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data/en.wikipedia.org/wiki/99_Points_of_Intersection-0.md
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data/en.wikipedia.org/wiki/99_Points_of_Intersection-0.md
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title: "99 Points of Intersection"
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source: "https://en.wikipedia.org/wiki/99_Points_of_Intersection"
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category: "reference"
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99 Points of Intersection: Examples—Pictures—Proofs is a book on constructions in Euclidean plane geometry in which three or more lines or curves meet in a single point of intersection. This book was originally written in German by Hans Walser as 99 Schnittpunkte (Eagle / Ed. am Gutenbergplatz, 2004), translated into English by Peter Hilton and Jean Pedersen, and published by the Mathematical Association of America in 2006 in their MAA Spectrum series (ISBN 978-0-88385-553-9).
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== Topics and organization ==
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The book is organized into three sections. The first section provides introductory material, describing different mathematical situations in which multiple curves might meet, and providing different possible explanations for this phenomenon, including symmetry, geometric transformations, and membership of the curves in a pencil of curves. The second section shows the 99 points of intersection of the title. Each is given on its own page, as a large figure with three smaller figures showing its construction, with a one-line caption but no explanatory text. The third section provides background material and proofs for some of these points of intersection, as well as extending and generalizing some of these results.
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Some of these points of intersection are standard; for instance, these include the construction of the centroid of a triangle as the point where its three median lines meet, the construction of the orthocenter as the point where the three altitudes meet, and the construction of the circumcenter as the point where the three perpendicular bisectors of the sides meet, as well as two versions of Ceva's theorem. However, others are new to this book, and include intersections related to silver rectangles, tangent circles, the Pythagorean theorem, and the nine-point hyperbola.
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== Audience ==
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John Jensen writes that "the clear and uncluttered illustrations of intersection make for a rich source for geometric investigation by high school geometry students". Jensen suggests that its examples would make a good complement to coursework both in exploratory geometry using interactive geometry software and in a geometry course focused on the formal proof of geometry propositions. He adds that the book itself is a proof of the possibility of presenting geometry without detailed explanations, and of introducing students to the beauty of the subject.
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Conversely, Gerry Leversha (writing in The Mathematical Gazette) calls the book "eccentric" and states that it "is clearly nothing to do with any syllabus anywhere".
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== References ==
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== External links ==
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Schnittpunkte, web site with a larger collection of points of intersection, by Hans Walser
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data/en.wikipedia.org/wiki/Adventures_Among_the_Toroids-0.md
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data/en.wikipedia.org/wiki/Adventures_Among_the_Toroids-0.md
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title: "Adventures Among the Toroids"
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Adventures Among the Toroids: A study of orientable polyhedra with regular faces is a book on toroidal polyhedra that have regular polygons as their faces. It was written, hand-lettered, and illustrated by mathematician Bonnie Stewart, and self-published under the imprint "Number One Tall Search Book" in 1970. Stewart put out a second edition, again hand-lettered and self-published, in 1980. Although out of print, the Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.
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== Topics ==
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The Platonic solids, known to antiquity, have all faces regular polygons, all symmetric to each other (each face can be taken to each other face by a symmetry of the polyhedron). However, if less symmetry is required, a greater number of polyhedra can be formed while having all faces regular. The convex polyhedra with all faces regular were catalogued in 1966 by Norman Johnson (after earlier study e.g. by Martyn Cundy and A. P. Rollett), and have come to be known as the Johnson solids. Adventures Among the Toroids extends the investigation of polyhedra with regular faces to non-convex polyhedra, and in particular to polyhedra of higher genus than the sphere. Many of these polyhedra can be formed by gluing together smaller polyhedral pieces, carving polyhedral tunnels through them, or piling them into elaborate towers. The toroidal polyhedra described in this book, formed from regular polygons with no self-intersections or flat angles, have come to be called Stewart toroids.
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The second edition is rewritten in a different page format, letter sized in landscape mode compared to the tall and narrow 5 inches (13 cm) by 13 inches (33 cm) page size of the first edition, with two columns per page. It includes new material on knotted polyhedra and on rings of regular octahedra and regular dodecahedra; as the ring of dodecahedra forms the outline of a golden rhombus, it can be extended to make skeletal pentagon-faced versions of the convex polyhedra formed from the golden rhombus, including the Bilinski dodecahedron, rhombic icosahedron, and rhombic triacontahedron. The second edition also includes the Császár polyhedron and Szilassi polyhedron, toroidal polyhedra with non-regular faces but with pairwise adjacent vertices and faces respectively, and constructions by Alaeglu and Giese of polyhedra with irregular but congruent faces and with the same numbers of edges at every vertex.
