From 23ef0f95c06c284513ba98ccbe2682a28374736e Mon Sep 17 00:00:00 2001 From: turtle89431 Date: Tue, 5 May 2026 01:48:09 -0700 Subject: [PATCH] Scrape wikipedia-science: 3705 new, 2848 updated, 6714 total (kb-cron) --- _index.db | Bin 66826240 -> 67821568 bytes .../wiki/AHFS_Drug_Information_Book-0.md | 23 ++ .../wiki/A_Colour_Atlas_of_Urology-0.md | 31 +++ .../wiki/A_Letter_to_a_Friend-0.md | 30 +++ .../wiki/A_Topological_Picturebook-0.md | 24 +++ ...Treatise_on_the_Circle_and_the_Sphere-0.md | 87 ++++++++ data/en.wikipedia.org/wiki/Aequanimitas-0.md | 53 +++++ data/en.wikipedia.org/wiki/Al-Hawi-0.md | 36 ++++ ...nical_Medicine,_Surgery_and_Pathology-0.md | 14 ++ data/en.wikipedia.org/wiki/Articella-0.md | 31 +++ data/en.wikipedia.org/wiki/Autism | 0 .../wiki/Autism's_False_Prophets-0.md | 43 ++++ data/en.wikipedia.org/wiki/Bad_Pharma-0.md | 2 +- data/en.wikipedia.org/wiki/Bad_Pharma-1.md | 2 +- data/en.wikipedia.org/wiki/Bad_Pharma-2.md | 2 +- 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z|ISF~AHV<0)OJ{SaHyHr(`E!cq|Nbs?F6%Qi#NuvxF=KALvtq+9Kpn0P$5|`c6%dO z;BAyKqkRCd59<;9csQ9+AjxWnpJKjf88t}q1NgOQCl8ZEFi+5mN~@U^3bSH8uOm1h zE$?PIyC`%BVe<>f-T-Qc1uokEE|ORT<@mXZ^I6&3jBn8;gnh<;7j!Cv*P)>VzP#Sq z;*>*@ie){{T4!I=-;WdcONO1f9kCMV z_B-Vayej9V(KVcR(X z1+N}O-wfds_u!89)bmqmh5IgG50gf49kPVPV5w}p#}CmS&`0Y4KPNJl00CYc8Z zmTKSXvBOAuIYi>735>cagQhr4Ul-U%B=9`|6`0Z_ad#uR)quW!DxtV8MD ztNARIgv#r`9c)mkkbJSeDGK=(@YYefh{I+-WeJK@D z&QcwAv?Tt;LSvgAY>?oFyRZTjX|*vn#`(C^= + 0 + + + {\displaystyle a>0} + + is an irrational number, there is a zero-one law for the first-order properties of the random graphs + + + + G + ( + n + , + + n + + − + a + + + ) + + + {\displaystyle G(n,n^{-a})} + +. A key tool in the proof is the Ehrenfeucht–Fraïssé game. + + +== Audience and reception == +Although it is essentially the proof of a single theorem, aimed at specialists in the area, the book is written in a readable style that introduces the reader to many important topics in finite model theory and the theory of random graphs. Reviewer Valentin Kolchin, himself the author of another book on random graphs, writes that the book is "self-contained, easily read, and is distinguished by elegant writing", recommending it to probability theorists and logicians. Reviewer Alessandro Berarducci calls the book "beautifully written" and its subject "fascinating". + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/The_Tower_of_Hanoi_–_Myths_and_Maths-0.md b/data/en.wikipedia.org/wiki/The_Tower_of_Hanoi_–_Myths_and_Maths-0.md new file mode 100644 index 000000000..55567ed29 --- /dev/null +++ b/data/en.wikipedia.org/wiki/The_Tower_of_Hanoi_–_Myths_and_Maths-0.md @@ -0,0 +1,30 @@ +--- +title: "The Tower of Hanoi – Myths and Maths" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/The_Tower_of_Hanoi_–_Myths_and_Maths" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:03.324829+00:00" +instance: "kb-cron" +--- + +The Tower of Hanoi – Myths and Maths is a book in recreational mathematics, on the tower of Hanoi, baguenaudier, and related puzzles. It was written by Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, and published in 2013 by Birkhäuser, with an expanded second edition in 2018. The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries. + + +== Topics == +Although this book is in recreational mathematics, it takes its subject seriously, and brings in material from automata theory, computational complexity, the design and analysis of algorithms, graph theory, and group theory, topology, fractal geometry, chemical graph theory, and even psychology (where related puzzles have applications in psychological testing). +The 1st edition of the book had 10 chapters, and the 2nd edition has 11. In both cases they begin with chapter zero, on the background and history of the Tower of Hanoi puzzle, covering its real-world invention by Édouard Lucas and in the mythical backstory he invented for it. Chapter one considers the Baguenaudier puzzle (or, as it is often called, the Chinese rings), related to the tower of Hanoi both in the structure of its state space and in the fact that it takes an exponential number of moves to solve, and likely the inspiration for Lucas. Chapter two introduces the main topic of the book, the tower of Hanoi, in its classical form in which one must move disks one-by-one between three towers, always keeping the disks on each tower sorted by size. It provides several different algorithms for solving the classical puzzle (in which the disks begin and end all on a single tower) in as few moves as possible, and for collecting all disks on a single tower when they begin in other configurations, again as quickly as possible. It introduces the Hanoi graphs describing the state space of the puzzle, and relates numbers of puzzle steps to distances within this graph. After a chapter on "irregular" puzzles in which the initial placement of disks on their towers is not sorted, chapter four discusses the "Sierpiński graphs" derived from the Sierpiński triangle; these are closely related to the three-tower Hanoi graphs but diverge from them for higher numbers of towers of Hanoi or higher-dimensional Sierpinski fractals. +The next four chapters concern additional variants of the tower of Hanoi, in which more than three towers are used, the disks are only allowed to move between some of the towers or in restricted directions between the towers, or the rules for which disks can be placed on which are modified or relaxed. A particularly important case is the Reve's puzzle, in which the rules are unchanged except that there are four towers instead of three. An old conjecture concerning the minimum possible number of moves between two states with all disks on a single tower was finally proven in 2014, after the publication of the first edition of the book, and the second edition includes this material. +Some of the definitions and proofs are extended into the book's many exercises. A new chapter in the second edition provides hints and partial solutions, and the final chapter collects open problems and (in the second edition) provides updates to previously-listed problems. Many color illustrations and photographs are included throughout the book. + + +== Audience == +The book can be read both by mathematicians working on topics related to the tower of Hanoi puzzle, and by a general audience interested in recreational mathematics. Reviewer László Kozma describes the book as essential reading for the first type of audience and (despite occasional heavy notation and encyclopedic detail) accessible and interesting to the second type, even for readers with only a high school level background in mathematics. On the other hand, reviewer Cory Palmer cautions that "this book is not for a casual reader", adding that a good understanding of combinatorics is necessary to read it, and reviewer Charles Ashbacher suggests that it has enough depth of content to be the topic of an advanced undergraduate elective course. +Although generally positive, reviewer S. V. Nagaraj complains about a "significant number of errors" in the book. Reviewer Andrew Percy calls it "an enjoyable adventure", "humorous, and very thorough". Reviewer Martin Klazar calls the book "wonderful", recommending it to anyone interested in recreational mathematics or mathematics more generally. + + +== References == + + +== External links == +Home page \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/The_Whetstone_of_Witte-0.md b/data/en.wikipedia.org/wiki/The_Whetstone_of_Witte-0.md new file mode 100644 index 000000000..103757fdc --- /dev/null +++ b/data/en.wikipedia.org/wiki/The_Whetstone_of_Witte-0.md @@ -0,0 +1,46 @@ +--- +title: "The Whetstone of Witte" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/The_Whetstone_of_Witte" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:18.369323+00:00" +instance: "kb-cron" +--- + +The Whetstone of Witte is the shortened title of Robert Recorde's mathematics book published in 1557, the full title being The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers. The book covers topics including whole numbers, the extraction of roots and irrational numbers. The work is notable for containing the first recorded use of the equals sign and also for being the first book in English to use the plus and minus signs. +Recordian notation for exponentiation, however, differed from the later Cartesian notation + + + + + p + + q + + + = + p + × + p + × + p + ⋯ + × + p + + + {\displaystyle p^{q}=p\times p\times p\cdots \times p} + +. Recorde expressed indices and surds larger than 3 in a systematic form based on the prime factorization of the exponent: a factor of two he termed a zenzic, and a factor of three, a cubic. Recorde termed the larger prime numbers appearing in this factorization sursolids, distinguishing between them by use of ordinal numbers: that is, he defined 5 as the first sursolid, written as ʃz and 7 as the second sursolid, written as Bʃz. + +He also devised symbols for these factors: a zenzic was denoted by z, and a cubic by &. For instance, he referred to p8=p2×2×2 as zzz (the zenzizenzizenzic), and q12=q2×2×3 as zz& (the zenzizenzicubic).Later in the book he includes a chart of exponents all the way up to p80=p2×2×2×2×5 written as zzzzʃz. There is an error in the chart, however, writing p69 as Sʃz, despite it not being a prime. It should be p3×23 or &Gʃz. +Page images have been made available by Victor Katz and Frank Swetz through Convergence, a publication of Mathematical Association of America. + + +== References == + + +== External links == +The Whetstone of Witte at The Internet Archive +The Whetstone of Witte at The Library of Congres \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Theory_of_Lie_groups-0.md b/data/en.wikipedia.org/wiki/Theory_of_Lie_groups-0.md new file mode 100644 index 000000000..632a9906c --- /dev/null +++ b/data/en.wikipedia.org/wiki/Theory_of_Lie_groups-0.md @@ -0,0 +1,20 @@ +--- +title: "Theory of Lie groups" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Theory_of_Lie_groups" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:46:59.850097+00:00" +instance: "kb-cron" +--- + +In mathematics, Theory of Lie groups is a series of books on Lie groups by Claude Chevalley (1946, 1951, 1955). The first in the series was one of the earliest books on Lie groups to treat them from the global point of view, and for many years was the standard text on Lie groups. The second and third volumes, on algebraic groups and Lie algebras, were written in French, and later reprinted bound together as one volume. Apparently further volumes were planned but not published, though his lectures (Chevalley 2005) on the classification of semisimple algebraic groups could be considered as a continuation of the series. + + +== References == +Chevalley, Claude (1946), Theory of Lie Groups. I, Princeton Mathematical Series, vol. 8, Princeton University Press, ISBN 978-0-691-04990-8, MR 0015396 {{citation}}: ISBN / Date incompatibility (help) +Chevalley, Claude (1951), Théorie des groupes de Lie. Tome II. Groupes algébriques, Actualités Sci. Ind., vol. 1152, Hermann & Cie., Paris, MR 0051242 +Chevalley, Claude (1955), Théorie des groupes de Lie. Tome III. Théorèmes généraux sur les algèbres de Lie, Actualités Sci. Ind., vol. 1226, Hermann & Cie, Paris, MR 0068552 +Chevalley, Claude (1968), Théorie des groupes de Lie : Groupes algébriques, théorèmes généraux sur les algèbres de Lie (in French), vol. 8, Paris: Hermann, Reprint of volumes II and III bound as one volume +Chevalley, Claude (2005) [1958], Cartier, P. (ed.), Classification des groupes algébriques semi-simples, Collected works., vol. 3, Berlin, New York: Springer-Verlag, ISBN 978-3-540-23031-1, MR 0106966 +Smith, P. A. (1947), "Review: Claude Chevalley, The theory of Lie groups, I", Bull. Amer. Math. Soc., 53 (9): 884–887, doi:10.1090/s0002-9904-1947-08876-5 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Tilings_and_patterns-0.md b/data/en.wikipedia.org/wiki/Tilings_and_patterns-0.md new file mode 100644 index 000000000..9b4d550ed --- /dev/null +++ b/data/en.wikipedia.org/wiki/Tilings_and_patterns-0.md @@ -0,0 +1,51 @@ +--- +title: "Tilings and patterns" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Tilings_and_patterns" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:01.029296+00:00" +instance: "kb-cron" +--- + +Tilings and patterns is a book by mathematicians Branko Grünbaum and Geoffrey Colin Shephard published in 1987 by W.H. Freeman. The book was 10 years in development, and upon publication it was widely reviewed and highly acclaimed. + + +== Structure and topics == +The book is concerned with tilings—a partition of the plane into regions (the tiles)—and patterns—repetitions of a motif in the plane in a regular manner. +The book is divided into two parts. The first seven chapters define concepts and terminology, establish the general theory of tilings, survey tilings by regular polygons, review the theory of patterns, and discuss tilings in which all the tiles, or all the edges, or all the vertices, play the same role. +The last five chapters survey a variety of advanced topics in tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings with unusual kinds of tiles. +Each chapter open with an introduction to the topic, this is followed by the detailed material of the chapter, much previously unpublished, which is always profusely illustrated, and normally includes examples and proofs. Chapters close with exercises, and a section of notes and references which detail the historical development of the topic. These notes sections are interesting and entertaining, as they discuss the efforts of the previous workers in the field and detail the good (and bad) approaches to the topic. The notes also identify unsolved problems, point out areas of potential application, and provide connections to other disciplines in mathematics, science, and the arts. +The book has 700 pages, including a 40-page, 800-entry bibliography, and an index. The book is used as a source on numerous Wikipedia pages. + + +== Audience == +In their preface the authors state "We have written this book with three main groups of readers in mind—students, professional mathematicians and non-mathematicians whose interests include patterns and shapes (such as artists, architects, crystallographers and others). +Other reviewers commented as follows: + +"The most striking feature of the book is its extensive collection of figures, including hundreds of examples of tilings and patterns. The sheer abundance is perhaps one reason why artists and designers have been drawn to it