unitary(x) equals to half of width
+we get the equation:
+
+  
+    
+      
+        
+          x
+          
+            4
+          
+        
+        +
+        360
+        
+          x
+          
+            3
+          
+        
+        −
+        270000
+        
+          x
+          
+            2
+          
+        
+        +
+        20736000
+        x
+        +
+        1866240000
+        =
+        0
+      
+    
+    {\displaystyle x^{4}+360x^{3}-270000x^{2}+20736000x+1866240000=0}
+  
+
+Solve it to obtain x=180 paces, length =360 paces =one li.
+
+Problem 7
+Identical to The depth of a ravine (using hence-forward cross-bars) in Haidao Suanjing.
+Problem 8
+Identical to The depth of a transparent pool  in Haidao Suanjing.
+
+=== Hay Stacks ===
+
+=== Bundles of Arrows ===
+
+=== Land Measurement ===
+
+=== Summon Men According to Need ===
+Problem No 5 is the earliest 4th order interpolation formula in the world
+men summoned :
+  
+    
+      
+        n
+        a
+        +
+        
+          
+            
+              1
+              
+                2
+                !
+              
+            
+          
+        
+        n
+        (
+        n
+        −
+        1
+        )
+        b
+        +
+        
+          
+            
+              1
+              
+                3
+                !
+              
+            
+          
+        
+        n
+        (
+        n
+        −
+        1
+        )
+        (
+        n
+        −
+        2
+        )
+        c
+        +
+        
+          
+            
+              1
+              
+                4
+                !
+              
+            
+          
+        
+        n
+        (
+        n
+        −
+        1
+        )
+        (
+        n
+        −
+        2
+        )
+        (
+        n
+        −
+        3
+        )
+        d
+      
+    
+    {\displaystyle na+{\tfrac {1}{2!}}n(n-1)b+{\tfrac {1}{3!}}n(n-1)(n-2)c+{\tfrac {1}{4!}}n(n-1)(n-2)(n-3)d}
+  
+
+In which
+
+a=1st order difference
+b=2nd order difference
+c=3rd order difference
+d=4th order difference
+
+== Book III ==
+
+=== Fruit pile ===
+This section contains 20 problems dealing with triangular piles, rectangular piles
+Problem 1
+Find the sum of triangular pile
+
+  
+    
+      
+        1
+        +
+        3
+        +
+        6
+        +
+        10
+        +
+        .
+        .
+        .
+        +
+        
+          
+            1
+            2
+          
+        
+        n
+        (
+        n
+        +
+        1
+        )
+      
+    
+    {\displaystyle 1+3+6+10+...+{\frac {1}{2}}n(n+1)}
+  
+
+and value of the fruit pile is:
+
+  
+    
+      
+        v
+        =
+        2
+        +
+        9
+        +
+        24
+        +
+        50
+        +
+        90
+        +
+        147
+        +
+        224
+        +
+        ⋯
+        +
+        
+          
+            1
+            2
+          
+        
+        n
+        (
+        n
+        +
+        1
+        
+          )
+          
+            2
+          
+        
+      
+    
+    {\displaystyle v=2+9+24+50+90+147+224+\cdots +{\frac {1}{2}}n(n+1)^{2}}
+  
+
+Zhu Shijie use Tian yuan shu to solve this problem by letting x=n
+and obtained the formular
+
+  
+    
+      
+        v
+        =
+        
+          
+            1
+            
+              2
+              ⋅
+              3
+              ⋅
+              4
+            
+          
+        
+        (
+        3
+        x
+        +
+        5
+        )
+        x
+        (
+        x
+        +
+        1
+        )
+        (
+        x
+        +
+        2
+        )
+      
+    
+    {\displaystyle v={\frac {1}{2\cdot 3\cdot 4}}(3x+5)x(x+1)(x+2)}
+  
+
+From given condition 
+  
+    
+      
+        v
+        =
+        1320
+      
+    
+    {\displaystyle v=1320}
+  
+, hence
+
+  
+    
+      
+        3
+        
+          x
+          
+            4
+          
+        
+        +
+        14
+        
+          x
+          
+            3
+          
+        
+        +
+        21
+        
+          x
+          
+            2
+          
+        
+        +
+        10
+        x
+        −
+        31680
+        =
+        0
+      
+    
+    {\displaystyle 3x^{4}+14x^{3}+21x^{2}+10x-31680=0}
+  
+
+Solve it to obtain 
+  
+    
+      
+        x
+        =
+        n
+        =
+        9
+      
+    
+    {\displaystyle x=n=9}
+  
+.
+Therefore,
+
+  
+    
+      
+        v
+        =
+        2
+        +
+        9
+        +
+        24
+        +
+        50
+        +
+        90
+        +
+        147
+        +
+        224
+        +
+        324
+        +
+        450
+        =
+        1320
+      
+    
+    {\displaystyle v=2+9+24+50+90+147+224+324+450=1320}
+  
+。
+
+=== Figures within Figure ===
+
+=== Simultaneous Equations ===
+
+=== Equation of two unknowns ===
+
+=== Left and Right ===
+
+=== Equation of Three Unknowns ===
+
+=== Equation of Four Unknowns ===
+Six problems of four unknowns.
+Question 2
+Yield a set of equations in four unknowns: .
+
+  
+    
+      
+        
+          
+            {
+            
+              
+                
+                  −
+                  3
+                  
+                    y
+                    
+                      2
+                    
+                  
+                  +
+                  8
+                  y
+                  −
+                  8
+                  x
+                  +
+                  8
+                  z
+                  =
+                  0
+                
+              
+              
+                
+                  4
+                  
+                    y
+                    
+                      2
+                    
+                  
+                  −
+                  8
+                  x
+                  y
+                  +
+                  3
+                  
+                    x
+                    
+                      2
+                    
+                  
+                  −
+                  8
+                  y
+                  z
+                  +
+                  6
+                  x
+                  z
+                  +
+                  3
+                  
+                    z
+                    
+                      2
+                    
+                  
+                  =
+                  0
+                
+              
+              
+                
+                  
+                    y
+                    
+                      2
+                    
+                  
+                  +
+                  
+                    x
+                    
+                      2
+                    
+                  
+                  −
+                  
+                    z
+                    
+                      2
+                    
+                  
+                  =
+                  0
+                
+              
+              
+                
+                  2
+                  y
+                  +
+                  4
+                  x
+                  +
+                  2
+                  z
+                  −
+                  w
+                  =
+                  0
+                
+              
+            
+            
+          
+        
+      
+    
+    {\displaystyle {\begin{cases}-3y^{2}+8y-8x+8z=0\\4y^{2}-8xy+3x^{2}-8yz+6xz+3z^{2}=0\\y^{2}+x^{2}-z^{2}=0\\2y+4x+2z-w=0\end{cases}}}
+  
+
+== References ==
+
+Sources

data/en.wikipedia.org/wiki/Journey_into_Geometries-0.md

@@ -0,0 +1,28 @@
+---
+title: "Journey into Geometries"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Journey_into_Geometries"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:25.051504+00:00"
+instance: "kb-cron"
+---
+
+Journey into Geometries is a book on non-Euclidean geometry. It was written by Hungarian-Australian mathematician Márta Svéd and published in 1991 by the Mathematical Association of America in their MAA Spectrum book series.
+
+
+== Topics ==
+Journey into Geometries is written as a conversation between three characters: Alice, from Alice's Adventures in Wonderland (but older and familiar with Euclidean geometry), Lewis Carroll, the author of Alice's adventures, and a modern mathematician named "Dr. Whatif". Its topics include hyperbolic geometry, inversive geometry, and projective geometry, following an arrangement of these topics credited to Australian mathematician Carl Moppert, and possibly based on an earlier German-language textbook on similar topics by F. Gonseth and P. Marti.
+As in Alice's original adventures, the first part of the book is arranged as a travelogue. This part of the book has six chapters, each ending with a set of exercises. Following these chapters, more conventionally written material covers geometric axiom systems and provides solutions to the exercises.
+
+
+== Audience and reception ==
+Reviewer William E. Fenton is unsure of the audience of the book, writing that it is not suitable as a textbook and would scare most undergraduates, but is too unserious for graduate students. David A. Thomas identifies the audience as "people who like to play with mathematical ideas".
+Fenton criticizes the book's style as a little too glib and lead-footed, and its illustrations as amateurish. H. W. Guggenheimer faults the treatment of projective geometry as "rather sketchy". Nevertheless, Fenton writes that he found the book engrossing and well-organized, particularly praising its exercises. Both Fenton and Guggenheimer recommend the book to talented students of mathematics, and both Fenton and David A. Thomas suggest it as auxiliary reading for geometry courses.
+
+
+== References ==
+
+
+== External links ==
+Journey into Geometries on the Internet Archive

data/en.wikipedia.org/wiki/La_Géométrie-0.md

@@ -0,0 +1,54 @@
+---
+title: "La Géométrie"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/La_Géométrie"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:56.902470+00:00"
+instance: "kb-cron"
+---
+
+La Géométrie (French pronunciation: [la ʒeɔmetʁi]) was published in 1637 as an appendix to Discours de la méthode (Discourse on the Method), written by René Descartes. In the Discourse, Descartes presents his method for obtaining clarity on any subject. La Géométrie and two other appendices, also by Descartes, La Dioptrique (Optics) and Les Météores (Meteorology), were published with the Discourse to give examples of the kinds of successes he had achieved following his method (as well as, perhaps, considering the contemporary European social climate of intellectual competitiveness, to show off a bit to a wider audience).
+
+The work was the first to propose the idea of uniting algebra and geometry into a single subject and invented an algebraic geometry called analytic geometry, which involves reducing geometry to a form of arithmetic and algebra and translating geometric shapes into algebraic equations. For its time this was ground-breaking. It also contributed to the mathematical ideas of Leibniz and Newton and was thus important in the development of calculus.
+
+
+== The text ==
+This appendix is divided into three "books". 
+Book I is titled Problems Which Can Be Constructed by Means of Circles and Straight Lines Only. In this book he introduces algebraic notation that is still in use today. The letters at the end of the alphabet, viz., x, y, z, etc. are to denote unknown variables, while those at the start of the alphabet, a, b, c, etc. denote constants. He introduces modern exponential notation for powers (except for squares, where he kept the older tradition of writing repeated letters, such as, aa). He also breaks with the Greek tradition of associating powers with geometric referents, a2 with an area, a3 with a volume and so on, and treats them all as possible lengths of line segments. These notational devices permit him to describe an association of numbers to lengths of line segments that could be constructed with straightedge and compass. The bulk of the remainder of this book is occupied by Descartes's solution to "the locus problems of Pappus." According to Pappus, given three or four lines in a plane, the problem is to find the locus of a point that moves so that the product of the distances from two of the fixed lines (along specified directions) is proportional to the square of the distance to the third line (in the three line case) or proportional to the product of the distances to the other two lines (in the four line case). In solving these problems and their generalizations, Descartes takes two line segments as unknown and designates them x and y. Known line segments are designated a, b, c, etc. The germinal idea of a Cartesian coordinate system can be traced back to this work.
+In the second book, called On the Nature of Curved Lines, Descartes described two kinds of curves, called by him geometrical and mechanical. Geometrical curves are those which are now described by algebraic equations in two variables, however, Descartes described them kinematically and an essential feature was that all of their points could be obtained by construction from lower order curves. This represented an expansion beyond what was permitted by straightedge and compass constructions. Other curves like the quadratrix and spiral, where only some of whose points could be constructed, were termed mechanical and were not considered suitable for mathematical study. Descartes also devised an algebraic method for finding the normal at any point of a curve whose equation is known. The construction of the tangents to the curve then easily follows and Descartes applied this algebraic procedure for finding tangents to several curves. 
+The third book, On the Construction of Solid and Supersolid Problems, is more properly algebraic than geometric and concerns the nature of equations and how they may be solved. He recommends that all terms of an equation be placed on one side and set equal to 0 to facilitate solution. He points out the factor theorem for polynomials and gives an intuitive proof that a polynomial of degree n has n roots. He systematically discussed negative and imaginary roots of equations and explicitly used what is now known as Descartes' rule of signs.
+
+
+== Aftermath ==
+Descartes wrote La Géométrie in French rather than the language used for most scholarly publication at the time, Latin. His exposition style was far from clear, the material was not arranged in a systematic manner and he generally only gave indications of proofs, leaving many of the details to the reader. His attitude toward writing is indicated by statements such as "I did not undertake to say everything," or "It already wearies me to write so much about it," that occur frequently. Descartes justifies his omissions and obscurities with the remark that much was deliberately omitted "in order to give others the pleasure of discovering [it] for themselves."
+Descartes is often credited with inventing the coordinate plane because he had the relevant concepts in his book, however, nowhere in La Géométrie does the modern rectangular coordinate system appear. This and other improvements were added by mathematicians who took it upon themselves to clarify and explain Descartes' work.
+This enhancement of Descartes' work was primarily carried out by Frans van Schooten, a professor of mathematics at Leiden and his students. Van Schooten published a Latin version of La Géométrie in 1649 and this was followed by three other editions in 1659−1661, 1683 and 1693. The 1659−1661 edition was a two volume work more than twice the length of the original filled with explanations and examples provided by van Schooten and his students. One of these students, Johannes Hudde provided a convenient method for determining double roots of a polynomial, known as Hudde's rule, that had been a difficult procedure in Descartes's method of tangents. These editions established analytic geometry in the seventeenth century.
+
+
+== See also ==
+Claude Rabuel
+
+
+== Notes ==
+
+
+== References ==
+Boyer, Carl B. (2004) [1956], History of Analytic Geometry, Dover, ISBN 978-0-486-43832-0
+Burton, David M. (2011), The History of Mathematics / An Introduction (7th ed.), McGraw Hill, ISBN 978-0-07-338315-6
+Descartes, René (2006) [1637]. A discourse on the method of correctly conducting one's reason and seeking truth in the sciences. Translated by Ian Maclean. Oxford University Press. ISBN 0-19-282514-3.
+
+
+== Further reading ==
+Grosholz, Emily (1998). "Chapter 4: Cartesian method and the Geometry". In Georges J. D. Moyal (ed.). René Descartes: critical assessments. Routledge. ISBN 0-415-02358-0.
+Hawking, Stephen W. (2005). "René Descartes". God created the integers: the mathematical breakthroughs that changed history. Running Press. pp. 285 ff. ISBN 0-7624-1922-9.
+Serfati, M. (2005). "Chapter 1: René Descartes, Géométrie, Latin edition (1649), French edition (1637)". In I. Grattan-Guinness; Roger Cooke (eds.). Landmark writings in Western mathematics 1640-1940. Elsevier. ISBN 0-444-50871-6.
+Smith, David E.; Latham, M. L. (1954) [1925]. The Geometry of René Descartes. Dover Publications. ISBN 0-486-60068-8. {{cite book}}: ISBN / Date incompatibility (help)
+
+
+== External links ==
+ Quotations related to La Géométrie at Wikiquote
+Project Gutenberg copy of La Géométrie
+Bad OCR: Cornell University Library copy of La Géométrie
+Archive.org: The Geometry of Rene Descartes
+Facsimile (fr) : La Géométrie Archived 2019-11-28 at the Wayback Machine

data/en.wikipedia.org/wiki/Letters_to_a_German_Princess-0.md

@@ -0,0 +1,47 @@
+---
+title: "Letters to a German Princess"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Letters_to_a_German_Princess"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:26.216400+00:00"
+instance: "kb-cron"
+---
+
+Letters to a German Princess, On Different Subjects in Physics and Philosophy (French: Lettres à une princesse d'Allemagne sur divers sujets de physique et de philosophie) were a series of 234 letters written by the mathematician Leonhard Euler between 1760 and 1762 addressed to Friederike Charlotte of Brandenburg-Schwedt and her younger sister Louise.
+
+
+== Contents ==
+Euler started the first letter with an explanation of the concept of "size". Starting with the definition of a foot, he defined the mile and the diameter of the earth as a unit in terms of foot and then calculated the distance of the planets of the Solar System in terms of the diameter of the earth.
+
+
+== Publication ==
+The first two volumes of the 234 letters originally written in French appeared in print in Saint Petersburg in 1768 and the third in Frankfurt in 1774. The letters were later reprinted in Paris with the first volume in 1787, the second in 1788 and the third in 1789.
+The publication of the book was supported by empress Catherine II with her personally writing to Count Vorontsov in January 1766:
+
+I am certain that the academy will be resurrected from its ashes by such an important acquisition, and congratulate myself in advance in having restored this great man to Russia.
+Russian translation of the letters followed in Saint Petersburg by Euler's student Stepan Rumovsky between 1768 and 1774 in 3 volumes.
+
+
+== Translations ==
+The first English translation of the Letters were done by the Scottish minister Henry Hunter in 1795.  Hunter targeted the translation at British women, believing that Euler intended to educate women through his work.
+The translation of Hunter was based on the 1787 Paris Edition, of Marquis de Condorcet and Sylvestre François Lacroix. The translation differed from the original letters of Euler in its omission of "... the frequent, tiresome, courtly address of YOUR HIGHNESS". 
+The Marquis de Condorcet's translation, made during the Age of Enlightenment,  was notable for its omission of Euler's theological references which Condorcet found as "anathema" to teaching science and rationalism.
+Translations followed in other languages including Spanish (1798) which differed from the original book by a footnote describing the newly discovered planet Uranus. Subsequent German edition (1847), and French editions (1812 and 1829) were also noted for their reference to Uranus and four minor planets respectively.
+
+
+== See also ==
+On the Connexion of the Physical Sciences
+
+
+== References and notes ==
+
+Attribution
+ This article incorporates text from a publication now in the public domain: Lee, Sidney, ed. (1891). "Hunter, Henry". Dictionary of National Biography. Vol. 28. London: Smith, Elder & Co. 
+
+
+== External links ==
+ Letters on Different Subjects in Natural Philosophy, Volume 1 public domain audiobook at LibriVox
+
+
+== Further reading ==

data/en.wikipedia.org/wiki/Letters_to_a_Young_Mathematician-0.md

@@ -0,0 +1,16 @@
+---
+title: "Letters to a Young Mathematician"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Letters_to_a_Young_Mathematician"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:27.375942+00:00"
+instance: "kb-cron"
+---
+
+Letters to a Young Mathematician  (ISBN 0-465-08231-9) is a 2006 book by Ian Stewart, and is part of Basic Books' Art of Mentoring series. Stewart mentions in the preface that he considers this book an update to G.H. Hardy's A Mathematician's Apology.
+The book is made up of letters to a fictional correspondent of Stewart's, an aspiring mathematician named Meg. The roughly chronological letters follow Meg from her high school years up to her receiving tenure from an American university.
+Reviews of the book were generally positive. Fernando Q. Gouvêa's review for the MAA calls it "full of good advice, much of it direct and to the point" and later, that "while it won't change the world, it may well help some young people decide to be (or not to be) mathematicians." In Emma Carberry's review for the AMS, reacted differently, saying that "one does not so much feel the benefit of a ream of practical advice, but rather of exposure to the inner realm of mathematics". A review in Nature was harsher, however, saying that "there is a general lack of information ... [and] too much jargon" and that it "suffers from being written entirely for a US audience", but even this review finds a bright note, "The letter in which Stewart tells Meg how to teach undergraduates should be compulsory reading for all lecturers and tutors."
+
+
+== References ==

data/en.wikipedia.org/wiki/Liber_Abaci-0.md

@@ -0,0 +1,25 @@
+---
+title: "Liber Abaci"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Liber_Abaci"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:28.580605+00:00"
+instance: "kb-cron"
+---
+
+The Liber Abaci or Liber Abbaci (Latin for "The Book of Calculation") was a 1202 Latin work on arithmetic by Leonardo of Pisa, posthumously known as Fibonacci. It is primarily famous for introducing both base-10 positional notation and the symbols known as Arabic numerals in Europe.
+
+== Premise ==
+Liber Abaci was among the first Western books to describe the Hindu–Arabic numeral system and to use symbols resembling modern "Arabic numerals". By addressing the applications of both commercial tradesmen and mathematicians, it promoted the superiority of the system and the use of these glyphs.
+Although the book's title is sometimes translated as "The Book of the Abacus", Sigler (2002) notes that the word in title does not refer to the abacus as a calculating device. Rather, the word "abacus" was used at the time to refer to calculation in any form; the spelling "abbacus" with two "b"s was, and still is in Italy, used to refer to calculation using Hindu-Arabic numerals. The book describes methods of doing calculations without aid of an abacus, and as Ore (1948) confirms, for centuries after its publication the algorismists (followers of the style of calculation demonstrated in Liber Abaci) remained in conflict with the abacists (traditionalists who continued to use the abacus in conjunction with Roman numerals). Carl Boyer emphasizes in his History of Mathematics that although "Liber abaci...is not on the abacus" per se, nevertheless "...it is a very thorough treatise on algebraic methods and problems in which the use of the Hindu-Arabic numerals is strongly advocated."
+
+== Summary of sections ==
+The first section introduces the Hindu–Arabic numeral system, including its arithmetic and methods for converting between different representation systems. This section also includes the first known description of trial division for testing whether a number is composite and, if so, factoring it.
+The second section presents examples from commerce, such as conversions of currency and measurements, and calculations of profit and interest.
+The third section discusses a number of mathematical problems; for instance, it includes the Chinese remainder theorem, perfect numbers and Mersenne primes as well as formulas for arithmetic series and for square pyramidal numbers. Another example in this chapter involves the growth of a population of rabbits, where the solution requires generating a numerical sequence. Although the resulting Fibonacci sequence dates back long before Leonardo, its inclusion in his book is why the sequence is named after him today.
+The fourth section derives approximations, both numerical and geometrical, of irrational numbers such as square roots.
+The book also includes proofs in Euclidean geometry. Fibonacci's method of solving algebraic equations shows the influence of the early 10th-century Egyptian mathematician Abū Kāmil Shujāʿ ibn Aslam.
+
+== Fibonacci's notation for fractions ==
+In Liber Abaci, Fibonacci's notation for rational numbers is intermediate in form between the Egyptian fractions commonly used until that time and the vulgar fractions still in use today. It differs from modern fraction notation in three key ways:

data/en.wikipedia.org/wiki/Liber_Abaci-1.md

@@ -0,0 +1,371 @@
+---
+title: "Liber Abaci"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Liber_Abaci"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:28.580605+00:00"
+instance: "kb-cron"
+---
+
+Modern notation generally writes a fraction to the right of the whole number to which it is added, for instance 
+  
+    
+      
+        2
+        
+        
+          
+            
+              1
+              3
+            
+          
+        
+      
+    
+    {\displaystyle 2\,{\tfrac {1}{3}}}
+  
+ for 7/3. Fibonacci instead would write the same fraction to the left, i.e., 
+  
+    
+      
+        
+          
+            
+              1
+              3
+            
+          
+        
+        
+        2
+      
+    
+    {\displaystyle {\tfrac {1}{3}}\,2}
+  
+.
+Fibonacci used a composite fraction notation in which a sequence of numerators and denominators shared the same fraction bar; each such term represented an additional fraction of the given numerator divided by the product of all the denominators below and to the right of it. That is, 
+  
+    
+      
+        
+          
+            
+              
+                b
+                
+                
+                a
+              
+              
+                d
+                
+                
+                c
+              
+            
+          
+        
+        =
+        
+          
+            
+              a
+              c
+            
+          
+        
+        +
+        
+          
+            
+              b
+              
+                c
+                d
+              
+            
+          
+        
+      
+    
+    {\displaystyle {\tfrac {b\,\,a}{d\,\,c}}={\tfrac {a}{c}}+{\tfrac {b}{cd}}}
+  
+, and 
+  
+    
+      
+        
+          
+            
+              
+                c
+                
+                
+                b
+                
+                
+                a
+              
+              
+                f
+                
+                
+                e
+                
+                
+                d
+              
+            
+          
+        
+        =
+        
+          
+            
+              a
+              d
+            
+          
+        
+        +
+        
+          
+            
+              b
+              
+                d
+                e
+              
+            
+          
+        
+        +
+        
+          
+            
+              c
+              
+                d
+                e
+                f
+              
+            
+          
+        
+      
+    
+    {\displaystyle {\tfrac {c\,\,b\,\,a}{f\,\,e\,\,d}}={\tfrac {a}{d}}+{\tfrac {b}{de}}+{\tfrac {c}{def}}}
+  
+.  The notation was read from right to left. For example, 29/30 could be written as 
+  
+    
+      
+        
+          
+            
+              
+                1
+                
+                
+                2
+                
+                
+                4
+              
+              
+                2
+                
+                
+                3
+                
+                
+                5
+              
+            
+          
+        
+      
+    
+    {\displaystyle {\tfrac {1\,\,2\,\,4}{2\,\,3\,\,5}}}
+  
+, representing the value 
+  
+    
+      
+        
+          
+            
+              4
+              5
+            
+          
+        
+        +
+        
+          
+            
+              2
+              
+                3
+                ×
+                5
+              
+            
+          
+        
+        +
+        
+          
+            
+              1
+              
+                2
+                ×
+                3
+                ×
+                5
+              
+            
+          
+        
+      
+    
+    {\displaystyle {\tfrac {4}{5}}+{\tfrac {2}{3\times 5}}+{\tfrac {1}{2\times 3\times 5}}}
+  
+. This can be viewed as a form of mixed radix notation and was very convenient for dealing with traditional systems of weights, measures, and currency. For instance, for units of length, a foot is 1/3 of a yard, and an inch is 1/12 of a foot, so a quantity of 5 yards, 2 feet, and 
+  
+    
+      
+        7
+        
+          
+            
+              3
+              4
+            
+          
+        
+      
+    
+    {\displaystyle 7{\tfrac {3}{4}}}
+  
+ inches could be represented as a composite fraction: 
+  
+    
+      
+        
+          
+            
+              
+                3
+                 
+                
+                7
+                
+                
+                2
+              
+              
+                4
+                
+                
+                12
+                
+                
+                3
+              
+            
+          
+        
+        
+        5
+      
+    
+    {\displaystyle {\tfrac {3\ \,7\,\,2}{4\,\,12\,\,3}}\,5}
+  
+ yards. However, typical notations for traditional measures, while similarly based on mixed radixes, do not write out the denominators explicitly; the explicit denominators in Fibonacci's notation allow him to use different radixes for different problems when convenient. Sigler also points out an instance where Fibonacci uses composite fractions in which all denominators are 10, prefiguring modern decimal notation for fractions.
+Fibonacci sometimes wrote several fractions next to each other, representing a sum of the given fractions. For instance, 1/3+1/4 = 7/12, so a notation like 
+  
+    
+      
+        
+          
+            
+              1
+              4
+            
+          
+        
+        
+        
+          
+            
+              1
+              3
+            
+          
+        
+        
+        2
+      
+    
+    {\displaystyle {\tfrac {1}{4}}\,{\tfrac {1}{3}}\,2}
+  
+ would represent the number that would now more commonly be written as the mixed number 
+  
+    
+      
+        2
+        
+        
+          
+            
+              7
+              12
+            
+          
+        
+      
+    
+    {\displaystyle 2\,{\tfrac {7}{12}}}
+  
+, or simply the improper fraction 
+  
+    
+      
+        
+          
+            
+              31
+              12
+            
+          
+        
+      
+    
+    {\displaystyle {\tfrac {31}{12}}}
+  
+. Notation of this form can be distinguished from sequences of numerators and denominators sharing a fraction bar by the visible break in the bar. If all numerators are 1 in a fraction written in this form, and all denominators are different from each other, the result is an Egyptian fraction representation of the number. This notation was also sometimes combined with the composite fraction notation: two composite fractions written next to each other would represent the sum of the fractions.
+The complexity of this notation allows numbers to be written in many different ways, and Fibonacci described several methods for converting from one style of representation to another. In particular, chapter II.7 contains a list of methods for converting an improper fraction to an Egyptian fraction, including the greedy algorithm for Egyptian fractions, also known as the Fibonacci–Sylvester expansion.
+
+== Modus Indorum ==
+In the Liber Abaci, Fibonacci wrote the following, introducing the affirmative Modus Indorum (the method of the Indians), today known as Hindu–Arabic numeral system or base-10 positional notation. It also introduced digits that greatly resembled the modern Arabic numerals.
+
+As my father was a public official away from our homeland in the Bugia customshouse established for the Pisan merchants who frequently gathered there, he had me in my youth brought to him, looking to find for me a useful and comfortable future; there he wanted me to be in the study of mathematics and to be taught for some days. There from a marvelous instruction in the art of the nine Indian figures, the introduction and knowledge of the art pleased me so much above all else, and I learnt from them, whoever was learned in it, from nearby Egypt, Syria, Greece, Sicily and Provence, and their various methods, to which locations of business I travelled considerably afterwards for much study, and I learnt from the assembled disputations. But this, on the whole, the algorithm and even the Pythagorean arcs, I still reckoned almost an error compared to the Indian method. Therefore strictly embracing the Indian method, and attentive to the study of it, from mine own sense adding some, and some more still from the subtle Euclidean geometric art, applying the sum that I was able to perceive to this book, I worked to put it together in xv distinct chapters, showing certain proof for almost everything that I put in, so that further, this method perfected above the rest, this science is instructed to the eager, and to the Italian people above all others, who up to now are found without a minimum. If, by chance, something less or more proper or necessary I omitted, your indulgence for me is entreated, as there is no one who is without fault, and in all things is altogether circumspect.
+
+In other words, he advocated the use of the digits 0–9, and of place value. Until this time Europe used Roman numerals, making modern mathematics almost impossible. The book thus made an important contribution to the spread of decimal numerals. The spread of the Hindu-Arabic system, however, as Ore writes, was "long-drawn-out", taking many more centuries to spread widely, and did not become complete until the later part of the 16th century, accelerating dramatically only in the 1500s with the advent of printing.
+
+== Textual history ==
+The first appearance of the manuscript was in 1202. No copies of this version are known. A revised version of Liber Abaci,  dedicated to Michael Scot, appeared in 1228. There are at least nineteen manuscripts extant containing parts of this text. There are three complete versions of this manuscript from the thirteenth and fourteenth centuries. There are a further nine incomplete copies known between the thirteenth and fifteenth centuries, and there may be more not yet identified.
+There were no known printed versions of Liber Abaci until Boncompagni's edition of 1857. The first complete English translation was Sigler's text of 2002.
+
+== See also ==
+The Book of Squares
+
+== References ==
+
+== External links ==
+
+Pisano, Leonardo (1202), Incipit liber Abbaci compositus to Lionardo filio Bonaccii Pisano in year Mccij [Manuscript], Museo Galileo.