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== Audience and reception ==
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The second edition describes its intended audience in an elaborate subtitle, a throwback to times when long subtitles were more common: "a study of Quasi-Convex, aplanar, tunneled orientable polyhedra of positive genus having regular faces with disjoint interiors, being an elaborate description and instructions for the construction of an enormous number or new and fascinating mathematical models of interest to students of euclidean geometry and topology, both secondary and collegiate, to designers, engineers and architects, to the scientific audience concerned with molecular and other structural problems, and to mathematicians, both professional and dilettante, with hundreds of exercises and search projects, many outlined for self-instruction".
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Reviewer H. S. M. Coxeter summarizes the book as "a remarkable combination of sound mathematics, art, instruction and humor", while Henry Crapo calls it "highly recommended" to others interested in polyhedra and their juxtapositions.
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Mathematician Joseph A. Troccolo calls a method of constructing physical models of polyhedra developed in the book, using cardboard and rubber bands, "of inestimable value in the classroom". One virtue of this technique is that it allows for the quick disassembly and reuse of its parts.
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== See also ==
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List of books about polyhedra
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== References ==
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== External links ==
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Virtual reality models of Stewart's polyhedra, Alex Doskey
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Bonnie Stewarts Hohlkörper (in German), Christoph Pöppe, on the German-language site of Scientific American
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title: "Algorithmic Combinatorics on Partial Words"
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Algorithmic Combinatorics on Partial Words is a book in the area of combinatorics on words, and more specifically on partial words. It was written by Francine Blanchet-Sadri, and published in 2007 by Chapman & Hall/CRC in their Discrete Mathematics and its Applications book series.
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== Topics ==
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A partial word is a string whose characters may either belong to a given alphabet or be a wildcard character. Such a word can represent a set of strings over the alphabet without wildcards, by allowing each wildcard character to be replaced by any single character of the alphabet, independently of the replacements of the other wildcard characters. Two partial words are compatible when they agree on their non-wildcard characters, or equivalently when there is a string that they both match; one partial word
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x
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{\displaystyle x}
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contains another partial word
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y
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{\displaystyle y}
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if they are compatible and the non-wildcard positions of
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x
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{\displaystyle x}
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contain those of
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y
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{\displaystyle y}
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; equivalently, the strings matched by
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x
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{\displaystyle x}
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are a subset of those matched by
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y
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{\displaystyle y}
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.
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The book has 12 chapters, which can be grouped into five larger parts. The first part consists of two introductory chapters defining partial words, compatibility and containment, and related concepts. The second part generalizes to partial words some standard results on repetitions in strings, and the third part studies the problem of characterizing and recognizing primitive partial words, the partial words that have no repetition. Part four concerns codes defined from sets of partial words, in the sense that no two distinct concatenations of partial words from the set can be compatible with each other. A final part includes three chapters on advanced topics including the construction of repetitions of given numbers of copies of partial words that are compatible with each other, enumeration of the possible patterns of repetitions of partial words, and sets of partial words with the property that every infinite string contains a substring matching the set. Each chapter includes a set of exercises, and the end of the book provides hints to some of these exercises.
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== Audience and reception ==
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Although Algorithmic Combinatorics on Partial Words is primarily aimed at the graduate level, reviewer Miklós Bóna writes that it is for the most part "remarkably easy to read" and suggests that it could also be read by advanced undergraduates. However, Bóna criticizes the book as being too focused on the combinatorics on words as an end in itself, with no discussion of how to translate mathematical structures of other types into partial words so that the methods of this book can be applied to them. Because of this lack of generality and application, he suggests that the audience for the book is likely to consist only of other researchers specializing in this area. Similarly, although Patrice Séébold notes that this area can be motivated by applications to gene comparison, he criticizes the book as being largely a catalog of its author's own research results in partial words, without the broader thematic overview or identification of the fundamental topics and theorems that one would expect of a textbook, and suggests that a textbook that accomplishes these goals is still waiting to be written.