over the years." +"Their idea was that the book should be accessible to any reader who is attracted to geometry." + + +== Reception == +Contemporary reviews of the book were overwhelming positive. The book was reviewed by 15 journals in the fields of crystallography, mathematics, and the sciences. Quotations from major reviews: + + +== Influence == +The book was praised in later journal articles by multiple authors: + +The book was also praised in later books by other authors: + + +== Editions == +The hardback original Tilings and patterns was published in 1987. +Tilings and patterns - an introduction, a paperback reprint of the first seven chapters of the 1987 original, was published in 1989. +In 2016 a second edition of the full text was published by Dover in paperback, with a new preface and an appendix describing progress in the subject since the first edition. The reviewer at MAA Reviews commented "Dover has once again done the mathematical community a service in bringing back such a notable volume." + + +== References == + + +== External links == +"Tilings and patterns". at the Internet Archive +"Selection of reviews". at the MacTutor History of Mathematics Archive \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Traité_de_mécanique_céleste-0.md b/data/en.wikipedia.org/wiki/Traité_de_mécanique_céleste-0.md new file mode 100644 index 000000000..6401030de --- /dev/null +++ b/data/en.wikipedia.org/wiki/Traité_de_mécanique_céleste-0.md @@ -0,0 +1,72 @@ +--- +title: "Traité de mécanique céleste" +chunk: 1/2 +source: "https://en.wikipedia.org/wiki/Traité_de_mécanique_céleste" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:04.499255+00:00" +instance: "kb-cron" +--- + +Traité de mécanique céleste (transl. "Treatise of celestial mechanics") is a five-volume treatise on celestial mechanics written by Pierre-Simon Laplace and published from 1798 to 1825 with a second edition in 1829. In 1842, the government of Louis Philippe gave a grant of 40,000 francs for a 7-volume national edition of the Oeuvres de Laplace (1843–1847); the Traité de mécanique céleste with its four supplements occupies the first 5 volumes. + +Newton laid the foundations of Celestial Mechanics, at the close of the seventeenth century, by the discovery of the principle of universal gravitation. Even in his own hands, this discovery led to important consequences, but it has required a century and a half, and a regular succession of intellects the most powerful, to fill up the outline sketched by him. Of these, Laplace himself was the last, and, perhaps after Newton, the greatest; and the task commenced in the Principia of the former, is completed in the Mécanique Céleste of the latter. In this last named work, the illustrious author has proposed to himself his object, to unite all the theories scattered throughout the various channels of publication, employed by his predecessors, to reduce them to one common method, and present them all in the same point of view. +If one were asked to name the two most important works in the progress of mathematics and physics, the answer would undoubtedly be, the Principia of Newton and the Mécanique Céleste of Laplace. In their historical and philosophical aspects these works easily outrank all others, and furnish thus the standard by which all others must be measured. The distinguishing feature of the Principia is its clear and exhaustive enunciation of fundamental principles. The Mécanique Céleste, on the other hand, is conspicuous for the development of principles and for the profound generality of its methods. The Principia gives the plans and specifications of the foundations; the Mécanique Céleste affords the key to the vast and complex superstructure. + +== Tome I. (1798) == + +=== Livre I. Des lois générales de l'équilibre et du mouvement === +Chap. I. De l'équilibre et de la composition des forces qui agissent sur un point matériel +Chap. II. Du mouvement d'un point matériel +Chap. III. De l'équilibre d'un système de corps +Chap. IV. De l'équilibre des fluides +Chap. V. Principes généraux du mouvement d'un système de corps +Chap. VI. Des lois du mouvement d'un système de corps, dans toutes les relations mathématiquement possibles entre la force et la vitesse +Chat. VII. Des mouvemens d'un corps solide de figure quelconque +Chap. VIII. Du mouvement des fluides + +=== Livre II. De la loi pesanteur universelle, et du mouvement des centres de gravité des corps célestes === + +== Tome II. (1798) == + +=== Livre III. De la figure des corps céleste === + +=== Livre IV. Des oscillations de la mer et de l'atmosphère === + +=== Livre V. Des mouvemens des corps célestes, autour de leurs propre centres de gravité === + +== Tome III. (1802) == + +=== Livre VI. Théorie particulières des mouvemens célestes === + +=== Livre VII. Théorie de la lune === + +== Tome IV. (1805) == + +=== Livre VIII. Théorie des satellites de Jupiter, de Saturne et d'Uranus === + +=== Livre IX. Théorie des comètes === + +=== Livre X. Sur différens points relatifs au système du monde === +This book contains a discussion of continued fractions and a computation of the complementary error function in terms that came to be called the Laplace continued fraction, +1/(1+q/(1+2q/(1+3q/(...))). + +== Tome V. (1825) == + +=== Livre XI. De la figure et de la rotation de la terre === + +=== Livre XII. De l'attraction et de la répulsion des sphères, et des lois de l'equilibre et du mouvement des fluides élastiques === + +=== Livre XIII. Des oscillations des fluides qui recouvrent les planètes === + +=== Livre XIV. Des mouvemens des corps célestes autour de leurs centres de gravité === + +=== Livre XV. Du mouvement des planètes et des comètes === + +=== Livre XVI. Du mouvement des satellites === + +== English translations == +During the early nineteenth century at least five English translations of Mécanique Céleste were published. In 1814 the Reverend John Toplis prepared a translation of Book 1 entitled The Mechanics of Laplace. Translated with Notes and Additions. In 1821 Thomas Young anonymously published a further translation into English of the first book; beyond just translating from French to English he claimed in the preface to have translated the style of mathematics: The translator flatters himself, however, that he has not expressed the author's meaning in English words alone, but that he has rendered it perfectly intelligible to any person, who is conversant with the English mathematicians of the old school only, and that his book will serve as a connecting link between the geometrical and algebraical modes of representation.The Reverend Henry Harte, a fellow at Trinity College, Dublin translated the entire first volume of Mécanique Céleste, with Book 1 published in 1822 and Book 2 published separately in 1827. Similarly to Bowditch (see below), Harte felt that Laplace's exposition was too brief, making his work difficult to understand:... it may be safely asserted, that the chief obstacle to a more general knowledge of the work, arises from the summary manner in which the Author passes over the intermediate steps in several of his most interesting investigations. + +=== Bowditch's translation === +The famous American mathematician Nathaniel Bowditch translated the first four volumes of the Traité de mécanique céleste but not the fifth volume; however, Bowditch did make use of relevant portions of the fifth volume in his extensive commentaries for the first four volumes. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Traité_de_mécanique_céleste-1.md b/data/en.wikipedia.org/wiki/Traité_de_mécanique_céleste-1.md new file mode 100644 index 000000000..c89e72c5e --- /dev/null +++ b/data/en.wikipedia.org/wiki/Traité_de_mécanique_céleste-1.md @@ -0,0 +1,26 @@ +--- +title: "Traité de mécanique céleste" +chunk: 2/2 +source: "https://en.wikipedia.org/wiki/Traité_de_mécanique_céleste" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:04.499255+00:00" +instance: "kb-cron" +--- + +The first four volumes of Dr. Bowditch's Translation and Commentary were published successively, in 1828, 1832, 1834, and 1839, at the sacrifice of one quarter of his whole property. The expense was largely increased by the voluminous commentary. This was really of the nature of an original work, and was rendered necessary by the frequent gaps which Laplace had left in his own publication. Mr. N. I. Bowditch says, in his biography of his father, that Dr. Bowditch was accustomed to remark, "Whenever I meet in Laplace with the words, Thus it plainly appears, I am sure that hours, and perhaps days, of hard study will alone enable me to discover how it plainly appears." +Bowditch's translation of the first four volumes of Laplace's Traité de mécanique céleste was completed by 1818 but he would not publish it for many years. Almost certainly the cost of publication caused the delay, but Bowditch did not just put the work on one side after 1818 but continued to improve it over the succeeding years. Bowditch was helped by Benjamin Peirce in this project and his commentaries doubled the length of the book. His purpose was more than just an English translation. He wanted to supply steps omitted in the original text; to incorporate later results into the translation; and to give credits omitted by Laplace. + +=== Somerville's translation === +In 1826, it was still felt by Henry Brougham, president of the Society for the Diffusion of Useful Knowledge, that the British reader was lacking a readable translation of Mécanique Céleste. He thus approached Mary Somerville, who began to prepare a translation which would "explain to the unlearned the sort of thing it is—the plan, the vast merit, the wonderful truths unfolded or methodized—and the calculus by which all this is accomplished". In 1830, John Herschel wrote to Somerville and enclosed a copy of Bowditch's 1828 translation of Volume 1 which Herschel had just received. Undeterred, Somerville decided to continue with the preparation of her own work as she felt the two translations differed in their aims; whereas Bowditch's contained an overwhelming number of footnotes to explain each mathematical step, Somerville instead wished to state and demonstrate the results as clearly as possible. +A year later, in 1831, Somerville's translation was published under the title Mechanism of the Heavens. It received great critical acclaim, with complimentary reviews appearing in the Quarterly Review, the Edinburgh Review, and the Monthly Notices of the Royal Astronomical Society. + +== References == + +== External links == +Translation by Nathaniel Bowditch + +Volume I, 1829 +Volume II, 1832 +Volume III, 1834 +Volume IV, 1839 with a memoir of the translator by his son \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Treatise_on_Analysis-0.md b/data/en.wikipedia.org/wiki/Treatise_on_Analysis-0.md new file mode 100644 index 000000000..148b82607 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Treatise_on_Analysis-0.md @@ -0,0 +1,91 @@ +--- +title: "Treatise on Analysis" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Treatise_on_Analysis" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:05.664536+00:00" +instance: "kb-cron" +--- + +Treatise on Analysis is a translation by Ian G. Macdonald of the nine-volume work Éléments d'analyse on mathematical analysis by Jean Dieudonné, and is an expansion of his textbook Foundations of Modern Analysis. It is a successor to the various Cours d'Analyse by Augustin-Louis Cauchy, Camille Jordan, and Édouard Goursat. + + +== Contents and publication history == + + +=== Volume I === +The first volume was originally a stand-alone graduate textbook with a different title. It was first written in English and later translated into French, unlike the other volumes which were first written in French. It has been republished several times and is much more common than the later volumes of the series. +The contents include + +Chapter I: Sets +Chapter II Real numbers +Chapter III Metric spaces +Chapter IV The real line +Chapter V Normed spaces +Chapter VI Hilbert spaces +Chapter VII Spaces of continuous functions +Chapter VIII Differential calculus (This uses the Cauchy integral rather than the more common Riemann integral of functions.) +Chapter IX Analytic functions (of a complex variable) +Chapter X Existence theorems (for ordinary differential equations) +Chapter XI Elementary spectral theory +Dieudonné, J. (1960), Foundations of modern analysis, Pure and Applied Mathematics, vol. X, New York-London: Academic Press, MR 0120319 +Dieudonné, J. (1963), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, vol. XXVIII, Paris: Gauthier-Villars, MR 0161945 +Dieudonné, J. (1968), Éléments d'analyse. Tome I: Fondements de l'analyse moderne, Cahiers Scientifiques, vol. XXVIII (2nd ed.), Paris: Gauthier-Villars, MR 0235945 +Dieudonné, J. (1969), Foundations of modern analysis., Pure and Applied Mathematics, vol. 10-I (2nd ed.), New York-London: Academic Press, ISBN 978-0122155505, MR 0349288 + + +=== Volume II === +The second volume includes + +Chapter XII Topology and topological algebra +Chapter XIII Integration +Chapter XIV Integration in locally compact groups +Chapter XV Normed algebras and spectral theory +Dieudonné, J. (1968), Éléments d'analyse. Tome II: Chapitres XII à XV, Cahiers Scientifiques, vol. XXXI, Paris: Gauthier-Villars, MR 0235946 +Dieudonné, J. (1970), Treatise on analysis. Vol. II, Pure and Applied Mathematics, vol. 10-II, New York-London: Academic Press, MR 0258551 +Dieudonné, J. (1976), Treatise on analysis. Vol. II, Pure and Applied Mathematics, vol. 10-II (2nd ed.), New York-London: Academic Press, ISBN 0-12-215502-5, MR 0530406 + + +=== Volume III === +The third volume includes chapter XVI on differential manifolds and chapter XVII on distributions and differential operators. + + +=== Volume IV === +The fourth volume includes + +Chapter XVIII Differential systems +Chapter XIX Lie groups +Chapter XX Riemannian geometry + + +=== Volume V === +Volume V consists of chapter XXI on compact Lie groups. + + +=== Volume VI === +Volume VI consists of chapter XXII on harmonic analysis (mostly on locally compact groups) + + +=== Volume VII === +Volume VII consists of the first part of chapter XXIII on linear functional equations. This chapter is considerably more advanced than most of the other chapters. + + +=== Volume VIII === +Volume VIII consists of the second part of chapter XXIII on linear functional equations. + + +=== Volume IX === +Volume IX contains chapter XXIV on elementary differential topology. Unlike the earlier volumes there is no English translation of