data/en.wikipedia.org/wiki/List_of_books_about_polyhedra-0.md

@@ -0,0 +1,59 @@
+---
+title: "List of books about polyhedra"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/List_of_books_about_polyhedra"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:29.884776+00:00"
+instance: "kb-cron"
+---
+
+This is a list of books about polyhedra.
+
+== Polyhedral models ==
+
+=== Cut-out kits ===
+Jenkins, Gerald; Bear, Magdalen (1998). Paper Polyhedra in Colour. Tarquin. ISBN 1-899618-23-6. Advanced Polyhedra 1: The Final Stellation, ISBN 1-899618-61-9. Advanced Polyhedra 2: The Sixth Stellation, ISBN 1-899618-62-7. Advanced Polyhedra 3: The Compound of Five Cubes, ISBN 978-1-899618-63-7.
+Jenkins, Gerald; Wild, Anne (2000). Mathematical Curiosities. Tarquin. ISBN 1-899618-35-X. More Mathematical Curiosities, Tarquin, ISBN 1-899618-36-8. Make Shapes 1, ISBN 0-906212-00-6. Make Shapes 2, ISBN 0-906212-01-4.
+Smith, A. G. (1986). Cut and Assemble 3-D Geometrical Shapes: 10 Models in Full Color. Dover. Cut and Assemble 3-D Star Shapes, 1997. Easy-To-Make 3D Shapes in Full Color, 2000.
+Torrence, Eve (2011). Cut and Assemble Icosahedra: Twelve Models in White and Color. Dover.
+
+=== Origami ===
+Fuse, Tomoko (1990). Unit Origami: Multidimensional Transformations. Japan Publications. ISBN 978-0-87040-852-6.
+Gurkewitz, Rona; Arnstein, Bennett (1996). 3D Geometric Origami: Modular Origami Polyhedra. Dover. ISBN 9780486135601. Multimodular Origami Polyhedra: Archimedeans, Buckyballs and Duality, 2002. Beginner's Book of Modular Origami Polyhedra: The Platonic Solids, 2008. Modular Origami Polyhedra, also with Lewis Simon, 2nd ed., 1999.
+Mitchell, David (1997). Mathematical Origami: Geometrical Shapes by Paper Folding. Tarquin. ISBN 978-1-899618-18-7.
+Montroll, John (2009). Origami Polyhedra Design. A K Peters. ISBN 9781439871065. A Plethora of Polyhedra in Origami, Dover, 2002.
+
+=== Other model-making ===
+Cundy, H. M.; Rollett, A. P. (1952). Mathematical Models. Clarendon Press. 2nd ed., 1961. 3rd ed., Tarquin, 1981, ISBN 978-0-906212-20-2.
+Hilton, Peter; Pedersen, Jean (1988). Build Your Own Polyhedra. Addison-Wesley.
+Wenninger, Magnus (1971). Polyhedron Models. Cambridge University Press. 2nd ed., Polyhedron Models for the Classroom, 1974. Spherical Models, 1979. Dual Models, 1983.
+
+== Mathematical studies ==
+
+=== Introductory level and general audience ===
+Akiyama, Jin; Matsunaga, Kiyoko (2024). Treks into Intuitive Geometry: The World of Polygons and Polyhedra (2nd ed.). Singapore: Springer. ISBN 978-981-99-8607-1.
+Alsina, Claudi (2017). The Thousand Faces of Geometric Beauty: The Polyhedra. Our Mathematical World. Vol. 23. National Geographic. ISBN 978-84-473-8929-2.
+Britton, Jill (2001). Polyhedra Pastimes. Dale Seymour Publishing. ISBN 0-7690-2782-2.
+Cromwell, Peter R. (1997). Polyhedra. Cambridge University Press.
+Fetter, Ann E. (1991). The Platonic Solids Activity Book. Key Curriculum Press.
+Holden, Alan (1971). Shapes, Space and Symmetry. Dover, 1991.
+le Masne, Roger (2013). Les polyèdres, ou la beauté des mathématiques (in French) (4th ed.). Self-published.
+Miyazaki, Koji (1983). Katachi to kūkan: Tajigen sekai no kiseki (in Japanese). Wiley. Translated into English as An Adventure in Multidimensional Space: The Art and Geometry of Polygons, Polyhedra, and Polytopes, Wiley, 1986, and into German as Polyeder und Kosmos: Spuren einer mehrdimensionalen Welt, Vieweg, 1987.
+Pearce, Peter; Pearce, Susan (1979). Polyhedra Primer. Van Nostrand Reinhold. ISBN 978-0-442-26496-3.
+Pugh, Anthony (1976). Polyhedra: A Visual Approach. University of California Press.
+Radin, Dan (2008). The Platonic Solids Book. Self-published.
+Sutton, Daud (2002). Platonic & Archimedean Solids: The Geometry of Space. Wooden Books. ISBN 978-0802713865.
+
+=== Textbooks ===
+Alexandrov, A. D. (2005). Convex Polyhedra. Springer. Translated from 1950 Russian edition.
+Beck, Matthias; Robins, Sinai (2007). Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Vol. 154. Springer. 2nd ed., 2015, ISBN 978-1-4939-2968-9.
+Brøndsted, Arne (1983). An Introduction to Convex Polytopes. Graduate Texts in Mathematics. Vol. 90. Springer.
+Coxeter, H. S. M. (1948). Regular Polytopes. Methuen. 2nd ed., Macmillan, 1963. 3rd ed., Dover, 1973.
+Fejes Tóth, László (1964). Regular Figures. Pergamon.
+Grünbaum, Branko (1967). Convex Polytopes. Wiley. 2nd ed., Springer, 2003.
+Lyusternik, Lazar (1956). Выпуклые фигуры и многогранники (in Russian). Gosudarstv. Izdat. Tehn.-Teor. Lit. Translated into English as Convex Figures and Polyhedra by T. Jefferson Smith, Dover, 1963 and by Donald L. Barnett, Heath, 1966.
+Pineda Villavicencio, Guillermo (2024). Polytopes and Graphs. Cambridge Studies in Advanced Mathematics. Vol. 211. Cambridge University Press. doi:10.1017/9781009257794. ISBN 978-1-009-25781-7.
+Roman, Tiberiu (1968). Reguläre und halbreguläre Polyeder [Regular and semiregular polyhedra] (in German). VEB Deutscher Verlag der Wissenschaften.
+Thomas, Rekha (2006). Lectures in Geometric Combinatorics. American Mathematical Society.
+Ziegler, Günter M. (1993). Lectures on Polytopes. Springer.

data/en.wikipedia.org/wiki/List_of_books_about_polyhedra-1.md

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+---
+title: "List of books about polyhedra"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/List_of_books_about_polyhedra"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:29.884776+00:00"
+instance: "kb-cron"
+---
+
+=== Monographs and special topics ===
+Coxeter, H. S. M.; du Val, P.; Flather, H. T.; Petrie, J. F. (1938). The Fifty-Nine Icosahedra. University of Toronto Studies, Mathematical Series. Vol. 6. University of Toronto Press. 2nd ed., Springer, 1982. 3rd ed., Tarquin, 1999.
+Coxeter, H. S. M. (1974). Regular Complex Polytopes. Cambridge University Press. 2nd ed., 1991.
+Demaine, Erik; O'Rourke, Joseph (2007). Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press.
+Deza, Michel; Grishukhin, Viatcheslav; Shtogrin, Mikhail (2004). Scale-Isometric Polytopal Graphs in Hypercubes and Cubic Lattices: Polytopes in Hypercubes and 
+  
+    
+      
+        
+          
+            Z
+          
+          
+            n
+          
+        
+      
+    
+    {\displaystyle \mathbb {Z} _{n}}
+  
+. London: Imperial College Press. doi:10.1142/9781860945489. ISBN 1-86094-421-3.
+Lakatos, Imre (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
+McMullen, Peter (2020). Geometric Regular Polytopes. Encyclopedia of Mathematics and its Applications. Vol. 172. Cambridge University Press.
+McMullen, Peter; Schulte, Egon (2002). Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications. Vol. 92. Cambridge University Press.
+McMullen, Peter; Shephard, G. C. (1971). Convex Polytopes and the Upper Bound Conjecture. London Mathematical Society Lecture Note Series. Vol. 3. Cambridge University Press.
+Nef, Walter (1978). Beiträge zur Theorie der Polyeder: Mit Anwendungen in der Computergraphik [Contributions to the theory of the polyhedron, with applications in computer graphics] (in German). Herbert Lang.
+O'Rourke, Joseph; Vîlcu, Costin (2024). Reshaping Convex Polyhedra. Springer. arXiv:2107.03153. ISBN 978-3-031-47511-5.
+Rajwade, A. R. (2001). Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem. Texts and Readings in Mathematics. Vol. 21. Hindustan Book Agency.
+Richter-Gebert, Jürgen (1996). Realization Spaces of Polytopes. Lecture Notes in Mathematics. Vol. 1643. Springer.
+Stewart, B. M. (1970). Adventures Among the Toroids. Self-published. 2nd ed., 1980.
+Wachman, Avraham; Burt, Michael; Kleinmann, M. (1974). Infinite Polyhedra. Technion. 2nd ed., 2005.
+Wu, Wen-tsün (1965). A Theory of Imbedding, Immersion, and Isotopy of Polytopes in a Euclidean Space. Science Press.
+Zalgaller, Viktor A. (1969). Convex Polyhedra with Regular Faces. Consultants Bureau. Translated and corrected from Zalgaller, V. A. (1967). Выпуклые многогранники с правильными гранями. Zapiski Naučnyh Seminarov Leningradskogo Otdelenija Matematičeskogo Instituta im. V. A. Steklova Akademii Nauk SSSR (LOMI) (in Russian). Vol. 2. Nauka.
+Zhizhin, Gennadiy Vladimirovich (2022). The Classes of Higher Dimensional Polytopes in Chemical, Physical, and Biological Systems. Advances in Chemical and Materials Engineering. IGI Global. ISBN 9781799883760.
+
+=== Edited volumes ===
+Avis, David; Bremner, David; Deza, Antoine, eds. (2009). Polyhedral Computation. CRM Proceedings and Lecture Notes. Vol. 48. American Mathematical Society.
+Gabriel, Jean-François, ed. (1997). Beyond the Cube: The Architecture of Space Frames and Polyhedra. Wiley.
+Kalai, Gil; Ziegler, Günter M., eds. (2012). Polytopes - Combinatorics and Computation. DMV Seminar. Vol. 29. Springer.
+Senechal, Marjorie; Fleck, G., eds. (1988). Shaping Space: A Polyhedral Approach. Birkhauser. ISBN 0-8176-3351-0. 2nd ed., Shaping Space: Exploring Polyhedra in Nature, Art, and the Geometrical Imagination, Springer, 2013.
+Viana, Vera; Matos, Helena Mena; Xavier, João Pedro, eds. (2022). Polyhedra and Beyond: Contributions from Geometrias’19, Porto, Portugal, September 05-07. Trends in Mathematics. Birkhäuser.
+
+== History ==
+
+=== Early works ===
+Listed in chronological order, and including some works shorter than book length:
+
+Plato. Timaeus (in Greek).
+Euclid. Elements (in Greek).
+Pappus of Alexandria (1589). Mathematicae collectiones, liber quintus. apud Franciscum de Franciscis Senensem.
+Della Francesca, Piero (1482–1492). De quinque corporibus regularibus [On the five regular bodies] (in Latin).
+Pacioli, Luca (1509). Divina proportione [Divine proportion] (in Italian).
+de Bovelles, Charles (1511). De mathematicis corporibus.
+Dürer, Albrecht (1525). Underweysung der Messung, mit dem Zirckel und Richtscheyt, in Linien, Ebenen und gantzen corporen, Viertes Buch (in German).
+Maurolico, Francesco (1537). Compaginationes solidorum regularium.
+Jamnitzer, Wenzel (1568). Perspectiva corporum regularium [Perspectives of the regular bodies].
+Kepler, Johannes (1619). Harmonices Mundi (in Latin). Translated into English as Harmonies of the World by C. G. Wallis (1939).
+Descartes, René (c. 1630). De solidorum elementis [On the elements of solids] (in Latin). Original manuscript lost; copy by Gottfried Wilhelm Leibniz reprinted and translated in Descartes on Polyhedra, Springer, 1982.
+Cowley, John Lodge (1758). An Appendix to Euclid's Elements in Seven Books, Containing Forty-two Copper-plates, In Which the Doctrine of Solids, Delivered in the XIth, XIIth, and XVth Books of Euclid, is Illustrated by New-invented Schemes Cut Out of Paste-Board. Watkins.
+Poinsot, Louis (1810). Mémoire sur les polygones et sur les polyèdres (in French).
+Marie, François-Charles-Michel (1835). Géométrie stéréographique, ou reliefs des polyèdres (in French). Paris. hdl:2027/ucm.531073766x.
+Schläfli, Ludwig (1901) [1852]. Graf, J. H. (ed.). Theorie der vielfachen Kontinuität. Republished by Cornell University Library historical math monographs 2010 (in German). Zürich, Basel: Georg & Co. ISBN 978-1-4297-0481-6. {{cite book}}: ISBN / Date incompatibility (help)
+Wiener, Christian (1864). Über Vielecke und Vielflache. Teubner.
+Catalan, Eugène (1865). "Mémoire sur la théorie des polyèdres". Journal de l'École Polytechnique (in French). 24. hdl:2268/194785.
+Hess, Edmund (1883). Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder (in German). Teubner.
+Klein, Felix (1884). Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom 5ten Grade [Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree] (in German).
+Fedorov, E. S. (1885). Начала учения о фигурах [Introduction to the Theory of Figures] (in Russian).
+Gorham, John (1888). A System for the Construction of Crystal Models on the Type of an Ordinary Plait: Exemplified by the Forms Belonging to the Six Axial Systems in Crystallography. Reprint, Tarquin, 2007, ISBN 978-1-899618-68-2.
+Eberhard, Victor (1891). Zur Morphologie der Polyeder [On the morphology of polyhedra]. Teubner.
+von Lindemann, Ferdinand (1897). Zur Geschichte der Polyeder und der Zahlzeichen [History of Polyhedra and Numeral Signs] (in German). Munich: F. Straub. Reprinted from Sitz. Bay. Akad. Wiss. 1896, pp. 625–758.
+Brückner, Max (1900). Vielecke und Vielflache: Theorie und Geschichte (in German). Treubner. Über die gleicheckig-gleichflächigen diskontinuierlichen und nichtkonvexen Polyeder, 1906.
+Steinitz, Ernst (1934). Rademacher, Hans (ed.). Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie (in German).
+
+=== Books about historical topics ===
+Andrews, Noam (2022). The Polyhedrists: Art and Geometry in the Long Sixteenth Century. MIT Press.
+Davis, Margaret Daly (1977). Piero della Francesca's Mathematical Treatises: The "Trattato d'abaco" and "Libellus de quinque corporibus regularibus". Longo.
+Dézarnaud-Dandine, Christine; Sevin, Alain (2009). Histoire des polyèdres: Quand la nature est géomètre (in French). Vuibert.
+Federico, Pasquale Joseph (1984). Descartes on Polyhedra: A Study of the "De solidorum elementis". Sources in the History of Mathematics and Physical Sciences. Vol. 4. Springer.
+Richeson, D. S. (2008). Euler's Gem: The Polyhedron Formula and the Birth of Topology. Princeton University Press.
+Sanders, Philip Morris (1990). The Regular Polyhedra in Renaissance Science and Philosophy. Warburg Institute, University of London.
+Wade, David (2012). Fantastic Geometry: Polyhedra and the Artistic Imagination in the Renaissance. Squeeze Press.
+
+== References ==

data/en.wikipedia.org/wiki/List_of_mathematics_books-0.md

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 source: "https://en.wikipedia.org/wiki/List_of_mathematics_books"
 category: "reference"
 tags: "science, encyclopedia"
-date_saved: "2026-05-05T08:15:30.833032+00:00"
+date_saved: "2026-05-05T08:45:31.111931+00:00"
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data/en.wikipedia.org/wiki/Making_Mathematics_with_Needlework-0.md

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+---
+title: "Making Mathematics with Needlework"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Making_Mathematics_with_Needlework"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:32.269533+00:00"
+instance: "kb-cron"
+---
+
+Making Mathematics with Needlework: Ten Papers and Ten Projects is an edited volume on mathematics and fiber arts. It was edited by Sarah-Marie Belcastro and Carolyn Yackel, and published in 2008 by A K Peters, based on a meeting held in 2005 in Atlanta by the American Mathematical Society.
+
+
+== Topics ==
+The book includes ten different mathematical fiber arts projects, by eight contributors. An introduction provides a history of the connections between mathematics, mathematics education, and the fiber arts. Each of its ten project chapters is illustrated by many color photographs and diagrams, and is organized into four sections: an overview of the project, a section on the mathematics connected to it, a section of ideas for using the project as a teaching activity, and directions for constructing the project. Although there are some connections between topics, they can be read independently of each other, in any order. The thesis of the book is that directed exercises in fiber arts construction can help teach both mathematical visualization and concepts from three-dimensional geometry.
+The book uses knitting, crochet, sewing, and cross-stitch, but deliberately avoids weaving as a topic already well-covered in mathematical fiber arts publications. Projects in the book include a quilt in the form of a Möbius strip, a "bidirectional hat" connected to the theory of Diophantine equations, a shawl with a fractal design, a knitted torus connecting to discrete approximations of curvature, a sampler demonstrating different forms of symmetry in wallpaper group, "algebraic socks" with connections to modular arithmetic and the Klein four-group, a one-sided purse sewn together following a description by Lewis Carroll, a demonstration of braid groups on a cable-knit pillow, an embroidered graph drawing of an Eulerian graph, and topological pants.
+Beyond belcastro and Yackel, the contributors to the book include Susan Goldstine, Joshua Holden, Lana Holden, Mary D. Shepherd, Amy F. Szczepański, and D. Jacob Wildstrom.
+
+
+== Audience and reception ==
+Reviewers had mixed opinions on the appropriate audience for the book and its success in targeting that audience. Ketty Peeva writes that the book is "of interest to mathematicians, mathematics educators and crafters", and Mary Fortune writes that a wide group of people would enjoy browsing its contents, However, Kate Atherley warns that it is "not for the faint-of-heart" (either among mathematicians or crafters), and Mary Goetting complains that the audience for the book is not clearly defined, and is inconsistent across the book, with some chapters written for professional mathematicians and others for mathematical beginners. She writes that most readers will have to pick and choose among the chapters for material appealing to them. Similarly, reviewer Michelle Sipics writes that in aiming at multiple audiences, the book "sacrifices some accessibility". And although reviewer Gwen Fisher downplays the potential pedagogical applications of this book, complaining that its teaching ideas do not provide enough detail to be usable, and are not a good fit for typical teaching curricula, Sipics calls mathematics teachers "perhaps the greatest beneficiaries of this text".
+Fortune writes that, though the book increased her appreciation of and understanding of needlework, she didn't gain much new mathematical insight from reading it. In contrast, Fisher argues that by using only "straightforward applications of traditional needlework skills" the book is accessible even to beginners in the fiber arts, and that the book is "much more about maths than about fibre technique". The real value of the book, she argues, is in the scholarly connection it forges between traditional women's activities and mathematics. Pao-Sheng Hsu says that it would be "a great coffee table book" for browsing. And Anna Lena Phillips calls the book "an excellent synthesis" of textile crafts and mathematics, providing inspiration to those interested in either topic.
+
+
+== References ==
+
+
+== External links ==
+Home page

data/en.wikipedia.org/wiki/Math_Curse-0.md

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+---
+title: "Math Curse"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Math_Curse"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:33.411156+00:00"
+instance: "kb-cron"
+---
+
+Math Curse is a children's picture book written by Jon Scieszka and illustrated by Lane Smith. Published in 1995 through Viking Press, the book tells the story of a student cursed by how mathematics is connected to everyday life. In 2009, Weston Woods Studios, Inc. released a film based on the book.
+
+
+== Plot summary ==
+The nameless student begins with a seemingly innocent statement by her math teacher: "you know, almost everything in life can be considered a math problem." The next morning, the hero finds herself thinking of the time she needs to get up, along the lines of algebra. Next comes the mathematical school of probability, followed by charts and statistics. As the narrator slowly turns into a "math zombie", everything in her life is transformed into a problem. A class treat of cupcakes becomes a study in fractions, while a trip to the store turns into a problem of money. Finally, she is left painstakingly calculating how many minutes of "math madness" will be in her life now that she is a "mathematical lunatic." Her sister asks her what her problem is, and she responds, "365 days x 24 hours x 60 minutes." Finally, she collapses on her bed and dreams of being trapped in a blackboard room covered in math problems. Armed with only a piece of chalk, she must escape, and she manages to do just that by breaking the chalk in half, because "two halves make a whole." She escapes through this "whole" and awakens the next morning with the ability to solve any problem. Her curse is broken until the next day, when her science teacher mentions that everything can be viewed as a science experiment in life.
+
+
+== Math problems ==
+The book contains actual math problems (and some rather unrelated questions, such as "What does this inkblot look like?"). Readers can try to solve the problems and check their answers on the back cover.
+
+
+== Stage adaptation ==
+The book was also adapted for the stage by Heath Corson and Kathleen Collins in 1997. It was first performed at the A Red Orchid Theatre in Chicago, Illinois, in 1997, with subsequent productions at other locations. Its West Coast premiere was in 2003 at the Powerhouse Theatre of Santa Monica, California. Collins directed it, and the cast included Kerry Lacy, Thomas Colby, Will Moran, Andrew David James, and Emily Marver. The play met with warm reviews and succeeded with its audiences and local school children.
+
+
+== In popular culture ==
+The book was featured in the first episode of the seventeenth season of the children's series Reading Rainbow, narrated by Michelle Trachtenberg.
+
+
+== Awards ==
+The book was critically acclaimed, receiving several awards and accolades, including Maine's Student Favorite Book Award, the Texas Bluebonnet Award, and New Hampshire's The Great Stone Face Book Award.[1][2]

data/en.wikipedia.org/wiki/Mathematical_Cranks-0.md

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+---
+title: "Mathematical Cranks"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mathematical_Cranks"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:34.587603+00:00"
+instance: "kb-cron"
+---
+
+Mathematical Cranks is a book on pseudomathematics and the cranks who create it, written by Underwood Dudley. It was published by the Mathematical Association of America in their MAA Spectrum book series in 1992 (ISBN 0-88385-507-0).
+
+
+== Topics ==
+Previously, Augustus De Morgan wrote in A Budget of Paradoxes about cranks in multiple subjects, and Dudley wrote a book about angle trisection. However, this book is the first to focus on mathematical crankery as a whole.
+The book consists of 57 essays, loosely organized by the most common topics in mathematics for cranks to focus their attention on. The "top ten" of these topics, as listed by reviewer Ian Stewart, are, in order:
+
+squaring the circle,
+angle trisection,
+Fermat's Last Theorem,
+non-Euclidean geometry and the parallel postulate,
+the golden ratio,
+perfect numbers,
+the four color theorem,
+advocacy for duodecimal and other non-standard number systems,
+Cantor's diagonal argument for the uncountability of the real numbers, and
+doubling the cube.
+Other common topics for crankery, collected by Dudley, include calculations for the perimeter of an ellipse, roots of quintic equations, Fermat's little theorem, Gödel's incompleteness theorems, Goldbach's conjecture, magic squares, divisibility rules, constructible polygons, twin primes, set theory, statistics, and the Van der Pol oscillator.
+As David Singmaster writes, many of these topics are the subject of mainstream mathematics "and only become crankery in extreme cases". The book omits or passes lightly over other topics that apply mathematics to crankery in other areas, such as numerology and pyramidology. Its attitude towards the cranks it covers is one of "sympathy and understanding", and in order to keep the focus on their crankery it names them only by initials. The book also attempts to analyze the motivation and psychology behind crankery, and to provide advice to professional mathematicians on how to respond to cranks.
+Despite his work on the subject, which has "become enshrined in academic folklore", Dudley has stated "I've been at this for a decade and still can't pin down exactly what it is that makes a crank a crank", adding that "It's like obscenity – you can tell a crank when you see one."
+
+
+== Lawsuit ==
+After the book was published, one of the cranks whose work was featured in the book, William Dilworth, sued Dudley for defamation in a federal court in Wisconsin. The court dismissed the Dilworth vs Dudley case on two grounds. First, it found that by publishing his work on Cantor's diagonal argument, Dilworth had made himself a public figure, creating a higher burden of proof for a defamation case. Second, it found that the word "crank" was "rhetorical hyperbole" rather than an actionably inaccurate description. The United States Court of Appeals for the Seventh Circuit concurred. After Dilworth repeated the lawsuit in a state court, he lost again and was forced to pay Dudley's legal expenses.
+
+
+== Reception and audience ==
+Reviewer John N. Fujii calls the book "humorous and charming" and "difficult to put down", and advocates it to "all readers interested in the human side of mathematics". Although complaining that famous mathematicians Niels Henrik Abel and Srinivasa Ramanujan might have been dismissed as cranks by the standards of the book, reviewer Robert Matthews finds it an accurate reflection of most crankery. And David Singmaster adds that it should be read by "anyone likely to deal with a crank", including professional mathematicians, journalists, and legislators.
+
+
+== References ==