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However, reviewer Jan Kratochvíl is more positive, calling this "the first reference book on the theory of partial words", praising its pacing from introductory material to more advanced topics, and writing that it well supports its underlying thesis that many of the main results in the combinatorics of words without wildcards can be extended to partial words. He summarizes it as "an excellent textbook as well as a reference book for interested researchers".
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== References ==
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title: "Analytic Combinatorics (book)"
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source: "https://en.wikipedia.org/wiki/Analytic_Combinatorics_(book)"
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Analytic Combinatorics is a book on the mathematics of combinatorial enumeration, using generating functions and complex analysis to understand the growth rates of the numbers of combinatorial objects. It was written by Philippe Flajolet and Robert Sedgewick, and published by the Cambridge University Press in 2009. It won the Leroy P. Steele Prize in 2019.
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== Topics ==
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The main part of the book is organized into three parts. The first part, covering three chapters and roughly the first quarter of the book, concerns the symbolic method in combinatorics, in which classes of combinatorial objects are associated with formulas that describe their structures, and then those formulas are reinterpreted to produce the generating functions or exponential generating functions of the classes, in some cases using tools such as the Lagrange inversion theorem as part of the reinterpretation process. The chapters in this part divide the material into the enumeration of unlabeled objects, the enumeration of labeled objects, and multivariate generating functions.
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The five chapters of the second part of the book, roughly half of the text and "the heart of the book", concern the application of tools from complex analysis to the generating function, in order to understand the asymptotics of the numbers of objects in a combinatorial class. In particular, for sufficiently well-behaved generating functions, Cauchy's integral formula can be used to recover the power series coefficients (the real object of study) from the generating function, and knowledge of the singularities of the function can be used to derive accurate estimates of the resulting integrals. After an introductory chapter and a chapter giving examples of the possible behaviors of rational functions and meromorphic functions, the remaining chapters of this part discuss the way the singularities of a function can be used to analyze the asymptotic behavior of its power series, apply this method to a large number of combinatorial examples, and study the saddle-point method of contour integration for handling some trickier examples.
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The final part investigates the behavior of random combinatorial structures, rather than the total number of structures, using the same toolbox. Beyond expected values for combinatorial quantities of interest, it also studies limit theorems and large deviations theory for these quantities. Three appendices provide background on combinatorics and asymptotics, in complex analysis, and in probability theory.
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The combinatorial structures that are investigated throughout the book range widely over sequences, formal languages, integer partitions and compositions, permutations, graphs and paths in graphs, and lattice paths. With these topics, the analysis in the book connects to applications in other areas including abstract algebra, number theory, and the analysis of algorithms.
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== Audience and reception ==
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Analytic Combinatorics is not primarily a textbook; for instance, it has no exercises. Nevertheless, it can be used as the textbook for an upper-level undergraduate elective, graduate course, or seminar, although reviewer Miklós Bóna writes that some selection is needed, as it "has enough material for three or more semesters". It can also be a reference for researchers in this subject.
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Reviewer Toufik Mansour calls it not only "a comprehensive theoretical treatment" but "an interesting read". Reviewer Christopher Hanusa writes that "the writing style is inviting, the subject material is contemporary and riveting", and he recommends the book to anyone "learning or working in combinatorics".
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Analytic Combinatorics won the Leroy P. Steele Prize for Mathematical Exposition of the American Mathematical Society in 2019 (posthumously for Flajolet). The award citation called the book "an authoritative and highly accessible compendium of its subject, which demonstrates the deep interface between combinatorial mathematics and classical analysis". Although the application of analytic methods in combinatorics goes back at least to the work of G. H. Hardy and Srinivasa Ramanujan on the partition function, the citation also quoted a review by Robin Pemantle stating that "This is one of those books that marks the emergence of a subfield," the subfield of analytic combinatorics.
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Similarly, Bóna concludes, "Analytic Combinatorics is now defined. The authors wrote the book on it."
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== References ==
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== External links ==
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Analytic Combinatorics author web site, including a full-text downloadable copy of the book
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data/en.wikipedia.org/wiki/Animal_Math-0.md
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Animal Math is an educational book series featuring baby animals that introduces reader to mathematic principles. The series consists of five books: Tiger Math (2000), Chimp Math (2002), Polar Bear Math (2004), Panda Math (2005), and Cheetah Math (2007). On the right-hand pages, the books tell stories of baby animals cared for at either the Denver Zoo or San Diego Zoo who needed extra care when they were young. The left-hand pages discuss basic mathematic concepts using the animals as examples in problems.