it. + +Dieudonné, J. (1982), Éléments d'analyse. Tome IX. Chapitre XXIV, Cahiers Scientifiques, vol. XL11, Paris: Gauthier-Villars, ISBN 2-04-011499-8, MR 0658305 + + +=== Volume X === +Dieudonne planned a final volume containing chapter XXV on nonlinear problems, but this was never published. + + +== References == +Nachbin, Leopoldo (1961), "Review: J. Dieudonné, Foundations of Modern Analysis", Bull. Amer. Math. Soc., 67 (3): 246–250, doi:10.1090/s0002-9904-1961-10566-1 +Frank, Peter (1960), "Book reviews: Foundations of Modern Analysis. J. Dieudonné. Academic Press, New York, 1960", Science, 132 (3441): 1759, doi:10.1126/science.132.3441.1759-a +Marsden, Jerrold E. (1980), "Review: Jean Dieudonné, Treatise on analysis", Bull. Amer. Math. Soc. (N.S.), 3 (1): 719–724, doi:10.1090/s0273-0979-1980-14804-1 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Treks_into_Intuitive_Geometry-0.md b/data/en.wikipedia.org/wiki/Treks_into_Intuitive_Geometry-0.md new file mode 100644 index 000000000..470dfbe87 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Treks_into_Intuitive_Geometry-0.md @@ -0,0 +1,27 @@ +--- +title: "Treks into Intuitive Geometry" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Treks_into_Intuitive_Geometry" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:07.944460+00:00" +instance: "kb-cron" +--- + +Treks into Intuitive Geometry: The World of Polygons and Polyhedra is a book on geometry, written as a discussion between a teacher and a student in the style of a Socratic dialogue. It was written by Japanese mathematician Jin Akiyama and science writer Kiyoko Matsunaga, and published by Springer-Verlag in 2015 (ISBN 978-4-431-55841-5), with an expanded second edition in 2024 (ISBN 978-981-99-8607-1). + + +== Topics == +The term "intuitive geometry" of the title was used by László Fejes Tóth to refer to results in geometry that are accessible to the general public, and the book concerns topics of this type. +The book has 16 self-contained chapters, each beginning with an illustrative puzzle or real-world application. +It includes material on tessellations, polyhedra, and honeycombs, unfoldings of polyhedra and tessellations of unfoldings, cross sections of polyhedra, measuring boxes, gift wrapping, packing problems, wallpaper groups, pentagonal tilings, the Conway criterion for prototiles and Escher-like tilings of the plane by animal-shaped figures, aperiodic tilings including the Penrose tiling, the art gallery theorem, the Euler characteristic, dissection problems and the Dehn invariant, and the Steiner tree problem. +The book is heavily illustrated. And although the results of the book are demonstrated in an accessible way, the book provides sequences of deductions leading to each major claim, and more-complete proofs and references are provided in an appendix. + + +== Audience and reception == +Although it was initially developed from course material offered to undergraduates at the Tokyo University of Science, the book is aimed at a broad audience, and assumes only a high-school level knowledge of geometry. It could be used to encourage children in mathematics as well as to provide material for teachers and public lecturers. There is enough depth of material to also retain the interest of readers with a more advanced mathematical background. +Reviewer Matthieu Jacquemet writes that the ordering of topics is unintuitive and the dialogue-based format "artificial", but reviewer Tricia Muldoon Brown instead suggests that this format allows the work to flow very smoothly, "more like a novel or a play than a textbook ... with the ease of reading purely for pleasure". Jacquemet assesses the book as "well illustrated and entertaining", and Brown writes that it "is a delightful read". +Reviewer Michael Fox disagrees, finding the dialogue irritating and the book overall "rather disappointing". He cites as problematic the book's cursory treatment of some of its topics, and in particular its treatment of tiling patterns as purely monochromatic, its omission of the frieze groups, and its use of demonstrations by special examples that do not have all the features of the general case. He also complains about idiosyncratic terminology, the use of decimal approximations instead of exact formulas for angles, the small scale of some figures, and an uneven level of difficulty of material. Nevertheless, he writes that "this is an interesting work, with much that cannot be found elsewhere". + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Treviso_Arithmetic-0.md b/data/en.wikipedia.org/wiki/Treviso_Arithmetic-0.md new file mode 100644 index 000000000..d551ac5f4 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Treviso_Arithmetic-0.md @@ -0,0 +1,49 @@ +--- +title: "Treviso Arithmetic" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Treviso_Arithmetic" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:09.079015+00:00" +instance: "kb-cron" +--- + +The Treviso Arithmetic, or Arte dell'Abbaco, is an anonymous textbook in commercial arithmetic written in vernacular Venetian and published in Treviso, Italy, in 1478. +The author explains the motivation for writing this textbook: + +I have often been asked by certain youths in whom I have much interest, and who look forward to mercantile pursuits, to put into writing the fundamental principles of arithmetic, commonly called abacus. +The Treviso Arithmetic is the earliest known printed mathematics book in the West, and one of the first printed European textbooks dealing with a science. + + +== The Arithmetic as an early printed book == +There appears to have been only one edition of the work. David Eugene Smith translated parts of the Treviso Arithmetic for educational purposes in 1907. Frank J. Swetz translated the complete work using Smith's notes in 1987 in his Capitalism & Arithmetic: The New Math of the 15th Century. Swetz used a copy of the Treviso housed in the Manuscript Library at Columbia University. The volume found its way to this collection via a curious route. Maffeo Pinelli (1785), an Italian bibliophile, is the first known owner. After his death his library was purchased by a London book-dealer and sold at auction on February 6, 1790. The book was obtained for three shillings by Mr. Wodhull. About 100 years later the Arithmetic appeared in the library of Brayton Ives, a New York lawyer. When Ives sold the collection of books at auction, George Arthur Plimpton, a New York publisher, acquired the Treviso and made it an acquisition to his extensive collection of early scientific texts. Plimpton donated his library to Columbia University in 1936. Original copies of the Treviso Arithmetic are extremely rare. +There are 123 pages of text with 32 lines of print to a page. The pages are unnumbered, untrimmed and have wide margins. Some of the margins contain written notes. The size of the book is 14.5 cm by 20.6 cm. +The book included information taken from the 1202 Liber Abaci, such as lattice multiplication. George G. Joseph in Crest of the Peacock suggests that John Napier read this book to create Napier's bones (or rods). + + +== Reasons for publication == +The Treviso Arithmetic is a practical book intended for self study and for use in Venetian trade. It is written in vernacular Venetian and communicated knowledge to a large population. +It helped to end the monopoly on mathematical knowledge and gave important information to the middle class. It was not written for a large audience, but was intended to teach mathematics of everyday currency. +The Treviso became one of the first mathematics books written for the expansion of human knowledge. It provided an opportunity for the common person, rather than only a privileged few, to learn the art of computation. The Treviso Arithmetic provided an early example of the Hindu–Arabic numeral system computational algorithms. + + +== See also == +Ars Magna (Gerolamo Cardano) (1510) + + +== Notes == + + +== References == +Boyer, Carl. 1991. A History of Mathematics. New York City: Wiley. +Buck-Morss, Susan (1 January 1995). "Envisioning Capital: Political Economy on Display". Critical Inquiry. 21 (2): 434–467. doi:10.1086/448759. JSTOR 1343930. +Carter, Baker. 2006. The Role of the History of Mathematics in Middle School. Presentation at East Tennessee University, August 28. +Gazale, Midhat, J. 2000. Number. Princeton: Princeton University Press. +Newman, J, R. 1956. The World of Mathematics. New York City: Simon & Schuster. +Peterson, Ivars. 1996. Old and New Arithmetic. Mathematical Association of America. http://www.maa.org/mathland/mathland_8_5.html (accessed October 11, 2006). +Swetz, Frank, J. 1987. Capitalism and Arithmetic. La Salle: Open Court. + + +== External links == +Full text of the Treviso Arithmetic +Treviso Arithmetic at Columbia University \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Trigonometric_Series-0.md b/data/en.wikipedia.org/wiki/Trigonometric_Series-0.md new file mode 100644 index 000000000..97020be79 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Trigonometric_Series-0.md @@ -0,0 +1,28 @@ +--- +title: "Trigonometric Series" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Trigonometric_Series" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:10.228449+00:00" +instance: "kb-cron" +--- + +Antoni Zygmund wrote a classic two-volume set of books entitled Trigonometric Series, which discusses many different aspects of trigonometric series. The first edition was a single volume, published in 1935 (under the slightly different title Trigonometrical Series). The second edition of 1959 was greatly expanded, taking up two volumes, though it was later reprinted as a single volume paperback. The third edition of 2002 is similar to the second edition, with the addition of a preface by Robert A. Fefferman on more recent developments, in particular Carleson's theorem about almost everywhere pointwise convergence for square-integrable functions. + + +== Publication history == +Zygmund, Antoni (1935). Trigonometrical series. Monogr. Mat. Vol. 5. Warszawa, Lwow: Subwencji Fundusz Kultury Narodowej. Zbl 0011.01703. At icm.edu.pl: original archived +Zygmund, Antoni (1952). Trigonometrical series. New York: Chelsea Publishing Co. MR 0076084. +Zygmund, Antoni (1955). Trigonometrical series. New York: Dover Publications. MR 0072976. +Zygmund, Antoni (1959). Trigonometric series (2nd ed.). Cambridge University Press. MR 0107776. Volume I, Volume II. +Zygmund, Antoni (1968). Trigonometric series. Second edition, reprinted with corrections and some additions. Vol. I and II (2nd ed.). Cambridge University Press. MR 0236587. +Zygmund, Antoni (1977). Trigonometric series. Vol. I and II. Cambridge University Press. ISBN 978-0-521-07477-3. MR 0617944. +Zygmund, Antoni (1988). Trigonometric series. Cambridge Mathematical Library. Vol. I and II. Cambridge University Press. ISBN 978-0-521-35885-9. MR 0933759. +Zygmund, Antoni (2002). Fefferman, Robert A. (ed.). Trigonometric series. Cambridge Mathematical Library. Vol. I and II (3rd ed.). Cambridge University Press. ISBN 978-0-521-89053-3. MR 1963498. + + +== Reviews == +Kahane, Jean-Pierre (2004), "Book review: Trigonometric series, Vols. I, II", Bulletin of the American Mathematical Society, 41 (3): 377–390, doi:10.1090/s0273-0979-04-01013-4, ISSN 0002-9904 +Salem, Raphael (1960), "Book Review: Trigonometric series", Bulletin of the American Mathematical Society, 66 (1): 6–12, doi:10.1090/S0002-9904-1960-10362-X, ISSN 0002-9904, MR 1566029 +Tamarkin, J. D. (1936), "Zygmund on Trigonometric Series", Bull. Amer. Math. Soc., 42 (1): 11–13, doi:10.1090/s0002-9904-1936-06235-x \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Two-Sided_Matching-0.md b/data/en.wikipedia.org/wiki/Two-Sided_Matching-0.md new file mode 100644 index 000000000..29e27c01a --- /dev/null +++ b/data/en.wikipedia.org/wiki/Two-Sided_Matching-0.md @@ -0,0 +1,26 @@ +--- +title: "Two-Sided Matching" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Two-Sided_Matching" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:12.555607+00:00" +instance: "kb-cron" +--- + +Two-Sided Matching: A Study in Game-Theoretic Modeling and Analysis is a book on matching markets in economics and game theory, particularly concentrating on the stable marriage problem. It was written by Alvin E. Roth and Marilda Sotomayor, with a preface by Robert Aumann, and published in 1990 by the Cambridge University Press as volume 18 in their series of Econometric Society monographs. For this work, Roth and Sotomayor won the 1990 Frederick W. Lanchester Prize of the Institute for Operations Research and the Management Sciences. + + +== Topics == +The book's introduction discusses the National Resident Matching Program and its use of stable marriage to assign medical students to hospital positions, and collects the problems in economics that the theory of matching markets is positioned to solve. Following this, it has three main sections. +The first of these sections discusses the stable matching problem in its simplest form, in which two equal-sized groups of agents are to be matched one-to-one. It discusses the stability of solutions (the property that no pair of agents both prefer being matched to each other to their assigned matches), the lattice of stable matchings, the Gale–Shapley algorithm for finding stable solutions, and two key properties of this algorithm: that among all stable solutions it chooses the one that gives one group of agents their most-preferred stable match, and that it is an honest mechanism that incentivizes this group of agents to report their preferences truthfully. +The second part of the book, which reviewer Ulrich Kamecke describes as its most central, concerns extensions of these results to the many-one matching needed for the National Resident Matching Program, and to the specific economic factors that made that program successful compared to comparable programs elsewhere, and that have impeded its success. One example concerns the two-body problem of married couples who would both prefer to be assigned to the same place, a constraint that adds considerable complexity to the matching problem and may prevent a stable solution from existing. +The third part of the book concerns a different direction in which these ideas have been extended, to matching markets such as those for real estate in which indivisible goods are traded, with money used to transfer utility. It includes results in auction theory, linear and nonlinear utility functions, and the assignment game of Lloyd Shapley and Martin Shubik. + + +== Audience and reception == +Two-Sided