data/en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics-0.md

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+---
+title: "Mathematical Foundations of Quantum Mechanics"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:35.818465+00:00"
+instance: "kb-cron"
+---
+
+Mathematical Foundations of Quantum Mechanics (German: Mathematische Grundlagen der Quantenmechanik) is a quantum mechanics book written by John von Neumann in 1932. It is an important early work in the development of the mathematical formulation of quantum mechanics. The book mainly summarizes results that von Neumann had published in earlier papers.
+Von Neumann formalized quantum mechanics using the concept of Hilbert spaces and linear operators. He acknowledged the previous work by Paul Dirac on the mathematical formalization of quantum mechanics, but was skeptical of Dirac's use of delta functions. He wrote the book in an attempt to be even more mathematically rigorous than Dirac. It was von Neumann's last book in German, afterwards he started publishing in English. 
+
+== Publication history ==
+The book was originally published in German in 1932 by Springer. It was translated into French by Alexandru Proca in 1946, and into Spanish in 1949. An English translation by Robert T. Beyer was published in 1955 by Princeton University Press. A Russian translation, edited by Nikolay Bogolyubov, was published by Nauka in 1964. A new English edition, edited by Nicholas A. Wheeler, was published in 2018 by Princeton University Press.
+
+== Table of contents ==
+According to the 2018 version, the main chapters are:
+
+Introductory considerations
+Abstract Hilbert space
+The quantum statistics
+Deductive development of the theory
+General considerations
+The measuring process
+
+== Measurement process ==
+In chapter 6, von Neumann develops the theory of quantum measurement. Von Neumann addresses measurement by outlining two kind of processes:
+
+Process I: during measurement a quantum state of a system evolves into a mixed state of eigenstates of the measured observable. This process is non-causal (the outcome of a single measurement does not depend only on the initial state) and irreversible.
+Process II: when the system is unobserved, the state evolves according to Schrödinger equation. This process is causal and reversible.
+Von Neumann was concerned that having two incompatible processes violated what he called the principle of psycho-physical parallelism, indicating the need that every mental process can be described as a physical process. Von Neumann argues that this issue does not appear in quantum mechanics as it set the border between observed and observer arbitrarily along a sequence of subsystems.
+The sequence begins with a quantum system whose observable is to be measured. When the system interacts with a measuring device, they become entangled. As a result, the system does not end up in a definite eigenstate of the observable, and the measuring device does not display a specific value. When the observer is added to the picture, the description implies that their body (including the brain) are also entangled with the measuring apparatus and the system. This sequence is known as the von Neumann chain. The problem then becomes understanding how collapse to one of the eigenstates emerges from this chain. Von Neumann demonstrated that, when it comes to the final outcomes, the chain can be interrupted at any and a wave function collapse can be introduced at any point to explain the results.
+
+=== Interpretations ===
+Von Neumann measurement scheme is part of the orthodox Copenhagen interpretation which postulates a collapse, however alternative interpretations of quantum mechanics have come out of this idea. Eugene Wigner considered that the von Neumann chain implied that consciousness causes collapse of the wave function. However Wigner rejected this idea after the formalism of quantum decoherence was developed. Hugh Everett III developed the many-worlds interpretation based on von Neumann's processes, by keeping only process II.
+
+== No hidden variables proof ==
+
+One significant passage is its mathematical argument against the idea of hidden variables. Von Neumann's claim rested on the assumption that any linear combination of Hermitian operators represents an observable and the expectation value of such combined operator follows the combination of the expectation values of the operators themselves.  
+Von Neumann's makes the following assumptions:
+
+For an observable 
+  
+    
+      
+        R
+      
+    
+    {\displaystyle R}
+  
+, a function 
+  
+    
+      
+        f
+      
+    
+    {\displaystyle f}
+  
+ of that observable is represented by 
+  
+    
+      
+        f
+        (
+        R
+        )
+      
+    
+    {\displaystyle f(R)}
+  
+.
+For the sum of  observables 
+  
+    
+      
+        R
+      
+    
+    {\displaystyle R}
+  
+ and 
+  
+    
+      
+        S
+      
+    
+    {\displaystyle S}
+  
+ is represented by the operation 
+  
+    
+      
+        R
+        +
+        S
+      
+    
+    {\displaystyle R+S}
+  
+, independently of the mutual commutation relations.
+The correspondence between observables and Hermitian operators is one to one.
+If the observable 
+  
+    
+      
+        R
+      
+    
+    {\displaystyle R}
+  
+ is a non-negative operator, then its expected value 
+  
+    
+      
+        ⟨
+        R
+        ⟩
+        ≥
+        0
+      
+    
+    {\displaystyle \langle R\rangle \geq 0}
+  
+.
+Additivity postulate: For arbitrary observables 
+  
+    
+      
+        R
+      
+    
+    {\displaystyle R}
+  
+ and 
+  
+    
+      
+        S
+      
+    
+    {\displaystyle S}
+  
+, and real numbers 
+  
+    
+      
+        a
+      
+    
+    {\displaystyle a}
+  
+ and 
+  
+    
+      
+        b
+      
+    
+    {\displaystyle b}
+  
+, we have 
+  
+    
+      
+        ⟨
+        a
+        R
+        +
+        b
+        S
+        ⟩
+        =
+        a
+        ⟨
+        R
+        ⟩
+        +
+        b
+        ⟨
+        S
+        ⟩
+      
+    
+    {\displaystyle \langle aR+bS\rangle =a\langle R\rangle +b\langle S\rangle }
+  
+ for all possible ensembles.
+Von Neumann then shows that one can write 
+
+  
+    
+      
+        ⟨
+        R
+        ⟩
+        =
+        
+          ∑
+          
+            m
+            ,
+            n
+          
+        
+        
+          ρ
+          
+            n
+            m
+          
+        
+        
+          R
+          
+            m
+            n
+          
+        
+        =
+        
+          T
+          r
+        
+        (
+        ρ
+        R
+        )
+      
+    
+    {\displaystyle \langle R\rangle =\sum _{m,n}\rho _{nm}R_{mn}=\mathrm {Tr} (\rho R)}
+  
+
+for some 
+  
+    
+      
+        ρ
+      
+    
+    {\displaystyle \rho }
+  
+, where 
+  
+    
+      
+        
+          R
+          
+            m
+            n
+          
+        
+      
+    
+    {\displaystyle R_{mn}}
+  
+ and 
+  
+    
+      
+        
+          ρ
+          
+            n
+            m
+          
+        
+      
+    
+    {\displaystyle \rho _{nm}}
+  
+are the matrix elements in some basis. The proof concludes by noting that 
+  
+    
+      
+        ρ
+      
+    
+    {\displaystyle \rho }
+  
+ must be Hermitian and non-negative definite (
+  
+    
+      
+        ⟨
+        ρ
+        ⟩
+        ≥
+        0
+      
+    
+    {\displaystyle \langle \rho \rangle \geq 0}
+  
+) by construction. For von Neumann, this meant that the statistical operator representation of states could be deduced from the postulates. Consequently, there are no "dispersion-free" states: it is impossible to prepare a system in such a way that all measurements have predictable results. But if hidden variables existed, then knowing the values of the hidden variables would make the results of all measurements predictable, and hence there can be no hidden variables. Von Neumann's argues that if dispersion-free states were found, assumptions 1 to 3 should be modified.
+Von Neumann's concludes:

data/en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics-1.md

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+---
+title: "Mathematical Foundations of Quantum Mechanics"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Mathematical_Foundations_of_Quantum_Mechanics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:35.818465+00:00"
+instance: "kb-cron"
+---
+
+if there existed other, as yet undiscovered, physical quantities, in addition to those represented by the operators in quantum mechanics, because the relations assumed by quantum mechanics would have to fail already for the by now known quantities, those that we discussed above. It is therefore not, as is often assumed, a question of a re-interpretation of quantum mechanics, the present system of quantum mechanics would have to be objectively false, in order that another description of the elementary processes than the statistical one be possible.
+
+=== Rejection ===
+This proof was rejected as early as 1935 by Grete Hermann who found a flaw in the proof. The additive postulate above holds for quantum states, but it does not need to apply for measurements of dispersion-free states, specifically when considering non-commuting observables. Dispersion-free states only require to recover additivity when averaging over the hidden parameters. For example, for a spin-1/2 system, measurements of 
+  
+    
+      
+        (
+        
+          σ
+          
+            x
+          
+        
+        +
+        
+          σ
+          
+            y
+          
+        
+        )
+      
+    
+    {\displaystyle (\sigma _{x}+\sigma _{y})}
+  
+ can take values 
+  
+    
+      
+        ±
+        
+          
+            2
+          
+        
+      
+    
+    {\displaystyle \pm {\sqrt {2}}}
+  
+ for a dispersion-free state, but independent measurements of 
+  
+    
+      
+        
+          σ
+          
+            x
+          
+        
+      
+    
+    {\displaystyle \sigma _{x}}
+  
+ and 
+  
+    
+      
+        
+          σ
+          
+            y
+          
+        
+      
+    
+    {\displaystyle \sigma _{y}}
+  
+ can only take values of 
+  
+    
+      
+        ±
+        1
+      
+    
+    {\displaystyle \pm 1}
+  
+ (their sum can be 
+  
+    
+      
+        ±
+        2
+      
+    
+    {\displaystyle \pm 2}
+  
+ or ⁠
+  
+    
+      
+        0
+      
+    
+    {\displaystyle 0}
+  
+⁠). Thus there still the possibility that a hidden variable theory could reproduce quantum mechanics statistically.
+However, Hermann's critique remained relatively unknown until 1974 when it was rediscovered by Max Jammer. In 1952, David Bohm constructed the Bohmian interpretation of quantum mechanics in terms of statistical argument, suggesting a limit to the validity of von Neumann's proof. The problem was brought back to wider attention by John Stewart Bell in 1966. Bell showed that the consequences of that assumption are at odds with results of incompatible measurements, which are not explicitly taken into von Neumann's considerations.
+
+== Reception ==
+It was considered the most complete book written in quantum mechanics at the time of release. It was praised for its axiomatic approach. A review by Jacob Tamarkin compared von Neumann's book to what the works on Niels Henrik Abel or Augustin-Louis Cauchy did for mathematical analysis in the 19th century, but for quantum mechanics.
+Freeman Dyson said that he learned quantum mechanics from the book. Dyson remarks that in the 1940s, von Neumann's work was not very well cited in the English world, as the book was not translated into English until 1955, but also because the worlds of mathematics and physics were significantly distant at the time.
+Max Jammer observed that Paul Dirac's primary motivation in writing The Principles of Quantum Mechanics (1930) was creating an exposition in physics, treating mathematics as a tool. In this regard, John von Neumann's Mathematical Foundations of Quantum Mechanics, with its uncompromising emphasis on mathematical rigour, was a supplement to Dirac's book.
+
+== Works adapted in the book ==
+von Neumann, J. (1927). "Mathematische Begründung der Quantenmechanik [Mathematical Foundation of Quantum Mechanics]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 1–57.
+von Neumann, J. (1927). "Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik [Probabilistic Theory of Quantum Mechanics]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 245–272.
+von Neumann, J. (1927). "Thermodynamik quantenmechanischer Gesamtheiten [Thermodynamics of Quantum Mechanical Quantities]". Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse. 102: 273–291.
+von Neumann, J. (1929). "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren [General Eigenvalue Theory of Hermitian Functional Operators]". Mathematische Annalen: 49–131. doi:10.1007/BF01782338.
+von Neumann, J. (1931). "Die Eindeutigkeit der Schrödingerschen Operatoren [The uniqueness of Schrödinger operators]". Mathematische Annalen. 104: 570–578. doi:10.1007/bf01457956. S2CID 120528257.
+
+== See also ==
+
+Dirac–von Neumann axioms
+Heisenberg cut
+Timeline of quantum mechanics
+
+== Notes ==
+
+== References ==
+
+== External links ==
+
+Full online text of the 1932 German edition (facsimile) at the University of Göttingen.

data/en.wikipedia.org/wiki/Mathematical_Models_(Cundy_and_Rollett)-0.md

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+---
+title: "Mathematical Models (Cundy and Rollett)"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mathematical_Models_(Cundy_and_Rollett)"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:38.192969+00:00"
+instance: "kb-cron"
+---
+
+Mathematical Models is a book on the construction of physical models of mathematical objects for educational purposes. It was written by Martyn Cundy and A. P. Rollett, and published by the Clarendon Press in 1951, with a second edition in 1961. Tarquin Publications published a third edition in 1981.
+The vertex configuration of a uniform polyhedron, a generalization of the Schläfli symbol that describes the pattern of polygons surrounding each vertex, was devised in this book as a way to name the Archimedean solids, and has sometimes been called the Cundy–Rollett symbol as a nod to this origin.
+
+
+== Topics ==
+The first edition of the book had five chapters, including its introduction which discusses model-making in general and the different media and tools with which one can construct models. The media used for the constructions described in the book include "paper, cardboard, plywood, plastics, wire, string, and sheet metal".
+The second chapter concerns plane geometry, and includes material on the golden ratio, the Pythagorean theorem, dissection problems, the mathematics of paper folding, tessellations, and plane curves, which are constructed by stitching, by graphical methods, and by mechanical devices.
+The third chapter, and the largest part of the book, concerns polyhedron models, made from cardboard or plexiglass. It includes information about the Platonic solids, Archimedean solids, their stellations and duals, uniform polyhedron compounds, and deltahedra.
+The fourth chapter is on additional topics in solid geometry and curved surfaces, particularly quadrics but also including topological manifolds such as the torus, Möbius strip and Klein bottle, and physical models helping to visualize the map coloring problem on these surfaces. Also included are sphere packings. The models in this chapter are constructed as the boundaries of solid objects, via two-dimensional paper cross-sections, and by string figures.
+The fifth chapter, and the final one of the first edition, includes mechanical apparatus including harmonographs and mechanical linkages, the bean machine and its demonstration of the central limit theorem, and analogue computation using hydrostatics. The second edition expands this chapter, and adds another chapter on computational devices such as the differential analyser of Vannevar Bush.
+Much of the material on polytopes was based on the book Regular Polytopes by H. S. M. Coxeter, and some of the other material has been drawn from resources previously published in 1945 by the National Council of Teachers of Mathematics.
+
+
+== Audience and reception ==
+At the time they wrote the book, Cundy and Rollett were sixth form teachers in the UK, and they intended the book to be used by mathematics students and teachers for educational activities at that level. However, it may also be enjoyed by a general audience of mathematics enthusiasts.
+Reviewer Michael Goldberg notes some minor errors in the book's historical credits and its notation, and writes that for American audiences some of the British terminology may be unfamiliar, but concludes that it could still be valuable for students and teachers. Stanley Ogilvy complains about the inconsistent level of rigor of the mathematical descriptions, with some proofs given and others omitted, for no clear reason, but calls this issue minor and in general calls the book's presentation excellent. Dirk ter Haar is more enthusiastic, recommending it to anyone interested in mathematics, and suggesting that it should be required for mathematics classrooms. Similarly, B. J. F. Dorrington recommends it to all mathematical libraries, and The Basic Library List Committee of the Mathematical Association of America has given it their strong recommendation for inclusion in undergraduate mathematics libraries. By the time of its second edition, H. S. M. Coxeter states that Mathematical Models had become "well known".
+
+
+== References ==

data/en.wikipedia.org/wiki/Mathematical_Models_(Fischer)-0.md

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+---
+title: "Mathematical Models (Fischer)"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mathematical_Models_(Fischer)"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:39.351961+00:00"
+instance: "kb-cron"
+---
+
+Mathematical Models: From the Collections of Universities and Museums – Photograph Volume and Commentary is a book on the physical models of concepts in mathematics that were constructed in the 19th century and early 20th century and kept as instructional aids at universities. It credits Gerd Fischer as editor, but its photographs of models are also by Fischer. It was originally published by Vieweg+Teubner Verlag for their bicentennial in 1986, both in German (titled Mathematische Modelle. Aus den Sammlungen von Universitäten und Museen. Mit 132 Fotografien. Bildband und Kommentarband)  and (separately) in English translation, in each case as a two-volume set with one volume of photographs and a second volume of mathematical commentary. Springer Spektrum reprinted it in a second edition in 2017, as a single dual-language volume.
+
+
+== Topics ==
+The work consists of 132 full-page photographs of mathematical models, divided into seven categories, and seven chapters of mathematical commentary written by experts in the topic area of each category.
+These categories are:
+
+Wire and thread models, of hypercubes of various dimensions, and of hyperboloids, cylinders, and related ruled surfaces, described as "elementary analytic geometry" and explained by Fischer himself.
+Plaster and wood models of cubic and quartic algebraic surfaces, including Cayley's ruled cubic surface, the Clebsch surface, Fresnel's wave surface, the Kummer surface, and the Roman surface, with commentary by W. Barth and H. Knörrer.
+Wire and plaster models illustrating the differential geometry and curvature of curves and surfaces, including surfaces of revolution, Dupin cyclides, helicoids, and minimal surfaces including the Enneper surface, with commentary by M. P. do Carmo, G. Fischer, U. Pinkall, H. and Reckziegel.
+Surfaces of constant width including the surface of rotation of the Reuleaux triangle and the Meissner bodies, described by J. Böhm.
+Uniform star polyhedra, described by E. Quaisser.
+Models of the projective plane, including the Roman surface (again), the cross-cap, and Boy's surface, with commentary by U. Pinkall that includes its realization by Roger Apéry as a quartic surface (disproving a conjecture of Heinz Hopf).
+Graphs of functions, both with real and complex variables, including the Peano surface, Riemann surfaces, exponential function and Weierstrass's elliptic functions, with commentary by J. Leiterer.
+
+
+== Audience and reception ==
+This book can be viewed as a supplement to Mathematical Models by Martyn Cundy and A. P. Rollett (1950), on instructions for making mathematical models, which according to reviewer Tony Gardiner "should be in every classroom and on every lecturer's shelf" but in fact sold very slowly. Gardiner writes that the photographs may be useful in undergraduate mathematics lectures, while the commentary is best aimed at mathematics professionals in giving them an understanding of what each model depicts. Gardiner also suggests using the book as a source of inspiration for undergraduate research projects that use its models as starting points and build on the mathematics they depict. Although Gardiner finds the commentary at times overly telegraphic and difficult to understand, reviewer O. Giering, writing about the German-language version of the same commentary, calls it detailed, easy-to-read, and stimulating.
+By the time of the publication of the second edition, in 2017, reviewer Hans-Peter Schröcker evaluates the visualizations in the book as "anachronistic", superseded by the ability to visualize the same phenomena more easily with modern computer graphics, and he writes that some of the commentary is also "slightly outdated". Nevertheless, he writes that the photos are "beautiful and aesthetically pleasing", writing approvingly that they use color sparingly and aim to let the models speak for themselves rather than dazzling with many color images. And despite the fading strength of its original purpose, he finds the book valuable both for its historical interest and for what it still has to say about visualizing mathematics in a way that is both beautiful and informative.
+
+
+== References ==

data/en.wikipedia.org/wiki/Mathematicall_Magick-0.md

@@ -0,0 +1,48 @@
+---
+title: "Mathematicall Magick"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mathematicall_Magick"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:41.682100+00:00"
+instance: "kb-cron"
+---
+
+Mathematicall Magick. Or, The Wonders that may be Performed by Mechanicall Geometry is a treatise by the English clergyman, natural philosopher, polymath and author John Wilkins (1614–1672). It was first published in 1648 in London; another edition was printed in 1680 and further editions were published in 1691 and 1707. The work is dedicated to Charles I Louis, the Elector Palatine. 
+The first book describes traditional mechanical devices, speed, siege engines and the modern guns of Wilkins' era. The second book covers Wilkins' theories and observations on land yachts, submarines, flying machines, and perpetual motion. Wilkins thought that human aviation is feasible, if only sufficient exercise, research and development is directed towards it. He envisioned flying machines that would be large enough to carry several people. 
+The book repeats tales about early attempts at human flight, including Busbequius' 16th-century reports about Turkish flight experiments in Constantinople. Wilkins also mentions Anglo-Saxon flight experiments during the reign of Edward the Confessor. The researcher called "Elmerus" in the text is probably Eilmer of Malmesbury, who experimented with gliding flight in the 11th century.  
+
+
+== Abstract ==
+Wilkins dedicated his work to His Highness the Prince Elector Palatine (Charles I Louis) who was in London at the time. It is divided into two books, one headed Archimedes, because he was the chiefest in discovering of Mechanical powers, the other was called Daedalus because he was one of the first and most famous amongst the Ancients for his skill in making Automata. Wilkins sets out and explains the principles of mechanics in the first book and gives an outlook in the second book on future technical developments like flying which he anticipates as certain if only sufficient exercise, research and development would be directed to these topics. The treatise is an example of his general intention to disseminate scientific knowledge and method and of his attempts to persuade his readers to pursue further scientific studies.
+
+
+== First book ==
+In the 20 chapters of the first book, traditional mechanical devices are discussed such as the balance, the lever, the wheel or pulley and the block and tackle, the wedge, and the screw. The powers acting on them are compared to those acting in the human body. The book deals with the phrase attributed to Archimedes saying that if he did but know where to stand and fasten his instrument, he could move the world and shows the effect of a series of gear transmissions one linked to the other. It shows the importance of various speeds and the theoretical possibility to increase speed beyond the speed of the Earth at the equator. Finally, siege engines like catapults are compared with the cost and effect of then-modern guns.
+
+
+== Second book ==
+
+
+=== Various devices ===
+In the 15 chapters of the second book, various devices are examined which move independently of human interference like clocks and watches, water mills and wind mills. Wilkins explains devices being driven by the motion of air in a chimney or by pressurized air. A land yacht is proposed driven by two sails on  two masts, and a wagon powered by a vertical axis wind turbine. A number of independently moving small artificial figures representing men and animals are described. The possibilities are considered to improve the type of submarine designed and built by Cornelis Drebbel. The tales about various flying devices are related and doubts as to their truth are dissipated. Wilkins explains that it should be possible for a man, too, to fly by himself if a frame were built where the person could sit and if this frame was sufficiently pushed in the air.
+
+
+=== Art of flying ===
+In chapter VII, Wilkins discusses various methods how a man could fly, namely by the help of spirits and good or evil angels (as related on various occasions in the Bible), by the help of fowls, by wings fastened immediately to the body or by a flying chariot. The whole of this chapter (and of the following one) concern the possibilities of flying. In a single preliminary phrase, he refers to previous reports of flight attempts:
+
+Tis related of a certain English Monk called Elmerus [probably Eilmer of Malmesbury], about the Confessors time, that he did by such wings fly from a Tower above a furlong; and so another [probably Fausto Veranzio] from Saint Marks steeple in Venice; another at Norinberge; and Busbequius speaks of a Turk in Constantinople, who attempted something this way. Mt. Burton mentioning this quotation, doth believe that some new-fangled wit ('tis his Cynical phrase) will some time or other find out this art. Though the truth is, most of these Artists did unfortunately miscarry by falling down and breaking their arms or legs, yet that may be imputed to their want of experience ...
+He writes that sufficient practise should enable a man to fly, most probably by "a flying chariot, which may be so contrived as to carry a man within it" and equipped with a sort of engine, or else big enough to carry several people, each successively working to fly it. He used the next chapter to dissipate any doubts there may be as to the possibility of such a flying chariot, should a number of particular items be developed and tested.
+
+
+=== Perpetual motion and perpetual lamps ===
+In Chapters IX to XV, extensive discussions and deliberations are set out why a perpetual motion should be feasible, why the stories about lamps burning for hundreds of years were true and how such lamps could be made and perpetual motions created.
+
+
+== External links ==
+1st edition (1648) at the Internet Archive.
+1680 version at Google Books.
+4th edition (1691) at the Internet Archive.
+
+
+== References ==

data/en.wikipedia.org/wiki/Mathematics,_Form_and_Function-0.md

@@ -0,0 +1,37 @@
+---
+title: "Mathematics, Form and Function"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mathematics,_Form_and_Function"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:45.216076+00:00"
+instance: "kb-cron"
+---
+
+Mathematics, Form and Function, a book published in 1986 by Springer-Verlag, is a survey of the whole of mathematics, including its origins and deep structure, by the American mathematician Saunders Mac Lane.
+
+
+== Mathematics and human activities ==
+Throughout his book, and especially in chapter I.11, Mac Lane informally discusses how mathematics is grounded in more ordinary concrete and abstract human activities. The following table is adapted from one given on p. 35 of Mac Lane (1986). The rows are very roughly ordered from most to least fundamental. For a bullet list that can be compared and contrasted with this table, see section 3 of Where Mathematics Comes From.
+
+Also see the related diagrams appearing on the following pages of Mac Lane (1986): 149, 184, 306, 408, 416, 422-28.
+Mac Lane (1986) cites a related monograph by Lars Gårding (1977).
+
+
+== Mac Lane's relevance to the philosophy of mathematics ==
+Mac Lane cofounded category theory with Samuel Eilenberg, which enables a unified treatment of mathematical structures and of the relations among them, at the cost of breaking away from their cognitive grounding.  Nevertheless, his views—however informal—are a valuable contribution to the philosophy and anthropology of mathematics. His views anticipate, in some respects, the more detailed account of the cognitive basis of mathematics given by George Lakoff and Rafael E. Núñez in their Where Mathematics Comes From. Lakoff and Núñez argue that mathematics emerges via conceptual metaphors grounded in the human body, its motion through space and time, and in human sense perceptions.
+
+
+== See also ==
+1986 in philosophy
+
+
+== Notes ==
+
+
+== References ==
+Gårding, Lars, 1977. Encounter with Mathematics. Springer-Verlag.
+Reuben Hersh, 1997. What Is Mathematics, Really? Oxford Univ. Press.
+George Lakoff and Rafael E. Núñez, 2000. Where Mathematics Comes From. Basic Books.
+Mac Lane, Saunders (1986). Mathematics, Form and Function. Springer-Verlag. ISBN 0-387-96217-4.
+Leslie White, 1947, "The Locus of Mathematical Reality: An Anthropological Footnote," Philosophy of Science 14: 289-303. Reprinted in Hersh, R., ed., 2006. 18 Unconventional Essays on the Nature of Mathematics. Springer: 304–19.

data/en.wikipedia.org/wiki/Mathematics_Made_Difficult-0.md

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+---
+title: "Mathematics Made Difficult"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mathematics_Made_Difficult"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:44.083466+00:00"
+instance: "kb-cron"
+---
+
+Mathematics Made Difficult is a book by Carl E. Linderholm that uses advanced mathematical methods to prove results normally shown using elementary proofs.  Although the aim is largely satirical, it also shows the non-trivial mathematics behind operations normally considered obvious, such as numbering, counting, and factoring integers. Linderholm discusses these seemingly obvious ideas using concepts like categories and monoids.
+As an example, the proof that 2 is a prime number starts:
+
+It is easily seen that the only numbers between 0 and 2, including 0 but excluding 2, are 0 and 1.  Thus the remainder left by any number on division by 2 is either 0 or 1.  Hence the quotient ring Z/2Z, where 2Z is the ideal in Z generated by 2, has only the elements [0] and [1], where these are the images of 0 and 1 under the canonical quotient map.  Since [1] must be the unit of this ring, every element of this ring except [0] is a unit, and the ring is a field ...
+
+
+== References ==

data/en.wikipedia.org/wiki/Mathematics_and_Plausible_Reasoning-0.md

@@ -0,0 +1,41 @@
+---
+title: "Mathematics and Plausible Reasoning"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mathematics_and_Plausible_Reasoning"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:42.860670+00:00"
+instance: "kb-cron"
+---
+
+Mathematics and Plausible Reasoning is a two-volume book by the mathematician George Pólya describing various methods for being a good guesser of new mathematical results. In the Preface to Volume 1 of the book Pólya exhorts all interested students of mathematics thus: "Certainly, let us learn proving, but also let us learn guessing." P. R. Halmos reviewing the book summarised the central thesis of the book thus: ". . . a good guess is as important as a good proof."
+
+
+== Outline ==
+
+
+=== Volume I: Induction and analogy in mathematics ===
+Polya begins Volume I with a discussion on induction, not mathematical induction, but as a way of guessing new results. He shows how the chance observations of a few results of the form 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, etc., may prompt a sharp mind to formulate the conjecture that every even number greater than 4 can be represented as the sum of two odd prime numbers. This is the well known Goldbach's conjecture. The first problem in the first chapter is to guess the rule according to which the successive terms of the following sequence are chosen: 11, 31, 41, 61, 71, 101, 131, . . . In the next chapter the techniques of generalization, specialization and analogy are presented as possible strategies for plausible reasoning. In the remaining chapters, these ideas are illustrated by discussing the discovery of several results in  various fields of mathematics like number theory, geometry, etc. and also in physical sciences.
+
+
+=== Volume II: Patterns of Plausible Inference ===
+This volume attempts to formulate certain patterns of plausible reasoning. The relation of these patterns with the calculus of probability are also investigated. Their relation to mathematical invention and instruction are also discussed. The following are
+some of the patterns of plausible inference discussed by Polya.
+
+
+== Reviews ==
+Bernhart, Arthur (1958-01-01). "Review of Mathematics and Plausible Reasoning". The American Mathematical Monthly. 65 (6): 456–457. doi:10.2307/2310741. hdl:2027/mdp.39015008206248. JSTOR 2310741. S2CID 121427033.
+Rado, Tibor (1956-01-01). "Review of Mathematics and Plausible Reasoning". Philosophy of Science. 23 (2): 167. doi:10.1086/287478. JSTOR 185607.
+Van Dantzig, D. (1959-01-01). "Review of Mathematics and Plausible Reasoning, G. Pólya". Synthese. 11 (4): 353–358. doi:10.1007/bf00486196. JSTOR 20114312. S2CID 46957889.
+Broadbent, T. A. A. (1956-01-01). "Review of Mathematics and Plausible Reasoning". The Mathematical Gazette. 40 (333): 233–234. doi:10.2307/3608848. hdl:2027/mdp.39015008206248. JSTOR 3608848.
+Bush, Robert R. (1956-01-01). "Review of Mathematics and Plausible Reasoning". The American Journal of Psychology. 69 (1): 166–167. doi:10.2307/1418146. hdl:2027/mdp.39015008206248. JSTOR 1418146.
+Johansson, I. (1955-01-01). "Review of Mathematics and plausible reasoning, I and II". Nordisk Matematisk Tidskrift. 3 (1): 64–65. JSTOR 24524537.
+Prager, W. (1955-01-01). "Review of Mathematics and plausible reasoning. Volume I: Induction and analogy. Volume II: Patterns of plausible inference". Quarterly of Applied Mathematics. 13 (3): 344–345. JSTOR 43634251.
+Meserve, Bruce E. (1955-01-01). "Review of Induction and Analogy in Mathematics, Vol. I, and Patterns of Plausible Inference, Vol. II, of Mathematics and Plausible Reasoning". The Mathematics Teacher. 48 (4): 272. JSTOR 27954884.
+Savage, Leonard J. (1955-01-01). "Review of Mathematics and Plausible Reasoning. Volume I, Induction and Analogy in Mathematics. Volume II, Patterns of Plausible Inference". Journal of the American Statistical Association. 50 (272): 1352–1354. doi:10.2307/2281238. JSTOR 2281238.
+פ., א. י. י. (1957-01-01). "Review of Mathematics and Plausible Reasoning. Volume I: Induction and Analogy in Mathematics; Volume II: Patterns of Plausible Reasoning". Iyyun: The Jerusalem Philosophical Quarterly / עיון: רבעון פילוסופי. ח' (א'): 48–49. JSTOR 23301574.
+Stein, Robert G. (1991-01-01). "Review of Patterns of Plausible Inference. Vol. 2 of Mathematics and Plausible Reasoning (R), George Pólya". The Mathematics Teacher. 84 (7): 574. JSTOR 27967294.
+Alexanderson, G. L. (1979-01-01). "Review of Mathematics and Plausible Reasoning: Vol. I: Induction and Analogy in Mathematics; Mathematics and Plausible Reasoning: Vol. II: Patterns of Plausible Inference, George Polya". The Two-Year College Mathematics Journal. 10 (2): 119–122. doi:10.2307/3027025. JSTOR 3027025.
+
+
+== References ==

data/en.wikipedia.org/wiki/Mechanica-0.md

@@ -0,0 +1,24 @@
+---
+title: "Mechanica"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mechanica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:47.561752+00:00"
+instance: "kb-cron"
+---
+
+Mechanica (Latin: Mechanica sive motus scientia analytice exposita; 1736) is a two-volume work published by mathematician Leonhard Euler which describes analytically the mathematics governing movement.
+Euler both developed the techniques of analysis and applied them to numerous problems in mechanics, 
+notably in later publications the calculus of variations. Euler's laws of motion expressed scientific laws of Galileo and Newton in terms of points in reference frames and coordinate systems making them useful for calculation when the statement of a problem or example is slightly changed from the original. 
+Newton–Euler equations express the dynamics of a rigid body. Euler has been credited with contributing to the rise of Newtonian mechanics especially in topics other than gravity.
+
+
+== References ==
+
+
+== External links ==
+Mechanica Vol. 1 [E015] – Latin.
+Mechanica Vol. 1 – English translation by Ian Bruce.
+Mechanica Vol. 2 [E016] – Latin.
+Mechanica Vol. 2 – English translation by Ian Bruce.

data/en.wikipedia.org/wiki/Method_of_Fluxions-0.md

@@ -0,0 +1,53 @@
+---
+title: "Method of Fluxions"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Method_of_Fluxions"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:48.700102+00:00"
+instance: "kb-cron"
+---
+
+Method of Fluxions (Latin: De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed in 1671 and posthumously published in 1736.
+
+
+== Background ==
+Fluxion is Newton's term for a derivative. He originally developed the method at Woolsthorpe Manor during the closing of Cambridge due to the Great Plague of London from 1665 to 1667. Newton did not choose to make his findings known (similarly, his findings which eventually became the Philosophiae Naturalis Principia Mathematica were developed at this time and hidden from the world in Newton's notes for many years). Gottfried Leibniz developed his form of calculus independently around 1673, seven years after Newton had developed the basis for differential calculus, as seen in surviving documents like “the method of fluxions and fluents..." from 1666. Leibniz, however, published his discovery of differential calculus in 1684, nine years before Newton formally published his fluxion notation form of calculus in part during 1693.
+
+
+== Impact ==
+The calculus notation in use today is mostly that of Leibniz, although Newton's dot notation for differentiation 
+  
+    
+      
+        
+          
+            
+              x
+              ˙
+            
+          
+        
+      
+    
+    {\displaystyle {\dot {x}}}
+  
+ is frequently used to denote derivatives with respect to time.
+
+
+== Rivalry with Leibniz ==
+Newton's Method of Fluxions was formally published posthumously, but following Leibniz's publication of the calculus a bitter rivalry erupted between the two mathematicians over who had developed the calculus first, provoking Newton to reveal his work on fluxions.
+
+
+== Newton's development of analysis ==
+For a period of time encompassing Newton's working life, the discipline of analysis was a subject of controversy in the mathematical community. Although analytic techniques provided solutions to long-standing problems, including problems of quadrature and the finding of tangents, the proofs of these solutions were not known to be reducible to the synthetic rules of Euclidean geometry. Instead, analysts were often forced to invoke infinitesimal, or "infinitely small", quantities to justify their algebraic manipulations. Some of Newton's mathematical contemporaries, such as Isaac Barrow, were highly skeptical of such techniques, which had no clear geometric interpretation. Although in his early work Newton also used infinitesimals in his derivations without justifying them, he later developed something akin to the modern definition of limits in order to justify his work.
+
+
+== See also ==
+
+
+== References and notes ==
+
+
+== External links ==
+Method of Fluxions at the Internet Archive

data/en.wikipedia.org/wiki/Methoden_der_mathematischen_Physik-0.md

@@ -0,0 +1,21 @@
+---
+title: "Methoden der mathematischen Physik"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Methoden_der_mathematischen_Physik"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:49.831629+00:00"
+instance: "kb-cron"
+---
+
+Methoden der mathematischen Physik (translated into English with the title Methods of Mathematical Physics) is a 1924 book, in two volumes totalling around 1000 pages, published under the names of Richard Courant and David Hilbert. It was a comprehensive treatment of the "methods of mathematical physics" of the time. The second volume is devoted to the theory of partial differential equations. It contains presages of the finite element method, on which Courant would work subsequently, and which would eventually become basic to numerical analysis.
+The material of the book was worked up from the content of Hilbert's lectures. While Courant played the major editorial role, many at the University of Göttingen were involved in the writing-up, and in that sense it was a collective production.
+On its appearance in 1924 it apparently had little direct connection to the quantum theory questions at the centre of the theoretical physics of the time. That changed within two years, since the formulation of the Schrödinger equation made the Hilbert–Courant techniques of immediate relevance to the new wave mechanics.
+There was a second edition (1931/7), wartime edition in the USA (1943), and a third German edition (1968). The English version Methods of Mathematical Physics (1953) was revised by Courant, and the second volume had extensive work done on it by the faculty of the Courant Institute. The books quickly gained the reputation as classics, and are among most highly referenced books in advanced mathematical physics courses.
+
+
+== References ==
+Constance Reid (1986) Hilbert-Courant (separate biographies bound as one volume)
+Courant, R.; Hilbert, D. (2024) [1953], Methods of mathematical physics, vol. I, New York, NY: Interscience Publishers, ISBN 978-3-527-41447-5, MR 0065391
+Courant, R.; Hilbert, D. (2024) [1962], Methods of mathematical physics, vol. II, New York, NY: Interscience Publishers, doi:10.1002/9783527617234, ISBN 978-3-527-41448-2, MR 0140802
+Methoden der mathematischen Physik online reproduction of 1924 German edition.

data/en.wikipedia.org/wiki/Metric_Structures_for_Riemannian_and_Non-Riemannian_Spaces-0.md

@@ -0,0 +1,18 @@
+---
+title: "Metric Structures for Riemannian and Non-Riemannian Spaces"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Metric_Structures_for_Riemannian_and_Non-Riemannian_Spaces"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:52.170085+00:00"
+instance: "kb-cron"
+---
+
+Metric Structures for Riemannian and Non-Riemannian Spaces is a book in geometry by Mikhail Gromov. It was originally published in French in 1981 under the title Structures métriques pour les variétés riemanniennes, by CEDIC (Paris). 
+
+
+== History ==
+The 1981 edition was edited by Jacques Lafontaine and Pierre Pansu. The English version, considerably expanded, was published in 1999 by Birkhäuser Verlag, with appendices by Pierre Pansu, Stephen Semmes, and Mikhail Katz. The book was well received and has been reprinted several times.
+
+
+== References ==

data/en.wikipedia.org/wiki/Mishnat_ha-Middot-0.md

@@ -0,0 +1,112 @@
+---
+title: "Mishnat ha-Middot"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mishnat_ha-Middot"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:53.273259+00:00"
+instance: "kb-cron"
+---
+
+The Mishnat ha-Middot (Hebrew: מִשְׁנַת הַמִּדּוֹת, lit. 'Treatise of Measures') is the earliest known Hebrew treatise on geometry, composed of 49 mishnayot in six chapters. Scholars have dated the work to either the Mishnaic period or the early Islamic era.
+
+
+== History ==
+
+
+=== Date of composition ===
+Moritz Steinschneider dated the Mishnat ha-Middot to between 800 and 1200 CE. Sarfatti and Langermann have advanced Steinschneider's claim of Arabic influence on the work's terminology, and date the text to the early ninth century.
+On the other hand, Hermann Schapira argued that the treatise dates from an earlier era, most likely the Mishnaic period, as its mathematical terminology differs from that of the Hebrew mathematicians of the Arab period. Solomon Gandz conjectured that the text was compiled no later than 150 CE (possibly by Rabbi Nehemiah) and intended to be a part of the Mishnah, but was excluded from its final canonical edition because the work was regarded as too secular.  The content resembles both the work of Hero of Alexandria (c. 100 CE) and that of al-Khwārizmī (c. 800 CE) and the proponents of the earlier dating therefore see the Mishnat ha-Middot linking Greek and Islamic mathematics.
+
+
+=== Modern history ===
+The Mishnat ha-Middot was discovered in MS 36 of the Munich Library by Moritz Steinschneider in 1862. The manuscript, copied in Constantinople in 1480, goes as far as the end of Chapter V. According to the colophon, the copyist believed the text to be complete. Steinschneider published the work in 1864, in honour of the seventieth birthday of Leopold Zunz. The text was edited and published again by mathematician Hermann Schapira in 1880.
+After the discovery by Otto Neugebauer of a genizah-fragment in the Bodleian Library containing Chapter VI, Solomon Gandz published a complete version of the Mishnat ha-Middot in 1932, accompanied by a thorough philological analysis. A third manuscript of the work was found among uncatalogued material in the Archives of the Jewish Museum of Prague in 1965.
+
+
+== Contents ==
+Although primarily a practical work, the Mishnat ha-Middot attempts to define terms and explain both geometric application and theory. The book begins with a discussion that defines "aspects" for the different kinds of plane figures (quadrilateral, triangle, circle, and segment of a circle) in Chapter I (§1–5), and with the basic principles of measurement of areas (§6–9). In Chapter II, the work introduces concise rules for the measurement of plane figures (§1–4), as well as a few problems in the calculation of volume (§5–12). In Chapters III–V, the Mishnat ha-Middot explains again in detail the measurement of the four types of plane figures, with reference to numerical examples. The text concludes with a discussion of the proportions of the Tabernacle in Chapter VI.
+The treatise argues against the common belief that the Tanakh defines the geometric ratio π as being exactly equal to 3 and defines it as 22⁄7 instead. The book arrives at this approximation by calculating the area of a circle according to the formulae
+
+  
+    
+      
+        A
+        =
+        
+          d
+          
+            2
+          
+        
+        −
+        
+          
+            
+              
+                d
+                
+                  2
+                
+              
+              7
+            
+          
+        
+        −
+        
+          
+            
+              
+                d
+                
+                  2
+                
+              
+              14
+            
+          
+        
+      
+    
+    {\displaystyle A=d^{2}-{\tfrac {d^{2}}{7}}-{\tfrac {d^{2}}{14}}}
+  
+ and 
+  
+    
+      
+        A
+        =
+        
+          
+            
+              c
+              2
+            
+          
+        
+        ⋅
+        
+          
+            
+              d
+              2
+            
+          
+        
+      
+    
+    {\displaystyle A={\tfrac {c}{2}}\cdot {\tfrac {d}{2}}}
+  
+.
+
+
+== See also ==
+Baraita of the Forty-nine Rules
+
+
+== References ==
+
+
+== External links ==
+MS Heb. c. 18, Catalogue of the Genizah Fragments in the Bodleian Libraries.

data/en.wikipedia.org/wiki/Murderous_Maths-0.md

@@ -0,0 +1,34 @@
+---
+title: "Murderous Maths"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Murderous_Maths"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:54.477242+00:00"
+instance: "kb-cron"
+---
+
+Murderous Maths is a series of British educational books by author Kjartan Poskitt. Most of the books in the series are illustrated by illustrator Philip Reeve, with the exception of "The Secret Life of Codes", which is illustrated by Ian Baker, "Awesome Arithmetricks" illustrated by Daniel Postgate and Rob Davis, and "The Murderous Maths of Everything", also illustrated by Rob Davis.
+The Murderous Maths books have been published in over 25 countries. The books, which are aimed at children aged 8 and above, teach maths, spanning from basic arithmetic to relatively complex concepts such as the quadratic formula and trigonometry. The books are written in an informal similar style to the Horrible Histories, Horrible Science and Horrible Geography series, involving evil geniuses, gangsters, and a generally comedic tone.
+
+== Development ==
+The first two books of the series were originally part of "The Knowledge" (now "Totally") series, itself a spin-off of Horrible Histories. However, these books were eventually redesigned and they, as well as the rest of the titles in the series, now use the Murderous Maths banner. According to Poskitt, "these books have even found their way into schools and proved to be a boost to GCSE studies". The books are also available in foreign editions, including: German, Spanish, Polish, Czech, Greek, Dutch, Norwegian, Turkish, Croatian, Italian, Lithuanian, Korean, Danish, Hungarian, Finnish, Thai and Portuguese (Latin America). In 2009, the books were redesigned again, changing the cover art style and the titles of most of the books in the series.
+Poskitt's goal, according to the Murderous Maths website, is to write books that are "something funny to read", have "good amusing illustrations", include "tricks", and "explaining the maths involved as clearly as possible". He adds that although he doesn't "work to any government imposed curriculum or any stage achievement levels", he has "been delighted to receive many messages of support and thanks from parents and teachers in the UK, the United States and elsewhere".
+
+== Titles ==
+The following are the thirteen books that are available in the series.
+
+Guaranteed to Bend Your Brain (previously Murderous Maths) (1997), ISBN 0-439-01156-6 - (addition, subtraction, multiplication, division, percentages, powers, tessellation, Roman numerals, the development of the "10" and the place system, shortcomings of calculators, prime numbers, time - how the year and day got divided, digital/analogue clocks, angles, introduction to real Mathematicians, magic squares, mental arithmetic, card trick with algebra explanation, rounding and symmetry.)
+Guaranteed to Mash your Mind (previously More Murderous Maths) (1998), ISBN 0-439-01153-1 (the monomino, domino, tromino, tetromino, pentomino, hexomino and heptomino, length area and volume, dimensions, measuring areas and volumes, basic rectangle and triangle formulas, speed, conversion of units, Möbius strip, Pythagoras, right-angled triangles, irrational numbers, pi, area and perimeter, bisecting angles, triangular numbers, topology networks, magic squares.)
+Awesome Arithmetricks (previously The Essential Arithmetricks: How to + - × ÷) (1998), ISBN 0-439-01157-4 - (counting, odd even and negative numbers, signs of maths, place value and rounding off, manipulating equations, + - x ÷ %, long division, times tables, estimation, decimal signs, QED.)
+The Mean & Vulgar Bits (previously The Mean & Vulgar Bits: Fractions and Averages) (2000), ISBN 0-439-01270-8 (fractions, converting improper and mixed fractions, adding subtracting multiplying and dividing fractions, primes and prime factors, reducing fractions, highest common factor and lowest common denominators, Egyptian fractions, comparing fractions, cancelling out fractions, converting fractions to decimals, decimal place system, percentages: increase and decrease, averages: mean mode and median.)
+Desperate Measures (previously Desperate Measures: Length, Area and Volume) (2000), ISBN 0-439-01370-4 (measuring lines: units and accuracy, old measuring systems, the development of metric, the SI system and powers of ten, shapes, measuring areas and area formulas, weight, angles, measuring volume, Archimedes Principle, density, time and how the modern calendar developed.)
+Do You Feel Lucky? (previously Do You Feel Lucky: The Secrets of Probability) (2001), ISBN 0-439-99607-4 (chance, tree diagrams, mutually exclusive and independent chances, Pascal's Triangle, permutations and combinations, sampling.)
+Savage Shapes (previously Vicious Circles and Other Savage Shapes) (2002), ISBN 0-439-99747-X (signs in geometric diagrams, Loci, constructions: perpendicular bisectors; dropping perpendiculars; bisecting angles, triangles: similar; congruent; equal areas, polygons: regular; irregular; angle sizes and construction, tessellations and Penrose Tiles, origami, circles: chord; tangent; angle theorems, regular solids, Euler's formula, ellipses, Geometric proof of Pythagoras' Theorem.)
+The Key To The Universe (previously Numbers: The Key To The Universe) (2002), ISBN 0-439-98116-6 (phi, Fibonacci Series, Golden Ratio, properties of Square, Triangle, Cube, Centred Hexagon and Tetrahedral numbers, "difference of two squares", number superstitions, prime numbers, Mersenne primes, tests to see if a number will divide by anything from 2-13 and 19, finger multiplication, binary, octal, and hexadecimal, perfect numbers, tricks of the nine times table, irrational transcendental and imaginary numbers, infinity.)
+The Phantom X (previously The Phantom X: Algebra) (2003), ISBN 0-439-97729-0 (variables, elementary algebra, brackets, factorising, expanding, and simplifying expressions, solving quadratics and the quadratic formula, "Think of a number" tricks, difference of two squares, coefficients of (a-b)n, linear graphs: co-ordinates; gradients; y intercept, non-linear function graphs including parabolas, simultaneous equations: substitution and elimination, dividing by zero!.)
+The Fiendish Angletron (previously  The Fiendish Angletron: Trigonometry) (2004), ISBN 0-439-96859-3 (scales and ratios in maps and diagrams, protractor and compass, SIN, COS and TAN ratios in right angled triangles, trig on a calculator; normal and inverse, sine and cosine formulas for non-right-angled triangles, triangulation, parallax angles and parsecs, sin/cos/tan relationships, sin wave, bearings.)
+The Perfect Sausages (previously The Perfect Sausage and other Fundamental Formulas) (2005), ISBN 0-439-95901-2 (areas and volumes, ellipsoids and toruses, number formulas (e.g. triangle, hexagonal), speed, acceleration, stopping time, distance, force, gravity, projectiles, Money: percentages; simple and compound interest, permutations and combinations.)
+The 5ecret L1fe of Code5 (previously Codes: How to Make Them and Break Them) (2007), ISBN 978-1-4071-0715-8 (patterns, logic and deduction, prime numbers, high powers, modular arithmetic.)
+Easy Questions, Evil Answers (2010), ISBN 1-407-11451-4 (formulas, working out square roots by hand, π, Pythagoras, paradoxes, problem solving, metric prefixes, large numbers, vectors.)
+Related puzzle books have been published also:

data/en.wikipedia.org/wiki/Murderous_Maths-1.md

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+---
+title: "Murderous Maths"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Murderous_Maths"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:54.477242+00:00"
+instance: "kb-cron"
+---
+
+Professor Fiendish's Book of Diabolical Brain-benders (2002), ISBN 0-439-98226-X (mazes, logic, coin problems, number crosswords, shape cutting/rearranging, number squares.)
+Professor Fiendish's Book of Brain-benders (a smaller version of the above) (same as above)
+Sudoku: 100 Fun Number Puzzles (2005), ISBN 0-439-84570-X
+Kakuro and Other Fiendish Number Puzzles (2006), ISBN 0-439-95164-X
+One title that covers many different areas of mathematics has also been released:
+
+The Most Epic Book of Maths Ever (previously The Murderous Maths of Everything) (2010), ISBN 1-407-10367-9 (prime numbers, Sieve of Eratosthenes, Pythagoras' Theorem, triangular numbers, square numbers, the International Date Line, geometry, geometric constructions, topology, Möbius strips, curves (conic sections and cycloids Golomb Rulers, four-dimensional "Tic Tac Toe", The Golden Ratio, Fibonacci sequence, Logarithmic spirals, musical ratios, Theorems (including Ham sandwich theorem and Fixed point theorem), probability (cards, dice, cluedo etc.), Pascal's Triangle, Sierpinski Triangle, chess board, light years, size and distance of moon and planets, orbit, size of stars, shape of galaxy.)
+Kjartan has also written a book entitled Everyday Maths for Grown-Ups (2011).
+
+== Reviews ==
+A recommendation of the series by Scientific American includes a quote from a Stanford engineer named Stacy F. Bennet, who described the series as "very humorous and engaging introductions to such topics as algebra, geometry and probability". On 22 November 1997, that same publication said of the series, "Have a look at Murderous Maths by Kjartan Poskitt. It is a truly addictive reading book, and was leapt on and devoured by my children. The book is full of awful jokes, fascinating facts, real murders and yes, the maths is good too. This is a brilliant book."
+The Primary Times released a review of Professor Fiendish's Book of Diabolical Brain-benders on November 25, 2002, describing the title as "intriguing, fun to do, and not at all dry", and adding "I warn you, once you start, you'll be 'hooked'!". The Times Educational Supplement also published a review on the book on December 6, 2002, describing the title as being "action-packed" and reasoning that "behind the non-stop fun, serious mathematical principles are being investigated".
+Kjartan did a presentation for 350 kids and 10 teachers at Wolfreton School, Hull in June 2004. Reporter Linda Blackbourne described it as a "stand-up maths routine [that] has children - and teachers - in fits of laughter". Carousel issue 16 (the guide to children's books) commented on the event: "...he possesses a prodigious gift of the (Yorkshire) gab, appears to be incapable of not enjoying himself, and plays his audience with the finesse of a maestro. Maths will never seem the same again".
+The Times Educational Supplement described Murderous Maths as "A stand-up maths routine has children and teachers in fits of laughter... maths has never been so much fun". The Western Gazette said: "It is not often that you see a grown maths teacher cry with laughter...", while The Worthing Gazette said: "The kids went wild, they absolutely loved it...". The Stockton Evening Gazette said: "Headteacher Barry Winter said it was a stroke of genius inviting the quick-witted author to open the resource centre". The GCSE book in the Guardian said: "Those who have experienced Poskitt "live" will recognise his commitment to getting readers involved with the learning process" (Nov 6th 2001), and The Press (York) described it as "...charismatic..."
+
+A review by science writer Brian Clegg described his views on Murderous Maths: Desperate Measures: It's the usual clever mix of light historical context − mostly ancient from Israelites and Archimedes to the Romans − and real insights into fascinating aspects of something that sits nicely between maths and practical science. There's plenty to keep the reader and interested, and even adults perusing it will have one or two surprises along the way. Because it is very much applied maths, there is also a lot more opportunity to have fun with practical things to try out than has been the case with some of the Murderous Maths series. All in all this is a great addition to the fold.
+
+== Spin-offs ==
+Killer Puzzles (Written by Kjartan Poskitt)
+The Urgum The Axeman series (by Kjartan Poskitt and illustrated by Philip Reeve)
+
+== See also ==
+Horrible Histories
+Horrible Science
+
+== References ==
+
+== External links ==
+The official Murderous Maths website
+Horrible Books and Magazines United States
+archived Daily Telegraph article

data/en.wikipedia.org/wiki/Mécanique_analytique-0.md

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+---
+title: "Mécanique analytique"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Mécanique_analytique"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:46.353463+00:00"
+instance: "kb-cron"
+---
+
+Mécanique analytique (1788–89) is a two volume French treatise on analytical mechanics, written by Joseph-Louis Lagrange, and published 101 years after Isaac Newton's Philosophiæ Naturalis Principia Mathematica.
+
+
+== Treatise ==
+It consolidated into one unified and harmonious system, the scattered developments of contributors such as Alexis Clairaut, Jean le Rond d'Alembert, Pierre-Simon Laplace, Leonhard Euler, and Johann and Jacob Bernoulli in the historical transition from geometrical methods, as presented in Newton's Principia, to the methods of the calculus. The treatise expounds a great labor-saving and thought-saving general analytical method by which every mechanical question may be stated in a single differential equation.
+
+Lagrange wrote that this work was entirely new and that his intent was to reduce the theory and the art of solving mechanics problems to general formulae, providing all the equations necessary for the solution of each problem. He stated that:No diagrams will be found in this work. The methods that I explain require neither geometrical, nor mechanical, constructions or reasoning, but only algebraical operations in accordance with regular and uniform procedure. Those who love Analysis will see with pleasure that Mechanics has become a branch of it, and will be grateful to me for having thus extended its domain.
+Ernst Mach describes the work as follows:
+Analytic mechanics... was brought to the highest degree of perfection... Lagrange's aim is... to dispose, once and for all, of the reasoning necessary to resolve mechanical problems, by embodying as much as possible of it in a single formula. This he did. Every case... can now be dealt with by a very simple... schema; and whatever reasoning is left is performed by purely mechanical methods. The mechanics of Lagrange is a stupendous contribution to the economy of thought.
+
+
+== Publication history ==
+The work was first published in 1788 (volume 1) and 1789 (volume 2). Lagrange issued a substantially enlarged second edition of volume 1 in 1811, toward the end of his life. His revision of volume 2 was substantially complete at the time of his death in 1813, but was not published until 1815.
+
+The second edition of 1811/15 has been translated into English, and is available online at archive.org.
+
+
+== See also ==
+
+Calculus of variations
+Lagrangian mechanics
+Hamiltonian mechanics
+
+
+== References ==
+
+
+== External links ==
+Lagrange, J. L. (1811). Mécanique analytique. Vol. 1 (2d ed.). Paris: Courcier.
+Lagrange, J. L. (1815). Mécanique analytique. Vol. 2 (2d ed.). Paris: Courcier.
+Lagrange, J. L. (1997). Analytical mechanics. Vol. 1 (2d ed.). ISBN 9789401589031. English translation of the 1811 edition

data/en.wikipedia.org/wiki/Numerical_Recipes-0.md

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+date_saved: "2026-05-05T08:46:00.240449+00:00"
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+date_saved: "2026-05-05T08:46:00.240449+00:00"
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data/en.wikipedia.org/wiki/On_Numbers_and_Games-0.md