Matching presents known material on its topics, rather than introducing new research, but it is not a textbook. Instead, its aim is to provide a survey of this area aimed at economic practitioners, with arguments for the importance of its material based on its pragmatic significance rather than its mathematical beauty. Nevertheless, it also has material of interest to researchers, including an extensive bibliography and a concluding list of open problems for future research. Compared to other books on stable matching, including Marriages Stables by Donald Knuth and The Stable Marriage Problem: Structure and Algorithms by Dan Gusfield and Robert W. Irving, Two-Sided Matching focuses much more on the economic, application-specific, and strategic issues of stable matching, and much less on its algorithmic issues. +Alan Kirman calls the book a "clear and elegant account" of its material, writing that its focus on practical applications makes it "of particular interest". Theodore Bergstrom writes that it will also "delight economists who want to think beautiful thoughts about important practical problems". Benny Moldovanu predicts that it "will become the standard source of reference" for its material. And Uriel Rothblum calls it the kind of once-a-generation book that can "change the way in which an entire field of study is viewed." + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Two_New_Sciences-0.md b/data/en.wikipedia.org/wiki/Two_New_Sciences-0.md new file mode 100644 index 000000000..b434311d0 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Two_New_Sciences-0.md @@ -0,0 +1,24 @@ +--- +title: "Two New Sciences" +chunk: 1/6 +source: "https://en.wikipedia.org/wiki/Two_New_Sciences" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:11.405455+00:00" +instance: "kb-cron" +--- + +The Discourses and Mathematical Demonstrations Relating to Two New Sciences (Italian: Discorsi e dimostrazioni matematiche intorno a due nuove scienze pronounced [diˈskorsi e ddimostratˈtsjoːni mateˈmaːtike inˈtorno a dˈduːe ˈnwɔːve ʃˈʃɛntse]) published in 1638 was Galileo Galilei's final book and a scientific testament covering much of his work in physics over the preceding thirty years. It was written partly in Italian and partly in Latin. +After his Dialogue Concerning the Two Chief World Systems, the Roman Inquisition had banned the publication of any of Galileo's works, including any he might write in the future. After the failure of his initial attempts to publish Two New Sciences in France, Germany, and Poland, it was published by Lodewijk Elzevir who was working in Leiden, South Holland, where the writ of the Inquisition was of less consequence (see House of Elzevir). Fra Fulgenzio Micanzio, the official theologian of the Republic of Venice, had initially offered to help Galileo publish the new work there, but he pointed out that publishing the Two New Sciences in Venice might cause Galileo unnecessary trouble; thus, the book was eventually published in Holland. Galileo did not seem to suffer any harm from the Inquisition for publishing this book since in January 1639, the book reached Rome's bookstores, and all available copies (about fifty) were quickly sold. +Discourses was written in a style similar to Dialogues, in which three men (Simplicio, Sagredo, and Salviati) discuss and debate the various questions Galileo is seeking to answer. There is a notable change in the men, however; Simplicio, in particular, is no longer quite as simple-minded, stubborn and Aristotelian as his name implies. His arguments are representative of Galileo's own early beliefs, as Sagredo represents his middle period, and Salviati proposes Galileo's newest models. + +== Introduction == +The book is divided into four days, each addressing different areas of physics. Galileo dedicates Two New Sciences to Lord Count of Noailles. + +In the First Day, Galileo addressed topics that were discussed in Aristotle's Physics and the Aristotelian school's Mechanics. It also provides an introduction to the discussion of both of the new sciences. The likeness between the topics discussed, specific questions that are hypothesized, and the style and sources throughout give Galileo the backbone to his First Day. The First Day introduces the speakers in the dialogue: Salviati, Sagredo, and Simplicio, the same as in the Dialogue. These three people are all Galileo just at different stages of his life, Simplicio the youngest and Salviati, Galileo's closest counterpart. The Second Day addresses the question of the strength of materials. +The Third and Fourth days address the science of motion. The Third day discusses uniform and naturally accelerated motion, the issue of terminal velocity having been addressed in the First day. The Fourth day discusses projectile motion. +In Two Sciences uniform motion is defined as a motion that, over any equal periods of time, covers equal distance. With the use of the quantifier ″any″, uniformity is introduced and expressed more explicitly than in previous definitions. +Galileo had started an additional day on the force of percussion, but was not able to complete it to his own satisfaction. This section was referenced frequently in the first four days of discussion. It finally appeared only in the 1718 edition of Galilei's works. and it is often quoted as "Sixth Day" following the numbering in the 1898 edition. During this additional day Simplicio was replaced by Aproino, a former scholar and assistant of Galileo in Padua. + +== Summary == +Page numbers at the start of each paragraph are from the 1898 version, presently adopted as standard, and are found in the Crew and Drake translations. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Two_New_Sciences-1.md b/data/en.wikipedia.org/wiki/Two_New_Sciences-1.md new file mode 100644 index 000000000..776b90da2 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Two_New_Sciences-1.md @@ -0,0 +1,24 @@ +--- +title: "Two New Sciences" +chunk: 2/6 +source: "https://en.wikipedia.org/wiki/Two_New_Sciences" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:11.405455+00:00" +instance: "kb-cron" +--- + +=== Day one: Resistance of bodies to separation === +[50] Preliminary discussions. +Sagredo (taken to be the younger Galileo) cannot understand why with machines one cannot argue from the small to the large: "I do not see that the properties of circles, triangles and...solid figures should change with their size". Salviati (speaking for Galileo) says the common opinion is wrong. Scale matters: a horse falling from a height of 3 or 4 cubits will break its bones whereas a cat falling from twice the height won't, nor will a grasshopper falling from a tower. +[56] The first example is a hemp rope which is constructed from small fibres which bind together in the same way as a rope round a windlass to produce something much stronger. Then the vacuum that prevents two highly polished plates from separating even though they slide easily gives rise to an experiment to test whether water can be expanded or whether a vacuum is caused. In fact, Sagredo had observed that a suction pump could not lift more than 18 cubits of water and Salviati observes that the weight of this is the amount of resistance to a void. The discussion turns to the strength of a copper wire and whether there are minute void spaces inside the metal or whether there is some other explanation for its strength. +[68] This leads into a discussion of infinites and the continuum and thence to the observation that the number of squares equal the number of roots. He comes eventually to the view that "if any number can be said to be infinite, it must be unity" and demonstrates a construction in which an infinite circle is approached and another to divide a line. +[85] The difference between a fine dust and a liquid leads to a discussion of light and how the concentrated power of the sun can melt metals. He deduces that light has motion and describes an (unsuccessful) attempt to measure its speed. +[106] Aristotle believed that bodies fell at a speed proportional to weight but Salviati doubts that Aristotle ever tested this. He also did not believe that motion in a void was possible, but since air is much less dense than water Salviati asserts that in a medium devoid of resistance (a vacuum) all bodies—a lock of wool or a bit of lead—would fall at the same speed. Large and small bodies fall at the same speed through air or water providing they are of the same density. Since ebony weighs a thousand times as much as air (which he had measured), it will fall only a very little more slowly than lead which weighs ten times as much. But shape also matters—even a piece of gold leaf (the densest of all substances [asserts Salviati]) floats through the air and a bladder filled with air falls much more slowly than lead. +[128] Measuring the speed of a fall is difficult because of the small time intervals involved and his first way round this used pendulums of the same length but with lead or cork weights. The period of oscillation was the same, even when the cork was swung more widely to compensate for the fact that it soon stopped. +[139] This leads to a discussion of the vibration of strings and he suggests that not only the length of the string is important for pitch but also the tension and the weight of the string. + +=== Day two: Cause of cohesion === +[151] Salviati proves that a balance can be used not only with equal arms but with unequal arms with weights inversely proportional to the distances from the fulcrum. Following this he shows that the moment of a weight suspended by a beam supported at one end is proportional to the square of the length. The resistance to fracture of beams of various sizes and thicknesses is demonstrated, supported at one or both ends. +[169] He shows that animal bones have to be proportionately larger for larger animals and the length of a cylinder that will break under its own weight. He proves that the best place to break a stick placed upon the knee is the middle and shows how far along a beam that a larger weight can be placed without breaking it. +[178] He proves that the optimum shape for a beam supported at one end and bearing a load at the other is parabolic. He also shows that hollow cylinders are stronger than solid ones of the same weight. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Two_New_Sciences-2.md b/data/en.wikipedia.org/wiki/Two_New_Sciences-2.md new file mode 100644 index 000000000..69d786ccc --- /dev/null +++ b/data/en.wikipedia.org/wiki/Two_New_Sciences-2.md @@ -0,0 +1,39 @@ +--- +title: "Two New Sciences" +chunk: 3/6 +source: "https://en.wikipedia.org/wiki/Two_New_Sciences" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:11.405455+00:00" +instance: "kb-cron" +--- + +=== Day three: Naturally accelerated motion === +[191] He first defines uniform (steady) motion and shows the relationship between speed, time and distance. He then defines uniformly accelerated motion where the speed increases by the same amount in increments of time. Falling bodies start very slowly and he sets out to show that their velocity increases in simple proportionality to time, not to distance which he shows is impossible. +[208] He shows that the distance travelled in naturally accelerated motion is proportional to the square of the time. He describes an experiment in which a steel ball was rolled down a groove in a piece of wooden moulding 12 cubits long (about 5.5m) with one end raised by one or two cubits. This was repeated, measuring times by accurately weighing the amount of water that came out of a thin pipe in a jet from the bottom of a large jug of water. By this means he was able to verify the uniformly accelerated motion. He then shows that whatever the inclination of the plane, the square of the time taken to fall a given vertical height is proportional to the inclined distance. +[221] He next considers descent along the chords of a circle, showing that the time is the same as that falling from the vertex, and various other combinations of planes. He gives an erroneous solution to the brachistochrone problem, claiming to prove that the arc of the circle is the fastest descent. 16 problems with solutions are given. + +=== Day four: The motion of projectiles === + +[268] The motion of projectiles consists of a combination of uniform horizontal motion and a naturally accelerated vertical motion which produces a parabolic curve. Two motions at right angles can be calculated using the sum of the squares. He shows in detail how to construct the parabolas in various situations and gives tables for altitude and range depending on the projected angle. +[274] Air resistance shows itself in two ways: by affecting less dense bodies more and by offering greater resistance to faster bodies. A lead ball will fall slightly faster than an oak ball, but the difference with a stone ball is negligible. However the speed does not go on increasing indefinitely but reaches a maximum. Though at small speeds the effect of air resistance is small, it is greater when considering, say, a ball fired from a cannon. +[292] The effect of a projectile hitting a target is reduced if the target is free to move. The velocity of a moving body can overcome that of a larger body if its speed is proportionately greater than the resistance. +[310] A cord or chain stretched out is never level but also approximates to a parabola. (But see also catenary.) + +=== Additional day: The force of percussion === +[323] What is the weight of water falling from a bucket hanging on a balance arm onto another bucket suspended to the same arm? +[325] Piling of wooden poles for foundations; hammers and the force of percussion. +[336] Speed of fall along inclined planes; again on the principle of inertia. + +== Methodology == +Many contemporary scientists, such as Gassendi, dispute Galileo's methodology for conceptualizing his law of falling bodies. Two of the main arguments are that his epistemology followed the example of Platonist thought or hypothetico-deductivist. It has now been considered to be ex suppositione, or knowing the how and why effects from past events in order to determine the requirements for the production of similar effects in the future. Galilean methodology mirrored that of Aristotelian and Archimedean epistemology. Following a letter from Cardinal Bellarmine in 1615 Galileo distinguished his arguments and Copernicus' as natural suppositions as opposed to the "fictive" that are "introduced only for the sake of astronomical computations," such as Ptolemy's hypothesis on eccentrics and equants. +Galileo's earlier writing considered Juvenilia, or youthful writings, are considered his first attempts at creating lecture notes for his course "hypothesis of the celestial motions" while teaching in at the University of Padua. These notes mirrored those of his contemporaries at the Collegio as well as contained an "Aristotelian context with decided Thomistic (St. Thomas Aquinas) overtones." These earlier papers are believed to have encouraged him to apply demonstrative proof in order to give validity to his discoveries on motion. +Discovery of folio 116v gives evidence of experiments that had previously not been reported and therefore demonstrated Galileo's actual calculations for the Law of Falling Bodies. +His methods of experimentation have been proved by the recording and recreation done by scientists such as James MacLachlan, Stillman Drake, R.H. Taylor and others in order to prove he did not merely imagine his ideas as historian Alexandre Koyré argued, but sought to prove them mathematically. +Galileo believed that knowledge could be acquired through reason, and reinforced through observation and experimentation. Thus, it can be argued that Galileo was a rationalist, and also that he was an empiricist. + +== The two new sciences == +The two sciences mentioned in the title are the strength of materials and the motion of objects (the forebears of modern material engineering and kinematics). In the title of the book "mechanics" and "motion" are separate, since at Galileo's time "mechanics" meant only statics and strength of materials. + +=== The science of materials === +The discussion begins with a demonstration of the reasons that a large structure proportioned in exactly the same way as a smaller one must necessarily be weaker known as the square–cube law. Later in the discussion this principle is applied to the thickness required of the bones of a large animal, possibly the first quantitative result in biology, anticipating J. B. S. Haldane's work On Being the Right Size, and other essays, edited by John Maynard Smith. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Two_New_Sciences-3.md b/data/en.wikipedia.org/wiki/Two_New_Sciences-3.md new file mode 100644 index 000000000..25130ac49 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Two_New_Sciences-3.md @@ -0,0 +1,35 @@ +--- +title: "Two New Sciences" +chunk: 4/6 +source: "https://en.wikipedia.org/wiki/Two_New_Sciences" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:11.405455+00:00" +instance: "kb-cron" +--- + +=== The motion of objects === +Galileo expresses clearly for the first time the constant acceleration of a falling body which he was able to measure accurately by slowing it down using an inclined plane. +In Two New Sciences, Galileo (Salviati speaks for him) used a wood molding, "12 cubits long, half a cubit wide and three finger-breadths thick" as a ramp with a straight, smooth, polished groove to study rolling balls ("a hard, smooth and very round bronze ball"). He lined the groove with "parchment, also smooth and polished as possible". He inclined the ramp at various angles, effectively slowing down the acceleration enough so that he could measure the elapsed time. He would let the ball roll a known distance down the ramp, and use a water clock to measure the time taken to move the known distance. This clock was + +a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length. The water collected was weighed, and after each descent on a very accurate balance, the differences and ratios of these weights gave him the differences and ratios of the times. This was done with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results. + +==== The law of falling bodies ==== +While Aristotle had observed that heavier objects fall more quickly than lighter ones, in Two New Sciences Galileo postulated that this was due not to inherently stronger forces acting on the heavier objects, but to the countervailing forces of air resistance and friction. To compensate, he conducted experiments using a shallowly inclined ramp, smoothed so as to eliminate as much friction as possible, on which he rolled down balls of different weights. In this manner, he was able to provide empirical evidence that matter accelerates vertically downward at a constant rate, regardless of mass, due to the effects of gravity. +The unreported experiment found in folio 116V tested the constant rate of acceleration in falling bodies due to gravity. This experiment consisted of dropping a ball from specified heights onto a deflector in order to transfer its motion from vertical to horizontal. The data from the inclined plane experiments were used to calculate the expected horizontal motion. However, discrepancies were found in the results of the experiment: the observed horizontal distances disagreed with the calculated distances expected for a constant rate of acceleration. Galileo attributed the discrepancies to air resistance in the unreported experiment, and friction in the inclined plane experiment. These discrepancies forced Galileo to assert that the postulate held only under "ideal conditions", i.e., in the absence of friction and/or air resistance. + +==== Bodies in motion ==== +Aristotelian physics argued that the Earth must not move as humans are unable to perceive the effects of this motion. A popular justification of this is the experiment of an archer shooting an arrow straight up into the air. If the Earth were moving, Aristotle argued, the arrow should fall in a different location than the launch point. Galileo refuted this argument in Dialogues Concerning the Two Chief World Systems. He provided the example of sailors aboard a boat at sea. The boat is obviously in motion, but the sailors are unable to perceive this motion. If a sailor were to drop a weighted object from the mast, this object would fall at the base of the mast rather than behind it (due to the ship's forward motion). This was the result of simultaneously the horizontal and vertical motion of the ship, sailors, and ball. + +==== Relativity of motions ==== + +One of Galileo's experiments regarding falling bodies was that describing the relativity of motions, explaining that, under the right circumstances, "one motion may be superimposed upon another without effect upon either...". In Two New Sciences, Galileo made his case for this argument and it would become the basis of Newton's first law, the law of inertia. +He poses the question of what happens to a ball dropped from the mast of a sailing ship or an arrow fired into the air on the deck. According to Aristotle's physics, the ball dropped should land at the stern of the ship as it falls straight down from the point of origin. Likewise the arrow when fired straight up should not land in the same spot if the ship is in motion. Galileo offers that there are two independent motions at play. One is the accelerating vertical motion caused by gravity while the other is the uniform horizontal motion caused by the moving ship which continues to influence the trajectory of the ball through the principle of inertia. The combination of these two motions results in a parabolic curve. The observer cannot identify this parabolic curve because the ball and observer share the horizontal movement imparted to them by the ship, meaning only the perpendicular, vertical motion is perceivable. Surprisingly, nobody had tested this theory with the simple experiments needed to gain a conclusive result until Pierre Gassendi published the results of said experiments in his letters entitled De Motu Impresso a Motore Translato (1642). + +== Infinity == + +The book also contains a discussion of infinity. Galileo considers the example of numbers and their squares. He starts by noting that: + +It cannot be denied that there are as many [squares] as there are numbers because every number is a [square] root of some square: +1 ↔ 1, 2 ↔ 4, 3 ↔ 9, 4 ↔ 16, and so on. +But he notes what appears to be a contradiction: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Two_New_Sciences-4.md b/data/en.wikipedia.org/wiki/Two_New_Sciences-4.md new file mode 100644 index 000000000..6dd17e3a4 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Two_New_Sciences-4.md @@ -0,0 +1,34 @@ +--- +title: "Two New Sciences" +chunk: 5/6 +source: "https://en.wikipedia.org/wiki/Two_New_Sciences" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:11.405455+00:00" +instance: "kb-cron" +--- + +Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers. +(In modern language, there is a bijection between the set of positive integers N and the set of squares S, and S is a proper subset of N of density zero.) He resolves the contradiction by denying the possibility of comparing infinite numbers (and of comparing infinite and finite numbers): + +We can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal", greater", and "less", are not applicable to infinite, but only to finite, quantities. +This conclusion, that ascribing sizes to infinite sets should be ruled impossible, owing to the contradictory results obtained from these two ostensibly natural ways of attempting to do so, is a resolution to the problem that is consistent with, but less powerful than, the methods used in modern mathematics. The resolution to the problem may be generalized by considering Galileo's first definition of what it means for sets to have equal sizes, that is, the ability to put them in one-to-one correspondence. This turns out to yield a way of comparing the sizes of infinite sets that is free from contradictory results. +These issues of infinity arise from problems of rolling circles. If two concentric circles of different radii roll along lines, then if the larger does not slip it appears clear that the smaller must slip. But in what way? Galileo attempts to clarify the matter by considering hexagons and then extending to rolling 100 000-gons, or n-gons, where he shows that a finite number of finite slips occur on the inner shape. Eventually, he concludes "the line traversed by the larger circle consists then of an infinite number of points which completely fill it; while that which is traced by the smaller circle consists of an infinite number of points which leave empty spaces and only partly fill the line," which would not be considered satisfactory now. + +== Reactions by commentators == +So great a contribution to physics was Two New Sciences that scholars have long maintained that the book anticipated Isaac Newton's laws of motion. +Galileo ... is the father of modern physics—indeed of modern science +Part of Two New Sciences was pure mathematics, as has been pointed out by the mathematician Alfréd Rényi, who said that it was the most significant book on mathematics in over 2000 years: Greek mathematics did not deal with motion, and so they never formulated mathematical laws of motion, even though Archimedes developed differentiation and integration. Two New Sciences opened the way to treating physics mathematically by treating motion mathematically for the first time. The Greek mathematician Zeno had designed his paradoxes to prove that motion could not be treated mathematically, and that any attempt to do so would lead to paradoxes. (He regarded this as an inevitable limitation of mathematics.) Aristotle reinforced this belief, saying that mathematic could only deal with abstract objects that were immutable. Galileo used the very methods of the Greeks to show that motion could indeed be treated mathematically. His idea was to separate out the paradoxes of the infinite from Zeno's paradoxes. He did this in several steps. First, he showed that the infinite sequence S of the squares 1, 4, 9, 16, ...contained as many elements as the sequence N of all positive integers (infinity); this is now referred to as Galileo's paradox. Then, using Greek style geometry, he showed a short line interval contained as many points as a longer interval. At some point he formulates the general principle that a smaller infinite set can have just as many points as a larger infinite set containing it. It was then clear that Zeno's paradoxes on motion resulted entirely from this paradoxical behavior of infinite quantities. Renyi said that, having removed this 2000-year-old stumbling block, Galileo went on to introduce his mathematical laws of motion, anticipating Newton. + +=== Gassendi's thoughts === +Pierre Gassendi defended Galileo's opinions in his book, De Motu Impresso a Motore Translato. In Howard Jones' article, Gassendi's Defence of Galileo: The Politics of Discretion, Jones says Gassendi displayed an understanding of Galileo's arguments and a clear grasp of their implications for the physical objections to the earth's motion. + +=== Koyré's thoughts === +The law of falling bodies was published by Galileo in 1638. But in the 20th century some authorities challenged the reality of Galileo's experiments. In particular, the French historian of science Alexandre Koyré bases his doubt on the fact that the experiments reported in Two New Sciences to determine the law of acceleration of falling bodies, required accurate measurements of time which appeared to be impossible with the technology of 1600. According to Koyré, the law was created deductively, and the experiments were merely illustrative thought experiments. In fact, Galileo's water clock (described above) provided sufficiently accurate measurements of time to confirm his conjectures. +Later research, however, has validated the experiments. The experiments on falling bodies (actually rolling balls) were replicated using the methods described by Galileo, and the precision of the results was consistent with Galileo's report. Later research into Galileo's unpublished working papers from 1604 clearly showed the reality of the experiments and even indicated the particular results that led to the time-squared law. + +== See also == +De Motu Antiquiora (Galileo's earliest investigations of the motion of falling bodies) +Le Mecaniche (Galileo's early refinements of his investigation of mechanics) + +== Notes == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Two_New_Sciences-5.md b/data/en.wikipedia.org/wiki/Two_New_Sciences-5.md new file mode 100644 index 000000000..359789912 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Two_New_Sciences-5.md @@ -0,0 +1,26 @@ +--- +title: "Two New Sciences" +chunk: 6/6 +source: "https://en.wikipedia.org/wiki/Two_New_Sciences" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:11.405455+00:00" +instance: "kb-cron" +--- + +== References == +Drake, Stillman, translator (1974). Two New Sciences, University of Wisconsin Press, 1974. ISBN 0-299-06404-2. A new translation including sections on centers of gravity and the force of percussion. +Drake, Stillman (1978). Galileo At Work. Chicago: University of Chicago Press. ISBN 978-0-226-16226-3. +Henry Crew and Alfonso de Salvio, translators, [1914] (1954). Dialogues Concerning Two New Sciences, Dover Publications Inc., New York, NY. ISBN 978-0-486-60099-4. The classic source in English, originally published by McMillan (1914). +Jones, Howard, "Gassendi's Defense of Galileo: The Politics of Discretion", Medieval Renaissance Texts and Studies 58, 1988. +Titles of the first editions taken from Leonard C. Bruno 1989, The Landmarks of Science: from the Collections of the Library of Congress. ISBN 0-8160-2137-6 Q125.B87 +Galileo Galilei, Discorsi e dimostrazioni matematiche intorno a due nuove scienze attinenti la meccanica e i movimenti locali (pag.664, of Claudio Pierini) publication Cierre, Simeoni Arti Grafiche, Verona, 2011, ISBN 9788895351049. +Wallace, Willian, A. Galileo and Reasoning Ex Suppositione: The Methodology of the Two New Sciences. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association, Vol. 1974, (1974), pp. 79–104 +Salvia, Stafano (2014). "'Galileo's Machine': Late Notes on Free Fall, Projectile Motion, and the Force of Percussion (ca. 1638–1639)". Physics in Perspective. 16 (4): 440–460. Bibcode:2014PhP....16..440S. doi:10.1007/s00016-014-0149-1. S2CID 122967350. +De Angelis, Alessandro (2021). Discorsi e Dimostrazioni Matematiche di Galileo Galilei per il Lettore Moderno (in Italian). Torino: Codice. ISBN 978-8875789305. +De Angelis, Alessandro (2021). Galilei's Two New Sciences for Modern Readers. Heidelberg: Springer Nature. ISBN 978-3030719524. With prefaces by Ugo Amaldi and Telmo Pievani. + +== External links == + +(in Italian) Italian text with figures +English translation by Crew and de Salvio, with original figures \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Urania_Propitia-0.md b/data/en.wikipedia.org/wiki/Urania_Propitia-0.md new file mode 100644 index 000000000..4584928b3 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Urania_Propitia-0.md @@ -0,0 +1,73 @@ +--- +title: "Urania Propitia" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Urania_Propitia" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:13.709479+00:00" +instance: "kb-cron" +--- + +Urania Propitia (lit. 'kind/beneficent Urania') is a book of astronomical tables written by the Silesian astronomer Maria Cunitz and published in 1650. As Maria Cunitz was the daughter of both a physician and mathematician, it was her ability to grasp complex mathematics quickly and transcribe her findings as a polyglot that allowed her to do what few women had done before her. +The work served as a simplification of the Rudolphine Tables (1627) by Johannes Kepler, with Cunitz attempting to both increase the accuracy and to simplify the calculations from the original work. While she corrected a number of Kepler's computational errors, her equations also include inaccuracies. Cunitz's work is credited with making astrological tables accessible outside of the era's universities. Cunitz's cosmology combined elements from the cosmologies of both Kepler and Tycho Brahe. Like Brahe, she thought that the Sun and the Moon orbit around the planet Earth, while the rest of the planets in the Solar System orbit around the Sun. In her work, the physics within the cosmos involve ellipticals and aphelions. + + +== Introduction == +Urania Propitia was a simplification of the Rudolphine Tables written by Johannes Kepler in 1627. Kepler's dedication to Emperor Ferdinand II which was originally dedicated to Rudolf II was filled with complex and tedious logarithms. Cunitz found many errors within the Rudolphine Tables. The simplifications to these tables were published as the Urania Propitia. In Urania Propitia, Cunitz removed logarithms from Kepler's work, which increased the accuracy and simplified the calculations from the Rudolphine Tables. Cunitz omitted some coefficients from her equations, leading to inaccuracies within her work. Astronomical tables of the time all contained computational errors, so despite these errors, Urania Propitia is seen as more accurate than Kepler’s work. The tables are mostly astrological, but the instructions are completely astronomical. +Urania Propitia was published in Latin and German. The German publication is credited as a source that led to establishing German as a scientific language. The publishing of Urania Propitia is credited with making astrological tables more accessible outside of universities. +Beyond the instructions, Maria Cunitz's book is split into three parts. + + +=== Part 1: Tables for spherical astronomy === +The first part is the mathematical starting point. These include "sexagesimal sines, solutions of small right triangles in minutes and seconds, and tables for spherical astronomy for degrees of the ecliptic of: declination, right ascension, oblique ascension for latitudes 0 degrees to 72 degrees at 2 degree intervals..." + + +=== Part 2: Tables of average motions === +It is the second part that the heart of Maria Cunitz's simplification is brought forward. Using the geometry and spherical astronomy from part one, Cunitz brings the rotational motions of the planets and moons into light using various mathematical formulas. One of the formulas from the Rudolphine Tables, + + + + + M + = + E + − + e + sin + ⁡ + E + + + {\displaystyle M=E-e\sin E} + + +where e, M and E denote the orbital eccentricity, the mean anomaly and the eccentric anomaly. This equation is known as "Kepler's equation" which normally has no "geometrical or algebraic solution for E." However, when M is given it becomes more possible to but find "E from M to any degree of precision by iteration or interpolation." The major accomplishment Cunitz brought was the ability to compute + + + + υ + + + {\displaystyle \upsilon } + + (the true anomaly) from M without the necessity to use "E as a coefficient of interpolation". Thus, Cunitz was able to simplify the Rudolphine Tables and determine the position of a planet in its orbit as a function of time. + + +=== Part 3: Tables for computation of eclipses === +These tables were used for both the location of eclipses and the time of eclipses. This includes the use of the "golden astronomical number", the parallax of the moon and sun in a variety in longitude, latitude, altitude, etc..., and a catalog of other fixed stars in the universe. + + +== Cosmology == +Cunitz's cosmology has variations of both Tycho Brahe and Johannes Kepler. While the basic structure of her cosmology is like Tycho's with the Sun and Moon orbiting around the Earth while the rest of the planets orbit around the Sun, the physics within the cosmos involves ellipticals and "aphelions." + + +== Historical significance == +Beyond the rewards that came from a simplification of the Rudolphine Tables, a scientific advance written by a woman in the seventeenth century was an accomplishment in itself. It was described by Noel Swerdlow as "the earliest surviving scientific work by a woman on the highest technical level of its age." It was common for male scientists before Maria Cunitz to attribute their discoveries to muses. For Urania Propitia, Urania in Greek mythology was the muse of astronomy, and propitia is "favoring" or "beneficent" in Latin. This suggests that Maria Cunitz both saw Urania as her muse while also making strides for women as scientists, because she could be so easily compared to the ancient Greek astronomer. +Urania propitia was privately published and as of 2016 there are nine physical copies in the world along with multiple online copies. Physical copies can be found in the Library of the Astronomical Observatory of Paris, Library of the University of Florida, in the exhibit of Galileo and Kepler at the University Libraries of Norman, Oklahoma, and Bloomington Lilly Library of Indiana University. Prior to June 10, 2004 the first edition of Urania propitia was located at the Library of the Earls of Macclesfield in Shirburn Castle: Part 2 Science A-C section. The book was sold at the Sotheby's auction house for $19,827 USD. + + +== References == + + +== External links == +Digital scan of Urania Propitia - Wroclaw University Library \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Vectors_in_Three-dimensional_Space-0.md b/data/en.wikipedia.org/wiki/Vectors_in_Three-dimensional_Space-0.md new file mode 100644 index 000000000..a3e054b09 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Vectors_in_Three-dimensional_Space-0.md @@ -0,0 +1,26 @@ +--- +title: "Vectors in Three-dimensional Space" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Vectors_in_Three-dimensional_Space" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:14.876227+00:00" +instance: "kb-cron" +--- + +Vectors in Three-dimensional Space (1978) is a book concerned with physical quantities defined in "ordinary" 3-space. It was written by J. S. R. Chisholm, an English mathematical physicist, and published by Cambridge University Press. According to the author, such physical quantities are studied in Newtonian mechanics, fluid mechanics, theories of elasticity and plasticity, non-relativistic quantum mechanics, and many parts of solid state physics. The author further states that "the vector concept developed in two different ways: in a wide variety of physical applications, vector notation and techniques became, by the middle of this century, almost universal; on the other hand, pure mathematicians reduced vector algebra to an axiomatic system, and introduced wide generalisations of the concept of a three-dimensional 'vector space'." Chisholm explains that since these two developments proceeded largely independently, there is a need to show how one can be applied to the other. + + +== Summary == +Vectors in Three-Dimensional Space has six chapters, each divided into five or more subsections. The first on linear spaces and displacements including these sections: Introduction, Scalar multiplication of vectors, Addition and subtraction of vectors, Displacements in Euclidean space, Geometrical applications. The second on Scalar products and components including these sections: Scalar products, Linear dependence and dimension, Components of a vector, Geometrical applications, Coordinate systems. The third on Other products of vectors. The last three chapters round out Chisholm's integration of these two largely independent developments. + + +== References == + + +=== Footnotes === + + +=== Bibliography === +Vectors in Three-dimensional Space has been cited by the 2002 Encyclopedia Americana article on Vector Analysis +Chisholm, J. S. R. Vectors in Three-dimensional Space, Cambridge University Press, 1978, ISBN 0-521-29289-1 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Vorlesungen_über_Zahlentheorie-0.md b/data/en.wikipedia.org/wiki/Vorlesungen_über_Zahlentheorie-0.md new file mode 100644 index 000000000..e049273a8 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Vorlesungen_über_Zahlentheorie-0.md @@ -0,0 +1,74 @@ +--- +title: "Vorlesungen über Zahlentheorie" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Vorlesungen_über_Zahlentheorie" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:16.045939+00:00" +instance: "kb-cron" +--- + +Vorlesungen über Zahlentheorie (German pronunciation: [ˈfoːɐ̯ˌleːzʊŋən ˈyːbɐ ˈtsaːlənteoˌʁiː]; German for Lectures on Number Theory) is the name of several different textbooks of number theory. The best known was written by Peter Gustav Lejeune Dirichlet and Richard Dedekind, and published in 1863. Others were written by Leopold Kronecker, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics. + + +== Dirichlet and Dedekind's book == +Based on Dirichlet's number theory course at the University of Göttingen, the Vorlesungen were edited by Dedekind and published after Lejeune Dirichlet's death. Dedekind added several appendices to the Vorlesungen, in which he collected further results of Lejeune Dirichlet's and also developed his own original mathematical ideas. + + +=== Scope === +The Vorlesungen cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms. + + +=== Contents === +The contents of Professor John Stillwell's 1999 translation of the Vorlesungen are as follows + +Chapter 1. On the divisibility of numbers +Chapter 2. On the congruence of numbers +Chapter 3. On quadratic residues +Chapter 4. On quadratic forms +Chapter 5. Determination of the class number of binary quadratic forms +Supplement I. Some theorems from Gauss's theory of circle division +Supplement II. On the limiting value of an infinite series +Supplement III. A geometric theorem +Supplement IV. Genera of quadratic forms +Supplement V. Power residues for composite moduli +Supplement VI. Primes in arithmetic progressions +Supplement VII. Some theorems from the theory of circle division +Supplement VIII. On the Pell equation +Supplement IX. Convergence and continuity of some infinite series +This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals. +The German titles of supplements X and XI are: + +Supplement X: Über die Composition der binären quadratische Formen (On the composition of binary quadratic forms) +Supplement XI: Über die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers) +Chapters 1 to 4 cover similar ground to Gauss' Disquisitiones Arithmeticae, and Dedekind added footnotes which specifically cross-reference the relevant sections of the Disquisitiones. These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss's presentation, and introduces his own proofs in some places. +Chapter 5 contains Dirichlet's derivation of the class number formula for real and imaginary quadratic fields. Although other mathematicians had conjectured similar formulae, Dirichlet gave the first rigorous proof. +Supplement VI contains Dirichlet's proof that an arithmetic progression of the form a+nd where a and d are coprime contains an infinite number of primes. + + +=== Importance === +The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory. +The Vorlesungen contains two key results in number theory which were first proved by Dirichlet. The first of these is the class number formulae for binary quadratic forms. The second is a proof that arithmetic progressions contains an infinite number of primes (known as Dirichlet's theorem); this proof introduces Dirichlet L-series. These results are important milestones in the development of analytic number theory. + + +== Kronecker's book == +Leopold Kronecker's book was first published in 1901 in 2 parts and