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+---
+title: "On Numbers and Games"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/On_Numbers_and_Games"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:01.406278+00:00"
+instance: "kb-cron"
+---
+
+On Numbers and Games is a mathematics book by John Horton Conway first published in 1976. The book is  written by a pre-eminent mathematician, and is directed at other mathematicians. The material is, however, developed in a playful and unpretentious manner and many chapters are accessible to non-mathematicians.  Martin Gardner discussed the book at length, particularly Conway's construction of surreal numbers, in his Mathematical Games column in Scientific American in September 1976.
+The book is roughly divided into two sections: the first half (or Zeroth Part), on numbers, the second half (or First Part), on games. In the Zeroth Part, Conway provides axioms for arithmetic: addition, subtraction, multiplication, division and inequality. This allows an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals. The object to which these axioms apply takes the form {L|R}, which can be interpreted as a specialized kind of set; a kind of two-sided set. By insisting that L<R, this two-sided set resembles the Dedekind cut. The resulting construction yields a field, now called the surreal numbers. The ordinals are embedded in this field. The construction is rooted in axiomatic set theory, and is closely related to the Zermelo–Fraenkel axioms. In the original book, Conway simply refers to this field as "the numbers". The term "surreal numbers" is adopted later, at the suggestion of Donald Knuth.
+In the First Part, Conway notes that, by dropping the constraint that L<R, the axioms still apply and the construction goes through, but the resulting objects can no longer be interpreted as numbers. They can be interpreted as the class of all two-player games. The axioms for greater than and less than are seen to be a natural ordering on games, corresponding to which of the two players may win. The remainder of the book is devoted to exploring a number of different (non-traditional, mathematically inspired) two-player games, such as nim, hackenbush, and the map-coloring games col and snort. The development includes their scoring, a review of the Sprague–Grundy theorem, and the inter-relationships to numbers, including their relationship to infinitesimals.
+The book was first published by Academic Press in 1976, ISBN 0-12-186350-6, and a second edition was released by A K Peters in 2001 (ISBN 1-56881-127-6), containing a new prologue and an epilogue by Conway and several updates in the text. The currently available book by CRC Press, who acquired A K Peters in 2010, is printed in a notably bad quality, see the example at the end of this article.
+
+
+== Zeroth Part ... On Numbers ==
+
+In the Zeroth Part, Chapter 0, Conway introduces a specialized form of set notation, having the form {L|R}, where L and R are again of this form, built recursively, terminating in {|}, which is to be read as an analog of the empty set. Given this object, axiomatic definitions for addition, subtraction, multiplication, division and inequality may be given. As long as one insists that L<R (with this holding vacuously true when L or R are the empty set), then the resulting class of objects can be interpreted as numbers, the surreal numbers. The {L|R} notation then resembles the Dedekind cut.
+The ordinal 
+  
+    
+      
+        ω
+      
+    
+    {\displaystyle \omega }
+  
+ is built by transfinite induction. As with conventional ordinals, 
+  
+    
+      
+        ω
+        +
+        1
+      
+    
+    {\displaystyle \omega +1}
+  
+ can be defined. Thanks to the axiomatic definition of subtraction, 
+  
+    
+      
+        ω
+        −
+        1
+      
+    
+    {\displaystyle \omega -1}
+  
+ can also be coherently defined: it is strictly less than 
+  
+    
+      
+        ω
+      
+    
+    {\displaystyle \omega }
+  
+, and obeys the "obvious" equality 
+  
+    
+      
+        (
+        ω
+        −
+        1
+        )
+        +
+        1
+        =
+        ω
+        .
+      
+    
+    {\displaystyle (\omega -1)+1=\omega .}
+  
+ Yet, it is still larger than any natural number.
+The construction enables an entire zoo of peculiar numbers, the surreals, which form a field.  Examples include 
+  
+    
+      
+        ω
+        
+          /
+        
+        2
+      
+    
+    {\displaystyle \omega /2}
+  
+, 
+  
+    
+      
+        1
+        
+          /
+        
+        ω
+      
+    
+    {\displaystyle 1/\omega }
+  
+, 
+  
+    
+      
+        
+          
+            ω
+          
+        
+        =
+        
+          ω
+          
+            1
+            
+              /
+            
+            2
+          
+        
+      
+    
+    {\displaystyle {\sqrt {\omega }}=\omega ^{1/2}}
+  
+, 
+  
+    
+      
+        
+          ω
+          
+            1
+            
+              /
+            
+            ω
+          
+        
+      
+    
+    {\displaystyle \omega ^{1/\omega }}
+  
+ and similar.
+
+
+== First Part ... and Games ==
+
+In the First Part, Conway abandons the constraint that L<R, and then interprets the form {L|R} as a two-player game: a position in a contest between two players, Left and Right.  Each player has a set of games called options to choose from in turn.  Games are written {L|R} where L is the set of Left's options and R is the set of Right's options. At the start there are no games at all, so the empty set (i.e., the set with no members) is the only set of options we can provide to the players.  This defines the game {|}, which is called 0.  We consider a player who must play a turn but has no options to have lost the game.  Given this game 0 there are now two possible sets of options, the empty set and the set whose only element is zero.  The game {0|} is called 1, and the game {|0} is called -1.  The game {0|0} is called * (star), and is the first game we find that is not a number.
+All numbers are positive, negative, or zero, and we say that a game is positive if Left has a winning strategy, negative if Right has a winning strategy, or zero if the second player has a winning strategy.  Games that are not numbers have a fourth possibility: they may be fuzzy, meaning that the first player has a winning strategy.  * is a fuzzy game.
+
+
+== See also ==
+Winning Ways for Your Mathematical Plays
+
+
+== References ==

data/en.wikipedia.org/wiki/Opera_Omnia_Leonhard_Euler-0.md

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+---
+title: "Opera Omnia Leonhard Euler"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Opera_Omnia_Leonhard_Euler"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:02.590672+00:00"
+instance: "kb-cron"
+---
+
+Opera Omnia Leonhard Euler (Leonhardi Euleri Opera omnia) is the compilation of Leonhard Euler's scientific writings. The project of this compilation was undertaken by the Euler Committee of the Swiss Academy of Sciences, established in 1908, and is ongoing as of September 2022. The Committee decided on "the edition of the Collected Works of Leonhard Euler in the original languages, convinced of rendering the entire scientific world a service thereby", and, in 1919, it indicated to collect “All works from Leonhard Euler, hitherto unseen or already printed, coming from St Petersburg or elsewhere need to be integrated. This also includes the scientific letters of Euler”. The project has been supported by the international community, notably the Petersburg Academy of Sciences where Euler taught and which lent out its Euler materials in 1910. Publishing Euler's Opera Omnia has been termed "one of the most extraordinary projects in publishing".
+The Opera Omnia, excepting correspondences still being compiled in IVA9, was made available online in 2022 via the Opera-Bernoulli-Euler, which is working to make "the entire work of Euler, the Bernoulli family and their environment" freely available online.
+
+
+== Euler's writings ==
+During his life, Euler published about 560 writings. After his death in 1783, the Petersburg Academy published more of his manuscripts until 1830, increasing his number of publications to 756. Additional manuscripts were found later by his grandson Paul-Heinrich Fuss and published. Gustav Eneström established an inventory – the Eneström Index – between 1910 and 1913 listing 866 publications, namely E1–E866. Euler's writings are primarily in Latin, French, and German, though some are also in Russian and English. Previous attempts to compile all of Euler's writing had been made prior to the work of the Euler Committee.
+The committee has been publishing the 866 publications of Euler's work since 1911. The work was delayed by two world wars and economic issues. In the second part of the 20th century, it became more difficult to find qualified editors who could work with the Latin texts and German Kurrent handwriting. Recent editions contain more extensive footnotes. In 2005, the Committee decided that the final volumes would  be digitalized rather than printed, and that, eventually, the whole Opera Omnia would be online for easy access. The Opera-Bernoulli-Euler, begun in 2022, included the Opera Omnia, as well as the works of the Bernoulli family and other contemporary scientists. The Opera-Bernoulli-Euler project also planned to retrodigitize the already published content from the Opera Omnia and link it with the Eneström index so it would be easily accessible and searchable.
+
+
+== Series I - IV ==
+In 2018, the Committee indicated that 76 of 81 volumes had been published in four series: Earlier, it was planned to consist of 84 volumes with about 35,000 pages. As of September 2022, 80 of the 81 volumes have been completed with IVA9, the last volume of Euler's correspondence, being prepared for publication.
+
+Series I: Opera mathematica (Mathematics), 29 volumes, completed.
+Series II: Opera mechanica et astronomica (Mechanics and Astronomy), 31 volumes, completed.
+Series III: Opera physica, Miscellanea (Physics and Miscellaneous), 12 volumes, completed.
+Series IVA: Commercium epistolicum (Correspondence), 9 volumes
+Volume IVA/9 is the last volume to be published. Additional scientific correspondences will be made available online.
+Series IVB: Manuscripts, to be published online.
+
+
+== Major collaborating institutions ==
+Swiss National Science Foundation
+Petersburg Academy of Sciences, later Academy of Sciences of the Soviet Union, since 1991 Russian Academy of Sciences, Saint Petersburg, Russia
+University of Basel, Basel, Switzerland
+
+
+== References ==
+
+
+== External links ==
+Opera-Bernoulli-Euler (compiled works of Euler, Bernoulli family, and contemporary peers)
+The Euler Archive, with organization by Eneström Index
+Bernoulli-Euler-Zentrum (Bernoulli-Euler Center), University of Basel
+Series listing by Springer

data/en.wikipedia.org/wiki/Opticks-0.md

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+---
+title: "Opticks"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Opticks"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:03.830561+00:00"
+instance: "kb-cron"
+---
+
+Opticks: or, A Treatise of the Reflexions, Refractions, Inflexions and Colours of Light is a collection of three books by Isaac Newton that was published in English in 1704 (a scholarly Latin translation appeared in 1706). The treatise analyses the fundamental nature of light by means of the refraction of light with prisms and lenses, the diffraction of light by closely spaced sheets of glass, and the behaviour of colour mixtures with spectral lights or pigment powders. Opticks was Newton's second major work on physical science and it is considered one of the three major works on optics during the Scientific Revolution (alongside Johannes Kepler's Astronomiae Pars Optica and Christiaan Huygens' Treatise on Light).
+
+== Overview ==
+The publication of Opticks represented a major contribution to science, different from but in some ways rivalling the Principia, yet Newton's name did not appear on the cover page of the first edition. Opticks is largely a record of experiments and the deductions made from them, covering a wide range of topics in what was later to be known as physical optics. That is, this work is not a geometric discussion of catoptrics or dioptrics, the traditional subjects of reflection of light by mirrors of different shapes and the exploration of how light is "bent" as it passes from one medium, such as air, into another, such as water or glass. Rather, the Opticks is a study of the nature of light and colour and the various phenomena of diffraction, which Newton called the "inflexion" of light.
+Newton sets forth in full his experiments, first reported to the Royal Society of London in 1672, on dispersion, or the separation of light into a spectrum of its component colours. He demonstrates how the appearance of colour arises from selective absorption, reflection, or transmission of the various component parts of the incident light.
+The major significance of Newton's work is that it overturned the dogma, attributed to Aristotle or Theophrastus and accepted by scholars in Newton's time, that "pure" light (such as the light attributed to the Sun) is fundamentally white or colourless, and is altered into color by mixture with darkness caused by interactions with matter. Newton showed the opposite was true: light is composed of different spectral hues (he describes seven – red, orange, yellow, green, blue, indigo and violet), and all colours, including white, are formed by various mixtures of these hues. He demonstrates that colour arises from a physical property of light – each hue is refracted at a characteristic angle by a prism or lens – but he clearly states that colour is a sensation within the mind and not an inherent property of material objects or of light itself. For example, he demonstrates that a red violet (magenta) colour can be mixed by overlapping the red and violet ends of two spectra, although this colour does not appear in the spectrum and therefore is not a "colour of light". By connecting the red and violet ends of the spectrum, he organised all colours as a colour circle that both quantitatively predicts colour mixtures and qualitatively describes the perceived similarity among hues.
+Newton's contribution to prismatic dispersion was the first to outline multiple-prism arrays. Multiple-prism configurations, as beam expanders, became central to the design of the tunable laser more than 275 years later and set the stage for the development of the multiple-prism dispersion theory.
+
+== Comparison to the Principia ==
+
+Opticks differs in many respects from the Principia. It was first published in English rather than in the Latin used by European philosophers, contributing to the development of a vernacular science literature. The books were a model of popular science exposition: although Newton's English is somewhat dated—he shows a fondness for lengthy sentences with much embedded qualifications—the book can still be understood easily by a modern reader. In contrast, few readers of Newton's time found the Principia accessible or even comprehensible. His formal but flexible style shows colloquialisms and metaphorical word choice.
+Unlike the Principia, Opticks is not developed using the geometric convention of propositions proved by deduction from either previous propositions, lemmas or first principles (or axioms). Instead, axioms define the meaning of technical terms or fundamental properties of matter and light, and the stated propositions are demonstrated by means of specific, carefully described experiments. The first sentence of Book I declares "My Design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments." In an Experimentum crucis or "critical experiment" (Book I, Part II, Theorem ii), Newton showed that the colour of light corresponded to its "degree of refrangibility" (angle of refraction), and that this angle cannot be changed by additional reflection or refraction or by passing the light through a coloured filter.
+The work is a vade mecum of the experimenter's art, displaying in many examples how to use observation to propose factual generalisations about the physical world and then exclude competing explanations by specific experimental tests. Unlike the Principia, which vowed Non fingo hypotheses or "I make no hypotheses" outside the deductive method, the Opticks develops conjectures about light that go beyond the experimental evidence: for example, that the physical behaviour of light was due its "corpuscular" nature as small particles, or that perceived colours were harmonically proportioned like the tones of a diatonic musical scale.
+
+== Queries ==

data/en.wikipedia.org/wiki/Opticks-1.md

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+---
+title: "Opticks"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Opticks"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:03.830561+00:00"
+instance: "kb-cron"
+---
+
+Newton originally considered to write four books, but he dropped the last book on action at a distance. Instead he concluded Opticks with a set of unanswered questions and positive assertions referred to as queries in Book III. The first set of queries were brief, but the later ones became short essays, filling many pages. In the first edition, these were sixteen such queries; that number was increased to 23 in the Latin edition, published in 1706, and then in the revised English edition, published in 1717–18. In the fourth edition of 1730, there were 31 queries.
+These queries, especially the later ones, deal with a wide range of physical phenomena that go beyond the topic of optics. The queries concern the nature and transmission of heat; the possible cause of gravity; electrical phenomena; the nature of chemical action; the way in which God created matter; the proper way to do science; and even the ethical conduct of human beings. These queries are not really questions in the ordinary sense. These queries are almost all posed in the negative, as rhetorical questions. That is, Newton does not ask whether light "is" or "may be" a "body." Rather, he declares: "Is not Light a Body?" Stephen Hales, a firm Newtonian of the early eighteenth century, declared that this was Newton's way of explaining "by Quaere."
+
+=== Notable queries ===
+The first query reads: "Do not Bodies act upon Light at a distance, and by their action bend its Rays; and is not this action (caeteris paribus) strongest at the least distance?" suspecting on the effect of gravity on the trajectory of light rays. This query predates the prediction of gravitational lensing by Albert Einstein's general relativity by two centuries and later confirmed by the Eddington experiment in 1919.  
+Query 3, attempts to explain diffraction and Newton's rings by considering an "eel-like" motion of corpuscles of light when passing by edges. Newton explained earlier that this motion is produced by oscillations in the luminiferous aether creating some interaction between the corpuscles and their own medium. In the 19th century, wave theory superseded this theory, but in the advent of quantum mechanics, some authors consider that pilot wave theory, one of the interpretations of quantum mechanics, vindicates Newton's corpuscular theory in this regard.
+Query 6 of the book reads "Do not black Bodies conceive heat more easily from Light than those of other Colours do, by reason that the Light falling on them is not reflected outwards, but enters into the Bodies, and is often reflected and refracted within them, until it be stifled and lost?", thereby introducing the concept of a black body. The first part of query 30 reads "Are not gross Bodies and Light convertible into one another" thereby anticipating mass-energy equivalence.
+The last query (number 31) wonders if a corpuscular theory could explain how different substances react more to certain substances than to others, in particular how aqua fortis (nitric acid) reacts more with calamine that with iron. This 31st query has been often been linked to the origin of the concept of affinity in chemical reactions. Various 18th-century historians and chemists, such as William Cullen and Torbern Bergman, credited Newton for the development of affinity tables.
+
+== Reception ==
+The Opticks was widely read and debated in England and in continental Europe. The early presentation of the work to the Royal Society stimulated a bitter dispute between Newton and Robert Hooke over the corpuscular theory of light, which prompted Newton to postpone publication of the work until after Hooke's death in 1703. On the continent, and in France in particular, both the Principia and the Opticks were initially rejected by many natural philosophers, who continued to defend Cartesian natural philosophy and the Aristotelian version of colour, and claimed to find Newton's prism experiments difficult to replicate. Indeed, the Aristotelian theory of the fundamental nature of white light was defended into the 19th century, for example by the German writer Johann Wolfgang von Goethe in his 1810 Theory of Colours (German: Zur Farbenlehre).
+Newtonian science became a central issue in the assault waged by the philosophes in the Age of Enlightenment against a natural philosophy based on the authority of ancient Greek or ancient Roman naturalists or on deductive reasoning from first principles (the method advocated by the French philosopher René Descartes), rather than on the application of mathematical reasoning to experience or experiment. Voltaire popularised Newtonian science, including the content of both the Principia and the Opticks, in his Elements de la philosophie de Newton (1738), and after about 1750 the combination of the experimental methods exemplified by the Opticks and the mathematical methods exemplified by the Principia were established as a unified and comprehensive model of Newtonian science. Some of the primary adepts in this new philosophy were such prominent figures as Benjamin Franklin, Antoine Lavoisier and James Black.
+Subsequent to Newton, much has been amended. Thomas Young and Augustin-Jean Fresnel showed that the wave theory Christiaan Huygens described in his Treatise on Light (1690) could prove that colour is the visible manifestation of light's wavelength. Science also slowly came to recognise the difference between perception of colour and mathematisable optics. Goethe, with his epic diatribe Theory of Colours, could not shake the Newtonian foundation – but, John Tyndall writes in 1880, "one hole Goethe did find in Newton's armour.. Newton had committed himself to the doctrine that refraction without colour was impossible. He therefore thought that the object-glasses of telescopes must for ever remain imperfect, achromatism and refraction being incompatible. This inference was proved by Dollon to be wrong."
+
+== See also ==
+Color theory
+Luminiferous aether
+Prism (optics)
+Theory of Colours
+Book of Optics (Ibn al-Haytham)
+Elements of the Philosophy of Newton (Voltaire)
+Multiple-prism dispersion theory
+
+== Notes ==
+
+== References ==
+
+== External links ==
+
+Full and free online editions of Newton's Opticks
+
+Rarebookroom, First edition
+ETH-Bibliothek, First edition
+Gallica, First edition
+Internet Archive, Fourth edition
+Project Gutenberg digitized text & images of the Fourth Edition
+Cambridge University Digital Library, Papers on Hydrostatics, Optics, Sound and Heat – Manuscript papers by Isaac Newton containing draft of Opticks
+ Opticks public domain audiobook at LibriVox

data/en.wikipedia.org/wiki/Origami_Polyhedra_Design-0.md

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+---
+title: "Origami Polyhedra Design"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Origami_Polyhedra_Design"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:04.980941+00:00"
+instance: "kb-cron"
+---
+
+Origami Polyhedra Design is a book on origami designs for constructing polyhedra. It was written by the origami artist and mathematician John Montroll, and it was published in 2009 by A K Peters.
+
+
+== Topics ==
+There are two traditional methods for making polyhedra out of paper: polyhedral nets and modular origami. In the net method, the faces of the polyhedron are placed to form an irregular shape on a flat sheet of paper, with some of these faces connected to each other within this shape; it is cut out and folded into the shape of the polyhedron, and the remaining pairs of faces are attached together. In the modular origami method, many similarly-shaped "modules" are each folded from a single sheet of origami paper, and then assembled to form a polyhedron, with pairs of modules connected by the insertion of a flap from one module into a slot in another module. This book does neither of those two things. Instead, it provides designs for folding polyhedra, each out of a single uncut sheet of origami paper.
+After a brief introduction to the mathematics of polyhedra and the concepts used to design origami polyhedra, book presents designs for folding 72 different shapes, organized by their level of difficulty. These include the regular polygons and the Platonic solids, Archimedean solids, and Catalan solids, as well as less-symmetric convex polyhedra such as dipyramids and non-convex shapes such as a "sunken octahedron" (a compound of three mutually-perpendicular squares). An important constraint used in the designs was that the visible faces of each polyhedron should have few or no creases; additionally, the symmetries of the polyhedron should be reflected in the folding pattern, to the extent possible, and the resulting polyhedron should be large and stable.
+
+
+== Audience and reception ==
+Reviewer Tom Hagedorn writes that "The book is well designed and organized and makes you want to start folding polyhedra," and that its instructions are "clear and easy to understand"; he recommends it to anyone interested in origami, polyhedra, or both. Reviewer Rachel Thomas recommends it to origami folders, to demonstrate to them the beauty of geometric forms, and to mathematicians, to show these forms in a new light and demonstrate the creativity of origami design. The book can also be used as a source for mathematical school projects, and to provide hands-on experience with geometry concepts such as length, angles, surface area, and volume; some of its designs are suitable for students as young as middle school, although others require more experience as an origami folder.
+
+
+== See also ==
+List of books about polyhedra
+
+
+== References ==

data/en.wikipedia.org/wiki/Origamics-0.md

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+---
+title: "Origamics"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Origamics"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:06.100462+00:00"
+instance: "kb-cron"
+---
+
+Origamics: Mathematical Explorations Through Paper Folding is a book on the mathematics of paper folding by Kazuo Haga, a Japanese retired biology professor. It was edited and translated into English by Josefina C. Fonacier and Masami Isoda, based on material published in several Japanese-language books by Haga, and published in 2008 by World Scientific. The title is a portmanteau of "origami" and "mathematics", coined in the 1990s by Haga to describe the type of paper-folding mathematical exploration that would later be described in this book.
+
+
+== Topics ==
+Although much of its content involves folding square sheets of origami paper, the book focuses on mathematical explorations developing from folding and unfolding paper rather than on the traditional use of origami to create paper figures and artworks. It is divided into ten chapters, exploring concepts in paper folding that are "so simple that they could be discovered by middle- or high-school students".
+The book begins with the exploration of a single fold of a corner of a square to a midpoint of an opposite edge, and its analysis involving the geometry of the 3–4–5 right triangle. Later explorations (sometimes presented with colorful stories of knights and princesses as motivation)  concern folding one or more corners of the square to other points on the square, similar folds on paper with the shape of a silver rectangle (such as A4 letter paper), the interactions of the fold lines produced in this way, and the use of these folds to obtain subdivisions of the interval into different numbers of parts.
+
+
+== Audience and reception ==
+The book is primarily aimed at secondary-school mathematics teachers, and reviewer Gertraud Ehrig suggests that this book would be particularly helpful for them in providing inspiration for activities for their students.
+Although the many activities discussed throughout the book are suitable for discovery learning by students, it also includes more technical material proving the mathematical insights found through these activities. These parts use only elementary methods in Euclidean geometry, such as the Pythagorean theorem and the use of triangle centers, and may be best omitted when presenting this material to students.
+
+
+== References ==

data/en.wikipedia.org/wiki/Pasta_by_Design-0.md

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+---
+title: "Pasta by Design"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Pasta_by_Design"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:07.271123+00:00"
+instance: "kb-cron"
+---
+
+Pasta by Design is a book by George L. Legendre, with a foreword by Paola Antonelli, and photography by Stefano Graziani. It is based on an idea by Marco Guarnieri.
+
+
+== Overview ==
+Pasta by Design spans the fields of architecture, food, and popular science. The book features 92 pasta shapes, each depicted by a photograph, a mathematical equation, a 3D visual, and a short paragraph on geographic provenance and cooking etiquette. It was first published in 2011 by Thames & Hudson, while a German translation was published in 2012 by Springer Verlag.
+
+
+== Taxonomy ==
+Pasta by Design is primarily a work of taxonomy, or classification. The critical inventory of shapes recalls the compilations of building-related knowledge known in the nineteenth century as architectural treatises, in which the source material was systematically drawn and formatted as a catalogue. To organise and classify the large variety of pasta shapes, the book employed principles of phylogenetics. The first application of phylogenetics to architectural criticism appeared in Phylogenesis: FOA’s Ark (2003), an architectural monograph by Foreign Office Architects, the first such publication to catalogue projects exclusively by design property, instead of the commonly used markers of programme, location, and client. Pasta by Design opens with a phylogenetic chart  and uses it to classify 92 pasta shapes. Unlike FOA’s Ark its criteria of classification are couched in analytic mathematics.
+
+
+== Morphology ==
+The organizing principle of classification is the morphology of each pasta shape, reduced to its elemental characteristics and expressed by simple mathematical relationships. Shapes which may look dissimilar at first glance, such as Sagne Incannulate and Cappelletti, may still be described with the same mathematical relationships and hence may turn out to be more closely related than is immediately apparent. Technically, each shape is depicted by three parametric equations of two mathematical functions, the sine and cosine of two angles. Parametric equations can describe any three-dimensional form. The mathematics of the book are neither exclusive to the modelling of pasta, nor innovative in a technical sense.
+
+
+== Design, science, and culture ==
+Pasta by Design reflects the recent anointment of pasta as the subject of theoretical   and historical  investigations, its embrace by the public as both foodstuff and design icon, and the prominence of food in society and culture at large. It is also indicative of a broader, post-millennium cultural trend summarised by design curator Paola Antonelli in those words: In our contemporary world (...) where programmers talk about the beauty of code and architects and designers tinker with algorithms and software to achieve organic formal and structural behaviours, seeing mathematics in fusilli makes perfect sense. The application of geometry and phylogenetics to design and illustration is symptomatic of the wholesale identification of many creatives with the ambitions of mathematicians, biologists and scientists at large. This cultural trend is in evidence in architecture’s decade-long pursuit of morphogenetics, the mathematical and biomorphic modelling of form. In the design arts, it is the subject of MoMA’s Design and the Elastic Mind (2008), a curatorial exploration of the designers’ ability to grasp momentous changes in technology, science, and social mores. In the field of cookery, it is found (among other places) in Nathan Myhrvold's Modernist Cuisine: The Art and Science of Cooking (2011)  and the rise of molecular gastronomy.
+
+
+== References ==

data/en.wikipedia.org/wiki/Perspectiva_corporum_regularium-0.md

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+---
+title: "Perspectiva corporum regularium"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Perspectiva_corporum_regularium"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:08.403606+00:00"
+instance: "kb-cron"
+---
+
+Perspectiva corporum regularium (from Latin: Perspective of the Regular Solids) is a book of perspective drawings of polyhedra by German Renaissance goldsmith Wenzel Jamnitzer, with engravings by Jost Amman, published in 1568.
+Despite its Latin title, Perspectiva corporum regularium is written mainly in the German language. It was "the most lavish of the perspective books published in Germany in the late sixteenth century" and was included in several royal art collections. It may have been the first work to depict chiral icosahedral symmetry.
+
+
+== Topics ==
+
+The book focuses on the five Platonic solids, with the subtitles of its title page citing Plato's Timaeus and Euclid's Elements for their history. Each of these five shapes has a chapter, whose title page relates the connection of its polyhedron to the classical elements in medieval cosmology: fire for the tetrahedron, earth for the cube, air for the octahedron, and water for the icosahedron, with the dodecahedron representing the heavens, its 12 faces corresponding to the 12 symbols of the zodiac. Each chapter includes four engravings of polyhedra, each showing six variations of the shape including some of their stellations and truncations, for a total of 120 polyhedra. This great amount of variation, some of which obscures the original Platonic form of each polyhedron, demonstrates the theory of the time that all the variation seen in the physical world comes from the combination of these basic elements.
+Following these chapters, additional engravings depict additional polyhedral forms, including polyhedral compounds such as the stella octangula, polyhedral variations of spheres and cones, and outlined skeletons of polyhedra following those drawn by Leonardo da Vinci for Luca Pacioli's earlier book Divina proportione. In this part of the book, the shapes are arranged in a three-dimensional setting and often placed on smaller polyhedral pedestals.
+
+
+== Creation process ==
+
+The roughly 50 engravings for the book were made by Jost Amman, a German woodcut artist, based on drawings by Jamnitzer. As Jamnitzer describes in his prologue, he built models of polyhedra out of paper and wood and used a mechanical device to help trace their perspective. This process was depicted in another engraving by Amman from around 1565, showing Jamnitzer at work on his drawings. Amman included this engraving in another book, Das Ständebuch (1658).
+
+
+== Related works ==
+A later work on perspective, Artes Excelençias de la Perspectiba (1688) by P. Gómez de Alcuña, was heavily influenced by Jamnitzer.
+A 2008 German postage stamp, issued to commemorate the 500th anniversary of Jamnitzer's birth, included a reproduction of one of the pages of the book, depicting two polyhedral cones tilted towards each other. The full sheet of ten stamps also includes another figure from the book, a skeletal icosahedron.
+A French edition of Perspectiva corporum regularium, edited by Albert Flocon, was published by Brieux in 1964. Gutenberg Reprints republished it both in the original German and in the French edition in 1981. A Spanish translation of Perspectiva corporum regularium was published in 2006.
+
+
+== See also ==
+List of books about polyhedra
+
+
+== References ==
+
+
+== Further reading ==
+Flocon, Albert (1980), "Wenzel Jamnitzer: Perspectiva corporum regularium. Un des plus beaux livres gravés du 16siècle", Nouvelles de l'Estampe Paris (in French), 52–53: 24–25
+
+
+== External links ==
+
+Perspectiva corporum regularium on the Internet Archive
+Wentzel Jamnitzer's Polyhedra, George W. Hart's Virtual Polyhedra

data/en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica-0.md

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+---
+title: "Philosophiæ Naturalis Principia Mathematica"
+chunk: 1/11
+source: "https://en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:10.722464+00:00"
+instance: "kb-cron"
+---
+
+Philosophiæ Naturalis Principia Mathematica (English: The Mathematical Principles of Natural Philosophy), often called simply the Principia (), is a book by Sir Isaac Newton that expounds Newton's laws of motion and his law of universal gravitation.
+The Principia is written in Latin and comprises three volumes, and was authorized (imprimatur) by Samuel Pepys, then-President of the Royal Society on 5 July 1686 and first published in 1687. After annotating and correcting his personal copy of the first edition, Newton published two further editions, during 1713 with errors in the 1687 version corrected, and an improved version of 1726.
+The Principia forms a mathematical foundation for the theory of classical mechanics, and is generally considered to be one of the most important works in the history of science. It has been referred to as "the greatest scientific work in history" and "the supreme expression in human thought of the mind's ability to hold the universe fixed as an object of contemplation". 
+In formulating his physical laws, Newton developed and used mathematical methods now included in the field of calculus, expressing them in the form of geometric propositions about "vanishingly small" shapes. In a revised conclusion to the Principia (see § General Scholium), Newton emphasized the empirical nature of the work with the expression Hypotheses non fingo ("I frame/feign no hypotheses"). Among other achievements, Newton provides an explanation for Johannes Kepler's laws of planetary motion, which Kepler had obtained empirically.
+
+== Contents ==
+
+=== Expressed aim and topics covered ===
+
+The Preface of the work states:
+
+... Rational Mechanics will be the sciences of motion resulting from any forces whatsoever, and of the forces required to produce any motion, accurately proposed and demonstrated ... And therefore we offer this work as mathematical principles of his philosophy. For all the difficulty of philosophy seems to consist in this—from the phenomena of motions to investigate the forces of Nature, and then from these forces to demonstrate the other phenomena ...
+Newton situates himself within the contemporary scientific movement which had "omit[ed] substantial forms and the occult qualities" and instead endeavoured to explain the world by empirical investigation and outlining of empirical regularities.
+The Principia deals primarily with massive bodies in motion, initially under a variety of conditions and hypothetical laws of force in both non-resisting and resisting media, thus offering criteria to decide, by observations, which laws of force are operating in phenomena that may be observed. It attempts to cover hypothetical or possible motions both of celestial bodies and of terrestrial projectiles. It explores difficult problems of motions perturbed by multiple attractive forces. Its third and final book deals with the interpretation of observations about the movements of planets and their satellites.
+The book:
+
+shows how astronomical observations verify the inverse square law of gravitation (to an accuracy that was high by the standards of Newton's time);
+offers estimates of relative masses for the known giant planets and for the Earth and the Sun;
+defines the motion of the Sun relative to the Solar System barycenter;
+shows how the theory of gravity can account for irregularities in the motion of the Moon;
+identifies the oblateness of the shape of the Earth;
+accounts approximately for marine tides including phenomena of spring and neap tides by the perturbing (and varying) gravitational attractions of the Sun and Moon on the Earth's waters;
+explains the precession of the equinoxes as an effect of the gravitational attraction of the Moon on the Earth's equatorial bulge; and
+gives theoretical basis for numerous phenomena about comets and their elongated, near-parabolic orbits.
+The opening sections of the Principia contain, in revised and extended form, nearly all of the content of Newton's 1684 tract De motu corporum in gyrum.
+The Principia begin with "Definitions" and "Axioms or Laws of Motion", and continues in three books:
+
+=== Book 1, De motu corporum ===
+Book 1, subtitled De motu corporum (On the motion of bodies) concerns motion in the absence of any resisting medium. It opens with a collection of mathematical lemmas on "the method of first and last ratios", a geometrical form of infinitesimal calculus.
+
+The second section establishes relationships between centripetal forces and the law of areas now known as Kepler's second law (Propositions 1–3), and relates circular velocity and radius of path-curvature to radial force (Proposition 4), and relationships between centripetal forces varying as the inverse-square of the distance to the center and orbits of conic-section form (Propositions 5–10).
+Propositions 11–31 establish properties of motion in paths of eccentric conic-section form including ellipses, and their relationship with inverse-square central forces directed to a focus and include Newton's theorem about ovals (lemma 28).
+Propositions 43–45 are demonstration that in an eccentric orbit under centripetal force where the apse may move, a steady non-moving orientation of the line of apses is an indicator of an inverse-square law of force.
+Book 1 contains some proofs with little connection to real-world dynamics. But there are also sections with far-reaching application to the solar system and universe:
+Propositions 57–69 deal with the "motion of bodies drawn to one another by centripetal forces". This section is of primary interest for its application to the Solar System, and includes Proposition 66 along with its 22 corollaries: here Newton took the first steps in the definition and study of the problem of the movements of three massive bodies subject to their mutually perturbing gravitational attractions, a problem which later gained name and fame (among other reasons, for its great difficulty) as the three-body problem.
+Propositions 70–84 deal with the attractive forces of spherical bodies. The section contains Newton's proof that a massive spherically symmetrical body attracts other bodies outside itself as if all its mass were concentrated at its centre. This fundamental result, called the Shell theorem, enables the inverse square law of gravitation to be applied to the real solar system to a very close degree of approximation.

data/en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica-1.md

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+---
+title: "Philosophiæ Naturalis Principia Mathematica"
+chunk: 2/11
+source: "https://en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:10.722464+00:00"
+instance: "kb-cron"
+---
+
+=== Book 2, part 2 of De motu corporum ===
+Part of the contents originally planned for the first book was divided out into a second book, which largely concerns motion through resisting mediums. Just as Newton examined consequences of different conceivable laws of attraction in Book 1, here he examines different conceivable laws of resistance; thus Section 1 discusses resistance in direct proportion to velocity, and Section 2 goes on to examine the implications of resistance in proportion to the square of velocity. Through Book II, Newton was an important pioneer of fluid mechanics, and a later analysis showed that of its 53 propositions almost all are correct, with only two or three open to question. Book 2 discusses (in Section 5) hydrostatics and the properties of compressible fluids; he also derives Boyle's law (incorrectly; for an ideal gas is not like an elastic fluid). The effects of air resistance on pendulums are studied in Section 6, along with Newton's account of experiments that he carried out, to try to find out some characteristics of air resistance in reality by observing the motions of pendulums under different conditions. Newton compares the resistance offered by a medium against motions of globes with different properties (material, weight, size). In Section 8, he derives rules to determine the speed of waves in fluids and relates them to the density and condensation (Proposition 48; this would become very important in acoustics). He assumes that these rules apply equally to light and sound and estimates that the speed of sound is around 1088 feet per second and can increase depending on the amount of water in air.
+Less of Book 2 has stood the test of time than of Books 1 and 3, and it has been said that Book 2 was largely written to refute a theory of Descartes which had some wide acceptance before Newton's work (and for some time after). According to Descartes's theory of vortices, planetary motions were produced by the whirling of fluid vortices that filled interplanetary space and carried the planets along with them. Newton concluded Book 2 by commenting that the hypothesis of vortices was completely at odds with the astronomical phenomena, and served not so much to explain as to confuse them.
+
+=== Book 3, De mundi systemate ===
+Book 3, subtitled De mundi systemate (On the system of the world), is an exposition of many consequences of universal gravitation, especially its consequences for astronomy. It builds upon the propositions of the previous books and applies them with greater specificity than in Book 1 to the motions observed in the Solar System. Here (introduced by Proposition 22, and continuing in Propositions 25–35) are developed several of the features and irregularities of the orbital motion of the Moon, especially the variation. Newton lists the astronomical observations on which he relies, and establishes in a stepwise manner that the inverse square law of mutual gravitation applies to Solar System bodies, starting with the satellites of Jupiter and going on by stages to show that the law is of universal application. He also gives starting at Lemma 4 and Proposition 40 the theory of the motions of comets, for which much data came from John Flamsteed and Edmond Halley, and accounts for the tides, attempting quantitative estimates of the contributions of the Sun and Moon to the tidal motions; and offers the first theory of the precession of the equinoxes. Book 3 also considers the harmonic oscillator in three dimensions, and motion in arbitrary force laws.
+In Book 3 Newton also made clear his heliocentric view of the Solar System, modified in a somewhat modern way, since already in the mid-1680s he recognised the "deviation of the Sun" from the centre of gravity of the Solar System. For Newton, "the common centre of gravity of the Earth, the Sun and all the Planets is to be esteem'd the Centre of the World", and that this centre "either is at rest, or moves uniformly forward in a right line". Newton rejected the second alternative after adopting the position that "the centre of the system of the world is immoveable", which "is acknowledg'd by all, while some contend that the Earth, others, that the Sun is fix'd in that centre". Newton estimated the mass ratios Sun:Jupiter and Sun:Saturn, and pointed out that these put the centre of the Sun usually a little way off the common center of gravity, but only a little, the distance at most "would scarcely amount to one diameter of the Sun".
+
+=== Commentary on the Principia ===
+
+The sequence of definitions used in setting up dynamics in the Principia is recognisable in many textbooks today. Newton first set out the definition of mass
+
+The quantity of matter is that which arises conjointly from its density and magnitude. A body twice as dense in double the space is quadruple in quantity. This quantity I designate by the name of body or of mass.
+This was then used to define the "quantity of motion" (today called momentum), and the principle of inertia in which mass replaces the previous Cartesian notion of intrinsic force. This then set the stage for the introduction of forces through the change in momentum of a body. Curiously, for today's readers, the exposition looks dimensionally incorrect, since Newton does not introduce the dimension of time in rates of changes of quantities.
+He defined space and time "not as they are well known to all". Instead, he defined "true" time and space as "absolute" and explained:

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+title: "Philosophiæ Naturalis Principia Mathematica"
+chunk: 11/11
+source: "https://en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:10.722464+00:00"
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+---
+
+In 1977, the spacecraft Voyager 1 and 2 left earth for the interstellar space carrying a picture of a page from Newton's Principia Mathematica, as part of the Golden Record, a collection of messages from humanity to extraterrestrials.
+In 2014, British astronaut Tim Peake named his upcoming mission to the International Space Station Principia after the book, in "honour of Britain's greatest scientist". Tim Peake's Principia launched on 15 December 2015 aboard Soyuz TMA-19M.
+
+== See also ==
+Atomism
+Elements of the Philosophy of Newton
+Isaac Newton's occult studies
+
+== References ==
+
+== Further reading ==
+
+== External links ==
+
+=== Latin versions ===
+
+First edition (1687)
+
+Trinity College Library, Cambridge High resolution digitised version of Newton's own copy of the first edition, with annotations.
+Cambridge University, Cambridge Digital Library High resolution digitised version of Newton's own copy of the first edition, interleaved with blank pages for his annotations and corrections.
+1687: Newton's Principia, first edition (1687, in Latin). High-resolution presentation of the Gunnerus Library copy.
+1687: Newton's Principia, first edition (1687, in Latin).
+Project Gutenberg.
+ETH-Bibliothek Zürich. From the library of Gabriel Cramer.
+Philosophiæ Naturalis Principia Mathematica From the Rare Book and Special Collection Division at the Library of Congress
+Second edition (1713)
+
+ETH-Bibliothek Zürich.
+ETH-Bibliothek Zürich (pirated Amsterdam reprint of 1723).
+Philosophiæ naturalis principia mathematica (Adv.b.39.2), a 1713 edition with annotations by Newton in the collections of Cambridge University Library and fully digitised in Cambridge Digital Library
+Third edition (1726)
+
+ETH-Bibliothek Zürich.
+Later Latin editions
+
+Principia (in Latin, annotated). 1833 Glasgow reprint (volume 1) with Books 1 and 2 of the Latin edition annotated by Leseur, Jacquier and Calandrini 1739–42 (described above).
+Archive.org (1871 reprint of the 1726 edition)
+
+=== English translations ===
+
+Andrew Motte, 1729, first English translation of third edition (1726)
+WikiSource, Partial
+Google books, vol. 1 with Book 1.
+Internet Archive, vol. 2 with Books 2 and 3. (Book 3 starts at p.200.) (Google's metadata wrongly labels this vol. 1).
+Partial HTML
+Robert Thorpe 1802 translation
+N. W. Chittenden, ed., 1846 "American Edition" a partly modernised English version, largely the Motte translation of 1729.
+Wikisource
+Archive.org #1
+Archive.org #2
+eBooks@Adelaide  eBooks@Adelaide
+Percival Frost 1863 translation with interpolations Archive.org
+Florian Cajori 1934 modernisation of 1729 Motte and 1802 Thorpe translations
+Ian Bruce has made a complete translation of the third edition, with notes, on his website.
+Charles Leedham-Green 2021 has published a complete and heavily annotated translation. Cambridge; Cambridge University Press.
+
+=== Other links ===
+David R. Wilkins of the School of Mathematics at Trinity College, Dublin has transcribed a few sections into TeX and METAPOST and made the source, as well as a formatted PDF available at Extracts from the Works of Isaac Newton.

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+
+Only I must observe, that the vulgar conceive those quantities under no other notions but from the relation they bear to perceptible objects. And it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. ... instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical discussions, we ought to step back from our senses, and consider things themselves, distinct from what are only perceptible measures of them.
+To some modern readers it can appear that some dynamical quantities recognised today were used in the Principia but not named. The mathematical aspects of the first two books were so clearly consistent that they were easily accepted; for example, John Locke asked Christiaan Huygens whether he could trust the mathematical proofs and was assured about their correctness.
+However, the concept of an attractive force acting at a distance received a cooler response. In his notes, Newton wrote that the inverse square law arose naturally due to the structure of matter. However, he retracted this sentence in the published version, where he stated that the motion of planets is consistent with an inverse square law, but refused to speculate on the origin of the law. Huygens and Gottfried Wilhelm Leibniz noted that the law was incompatible with the notion of the aether. From a Cartesian point of view, therefore, this was a faulty theory. Newton's defence has been adopted since by many famous physicists—he pointed out that the mathematical form of the theory had to be correct since it explained the data, and he refused to speculate further on the basic nature of gravity. The sheer number of phenomena that could be organised by the theory was so impressive that younger "philosophers" soon adopted the methods and language of the Principia.
+
+== Rules of Reason ==
+Perhaps to reduce the risk of public misunderstanding, Newton included at the beginning of Book 3 (in the second (1713) and third (1726) editions) a section titled "Rules of Reasoning in Philosophy". In the four rules, as they came finally to stand in the 1726 edition, Newton effectively offers a methodology for handling unknown phenomena in nature and reaching towards explanations for them. These rules have become the basis of the modern approaches to science. The four Rules of the 1726 edition run as follows (omitting some explanatory comments that follow each):
+
+We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.
+Therefore to the same natural effects we must, as far as possible, assign the same causes.
+The qualities of bodies, which admit neither intensification nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.
+In experimental philosophy we are to look upon propositions inferred by general induction from phenomena as accurately or very nearly true, not withstanding any contrary hypothesis that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.
+This section of Rules for philosophy is followed by a listing of "Phenomena", in which are listed a number of mainly astronomical observations, that Newton used as the basis for inferences later on, as if adopting a consensus set of facts from the astronomers of his time.
+Both the "Rules" and the "Phenomena" evolved from one edition of the Principia to the next. Rule 4 made its appearance in the third (1726) edition; Rules 1–3 were present as "Rules" in the second (1713) edition, and predecessors of them were also present in the first edition of 1687, but there they had a different heading: they were not given as "Rules", but rather in the first (1687) edition the predecessors of the three later "Rules", and of most of the later "Phenomena", were all lumped together under a single heading "Hypotheses" (in which the third item was the predecessor of a heavy revision that gave the later Rule 3).
+From this textual evolution, it appears that Newton wanted by the later headings "Rules" and "Phenomena" to clarify for his readers his view of the roles to be played by these various statements.
+In the third (1726) edition of the Principia, Newton explains each rule in an alternative way and/or gives an example to back up what the rule is claiming. The first rule is explained as a philosophers' principle of economy. The second rule states that if one cause is assigned to a natural effect, then the same cause so far as possible must be assigned to natural effects of the same kind: for example, respiration in humans and in animals, fires in the home and in the Sun, or the reflection of light whether it occurs terrestrially or from the planets. An extensive explanation is given of the third rule, concerning the qualities of bodies, and Newton discusses here the generalisation of observational results, with a caution against making up fancies contrary to experiments, and use of the rules to illustrate the observation of gravity and space.
+
+=== General Scholium ===

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+The General Scholium is a concluding essay added to the second edition, 1713 (and amended in the third edition, 1726). It is not to be confused with the General Scholium at the end of Book 2, Section 6, which discusses his pendulum experiments and resistance due to air, water, and other fluids.
+Here Newton used the expression hypotheses non fingo, "I formulate no hypotheses", in response to criticisms of the first edition of the Principia. ("Fingo" is sometimes nowadays translated "feign" rather than the traditional "frame," although "feign" does not properly translate "fingo"). Newton's gravitational attraction, an invisible force able to act over vast distances, had led to criticism that he had introduced "occult agencies" into science. Newton firmly rejected such criticisms and wrote that it was enough that the phenomena implied gravitational attraction, as they did; but the phenomena did not so far indicate the cause of this gravity, and it was both unnecessary and improper to frame hypotheses of things not implied by the phenomena: such hypotheses "have no place in experimental philosophy", in contrast to the proper way in which "particular propositions are inferr'd from the phenomena and afterwards rendered general by induction".
+Newton also underlined his criticism of the vortex theory of planetary motions, of Descartes, pointing to its incompatibility with the highly eccentric orbits of comets, which carry them "through all parts of the heavens indifferently".
+Newton also gave theological argument. From the system of the world, he inferred the existence of a god, along lines similar to what is sometimes called the argument from intelligent or purposive design. It has been suggested that Newton gave "an oblique argument for a unitarian conception of God and an implicit attack on the doctrine of the Trinity". The General Scholium does not address or attempt to refute the church doctrine; it simply does not mention Jesus, the Holy Ghost, or the hypothesis of the Trinity.
+
+== Publishing the book ==

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+
+=== Halley and Newton's initial stimulus ===
+In January 1684, Edmond Halley, Christopher Wren and Robert Hooke had a conversation in which Hooke claimed to not only have derived the inverse-square law but also all the laws of planetary motion. Wren was unconvinced, Hooke did not produce the claimed derivation although the others gave him time to do it, and Halley, who could derive the inverse-square law for the restricted circular case (by substituting Kepler's relation into Huygens' formula for the centrifugal force) but failed to derive the relation generally, resolved to ask Newton.
+Halley's visits to Newton in 1684 thus resulted from Halley's debates about planetary motion with Wren and Hooke, and they seem to have provided Newton with the incentive and spur to develop and write what became Philosophiae Naturalis Principia Mathematica. Halley was at that time a Fellow and Council member of the Royal Society in London (positions that in 1686 he resigned to become the Society's paid Clerk). Halley's visit to Newton in Cambridge in 1684 probably occurred in August. When Halley asked Newton's opinion on the problem of planetary motions discussed earlier that year between Halley, Hooke and Wren, Newton surprised Halley by saying that he had already made the derivations some time ago; but that he could not find the papers. (Matching accounts of this meeting come from Halley and the French mathematician Abraham de Moivre, in whom Newton had confided.) Halley then had to wait for Newton to "find" the results, and in November 1684 Newton sent Halley an amplified version of whatever previous work Newton had done on the subject. This took the form of a 9-page manuscript, De motu corporum in gyrum (Of the motion of bodies in an orbit): the title is shown on some surviving copies, although the (lost) original may have been without a title.
+Newton's tract De motu corporum in gyrum, which he sent to Halley in late 1684, derived what is now known as the three laws of Kepler, assuming an inverse square law of force, and generalised the result to conic sections. It also extended the methodology by adding the solution of a problem on the motion of a body through a resisting medium. The contents of De motu so excited Halley by their mathematical and physical originality and far-reaching implications for astronomical theory, that he immediately went to visit Newton again, in November 1684, to ask Newton to let the Royal Society have more of such work. The results of their meetings clearly helped to stimulate Newton with the enthusiasm needed to take his investigations of mathematical problems much further in this area of physical science, and he did so in a period of highly concentrated work that lasted at least until mid-1686.
+Newton's single-minded attention to his work generally, and to his project during this time, is shown by later reminiscences from his secretary and copyist of the period, Humphrey Newton. His account tells of Isaac Newton's absorption in his studies, how he sometimes forgot his food, or his sleep, or the state of his clothes, and how when he took a walk in his garden he would sometimes rush back to his room with some new thought, not even waiting to sit before beginning to write it down. Other evidence also shows Newton's absorption in the Principia: Newton for years kept up a regular programme of chemical or alchemical experiments, and he normally kept dated notes of them, but for a period from May 1684 to April 1686, Newton's chemical notebooks have no entries at all. So, it seems that Newton abandoned pursuits to which he was formally dedicated and did very little else for well over a year and a half, but concentrated on developing and writing what became his great work.
+The first of the three constituent books was sent to Halley for the printer in spring 1686, and the other two books somewhat later. The complete work, published by Halley at his own financial risk, appeared in July 1687. Newton had also communicated De motu to Flamsteed, and during the period of composition, he exchanged a few letters with Flamsteed about observational data on the planets, eventually acknowledging Flamsteed's contributions in the published version of the Principia of 1687.
+
+=== Preliminary version ===
+
+The process of writing that first edition of the Principia went through several stages and drafts: some parts of the preliminary materials still survive, while others are lost except for fragments and cross-references in other documents.
+Surviving materials show that Newton (up to some time in 1685) conceived his book as a two-volume work. The first volume was to be titled De motu corporum, Liber primus, with contents that later appeared in extended form as Book 1 of the Principia.
+A fair-copy draft of Newton's planned second volume De motu corporum, Liber Secundus survives, its completion dated to about the summer of 1685. It covers the application of the results of Liber primus to the Earth, the Moon, the tides, the Solar System, and the universe; in this respect, it has much the same purpose as the final Book 3 of the Principia, but it is written much less formally and is more easily read.

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+It is not known just why Newton changed his mind so radically about the final form of what had been a readable narrative in De motu corporum, Liber Secundus of 1685, but he largely started afresh in a new, tighter, and less accessible mathematical style, eventually to produce Book 3 of the Principia as we know it. Newton frankly admitted that this change of style was deliberate when he wrote that he had (first) composed this book "in a popular method, that it might be read by many", but to "prevent the disputes" by readers who could not "lay aside the[ir] prejudices", he had "reduced" it "into the form of propositions (in the mathematical way) which should be read by those only, who had first made themselves masters of the principles established in the preceding books". The final Book 3 also contained in addition some further important quantitative results arrived at by Newton in the meantime, especially about the theory of the motions of comets, and some of the perturbations of the motions of the Moon.
+The result was numbered Book 3 of the Principia rather than Book 2 because in the meantime, drafts of Liber primus had expanded and Newton had divided it into two books. The new and final Book 2 was concerned largely with the motions of bodies through resisting mediums.
+But the Liber Secundus of 1685 can still be read today. Even after it was superseded by Book 3 of the Principia, it survived complete, in more than one manuscript. After Newton's death in 1727, the relatively accessible character of its writing encouraged the publication of an English translation in 1728 (by persons still unknown, not authorised by Newton's heirs). It appeared under the English title A Treatise of the System of the World. This had some amendments relative to Newton's manuscript of 1685, mostly to remove cross-references that used obsolete numbering to cite the propositions of an early draft of Book 1 of the Principia. Newton's heirs shortly afterwards published the Latin version in their possession, also in 1728, under the (new) title De Mundi Systemate, amended to update cross-references, citations and diagrams to those of the later editions of the Principia, making it look superficially as if it had been written by Newton after the Principia, rather than before. The System of the World was sufficiently popular to stimulate two revisions (with similar changes as in the Latin printing), a second edition (1731), and a "corrected" reprint of the second edition (1740).
+
+=== Halley's role as publisher ===
+The text of the first of the three books of the Principia was presented to the Royal Society at the close of April 1686. Hooke made some priority claims (but failed to substantiate them), causing some delay. When Hooke's claim was made known to Newton, who hated disputes, Newton threatened to withdraw and suppress Book 3 altogether, but Halley, showing considerable diplomatic skills, tactfully persuaded Newton to withdraw his threat and let it go forward to publication. Samuel Pepys, as president, gave his imprimatur on 30 June 1686, licensing the book for publication. The Society had just spent its book budget on De Historia piscium, and the cost of publication was borne by Edmund Halley (who was also then acting as publisher of the Philosophical Transactions of the Royal Society): the book appeared in summer 1687. After Halley had personally financed the publication of Principia, he was informed that the society could no longer afford to provide him the promised annual salary of £50. Instead, Halley was paid with leftover copies of De Historia piscium.
+
+== Historical context ==
+
+=== Beginnings of the Scientific Revolution ===
+
+Nicolaus Copernicus had moved the Earth away from the center of the universe with the heliocentric theory for which he presented evidence in his book De revolutionibus orbium coelestium (On the revolutions of the heavenly spheres) published in 1543. Johannes Kepler wrote the book Astronomia nova (A new astronomy) in 1609, setting out the evidence that planets move in elliptical orbits with the Sun at one focus, and that planets do not move with constant speed along this orbit. Rather, their speed varies so that the line joining the centres of the sun and a planet sweeps out equal areas in equal times. To these two laws he added a third a decade later, in his 1619 book Harmonices Mundi (Harmonies of the world). This law sets out a proportionality between the third power of the characteristic distance of a planet from the Sun and the square of the length of its year.
+
+The foundation of modern dynamics was set out in Galileo's book Dialogo sopra i due massimi sistemi del mondo (Dialogue on the two main world systems) where the notion of inertia was implicit and used. In addition, Galileo's experiments with inclined planes had yielded precise mathematical relations between elapsed time and acceleration, velocity or distance for uniform and uniformly accelerated motion of bodies.
+Descartes' book of 1644 Principia philosophiae (Principles of philosophy) stated that bodies can act on each other only through contact: a principle that induced people, among them himself, to hypothesize a universal medium as the carrier of interactions such as light and gravity—the aether. Newton was criticized for apparently introducing forces that acted at distance without any medium. Not until the development of particle theory was Descartes' notion vindicated when it was possible to describe all interactions, like the strong, weak, and electromagnetic fundamental interactions, using mediating gauge bosons and gravity through hypothesized gravitons.