reprinted by Springer in 1978. It covers elementary and algebraic number theory, including Dirichlet's theorem. + + +== Landau's book == +Edmund Landau's book Vorlesungen über Zahlentheorie was first published as a 3-volume set in 1927. The first half of volume 1 was published as +Vorlesungen über Zahlentheorie. Aus der elementare Zahlentheorie in 1950, with an English translation in 1958 under the title Elementary number theory. In 1969 Chelsea republished the second half of volume 1 together with volumes 2 and 3 as a single volume. +Volume 1 on elementary and additive number theory includes the topics such as Dirichlet's theorem, Brun's sieve, binary quadratic forms, Goldbach's conjecture, Waring's problem, and the Hardy–Littlewood work on the singular series. Volume 2 covers topics in analytic number theory, such as estimates for the error in the prime number theorem, and topics in geometric number theory such as estimating numbers of lattice points. Volume 3 covers algebraic number theory, including ideal theory, quadratic number fields, and applications to Fermat's last theorem. Many of the results described by Landau were state of the art at the time but have since been superseded by stronger results. + + +== Hasse's book == +Helmut Hasse's book Vorlesungen über Zahlentheorie was published in 1950, and is different from and more elementary than his book Zahlentheorie. It covers elementary number theory, Dirichlet's theorem, and quadratic fields. + + +== References == + +P. G. Lejeune Dirichlet, R. Dedekind tr. John Stillwell: Lectures on Number Theory, American Mathematical Society, 1999 ISBN 0-8218-2017-6 The Göttinger Digitalisierungszentrum has a scanned copy of the original, 2nd edition text (in German) published in 1871 containing supplements I–X. Supplement XI can be found in volume three of Dedekind's complete works also at the Göttinger Digitalisierungszentrum. The 4th edition from 1894 which contains all of the supplements including Dedekind's XI is available at Internet Archive. +Hasse, Helmut (1950), Vorlesungen über Zahlentheorie, Die Grundlehren der mathematischen Wissenschaften, vol. LIX, Berlin-Göttingen-Heidelberg: Springer-Verlag, ISBN 978-3-642-88679-9, MR 0051844 {{citation}}: ISBN / Date incompatibility (help) +Kronecker, Leopold (1978) [1901], Vorlesungen über Zahlentheorie, Berlin-New York: Springer-Verlag, ISBN 3-540-08277-8, MR 0529431 +Landau, Edmund (1958) [1927], Elementary number theory., New York, N.Y.: Chelsea Publishing Co., MR 0092794 +Landau, Edmund (1969) [1927], Vorlesungen über Zahlentheorie. Erster Band, zweiter Teil; zweiter Band; dritter Band., New York: Chelsea Publishing Co., MR 0250844 \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-0.md b/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-0.md new file mode 100644 index 000000000..4bedcb1d9 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-0.md @@ -0,0 +1,45 @@ +--- +title: "Where Mathematics Comes From" +chunk: 1/3 +source: "https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:17.193195+00:00" +instance: "kb-cron" +--- + +Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (hereinafter WMCF) is a book by George Lakoff, a cognitive linguist, and Rafael E. Núñez, a psychologist. Published in 2000, WMCF seeks to found a cognitive science of mathematics, a theory of embodied mathematics based on conceptual metaphor. + +== WMCF definition of mathematics == +Mathematics makes up that part of the human conceptual system that is special in the following way: + +It is precise, consistent, stable across time and human communities, symbolizable, calculable, generalizable, universally available, consistent within each of its subject matters, and effective as a general tool for description, explanation, and prediction in a vast number of everyday activities, [ranging from] sports, to building, business, technology, and science. - WMCF, pp. 50, 377 +Nikolay Lobachevsky said "There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world." A common type of conceptual blending process would seem to apply to the entire mathematical procession. + +== Human cognition and mathematics == + +Lakoff and Núñez's avowed purpose is to begin laying the foundations for a truly scientific understanding of mathematics, one grounded in processes common to all human cognition. They find that four distinct but related processes metaphorically structure basic arithmetic: object collection, object construction, using a measuring stick, and moving along a path. +WMCF builds on earlier books by Lakoff (1987) and Lakoff and Johnson (1980, 1999), which analyze such concepts of metaphor and image schemata from second-generation cognitive science. Some of the concepts in these earlier books, such as the interesting technical ideas in Lakoff (1987), are absent from WMCF. +Lakoff and Núñez hold that mathematics results from the human cognitive apparatus and must therefore be understood in cognitive terms. WMCF advocates (and includes some examples of) a cognitive idea analysis of mathematics which analyzes mathematical ideas in terms of the human experiences, metaphors, generalizations, and other cognitive mechanisms giving rise to them. A standard mathematical education does not develop such idea analysis techniques because it does not pursue considerations of A) what structures of the mind allow it to do mathematics or B) the philosophy of mathematics. +Lakoff and Núñez start by reviewing the psychological literature, concluding that human beings appear to have an innate ability, called subitizing, to count, add, and subtract up to about 4 or 5. They document this conclusion by reviewing the literature, published in recent decades, describing experiments with infant subjects. For example, infants quickly become excited or curious when presented with "impossible" situations, such as having three toys appear when only two were initially present. +The authors argue that mathematics goes far beyond this very elementary level due to a large number of metaphorical constructions. For example, the Pythagorean position that all is number, and the associated crisis of confidence that came about with the discovery of the irrationality of the square root of two, arises solely from a metaphorical relation between the length of the diagonal of a square, and the possible numbers of objects. +Much of WMCF deals with the important concepts of infinity and of limit processes, seeking to explain how finite humans living in a finite world could ultimately conceive of the actual infinite. Thus much of WMCF is, in effect, a study of the epistemological foundations of the calculus. Lakoff and Núñez conclude that while the potential infinite is not metaphorical, the actual infinite is. Moreover, they deem all manifestations of actual infinity to be instances of what they call the "Basic Metaphor of Infinity", as represented by the ever-increasing sequence 1, 2, 3, ... +WMCF emphatically rejects the Platonistic philosophy of mathematics. They emphasize that all we know and can ever know is human mathematics, the mathematics arising from the human intellect. The question of whether there is a "transcendent" mathematics independent of human thought is a meaningless question, like asking if colors are transcendent of human thought—colors are only varying wavelengths of light, it is our interpretation of physical stimuli that make them colors. +WMCF (p. 81) likewise criticizes the emphasis mathematicians place on the concept of closure. Lakoff and Núñez argue that the expectation of closure is an artifact of the human mind's ability to relate fundamentally different concepts via metaphor. +WMCF concerns itself mainly with proposing and establishing an alternative view of mathematics, one grounding the field in the realities of human biology and experience. It is not a work of technical mathematics or philosophy. Lakoff and Núñez are not the first to argue that conventional approaches to the philosophy of mathematics are flawed. For example, they do not seem all that familiar with the content of Davis and Hersh (1981), even though the book warmly acknowledges Hersh's support. +Lakoff and Núñez cite Saunders Mac Lane (the inventor, with Samuel Eilenberg, of category theory) in support of their position. Mathematics, Form and Function (1986), an overview of mathematics intended for philosophers, proposes that mathematical concepts are ultimately grounded in ordinary human activities, mostly interactions with the physical world. + +== Examples of mathematical metaphors == +Conceptual metaphors described in WMCF, in addition to the Basic Metaphor of Infinity, include: + +Arithmetic is motion along a path, object collection/construction; +Change is motion; +Sets are containers, objects; +Continuity is gapless; +Mathematical systems have an "essence," namely their axiomatic algebraic structure; +Functions are sets of ordered pairs, curves in the Cartesian plane; +Geometric figures are objects in space; +Logical independence is geometric orthogonality; +Numbers are sets, object collections, physical segments, points on a line; +Recurrence is circular. +Mathematical reasoning requires variables ranging over some universe of discourse, so that we can reason about generalities rather than merely about particulars. WMCF argues that reasoning with such variables implicitly relies on what it terms the Fundamental Metonymy of Algebra. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-1.md b/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-1.md new file mode 100644 index 000000000..d0a69efdd --- /dev/null +++ b/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-1.md @@ -0,0 +1,115 @@ +--- +title: "Where Mathematics Comes From" +chunk: 2/3 +source: "https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:17.193195+00:00" +instance: "kb-cron" +--- + +== Example of metaphorical ambiguity == +WMCF (p. 151) includes the following example of what the authors term "metaphorical ambiguity." Take the set + + + + A + = + { + { + ∅ + } + , + { + ∅ + , + { + ∅ + } + } + } + . + + + {\displaystyle A=\{\{\emptyset \},\{\emptyset ,\{\emptyset \}\}\}.} + + Then recall two bits of standard terminology from elementary set theory: + +The recursive construction of the ordinal natural numbers, whereby 0 is + + + + ∅ + + + {\displaystyle \emptyset } + +, and + + + + n + + + 1 + + + {\displaystyle n+1} + + is + + + + n + ∪ + { + n + } + . + + + {\displaystyle n\cup \{n\}.} + + +The ordered pair (a,b), defined as + + + + { + { + a + } + , + { + a + , + b + } + } + . + + + {\displaystyle \{\{a\},\{a,b\}\}.} + + +By (1), A is the set {1,2}. But (1) and (2) together say that A is also the ordered pair (0,1). Both statements cannot be correct; the ordered pair (0,1) and the unordered pair {1,2} are fully distinct concepts. Lakoff and Johnson (1999) term this situation "metaphorically ambiguous." This simple example calls into question any Platonistic foundations for mathematics. +While (1) and (2) above are admittedly canonical, especially within the consensus set theory known as the Zermelo–Fraenkel axiomatization, WMCF does not let on that they are but one of several definitions that have been proposed since the dawning of set theory. For example, Frege, Principia Mathematica, and New Foundations (a body of axiomatic set theory begun by Quine in 1937) define cardinals and ordinals as equivalence classes under the relations of equinumerosity and similarity, so that this conundrum does not arise. In Quinian set theory, A is simply an instance of the number 2. For technical reasons, defining the ordered pair as in (2) above is awkward in Quinian set theory. Two solutions have been proposed: + +A variant set-theoretic definition of the ordered pair more complicated than the usual one; +Taking ordered pairs as primitive. + +== The Romance of Mathematics == +The "Romance of Mathematics" is WMCF's light-hearted term for a perennial philosophical viewpoint about mathematics which the authors describe and then dismiss as an intellectual myth: + +Mathematics is transcendent, namely it exists independently of human beings, and structures our actual physical universe and any possible universe. Mathematics is the language of nature, and is the primary conceptual structure we would have in common with extraterrestrial aliens, if any such there be. +Mathematical proof is the gateway to a realm of transcendent truth. +Reasoning is logic, and logic is essentially mathematical. Hence mathematics structures all possible reasoning. +Because mathematics exists independently of human beings, and reasoning is essentially mathematical, reason itself is disembodied. Therefore, artificial intelligence is possible, at least in principle. +It is very much an open question whether WMCF will eventually prove to be the start of a new school in the philosophy of mathematics. Hence the main value of WMCF so far may be a critical one: its critique of Platonism and romanticism in mathematics. + +== Critical response == + +Many working mathematicians resist the approach and conclusions of Lakoff and Núñez. Reviews of WMCF by mathematicians in professional journals, while often respectful of its focus on conceptual strategies and metaphors as paths for understanding mathematics, have taken exception to some of the WMCF's philosophical arguments on the grounds that mathematical statements have lasting 'objective' meanings. For example, Fermat's Last Theorem means exactly what it meant when Fermat initially proposed it in 1664. Other reviewers have pointed out that multiple conceptual strategies can be employed in connection with the same mathematically defined term, often by the same person (a point that is compatible with the view that we routinely understand the 'same' concept with different metaphors). The metaphor and the conceptual strategy are not the same as the formal definition which mathematicians employ. However, WMCF points out that formal definitions are built using words and symbols that have meaning only in terms of human experience. +Critiques of WMCF include the humorous: + +It's difficult for me to conceive of a metaphor for a real number raised to a complex power, but if there is one, I'd sure like to see it. — Joseph Auslander +and the physically informed: \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-2.md b/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-2.md new file mode 100644 index 000000000..0c5371bb8 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Where_Mathematics_Comes_From-2.md @@ -0,0 +1,58 @@ +--- +title: "Where Mathematics Comes From" +chunk: 3/3 +source: "https://en.wikipedia.org/wiki/Where_Mathematics_Comes_From" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:17.193195+00:00" +instance: "kb-cron" +--- + +But their analysis leaves at least a couple of questions insufficiently answered. For one thing, the authors ignore the fact that brains not only observe nature, but also are part of nature. Perhaps the math that brains invent takes the form it does because math had a hand in forming the brains in the first place (through the operation of natural laws in constraining the evolution of life). Furthermore, it's one thing to fit equations to aspects of reality that are already known. It's something else for that math to tell of phenomena never previously suspected. When Paul Dirac's equations describing electrons produced more than one solution, he surmised that nature must possess other particles, now known as antimatter. But scientists did not discover such particles until after Dirac's math told him they must exist. If math is a human invention, nature seems to know what was going to be invented. +Lakoff and Núñez tend to dismiss the negative opinions mathematicians have expressed about WMCF, because their critics do not appreciate the insights of cognitive science. Lakoff and Núñez maintain that their argument can only be understood using the discoveries of recent decades about the way human brains process language and meaning. They argue that any arguments or criticisms that are not grounded in this understanding cannot address the content of the book. +It has been pointed out that it is not at all clear that WMCF establishes that the claim "intelligent alien life would have mathematical ability" is a myth. To do this, it would be required to show that intelligence and mathematical ability are separable, and this has not been done. On Earth, intelligence and mathematical ability seem to go hand in hand in all life-forms, as pointed out by Keith Devlin among others. The authors of WMCF have not explained how this situation would (or even could) be different anywhere else. +Lakoff and Núñez also appear not to appreciate the extent to which intuitionists and constructivists have presaged their attack on the Romance of (Platonic) Mathematics. Brouwer, the founder of the intuitionist/constructivist point of view, in his dissertation On the Foundation of Mathematics, argued that mathematics was a mental construction, a free creation of the mind and totally independent of logic and language. He goes on to criticize the formalists for building verbal structures that are studied without intuitive interpretation. Symbolic language should not be confused with mathematics; it reflects, but does not contain, mathematical reality. +Educators have taken some interest in what WMCF suggests about how mathematics is learned, and why students find some elementary concepts more difficult than others. +However, even from an educational perspective, WMCF is still problematic. From the conceptual metaphor theory's point of view, metaphors reside in a different realm, the abstract, from that of 'real world', the concrete. In other words, despite their claim of mathematics being human, established mathematical knowledge — which is what we learn in school — is assumed to be and treated as abstract, completely detached from its physical origin. It cannot account for the way learners could access to such knowledge. +WMCF is also criticized for its monist approach. First, it ignores the fact that the sensori-motor experience upon which our linguistic structure — thus, mathematics — is assumed to be based may vary across cultures and situations. Second, the mathematics WMCF is concerned with is "almost entirely... standard utterances in textbooks and curricula", which is the most-well established body of knowledge. It is negligent of the dynamic and diverse nature of the history of mathematics. +WMCF's logo-centric approach is another target for critics. While it is predominantly interested in the association between language and mathematics, it does not account for how non-linguistic factors contribute to the emergence of mathematical ideas (e.g. See Radford, 2009; Rotman, 2008). + +== Summing up == +WMCF (pp. 378–79) concludes with some key points, a number of which follow. Mathematics arises from our bodies and brains, our everyday experiences, and the concerns of human societies and cultures. It is: + +The result of normal adult cognitive capacities, in particular the capacity for conceptual metaphor, and as such is a human universal. The ability to construct conceptual metaphors is neurologically based, and enables humans to reason about one domain using the language and concepts of another domain. Conceptual metaphor is both what enabled mathematics to grow out of everyday activities, and what enables mathematics to grow by a continual process of analogy and abstraction; +Symbolic, thereby enormously facilitating precise calculation; +Not transcendent, but the result of human evolution and culture, to which it owes its effectiveness. During experience of the world a connection to mathematical ideas is going on within the human mind; +A system of human concepts making extraordinary use of the ordinary tools of human cognition; +An open-ended creation of human beings, who remain responsible for maintaining and extending it; +One of the greatest products of the collective human imagination, and a magnificent example of the beauty, richness, complexity, diversity, and importance of human ideas. +The cognitive approach to formal systems, as described and implemented in WMCF, need not be confined to mathematics, but should also prove fruitful when applied to formal logic, and to formal philosophy such as Edward Zalta's theory of abstract objects. Lakoff and Johnson (1999) fruitfully employ the cognitive approach to rethink a good deal of the philosophy of mind, epistemology, metaphysics, and the history of ideas. + +== See also == +Abstract object +Cognitive science +Cognitive science of mathematics +Conceptual metaphor +Embodied philosophy +Foundations of mathematics +From Action to Mathematics per Mac Lane +Metaphor +Philosophy of mathematics +The Unreasonable Effectiveness of Mathematics in the Natural Sciences + +== Footnotes == + +== References == +Davis, Philip J., and Reuben Hersh, 1999 (1981). The Mathematical Experience. Mariner Books. First published by Houghton Mifflin. +George Lakoff, 1987. Women, Fire and Dangerous Things. Univ. of Chicago Press. +------ and Mark Johnson, 1999. Philosophy in the Flesh. Basic Books. +George Lakoff and Rafael Núñez, 2000, Where Mathematics Comes From. Basic Books. ISBN 0-465-03770-4 +John Randolph Lucas, 2000. The Conceptual Roots of Mathematics. Routledge. +Saunders Mac Lane, 1986. Mathematics: Form and Function. Springer Verlag. + +== External links == +WMCF web site. Archived 2011-07-18 at the Wayback Machine +Reviews of WMCF. +Joseph Auslander in American Scientist; +Bonnie Gold, MAA Reviews 2001 +Lakoff's response to Gold's MAA review. \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Word_Processing_in_Groups-0.md b/data/en.wikipedia.org/wiki/Word_Processing_in_Groups-0.md new file mode 100644 index 000000000..dac45cf57 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Word_Processing_in_Groups-0.md @@ -0,0 +1,27 @@ +--- +title: "Word Processing in Groups" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Word_Processing_in_Groups" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:19.528199+00:00" +instance: "kb-cron" +--- + +Word Processing in Groups is a monograph in mathematics on the theory of automatic groups, a type of abstract algebra whose operations are defined by the behavior of finite automata. The book's authors are David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Mike Paterson, and William Thurston. Widely circulated in preprint form, it formed the foundation of the study of automatic groups even before its 1992 publication by Jones and Bartlett Publishers (ISBN 0-86720-244-0). + + +== Topics == +The book is divided into two parts, one on the basic theory of these structures and another on recent research, connections to geometry and topology, and other related topics. +The first part has eight chapters. They cover automata theory and regular languages, and the closure properties of regular languages under logical combinations; the definition of automatic groups and biautomatic groups; examples from topology and "combable" structure in the Cayley graphs of automatic groups; abelian groups and the automaticity of Euclidean groups; the theory of determining whether a group is automatic, and its practical implementation by Epstein, Holt, and Sarah Rees; extensions to asynchronous automata; and nilpotent groups. +The second part has four chapters, on braid groups, isoperimetric inequalities, geometric finiteness, and the fundamental groups of three-dimensional manifolds. + + +== Audience and reception == +Although not primarily a textbook, the first part of the book could be used as the basis for a graduate course. More generally, reviewer Gilbert Baumslag recommends it "very strongly to everyone who is interested in either group theory or topology, as well as to computer scientists." +Baumslag was an expert in a related but older area of study, groups defined by finite presentations, in which research was eventually stymied by the phenomenon that many basic problems are undecidable. Despite tracing the origins of automatic groups to early 20th-century mathematician Max Dehn, he writes that the book studies "a strikingly new class of groups" that "conjures up the fascinating possibility that some of the exploration of these automatic groups can be carried out by means of high-speed computers" and that the book is "very likely to have a great impact". +Reviewer Daniel E. Cohen adds that two features of the book are unusual. First, that the mathematical results that it presents all have names, not just numbers, and second, that the cost of the book is low. +In 2009, mathematician Mark V. Lawson wrote that despite its "odd title," the book made automata theory more respectable among mathematicians stating that it became part of "a quiet revolution in the diplomatic relations between mathematics and computer science". + + +== References == \ No newline at end of file diff --git a/data/en.wikipedia.org/wiki/Zahlbericht-0.md b/data/en.wikipedia.org/wiki/Zahlbericht-0.md new file mode 100644 index 000000000..efccf0165 --- /dev/null +++ b/data/en.wikipedia.org/wiki/Zahlbericht-0.md @@ -0,0 +1,43 @@ +--- +title: "Zahlbericht" +chunk: 1/1 +source: "https://en.wikipedia.org/wiki/Zahlbericht" +category: "reference" +tags: "science, encyclopedia" +date_saved: "2026-05-05T08:47:20.658964+00:00" +instance: "kb-cron" +--- + +In mathematics, the Zahlbericht (number report) was a report on algebraic number theory by Hilbert (1897, 1998, (English translation)). + + +== History == +In 1893 the German Mathematical Society invited Hilbert and Minkowski to write reports on the theory of numbers. They agreed that Minkowski would cover the more elementary parts of number theory while Hilbert would cover algebraic number theory. Minkowski eventually abandoned his report, while Hilbert's report was published in 1897. It was reprinted in volume 1 of his collected works, and republished in an English translation in 1998. +Corry (1996) and Schappacher (2005) and the English introduction to (Hilbert 1998) give detailed discussions of the history and influence of Hilbert's Zahlbericht. +Some earlier reports on number theory include the report by H. J. S. Smith in 6 parts between 1859 and 1865, reprinted in Smith (1965), and the report by Brill & Noether (1894). Hasse (1926, 1927, 1930) wrote an update of Hilbert's Zahlbericht that covered class field theory (republished in 1 volume as (Hasse 1970)). + + +== Contents == +Part 1 covers the theory of general number fields, including ideals, discriminants, differents, units, and ideal classes. +Part 2 covers Galois number fields, including in particular Hilbert's theorem 90. +Part 3 covers quadratic number fields, including the theory of genera, and class numbers of quadratic fields. +Part 4 covers cyclotomic fields, including the Kronecker–Weber theorem (theorem 131), the Hilbert–Speiser theorem (theorem 132), and the Eisenstein reciprocity law for lth power residues (theorem 140) . +Part 5 covers Kummer number fields, and ends with Kummer's proof of Fermat's last theorem for regular primes. + + +== References == +Brill, A.; Noether, M. (1894), "Die Entwickelung der Theorie der algebraischen Functionen in älterer und neuerer Zeit", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 3: 107–566, ISSN 0012-0456 +Corry, Leo (1996), Modern algebra and the rise of mathematical structures, Science Networks. Historical Studies, vol. 17, Birkhäuser Verlag, ISBN 978-3-7643-5311-7, MR 1391720 +Hasse, H. (1926), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. I: Klassenkörpertheorie.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 35: 1–55 +Hasse, H. (1927), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil Ia: Beweise zu I.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 36: 233–311 +Hasse, H. (1930), "Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil II: Reziprozitätsgesetz", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), Ergänzungsband 6 +Hasse, Helmut (1970) [1930], Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil I: Klassenkörpertheorie. Teil Ia: Beweise zu Teil I. Teil II: Reziprozitätsgesetz (3rd ed.), Physica-Verlag, ISBN 978-3-7908-0010-4, MR 0266893 +Hilbert, David (1897), "Die Theorie der algebraischen Zahlkörper", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 4: 175–546, ISSN 0012-0456 +Hilbert, David (1998), The theory of algebraic number fields, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-662-03545-0, ISBN 978-3-540-62779-1, MR 1646901 +Schappacher, N. (2005), "Chapter 54. David Hilbert, Report on algebraic number fields", in Grattan-Guinness, Ivor (ed.), Landmark writings in western mathematics 1640–1940, Elsevier B. V., Amsterdam, ISBN 978-0-444-50871-3, MR 2169816 +Smith, Henry John Stephen (1965) [1894], Glaisher, J. W. L. (ed.), The Collected Mathematical Papers of Henry John Stephen Smith, vol. I, New York: AMS Chelsea Publishing, ISBN 978-0-8284-0187-6, volume 1volume 2, archived from the original on 2012-10-23, retrieved 2011-10-20 + + +== External links == + +Introduction to the English Edition of Hilbert's Zahlbericht \ No newline at end of file

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