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+
+=== Newton's role ===
+Newton had studied these books, or, in some cases, secondary sources based on them, and taken notes entitled Quaestiones quaedam philosophicae (Questions about philosophy) during his days as an undergraduate. During this period (1664–1666) he created the basis of calculus and performed the first experiments in the optics of colour. At this time, his proof that white light was a combination of primary colours (found via prismatics) replaced the prevailing theory of colours and received an overwhelmingly favourable response and occasioned bitter disputes with Robert Hooke and others, which forced him to sharpen his ideas to the point where he already composed sections of his later book Opticks by the 1670s in response. Work on calculus is shown in various papers and letters, including two to Leibniz. He became a fellow of the Royal Society and the second Lucasian Professor of Mathematics (succeeding Isaac Barrow) at Trinity College, Cambridge.
+
+=== Newton's early work on motion ===
+In the 1660s Newton studied the motion of colliding bodies and deduced that the centre of mass of two colliding bodies remains in uniform motion. Surviving manuscripts of the 1660s also show Newton's interest in planetary motion and that by 1669 he had shown, for a circular case of planetary motion, that the force he called "endeavour to recede" (now called centrifugal force) had an inverse-square relation with distance from the center. After his 1679–1680 correspondence with Hooke, described below, Newton adopted the language of inward or centripetal force. According to Newton scholar J. Bruce Brackenridge, although much has been made of the change in language and difference of point of view, as between centrifugal or centripetal forces, the actual computations and proofs remained the same either way. They also involved the combination of tangential and radial displacements, which Newton was making in the 1660s. The difference between the centrifugal and centripetal points of view, though a significant change of perspective, did not change the analysis. Newton also clearly expressed the concept of linear inertia in the 1660s: for this Newton was indebted to Descartes' work published 1644.

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+
+=== Controversy with Hooke ===
+Hooke published his ideas about gravitation in the 1660s and again in 1674. He argued for an attracting principle of gravitation in Micrographia of 1665, in a 1666 Royal Society lecture On gravity, and again in 1674, when he published his ideas about the System of the World in somewhat developed form, as an addition to An Attempt to Prove the Motion of the Earth from Observations. Hooke clearly postulated mutual attractions between the Sun and planets, in a way that increased with nearness to the attracting body, along with a principle of linear inertia. Hooke's statements up to 1674 made no mention, however, that an inverse square law applies or might apply to these attractions. Hooke's gravitation was also not yet universal, though it approached universality more closely than previous hypotheses. Hooke also did not provide accompanying evidence or mathematical demonstration. On these two aspects, Hooke stated in 1674: "Now what these several degrees [of gravitational attraction] are I have not yet experimentally verified" (indicating that he did not yet know what law the gravitation might follow); and as to his whole proposal: "This I only hint at present", "having my self many other things in hand which I would first compleat, and therefore cannot so well attend it" (i.e., "prosecuting this Inquiry").
+In November 1679, Hooke began an exchange of letters with Newton, of which the full text is now published. Hooke told Newton that Hooke had been appointed to manage the Royal Society's correspondence, and wished to hear from members about their researches, or their views about the researches of others; and as if to whet Newton's interest, he asked what Newton thought about various matters, giving a whole list, mentioning "compounding the celestial motions of the planets of a direct motion by the tangent and an attractive motion towards the central body", and "my hypothesis of the lawes or causes of springinesse", and then a new hypothesis from Paris about planetary motions (which Hooke described at length), and then efforts to carry out or improve national surveys, the difference of latitude between London and Cambridge, and other items. Newton's reply offered "a fansy of my own" about a terrestrial experiment (not a proposal about celestial motions) which might detect the Earth's motion, by the use of a body first suspended in air and then dropped to let it fall. The main point was to indicate how Newton thought the falling body could experimentally reveal the Earth's motion by its direction of deviation from the vertical, but he went on hypothetically to consider how its motion could continue if the solid Earth had not been in the way (on a spiral path to the centre). Hooke disagreed with Newton's idea of how the body would continue to move. A short further correspondence developed, and towards the end of it Hooke, writing on 6 January 1680 to Newton, communicated his "supposition ... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall, and Consequently that the Velocity will be in a subduplicate proportion to the Attraction and Consequently as Kepler Supposes Reciprocall to the Distance." (Hooke's inference about the velocity was actually incorrect.)
+In 1686, when the first book of Newton's Principia was presented to the Royal Society, Hooke claimed that Newton had obtained from him the "notion" of "the rule of the decrease of Gravity, being reciprocally as the squares of the distances from the Center". At the same time (according to Edmond Halley's contemporary report) Hooke agreed that "the Demonstration of the Curves generated therby" was wholly Newton's.
+A recent assessment about the early history of the inverse square law is that "by the late 1660s", the assumption of an "inverse proportion between gravity and the square of distance was rather common and had been advanced by a number of different people for different reasons". Newton himself had shown in the 1660s that for planetary motion under a circular assumption, force in the radial direction had an inverse-square relation with distance from the center. Newton, faced in May 1686 with Hooke's claim on the inverse square law, denied that Hooke was to be credited as author of the idea, giving reasons including the citation of prior work by others before Hooke. Newton also firmly claimed that even if it had happened that he had first heard of the inverse square proportion from Hooke, which it had not, he would still have some rights to it in view of his mathematical developments and demonstrations, which enabled observations to be relied on as evidence of its accuracy, while Hooke, without mathematical demonstrations and evidence in favour of the supposition, could only guess (according to Newton) that it was approximately valid "at great distances from the center".
+The background described above shows there was basis for Newton to deny deriving the inverse square law from Hooke. On the other hand, Newton did accept and acknowledge, in all editions of the Principia, that Hooke (but not exclusively Hooke) had separately appreciated the inverse square law in the Solar System. Newton acknowledged Wren, Hooke and Halley in this connection in the Scholium to Proposition 4 in Book 1. Newton also acknowledged to Halley that his correspondence with Hooke in 1679–80 had reawakened his dormant interest in astronomical matters, but that did not mean, according to Newton, that Hooke had told Newton anything new or original: "yet am I not beholden to him for any light into that business but only for the diversion he gave me from my other studies to think on these things & for his dogmaticalness in writing as if he had found the motion in the Ellipsis, which inclined me to try it ...".) Newton's reawakening interest in astronomy received further stimulus by the appearance of a comet in the winter of 1680/1681, on which he corresponded with John Flamsteed.
+In 1759, decades after the deaths of both Newton and Hooke, Alexis Clairaut, mathematical astronomer eminent in his own right in the field of gravitational studies, made his assessment after reviewing what Hooke had published on gravitation. "One must not think that this idea ... of Hooke diminishes Newton's glory", Clairaut wrote; "The example of Hooke" serves "to show what a distance there is between a truth that is glimpsed and a truth that is demonstrated".

data/en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica-8.md

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+---
+title: "Philosophiæ Naturalis Principia Mathematica"
+chunk: 9/11
+source: "https://en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:10.722464+00:00"
+instance: "kb-cron"
+---
+
+== Location of early edition copies ==
+
+It has been estimated that as many as 750 copies of the first edition were printed by the Royal Society, and "it is quite remarkable that so many copies of this small first edition are still in existence ... but it may be because the original Latin text was more revered than read". A survey published in 1953 located 189 surviving copies with nearly 200 further copies located by the most recent survey published in 2020, suggesting that the initial print run was larger than previously thought. However, more recent book historical and bibliographical research has examined those prior claims, and concludes that Macomber's earlier estimation of 500 copies is likely correct.
+
+Cambridge University Library has Newton's own copy of the first edition, with handwritten notes for the second edition.
+The Earl Gregg Swem Library at the College of William & Mary has a first edition copy of the Principia. Throughout are Latin annotations written by Thomas S. Savage. These handwritten notes are currently being researched at the college.
+The Frederick E. Brasch Collection of Newton and Newtoniana in Stanford University also has a first edition of the Principia.
+A first edition forms part of the Crawford Collection, housed at the Royal Observatory, Edinburgh.
+The Uppsala University Library owns a first edition copy, which was stolen in the 1960s and returned to the library in 2009.
+The Folger Shakespeare Library in Washington, D.C. owns a first edition, as well as a 1713 second edition.
+The Huntington Library in San Marino, California owns Isaac Newton's personal copy, with annotations in Newton's own hand.
+The Bodmer Library in Switzerland keeps a copy of the original edition that was owned by Leibniz. It contains handwritten notes by Leibniz, in particular concerning the controversy of who first formulated calculus (although he published it later, Newton argued that he developed it earlier).
+The Iron Library in Switzerland holds a first edition copy that was formerly in the library of the physicist Ernst Mach. The copy contains critical marginalia in Mach's hand.
+The University of St Andrews Library holds both variants of the first edition, as well as copies of the 1713 and 1726 editions.
+The Fisher Library in the University of Sydney has a first-edition copy, annotated by a mathematician of uncertain identity and corresponding notes from Newton himself.
+The Linda Hall Library holds the first edition, as well as a copy of the 1713 and 1726 editions.
+The Teleki-Bolyai Library of Târgu-Mureș holds a 2-line imprint first edition.
+One book is also located at Vasaskolan, Gävle, in Sweden.
+Dalhousie University in Canada has a copy as part of the William I. Morse collection.
+McGill University in Montreal has the copy once owned by Sir William Osler.
+The University of Toronto has a copy in the Thomas Fisher Rare Book Collection.
+University College London Special Collections has a copy previously owned by the lawyer and mathematician John T. Graves.
+In 2016, a first edition sold for $3.7 million.
+The second edition (1713) were printed in 750 copies, and the third edition (1726) were printed in 1,250 copies.
+A facsimile edition (based on the 3rd edition of 1726 but with variant readings from earlier editions and important annotations) was published in 1972 by Alexandre Koyré and I. Bernard Cohen.
+
+== Later editions ==
+
+=== Second edition, 1713 ===
+
+Two later editions were published by Newton: Newton had been urged to make a new edition of the Principia since the early 1690s, partly because copies of the first edition had already become very rare and expensive within a few years after 1687. Newton referred to his plans for a second edition in correspondence with Flamsteed in November 1694. Newton also maintained annotated copies of the first edition specially bound up with interleaves on which he could note his revisions; two of these copies still survive, but he had not completed the revisions by 1708. Newton had almost severed connections with one would-be editor, Nicolas Fatio de Duillier, and another, David Gregory seems not to have met with his approval and was also terminally ill, dying in 1708. Nevertheless, reasons were accumulating not to put off the new edition any longer. Richard Bentley, master of Trinity College, persuaded Newton to allow him to undertake a second edition, and in June 1708 Bentley wrote to Newton with a specimen print of the first sheet, at the same time expressing the (unfulfilled) hope that Newton had made progress towards finishing the revisions. It seems that Bentley then realised that the editorship was technically too difficult for him, and with Newton's consent he appointed Roger Cotes, Plumian professor of astronomy at Trinity, to undertake the editorship for him as a kind of deputy (but Bentley still made the publishing arrangements and had the financial responsibility and profit). The correspondence of 1709–1713 shows Cotes reporting to two masters, Bentley and Newton, and managing (and often correcting) a large and important set of revisions to which Newton sometimes could not give his full attention. Under the weight of Cotes' efforts, but impeded by priority disputes between Newton and Leibniz, and by troubles at the Mint, Cotes was able to announce publication to Newton on 30 June 1713. Bentley sent Newton only six presentation copies; Cotes was unpaid; Newton omitted any acknowledgement to Cotes.
+Among those who gave Newton corrections for the Second Edition were: Firmin Abauzit, Roger Cotes and David Gregory. However, Newton omitted acknowledgements to some because of the priority disputes. John Flamsteed, the Astronomer Royal, suffered this especially.
+The Second Edition was the basis of the first edition to be printed abroad, which appeared in Amsterdam in 1714.

data/en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica-9.md

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+---
+title: "Philosophiæ Naturalis Principia Mathematica"
+chunk: 10/11
+source: "https://en.wikipedia.org/wiki/Philosophiæ_Naturalis_Principia_Mathematica"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:10.722464+00:00"
+instance: "kb-cron"
+---
+
+=== Third edition, 1726 ===
+After his serious illness in 1722 and after the appearance of a reprint of the second edition in Amsterdam in 1723, the 80-year-old Newton began to revise once again the Principia in the autumn of 1723. The third edition was published 25 March 1726, under the stewardship of Henry Pemberton, M.D., a man of the greatest skill in these matters...; Pemberton later said that this recognition was worth more to him than the two hundred guinea award from Newton.
+In 1739–1742, two French priests, Pères Thomas LeSeur and François Jacquier (of the Minim order, but sometimes erroneously identified as Jesuits), produced with the assistance of J.-L. Calandrini an extensively annotated version of the Principia in the 3rd edition of 1726. Sometimes this is referred to as the Jesuit edition: it was much used, and reprinted more than once in Scotland during the 19th century.
+Émilie du Châtelet also made a translation of Newton's Principia into French. Unlike LeSeur and Jacquier's edition, hers was a complete translation of Newton's three books and their prefaces. She also included a Commentary section where she fused the three books into a much clearer and easier to understand summary. She included an analytical section where she applied the new mathematics of calculus to Newton's most controversial theories. Previously, geometry was the standard mathematics used to analyse theories. Du Châtelet's translation is the only complete one to have been done in French and hers remains the standard French translation to this day.
+
+== Translations ==
+
+Four full English translations of Newton's Principia have appeared, all based on Newton's 3rd edition of 1726. The first, from 1729, by Andrew Motte, was described by Newton scholar I. Bernard Cohen (in 1968) as "still of enormous value in conveying to us the sense of Newton's words in their own time, and it is generally faithful to the original: clear, and well written". The 1729 version was the basis for several republications, often incorporating revisions, among them a widely used modernised English version of 1934, which appeared under the editorial name of Florian Cajori (though completed and published only some years after his death). Cohen pointed out ways in which the 18th-century terminology and punctuation of the 1729 translation might be confusing to modern readers, but he also made severe criticisms of the 1934 modernised English version, and showed that the revisions had been made without regard to the original, also demonstrating gross errors "that provided the final impetus to our decision to produce a wholly new translation".
+The second full English translation, into modern English, is the work that resulted from this decision by collaborating translators I. Bernard Cohen, Anne Whitman, and Julia Budenz; it was published in 1999 with a guide by way of introduction.
+The third such translation is due to Ian Bruce, and appears, with many other translations of mathematical works of the 17th and 18th centuries, on his website.
+The fourth complete English translation is due to Charles Leedham-Green, professor emeritus of mathematics at Queen Mary University of London, and was published in 2021 by Cambridge University Press. Prof. Leedham-Green was motivated to produce that translation, on which he worked for twenty years, in part because of his dissatisfaction with the work of Cohen, Whitman, and Budenz, whose translation of the Principia he found unnecessarily obscure.  Leedham-Green's aim was to convey Newton's own reasoning and arguments in a way intelligible to a modern mathematical scientist.  His translation is heavily annotated and his explanatory notes make use of the modern secondary literature on some of the more difficult technical aspects of Newton's work.
+Dana Densmore and William H. Donahue have published a translation of the work's central argument, published in 1996, along with expansion of included proofs and ample commentary. The book was developed as a textbook for classes at St. John's College and the aim of this translation is to be faithful to the Latin text.
+
+== Legacy ==
+The French mathematical physicist Alexis Clairaut described The Principia in 1747: "The famous book of Mathematical Principles of Natural Philosophy marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton ... spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses."
+The French mathematician and physicist Joseph-Louis Lagrange described it as "the greatest production of the human mind".
+French polymath Pierre-Simon Laplace stated that "The Principia is pre-eminent above any other production of human genius". 
+Mathematician Eric Temple Bell wrote "It is no miracle then, when the power of Newton's mathematical genius is taken into account, the Principia is the unsurpassed masterpiece of both scientific coordination and the art of scientific prediction that it is."
+
+Mathematician George F. Simmons wrote of the immense impact and influence of Principia:In this one book–perhaps the greatest of all scientific treatises–his success in using mathematical methods to explain the most diverse natural phenomena was so profound and far-reaching that he essentially created the sciences of physics and astronomy where only a handful of disconnected observations and simple inferences had existed before. These achievements launched the modern age of science and technology and radically altered the direction of human history.
+A more recent assessment has been that while acceptance of Newton's laws was not immediate, by the end of the century after publication in 1687, "no one could deny that [out of the Principia] a science had emerged that, at least in certain respects, so far exceeded anything that had ever gone before that it stood alone as the ultimate exemplar of science generally". A 2020 study established a new census of owners of the first edition of the Principia, identifying 387 copies, significantly more than the long-held estimate of about 250. This new count suggests a much wider initial readership and a greater early impact on Enlightenment science, contrary to prior assumptions of its scarcity and limited early impact.
+
+== Varia ==

data/en.wikipedia.org/wiki/Photometria-0.md

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+---
+title: "Photometria"
+chunk: 1/2
+source: "https://en.wikipedia.org/wiki/Photometria"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:11.905861+00:00"
+instance: "kb-cron"
+---
+
+Photometria is a book on the measurement of light by Johann Heinrich Lambert published in 1760. It established a complete system of photometric quantities and principles; using them to measure the optical properties of materials, quantify aspects of vision, and calculate illumination.
+
+== Content of Photometria ==
+Written in Latin, the title of the book is a word Lambert devised from  Ancient Greek: φῶς, φωτος (transliterated phôs, photos) = light, and μετρια (transliterated metria) = measure. Lambert’s word has found its way into European languages as photometry, photometrie, and fotometria.  Photometria was the first work to accurately identify most fundamental photometric concepts, assemble them into a coherent system of photometric quantities, define these quantities with a precision sufficient for mathematical statements, and build from them a system of photometric principles. These concepts, quantities, and principles are still in use today.
+Lambert began with two simple axioms: light travels in a straight line in a uniform medium and rays that cross do not interact. Like Kepler before him, he recognized that "laws" of photometry are simply consequences and follow directly from these two assumptions.  In this way Photometria demonstrated (rather than assumed) that
+
+Illuminance varies inversely as the square of the distance from a point source of light,
+Illuminance on a surface varies as the cosine of the incidence angle measured from the surface perpendicular, and
+Light decays exponentially in an absorbing medium.
+In addition, Lambert postulated a surface that emits light (either as a source or by reflection) in a way such that the density of emitted light (luminous intensity) varies as the cosine of the angle measured from the surface perpendicular. In the case of a reflecting surface, this form of emission is assumed to be the case, regardless of the light's incident direction. Such surfaces are now referred to as "Perfectly Diffuse" or "Lambertian". See: Lambertian reflectance, Lambertian emitter.
+Lambert demonstrated these principles in the only way available at the time: by contriving often ingenious optical arrangements that could make two immediately adjacent luminous fields appear equally bright (something that could only be determined by visual observation) when two physical quantities that produced the two fields were unequal by some specific amount (things that could be directly measured, such as angle or distance). In this way, Lambert quantified purely visual properties (such as luminous power, illumination, transparency, reflectivity) by relating them to physical parameters (such as distance, angle, radiant power, and color). Today, this is known as "visual photometry." Lambert was among the first to accompany experimental measurements with estimates of uncertainties based on a theory of errors and what he experimentally determined as the limits of visual assessment.
+Although previous workers had pronounced photometric laws 1 and 3,  Lambert established the second and added the concept of perfectly diffuse surfaces. But more importantly, as Ernst Anding pointed out in his German translation of Photometria, "Lambert had incomparably clearer ideas about photometry" and with them established a complete system of photometric quantities.  Based on the three laws of photometry and the supposition of perfectly diffuse surfaces, Photometria developed and demonstrated the following:
+
+1. Just noticeable differences
+In the first section of Photometria, Lambert established and demonstrated the laws of photometry. He did this with visual photometry and to establish the uncertainties involved, described its approximate limits by determining how small a brightness difference the visual system could determine.
+2. Reflectance and transmittance of glass and other common materials
+Using visual photometry, Lambert presented the results of many experimental determinations of specular and diffuse reflectance, as well as the transmittance of panes of glass and lenses. Among the most ingenious experiments he conducted was to determine the reflectance of the interior surface of a pane of glass.
+3. Luminous radiative transfer between surfaces
+Assuming diffuse surfaces and the three laws of photometry, Lambert used Calculus to find the transfer of light between surfaces of various sizes, shapes, and orientations. He originated the concept of the per-unit transfer of flux between surfaces and in Photometria showed the closed form for many double, triple, and quadruple integrals which gave the equations for many different geometric arrangements of surfaces. Today, these fundamental quantities are called View Factors, Shape Factors, or Configuration Factors and are used in radiative heat transfer and in computer graphics.
+4. Brightness and pupil size
+Lambert measured his own pupil diameter by viewing it in a mirror. He measured the change in diameter as he viewed a larger or smaller part of a candle flame. This is the first known attempt to quantify pupillary light reflex.
+5. Atmospheric refraction and absorption
+Using the laws of photometry and a great deal of geometry, Lambert calculated the times and depths of twilight.
+6. Astronomic photometry
+Assuming that the planets had diffusely reflective surfaces, Lambert attempted to determine the amount of their reflectance, given their relative brightness and known distance from the sun. A century later, Zöllner studied Photometria and picked up where Lambert left off, and initiated the field of astrophysics.
+7. Demonstration of additive color mixing and colorimetry
+Lambert was the first to record the results of additive color mixing. By simultaneous transmission and reflection from a pane of glass, he superimposed the images of two different colored patches of paper and noted the resulting additive color.
+8. Daylighting calculations
+Assuming the sky was a luminous dome, Lambert calculated the illumination by skylight through a window, and the light occluded and interreflected by walls and partitions.

data/en.wikipedia.org/wiki/Photometria-1.md

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+---
+title: "Photometria"
+chunk: 2/2
+source: "https://en.wikipedia.org/wiki/Photometria"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:11.905861+00:00"
+instance: "kb-cron"
+---
+
+== Nature of Photometria ==
+Lambert's book is fundamentally experimental. The forty experiments described in Photometria were conducted by Lambert between 1755 and 1760, after he decided to write a treatise on light measurement. His interest in acquiring experimental data spanned several fields: optics, thermometry, pyrometry, hydrometry, and magnetics. This interest in experimental data and its analysis, so evident in Photometria, is also present in other articles and books Lambert produced. For his optics work, extremely limited equipment sufficed: a few panes of glass, convex and concave lenses, mirrors, prisms, paper and cardboard, pigments, candles, and the means to measure distances and angles.
+Lambert's book is also mathematical. Though he knew that the physical nature of light was unknown (it would be 150 years before the wave-particle duality was established) he was certain that light's interaction with materials and its effect on vision could be quantified. Mathematics was for Lambert not only indispensable for this quantification but also the indisputable sign of rigor.  He used linear algebra and calculus extensively with matter-of-fact confidence that was uncommon in optical works of the time. On this basis, Photometria is certainly uncharacteristic of mid-18th century works.
+
+== Writing and publishing of Photometria ==
+Lambert began conducting photometric experiments in 1755 and by August 1757 had enough material to begin writing. From the references in Photometria and the catalogue of his library auctioned after his death, it is clear that Lambert consulted the optical works of Isaac Newton, Pierre Bouguer, Leonhard Euler, Christiaan Huygens, Robert Smith, and Abraham Gotthelf Kästner. He finished Photometria in Augsburg in February 1760 and the printer had the book available by June 1760.
+Maria Jakobina Klett (1709–1795) was the owner of Eberhard Klett Verlag, one of the most important Augsburg “Protestant publishers.” She published many technical books, including Lambert’s Photometria, and 10 of his other works. Klett used Christoph Peter Detleffsen (1731–1774) to print Photometria. Its first and only printing was small, and within 10 years copies were difficult to obtain. In Joseph Priestley's survey of optics of 1772,  “Lambert’s Photometrie” appears in the list of books not yet procured.  Priestley makes a specific reference to Photometria; that it was an important book but unprocurable.
+An abridged German translation of Photometria appeared in 1892, a French translation in 1997, and an English translation in 2000.
+
+== Later influence ==
+Photometria presented significant advances and it was, perhaps, for that very reason that its appearance was greeted with general indifference. The central optical question in the middle of the 18th century was: what is the nature of light? Lambert's work was not related to this issue at all and so Photometria received no immediate systematic evaluation, and was not incorporated into the mainstream of optical science. The first appraisal of Photometria appeared in 1776 in Georg Simon Klügel’s German translation of Priestley’s 1772 survey of optics. An elaborate reworking and annotation appeared in 1777.  
+Photometria was not seriously evaluated and utilized until nearly a century after its publication, when the science of astronomy and the commerce of gas lighting needed photometry. Fifty years after that, Illuminating Engineering took up Lambert's results as the basis for lighting calculations that accompanied the great expanse of lighting early in the 20th century. Fifty years after that, computer graphics took up Lambert's results as the basis for radiosity calculations required to produce architectural renderings. Photometria had a significant, though long-delayed influence on technology and commerce once the Industrial Revolution was well underway, and is the reason that it was one of the books listed in Printing and the Mind of Man.
+
+== See also ==
+Beer–Lambert law (Lambert–Beer law, Beer–Lambert–Bouguer law)
+lambert (unit)
+Lambert's cosine law
+Lambertian reflectance
+
+== References ==
+
+== External links ==
+
+Lamberts Photometrie No. 31, 32, 33 of Ostwalds Klassiker der exakten Wissenschaften, Engelmann, Leipzig, 1892
+Photometria, Klett, Augsburg, 1760

data/en.wikipedia.org/wiki/Point_Processes-0.md

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+---
+title: "Point Processes"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Point_Processes"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:13.101692+00:00"
+instance: "kb-cron"
+---
+
+Point Processes is a book on the mathematics of point processes, randomly located sets of points on the real line or in other geometric spaces. It was written by David Cox and Valerie Isham, and published in 1980 by Chapman & Hall in their Monographs on Applied Probability and Statistics book series. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
+
+
+== Topics ==
+Although Point Processes covers some of the general theory of point processes, that is not its main focus, and it avoids any discussion of statistical inference involving these processes. Instead, its aim is to present the properties and descriptions of several specific processes arising in applications of this theory, which had not been previously collected in texts in this area.
+Three of its six chapters concern more general material, while the final three are more specific. The first chapter includes introductory material on standard processes: Poisson point processes, renewal processes, self-exciting processes, and doubly stochastic processes. The second chapter provides some general theory including stationarity, orderliness (meaning that the probability of multiple arrivals in short intervals is sublinear in the interval length), Palm distributions, Fourier analysis, and probability-generating functions. Chapter four (the third of the more general chapters) concerns point process operations, methods of modifying or combining point processes to generate other processes.
+Chapter three, the first of the three chapters on more specific models, is titled "Special models". The special models that it covers include non-stationary Poisson processes, compound Poisson processes, and the Moran process, along with additional treatment of  doubly stochastic processes and renewal processes. Until this point, the book focuses on point processes on the real line (possibly also with a time dimension), but the two final chapters concern multivariate processes and on point processes for higher dimensional spaces, including spatio-temporal processes and Gibbs point processes.
+
+
+== Audience and reception ==
+The book is primarily a reference for researchers. It could also be used to provide additional examples for a course on stochastic processes, or as the basis for an advanced seminar. Although it uses relatively little advanced mathematics, readers are expected to understand advanced calculus and have some familiarity with probability theory and Markov chains.
+Writing some ten years after its original publication, reviewer Fergus Daly of The Open University writes that his copy has been well used, and that it "still is a very good book: lucid, relevant and still not matched in its approach by any other text".
+
+
+== References ==

data/en.wikipedia.org/wiki/Polyhedra_(book)-0.md

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+---
+title: "Polyhedra (book)"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/Polyhedra_(book)"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:46:14.235573+00:00"
+instance: "kb-cron"
+---
+
+Polyhedra is a book on polyhedra, by Peter R. Cromwell. It was published by in 1997 by the Cambridge University Press, with an unrevised paperback edition in 1999.
+
+
+== Topics ==
+The book covers both the mathematics of polyhedra and its historical development, limiting itself only to three-dimensional geometry. The notion of what it means to be a polyhedron has varied over the history of the subject, as have other related definitions, an issue that the book handles largely by keeping definitions informal and flexible, and by pointing out problematic examples for these intuitive definitions. Many digressions help make the material readable, and the book includes many illustrations, including historical reproductions, line diagrams, and photographs of models of polyhedra.
+Polyhedra has ten chapters, the first four of which are primarily historical, with the remaining six more technical. The first chapter outlines the history of polyhedra from the ancient world up to Hilbert's third problem on the possibility of cutting polyhedra into pieces and reassembling them into different polyhedra. The second chapter considers the symmetries of polyhedra, the Platonic solids and Archimedean solids, and the honeycombs formed by space-filling polyhedra. Chapter 3 covers the history of geometry in medieval Islam and early Europe, including connections to astronomy and the study of visual perspective, and Chapter 4 concerns the contributions of Johannes Kepler to polyhedra and his attempts to use polyhedra to model the structure of the universe.
+Among the remaining chapters, Chapter 5 concerns angles and trigonometry, the Euler characteristic, and the Gauss–Bonnet theorem (including also some speculation on whether René Descartes knew about the Euler characteristic prior to Euler). Chapter 6 covers Cauchy's rigidity theorem and flexible polyhedra, and chapter 7 covers self-intersecting star polyhedra. Chapter 8 returns to the symmetries of polyhedra and the classification of possible symmetries, and chapter 9 concerns problems in graph coloring related to polyhedra such as the four color theorem. The final chapter includes material on polyhedral compounds and metamorphoses of polyhedra.
+
+
+== Audience and reception ==
+Most of the book requires little in the way of mathematical background, and can be read by interested amateurs; however, some of the material on symmetry towards the end of the book requires some background in group theory. Reviewer Bill Casselman writes that it would probably not be appropriate to use as a textbook in this area, but could be valuable as additional reference material for an undergraduate geometry class. Reviewer Thomas Bending writes that "The writing is clear and entertaining", and reviewer Ed Sandifer writes that Polyhedra is "solid and fascinating ... likely to become the classic book on the topic ... worthy of many readings".
+Despite complaints about vague referencing of its sources and credits for its historical images, missed connections to modern work in group theory, difficult-to-follow proofs, and occasionally-clumsy illustrations, and typographical errors, Casselman also reviews the book positively, calling it "valuable and a labor of love".
+However, two experts on the topics of the book who also reviewed it, polyhedral combinatorics specialist Peter McMullen and historian of mathematics Judith Grabiner, were much less positive. McMullen writes that "There appears to be some degree of carelessness in the preparation of the book", pointing to errors including calling the Dehn invariant a number, mis-dating Hilbert's problems, misspelling the name of artist Wenzel Jamnitzer and misattributing to Jamnitzer an image by M. C. Escher, and using idiosyncratic and occasionally incorrect names for polyhedra. McMullen writes of these errors that "every time I look at the book, I find more", casting into doubt the other less-familiar parts of the book's content. And Grabiner faults the book's history as naive or mistaken,
+citing as examples its claims that the discovery of irrational numbers ended Pythagorean mysticism, and that pre-Keplerian astronomy consisted only of observation and record-keeping. She accuses Cromwell of basing his narrative on secondary sources rather than checking the original sources he cites, points to sloppy sourcing of historical quotations, and complains about the book's minimal coverage of Islamic and medieval geometry. She writes that the book can be enjoyed as "a treasury" of "beautiful models" and "examples of the impact of polyhedra on the imagination of artists" but should not be relied on for historical insights.
+
+
+== See also ==
+List of books about polyhedra
+
+
+== References ==

data/en.wikipedia.org/wiki/The_Fifty-Nine_Icosahedra-0.md

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+---
+title: "The Fifty-Nine Icosahedra"
+chunk: 1/3
+source: "https://en.wikipedia.org/wiki/The_Fifty-Nine_Icosahedra"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:41.601047+00:00"
+instance: "kb-cron"
+---
+
+The Fifty-Nine Icosahedra is a book written and illustrated by H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic icosahedron, according to a set of rules put forward by J. C. P. Miller.
+It was first published by the University of Toronto in 1938, and a Second Edition reprint by Springer-Verlag followed in 1982. Tarquin's 1999 Third Edition included new reference material and photographs by K. and D. Crennell.
+
+== Stellating polyhedra ==
+
+In this book a polyhedron is stellated by extending the face planes of a polyhedron until they meet again to form a new polyhedron or compound. When the face planes of the polyhedron are extended indefinitely the space around the polyhedron is divided into unbounded sub spaces and often a number of bounded polyhedrons or cells.  Different sets of cells yield different stellations.
+For a symmetrical polyhedron, these cells will fall into groups of congruent cells, or sets – we say that the cells in such a set are of the same type. 
+This can still lead to a large number of possible forms, so further criteria are imposed to reduce the set to those stellations that are significant and distinct in some way.
+Among the Platonic solids, the tetrahedron and cube have no stellations, the octahedron has one (stella octangula), the dodecahedron has three (small stellated dodecahedron, great dodecahedron and great stellated dodecahedron) and the icosahedron has a much larger number (this book concludes 59).
+
+== Earlier publications on stellated icosahedra ==
+The Fifty-Nine Icosahedra is not the first work about stellated icosahedra.
+In 1809 Louis Poinsot discovered the first recognised examples, the great icosahedron (G in the list below) and the great dodecahedron, completing the set of what are nowadays known as the regular star or Kepler–Poinsot polyhedra.
+In 1876 Edmund Hess  used stellation diagrams and discovered  the remaining  mainline stellated icosahedra. (B to F and H in the list below) 
+In 1900 Max Brückner described and photographed many stellated icosahedra in his book Vielecke und Vielflache: Theorie und Geschichte (Polygons and polyhedra: Theory and History, Leipzig: B. G. Teubner, 1900).
+In 1924 A. Harry Wheeler gave a talk as an Invited Speaker of the ICM in 1924 at Toronto.
+In his talk he presented the method of selecting regions of the stellation diagram and combining their cells to form new polyhedral figures. Wheeler included hollow polyhedra and sets of discrete (non connected) cells.  
+Wheeler was initially to be a co-author of The Fifty-Nine Icosahedra, but he objected to Coxeter's approach, which he found so “involved and clumsy that I did not want to have anything to do with it. ... Coxeter has a way of taking a subject and tying it up into knots in such a way that I find it quite difficult to follow him and some times to even make sense.”
+
+== Authors' contributions ==
+
+=== Miller's rules ===
+Although J. C. P. Miller did not contribute to the book directly, he was a close colleague of Coxeter and Petrie. His contribution is immortalised in his set of rules  (now called Miller's rules) for defining which stellations should be considered "properly significant and distinct":
+
+(i) The faces must lie in twenty planes, viz., the bounding planes of the regular icosahedron.
+(ii) All parts composing the faces must be the same in each plane, although they may be quite disconnected.
+(iii) The parts included in any one plane must have trigonal symmetry, without or with reflection. This secures icosahedral symmetry for the whole solid.
+(iv) The parts included in any plane must all be "accessible" in the completed solid (i.e. they must be on the "outside". In certain cases we should require models of enormous size in order to see all the outside. With a model of ordinary size, some parts of the "outside" could only be explored by a crawling insect).
+
+(v) We exclude from consideration cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure. But we allow the combination of an enantiomorphous pair having no common part (which actually occurs in just one case).
+Rules (i) to (iii) are symmetry requirements for the face planes. Rule (iv) excludes buried holes, to ensure that no two stellations look outwardly identical. Rule (v) prevents any disconnected compound of simpler stellations.
+
+=== Coxeter ===
+Coxeter was the main driving force behind the work. He carried out the original analysis based on Miller's rules, adopting a number of techniques such as combinatorics and abstract graph theory whose use in a geometrical context was then novel.
+He observed that the stellation diagram comprised many line segments. He then developed procedures for manipulating combinations of the adjacent plane regions, to formally enumerate the combinations allowed under Miller's rules.
+
+His graph, reproduced here, shows the connectivity of the various faces identified in the stellation diagram (see below). The Greek symbols represent sets of possible alternatives:
+
+λ may be 3 or 4
+μ may be 7 or 8
+ν may be 11 or 12

data/en.wikipedia.org/wiki/The_Fifty-Nine_Icosahedra-1.md

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+---
+title: "The Fifty-Nine Icosahedra"
+chunk: 2/3
+source: "https://en.wikipedia.org/wiki/The_Fifty-Nine_Icosahedra"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:41.601047+00:00"
+instance: "kb-cron"
+---
+
+=== Du Val ===
+Du Val devised a symbolic notation for identifying sets of congruent cells, based on the observation that all the extended face (or boundary) planes together cut the space around the icosahedron in many different finite three dimensional regions that he called cells.
+A segment between a point in a cell and the centre of the icosahedron intersects a number of (extended) face planes; this number is the power of the point and the cell.
+All cells with the same power form a shell (or layer). The inner icosahedron (power = 0) is named A, the shell with power 1 b, the shell with power 2 c, and so on.
+If all cells of a shell are congruent, they are named as the shell itself; if there are different non-congruent cells in a shell, they are numbered like e1 and e2. If an enantiomorphic pair of cells is in a shell, if both are referred then f1 is used, if only one of them is used one is in roman and the other  in italic like f1 and f1.
+For example, there are 3 kinds of cells in the layer with power 5 (shell f): f1, f1 and f2.  f1 is used if only f1,  and f1 are used.
+A stellation consisting of a complete shell and all cells interior to it is named after the outer shell, capitalised, for example  B for A + b and De1 for A + b + c + d+ e1.
+Any combination of cells form a stellated icosahedron, except that the last two of Millers conditions (see above) rule out many combinations.
+
+With this scheme, Du Val tested all possible combinations against Miller's rules, confirming the result of Coxeter's approach.
+
+=== Flather ===
+Flather's contribution was indirect: he made card models of all 59. When he first met Coxeter he had already made many stellations, including some "non-Miller" examples. He went on to complete the series of fifty-nine, which are preserved in the mathematics library of Cambridge University, England. The library also holds some non-Miller models, but it is not known whether these were made by Flather or by Miller's later students.
+
+=== Petrie ===
+John Flinders Petrie was a lifelong friend of Coxeter and had a remarkable ability to visualise four-dimensional geometry. He and Coxeter had worked together on many mathematical problems. His direct contribution to the fifty-nine icosahedra was the exquisite set of three-dimensional drawings which provide much of the fascination of the published work.
+
+=== The Crennells ===
+For the Third Edition, Kate and David Crennell reset the text and redrew the diagrams. They also numbered the icosahedra, added a reference section containing tables, diagrams, and photographs of some of the Cambridge models (which at that time were all thought to be Flather's). Corrections to this edition have been published online.
+
+== Notes on the list ==
+
+Before Coxeter, only Brückner and Wheeler had recorded any significant sets of stellations, although a few such as the great icosahedron had been known for longer. Since publication of The 59, Wenninger published instructions on making models of many polyhedra, some being stellations of the icosahedra; the numbering scheme used in his book has become widely referenced, although he only recorded a few stellations of the icosahedra.
+Index
+
+In the index numbering was added to the Third Edition by the Crennells, the first 32 icosahedra  (indices 1–32) are reflective models, and the last 27 (indices 33–59) are chiral with only the right-handed forms listed. This index follows the order in which the stellations are depicted in the book (all editions).
+Cells
+
+In Du Val's notation (see above) , each shell is identified in bold type, working outwards, as a, b, c, ..., h with a being the original icosahedron. Some shells subdivide into two types of cell, for example e comprises e1 and  e2. The set f1 further subdivides into right- and left-handed forms, respectively f1 (plain type) and f1 (italic). Where a stellation has all cells present within an outer shell, the outer shell is capitalised and the inner omitted, for example a + b + c + e1 is written as Ce1.
+
+Faces
+
+All of the stellations can be specified by a stellation diagram. In the diagram shown here, the numbered colors indicate the regions of the stellation diagram which must occur together as a set, if full icosahedral symmetry is to be maintained. The diagram has 13 such sets. Some of these subdivide into chiral pairs (not shown), allowing stellations with rotational but not reflexive symmetry. In the table, faces which are seen from underneath (from the centre of the stellation)  are indicated by an apostrophe, for example 3'.
+Wenninger
+
+The index numbers and the numbered names were allocated arbitrarily by Wenninger's publisher according to their occurrence in his book Polyhedron models and bear no relation to any mathematical sequence. Only a few of his models were of icosahedra. His names are given in shortened form, with "... of the icosahedron" left off.
+Wheeler
+
+The numbers and names of Wheeler's icosahedra (which by no means were intended to be a complete enumeration) are from the talk he gave as an Invited Speaker of the ICM in 1924 at Toronto.
+Brückner
+
+Max Brückner made and photographed models of many polyhedra, only a few of which were icosahedra. Taf. is an abbreviation of Tafel, German for plate.
+Remarks
+
+No. 8 is sometimes called the echidnahedron after an imagined similarity to the spiny anteater or echidna. This usage is independent of Kepler's description of his regular star polyhedra as his echidnae.
+
+== List of the fifty-nine icosahedra ==
+Several more restrictive categories of stellated polyhedra have been identified, some are easily recognised: 

data/en.wikipedia.org/wiki/The_Fifty-Nine_Icosahedra-2.md

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+---
+title: "The Fifty-Nine Icosahedra"
+chunk: 3/3
+source: "https://en.wikipedia.org/wiki/The_Fifty-Nine_Icosahedra"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:41.601047+00:00"
+instance: "kb-cron"
+---
+
+mainline stellations are the stellations  whose name consist of a single capital letterA till (and including)  H
+fully supported stellations: stellations where there are no overhangs, and all visible parts of a face can be seen from the same side. these are those stellations where in the faces column there is no face that has  an apostrophe, for example 3'.
+The only Kepler–Poinsot polyhedron in the list is the great icosahedron G.
+The Great stellated dodecahedron is an edge-stellated icosahedron but only face-stellated icosahedra are in the list and it is therefore not included in the list.
+
+Some images illustrate the mirror-image icosahedron with the f1 rather than the f1 cell.
+
+== Subsequent debate ==
+There has subsequently been some debate on Miller's rules, with some writers questioning them. Flather himself made and exhibited models which were "non-Miller". Bridge (1974) obtained stellations of the icosahedron by dualising facettings of the dodecahedron, noting the significance of internal structure in distinguishing between stellations which Miller's rules treat as identical. Hudson and Kingston (1988) adopted a reduced rule set. Inchbald noted two non-Miller stellations, and went on to discuss various related issues.
+
+== See also ==
+List of Wenninger polyhedron models – Wenninger's book Polyhedron models included 21 of these stellations.
+Solids with icosahedral symmetry
+
+== Notes ==
+
+== References ==
+Brückner, Max (1900). Vielecke und Vielflache: Theorie und Geschichte [Polygons and Polyhedra: Theory and History]. Leipzig: B.G. Treubner. (in German). OCLC 25080888. (Plate scans at the Internet Archive, at bulatov.org. Reprinted 2004: Michigan Publishing. ISBN 978-1-4181-6590-1.)
+H. S. M. Coxeter, P. Du Val, H. T. Flather, J. F. Petrie (1938) The Fifty-nine Icosahedra, University of Toronto studies, mathematical series 6: 1–26.
+Third edition (1999) Tarquin ISBN 978-1-899618-32-3 MR 0676126
+Cromwell, Peter R. (1997). "Stars, Stellations and Skeletons". Polyhedra . Cambridge University Press. Ch. 7, pp. 249–287. ISBN 0-521-55432-2.
+Wenninger, Magnus J. (1983) Polyhedron models; Cambridge University Press, Paperback edition (2003). ISBN 978-0-521-09859-5.
+Wheeler, Albert Harry (1924) "Certain forms of the icosahedron and a method for deriving and designating higher polyhedra." Archived 2017-12-01 at the Wayback Machine In Proceedings of the International Mathematical Congress, Toronto, vol. 1, pp. 701–708.
+
+== External links ==
+Vladimir Bulatov (2001). "An Interactive Creation of Polyhedra Stellations with Various Symmetries" VisMath. Vol. 3, No. 2.
+The fifty nine stellations of the regular icosahedron
+Weisstein, Eric W. "Fifty nine icosahedron stellations". MathWorld.
+Weisstein, Eric W. "Echidnahedron". MathWorld.
+Stellations of the icosahedron
+George Hart, 59 Stellations of the Icosahedron - VRML 3D files.
+John L. Hudson (1987). "Set of 70 polyhedron models showing the icosahedron and its 58 stellations". Science Museum, London.
+A. Harry Wheeler. Models at the National Museum of American History. Collection search for "Harry Wheeler icosahedron"

data/en.wikipedia.org/wiki/The_Fractal_Dimension_of_Architecture-0.md

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+---
+title: "The Fractal Dimension of Architecture"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/The_Fractal_Dimension_of_Architecture"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:47.635701+00:00"
+instance: "kb-cron"
+---
+
+The Fractal Dimension of Architecture is a book that applies the mathematical concept of fractal dimension to the analysis of the architecture of buildings. It was written by Michael J. Ostwald and Josephine Vaughan, both of whom are architecture academics at the University of Newcastle (Australia); it was published in 2016 by Birkhäuser, as the first volume in their Mathematics and the Built Environment book series.
+
+
+== Topics ==
+The book applies the box counting method for computing fractal dimension, via the ArchImage software system, to compute a fractal dimension from architectural drawings (elevations and floor plans) of buildings, drawn at multiple levels of detail. The results of the book suggest that the results are consistent enough to allow for comparisons from one building to another, as long as the general features of the images (such as margins, line thickness, and resolution), parameters of the box counting algorithm, and statistical processing of the results are carefully controlled.
+The first five chapters of the book introduce fractals and the fractal dimension, and explain the methodology used by the authors for this analysis, also applying the same analysis to classical fractal structures including the Apollonian gasket, Fibonacci word, Koch snowflake, Minkowski sausage, pinwheel tiling, terdragon, and Sierpiński triangle. The remaining six chapters explain the authors' choice of buildings to analyze, apply their methodology to 625 drawings from 85 homes, built between 1901 and 2007, and perform a statistical analysis of the results.
+The authors use this technique to study three main hypotheses, with a fractal structure of subsidiary hypotheses depending on them. These are
+
+That the decrease in the complexity of social family units over the period of study should have led to a corresponding decrease in the complexity of their homes, as measured by a reduction in the fractal dimension.
+That distinctive genres and movements in architecture can be characterized by their fractal dimensions, and
+That individual architects can also be characterized by the fractal dimensions of their designs.
+The first and third hypotheses are not convincingly supported by the analysis, but the results suggest further work in these directions. The second hypothesis, on distinctive fractal descriptions of genres and movements, does not appear to be true, leading the authors to weaker replacements for it.
+
+
+== Audience and reception ==
+The book is aimed at architects and architecture students; its mathematical content is not deep, and it does not require much mathematical background of its readers. Reviewer Joel Haack suggests that it could also be used for general education courses in mathematics for liberal arts undergraduates.
+
+
+== Further reading ==
+
+
+== References ==

data/en.wikipedia.org/wiki/The_Fractal_Geometry_of_Nature-0.md

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+---
+title: "The Fractal Geometry of Nature"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/The_Fractal_Geometry_of_Nature"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:48.793264+00:00"
+instance: "kb-cron"
+---
+
+The Fractal Geometry of Nature is a 1982 book by the Franco-American mathematician Benoît Mandelbrot.
+
+
+== Overview ==
+The Fractal Geometry of Nature is a revised and enlarged version of his 1977 book entitled Fractals: Form, Chance and Dimension, which in turn was a revised, enlarged, and translated version of his 1975 French book, Les Objets Fractals: Forme, Hasard et Dimension. American Scientist put the book in its one hundred books of 20th century science.
+As technology has improved, mathematically accurate, computer-drawn fractals have become more detailed. Early drawings were low-resolution black and white; later drawings were higher resolution and in color. Many examples were created by programmers working with Mandelbrot, primarily at IBM Research. These visualizations have added to persuasiveness of the books and their impact on the scientific community.
+
+
+== See also ==
+Chaos theory
+
+
+== References ==

data/en.wikipedia.org/wiki/The_Geometry_of_the_Octonions-0.md

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+---
+title: "The Geometry of the Octonions"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/The_Geometry_of_the_Octonions"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:44:59.245218+00:00"
+instance: "kb-cron"
+---
+
+The Geometry of the Octonions is a mathematics book on the octonions, a system of numbers generalizing the complex numbers and quaternions, presenting its material at a level suitable for undergraduate mathematics students. It was written by Tevian Dray and Corinne Manogue, and published in 2015 by World Scientific. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
+
+
+== Topics ==
+The book is subdivided into three parts, with the second part being the most significant. Its contents combine both a survey of past work in this area, and much of its authors' own researches.
+The first part explains the Cayley–Dickson construction, which constructs the complex numbers from the real numbers, the quaternions from the complex numbers, and the octonions from the quaternions. Related algebras are also discussed, including the sedenions (a 16-dimensional real algebra formed in the same way by taking one more step past the octonions) and the split real unital composition algebras (also called Hurwitz algebras). A particular focus here is on interpreting the multiplication operation of these algebras in a geometric way. Reviewer Danail Brezov notes with disappointment that Clifford algebras, although very relevant to this material, are not covered.
+The second part of the book uses the octonions and the other division algebras associated with it to provide concrete descriptions of the Lie groups of geometric symmetries. These include rotation groups, spin groups, symplectic groups, and the exceptional Lie groups, which the book interprets as octonionic variants of classical Lie groups.
+The third part applies the octonions in geometric constructions including the Hopf fibration and its generalizations, the Cayley plane, and the E8 lattice. It also connects them to problems in physics involving the four-dimensional Dirac equation, the quantum mechanics of relativistic fermions, spinors, and the formulation of quantum mechanics using Jordan algebras. It also includes material on octonionic number theory, and concludes with a chapter on the Freudenthal magic square and related constructions.
+
+
+== Audience and reception ==
+Although presented at an undergraduate level, The Geometry of the Octonions is not a textbook: its material is likely too specialized for an undergraduate course, and it lacks exercises or similar material that would be needed to use it as a textbook. Readers should be familiar with linear algebra, and some experience with Lie groups would also be helpful. The later chapters on applications in physics are heavier going, and require familiarity with quantum mechanics.
+The book avoids a proof-heavy formal style of mathematical writing, so much so that reviewer Danail Brezov writes that at points it "seems to lack mathematical rigor".
+
+
+== Related reading ==
+Multiple reviewers suggest that this work would make a good introduction to the octonions, as a stepping stone to the more advanced material presented in other works on the same topic. Their suggestions include the following:
+
+Baez, John C. (2002), "The octonions", Bulletin of the American Mathematical Society, New Series, 39 (2): 145–205, doi:10.1090/S0273-0979-01-00934-X, MR 1886087, S2CID 586512
+Conway, John H.; Smith, Derek A. (2003), On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, Natick, Massachusetts: A K Peters, ISBN 1-56881-134-9, MR 1957212
+Kantor, I. L.; Solodovnikov, A. S. (1989), Hypercomplex Numbers: An Elementary Introduction to Algebras, New York: Springer-Verlag, doi:10.1007/978-1-4612-3650-4 (inactive February 1, 2026), ISBN 0-387-96980-2, MR 0996029{{citation}}:  CS1 maint: DOI inactive as of February 2026 (link)
+Salzmann, Helmut; Betten, Dieter; Grundhöfer, Theo; Hähl, Hermann; Löwen, Rainer; Stroppel, Markus (1995), Compact Projective Planes: With an Introduction to Octonion Geometry, De Gruyter Expositions in Mathematics, vol. 21, Walter de Gruyter & Co., Berlin, doi:10.1515/9783110876833, ISBN 3-11-011480-1, MR 1384300
+Springer, Tonny A.; Veldkamp, Ferdinand D. (2000), Octonions, Jordan Algebras and Exceptional Groups, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-12622-6, ISBN 3-540-66337-1, MR 1763974
+Ward, J. P. (1997), Quaternions and Cayley Numbers: Algebra and Applications, Mathematics and its Applications, vol. 403, Dordrecht: Kluwer Academic Publishers, doi:10.1007/978-94-011-5768-1, ISBN 0-7923-4513-4, MR 1458894
+
+
+== References ==

data/en.wikipedia.org/wiki/The_Ground_of_Arts-0.md

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+---
+title: "The Ground of Arts"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/The_Ground_of_Arts"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:05.145058+00:00"
+instance: "kb-cron"
+---
+
+Robert Recorde's Arithmetic: or, The Ground of Arts was one of the first printed English textbooks on arithmetic and the most popular of its time. The Ground of Arts appeared in London in 1543, and it was reprinted around 45 more editions until 1700. Editors and contributors of new sections included John Dee, John Mellis, Robert Hartwell, Thomas Willsford, and finally Edward Hatton.
+The text is in the format of a dialogue between master and student to facilitate learning arithmetic without a teacher.
+
+
+== References ==
+
+Bregman, Alvan (1 July 2005). "Alligation Alternate and the Composition of Medicines: Arithmetic and Medicine in Early Modern England". Med. Hist. 49 (3): 299–320. doi:10.1017/s0025727300008899. PMC 1172291. PMID 16092789.
+Karpinski, Louis (1925). The history of arithmetic. Rand McNally. LCC QA21.K3.
+Recorde, Robert (1543). The Grounde of Artes. London: Reynold Wolff. LCC QA33.R3 1542a.
+Recorde, Robert (1699) [1543]. Edward Hatton (ed.). Arithmetick, or, The ground of arts. London: J.H. for Charles Harper.
+
+
+== Further reading ==
+John Denniss & Fenny Smith, "Robert Recorde and his remarkable Arithmetic", pages 25 to 38 in Gareth Roberts & Fenny Smith (editors) (2012) Robert Recorde: The Life and Times of a Tudor Mathematician, Cardiff: University of Wales Press ISBN 978-0-7083-2526-1

data/en.wikipedia.org/wiki/The_Higher_Infinite-0.md

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+---
+title: "The Higher Infinite"
+chunk: 1/1
+source: "https://en.wikipedia.org/wiki/The_Higher_Infinite"
+category: "reference"
+tags: "science, encyclopedia"
+date_saved: "2026-05-05T08:45:09.938302+00:00"
+instance: "kb-cron"
+---
+
+The Higher Infinite: Large Cardinals in Set Theory from their Beginnings is a monograph in set theory by Akihiro Kanamori, concerning the history and theory of large cardinals, infinite sets characterized by such strong properties that their existence cannot be proven in Zermelo–Fraenkel set theory (ZFC). This book was published in 1994 by Springer-Verlag in their series Perspectives in Mathematical Logic, with a second edition in 2003 in their Springer Monographs in Mathematics series, and a paperback reprint of the second edition in 2009 (ISBN 978-3-540-88866-6).
+
+
+== Topics ==
+Not counting introductory material and appendices, there are six chapters in The Higher Infinite, arranged roughly in chronological order by the history of the development of the subject. The author writes that he chose this ordering "both because it provides the most coherent exposition of the mathematics and because it holds the key to any epistemological concerns".
+In the first chapter, "Beginnings", the material includes inaccessible cardinals, Mahlo cardinals, measurable cardinals, compact cardinals and indescribable cardinals. The chapter covers the constructible universe and inner models, elementary embeddings and ultrapowers, and a result of Dana Scott that measurable cardinals are inconsistent with the axiom of constructibility.
+The second chapter, "Partition properties", includes the partition calculus of Paul Erdős and Richard Rado, trees and Aronszajn trees, the model-theoretic study of large cardinals, and the existence of the set 0# of true formulae about indiscernibles. It also includes Jónsson cardinals and Rowbottom cardinals.
+Next are two chapters on "Forcing and sets of reals" and "Aspects of measurability". The main topic of the first of these chapters is forcing, a technique introduced by Paul Cohen for proving consistency and inconsistency results in set theory; it  also includes material in descriptive set theory. The second of these chapters covers the application of forcing by Robert M. Solovay to prove the consistency of measurable cardinals, and related results using stronger notions of forcing.
+Chapter five is "Strong hypotheses". It includes material on supercompact cardinals and their reflection properties, on huge cardinals, on Vopěnka's principle, on extendible cardinals, on strong cardinals, and on Woodin cardinals.
+The book concludes with the chapter "Determinacy", involving the axiom of determinacy and the theory of infinite games. Reviewer Frank R. Drake views this chapter, and the proof in it by Donald A. Martin of the Borel determinacy theorem, as central for Kanamori, "a triumph for the theory he presents".
+Although quotations expressing the philosophical positions of researchers in this area appear throughout the book, more detailed coverage of
+issues in the philosophy of mathematics regarding the foundations of mathematics are deferred to an appendix.
+
+
+== Audience and reception ==
+Reviewer Pierre Matet writes that this book "will no doubt serve for many years to come as the main reference for large cardinals", and reviewers Joel David Hamkins, Azriel Lévy and Philip Welch express similar sentiments. Hamkins writes that the book is "full of historical insight, clear writing, interesting theorems, and elegant proofs". Because this topic uses many of the important tools of set theory more generally, Lévy recommends the book "to anybody who wants to start doing research in set theory", and Welch recommends it to all university libraries.
+
+
+== References ==
+
+
+== External links ==
+The Higher Infinite(registration required) (1st edition) at the Internet Archive