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Published by Henry Holt and Company, the books are authored by Ann Whitehead Nagda, with support from Cindy Bickel, an assistant at the Denver Zoo, or in collaboration with the San Diego Zoo.
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== Books ==
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=== Tiger Math (2000) ===
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Tiger Math: Learning to Graph from a Baby Tiger (2000), co-written by Ann Whitehead Nagda and Cindy Bickel, teaches students about different types of graphs, including bar graphs, line graphs, picture graphs, and pie charts. The book features the Siberian tiger cub T. J., who was born at the Denver Zoo, then struggled to eat after his mother died.
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Tiger Math is a Junior Library Guild book. Kirkus Reviews said the book provided "a delightful way to learn math".
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=== Chimp Math (2002) ===
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Chimp Math: Learning about Time from a Baby Chimpanzee (2002), co-written by Ann Whitehead Nagda and Cindy Bickel, teaches students about time, such as timelines and telling time on a clock. The book features Jiggs, a baby chimpanzee at the Denver Zoo who was ignored by his mother. Jiggs is kept in the zoo's nursery during the day and stays with Bickel in the evenings. While in the nursery, he spends time with a jaguar cub named Giorgio, before returning to the chimpanzee enclosure.
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Kirkus Reviews praised Chimp Math, highlighting her strength of "combining multiple disciplines and teaching in a non-threatening, as-you-need-it manner".
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=== Polar Bear Math (2004) ===
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Polar Bear Math: Learning about Fractions from Klondike and Snow (2004), co-written by Ann Whitehead Nagda and Cindy Bickel, teaches students about fractions. The book features the polar bear cubs Klondike and Snow, who were cared for at the Denver Zoo after being abandoned. Along with a vet and another assistant at the zoo, Bickel learns how to care for the polar bears, who eventually thrive.
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Polar Bear Math is a Junior Library Guild book.
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=== Panda Math (2005) ===
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Panda Math: Learning about Subtraction from Hua Mei and Mei Sheng (2005), written by Ann Whitehead Nagda in collaboration with the San Diego Zoo, teaches students about subtraction. The mathematical text discusses different ways to subtract, including "regrouping, subtracting each place value, thinking about doubles and adding up". The story features Hua Mei and Mei Sheng, panda cubs at the San Diego Zoo.
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According to Kirkus Reviews, Nagda explains the different approaches to solving subtraction problems well, though they concede that "children will need additional support".
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=== Cheetah Math (2007) ===
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Cheetah Math: Learning about Division from Baby Cheetahs (2007), written by Ann Whitehead Nagda in collaboration with the San Diego Zoo, teaches students about division. The book features the baby cheetahs Majani and Kubali, who were raised at the San Diego Zoo.
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Cheetah Math is a Junior Library Guild book. Sally Woolsey called the book "well done" and it is a popular item in many elementary school libraries. Kirkus Reviews called the book "a great addition to both the math and wild-animal conservation bookshelves". The School Library Journal also gave a favorable review, saying Cheetah Math "is a wonderful cross-curricular book and an appealing way to introduce math".
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== Publication details ==
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Nagda, Ann Whitehead; Bickel, Cindy (2000). Tiger Math: Learning to Graph from a Baby Tiger. New York: Henry Holt and Company. ISBN 0-8050-6248-3.
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Nagda, Ann Whitehead; Bickel, Cindy (2002). Chimp Math: Learning about Time from a Baby Chimpanzee. New York: Henry Holt and Company. ISBN 0-8050-6674-8.
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Nagda, Ann Whitehead; Bickel, Cindy (2004). Polar Bear Math: Learning about Fractions from Klondike and Snow. New York: Henry Holt and Company. ISBN 978-0-8050-7301-0.
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Nagda, Ann Whitehead; San Diego Zoo (2005). Panda Math: Learning about Subtraction from Hua Mei and Mei Sheng. New York: Henry Holt and Company. ISBN 0-8050-7644-1.
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Nagda, Ann Whitehead; San Diego Zoo (2007). Cheetah Math: Learning about Division from Baby Cheetahs. Henry Holt and Company. ISBN 978-0-8050-7645-5.
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== References ==
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data/en.wikipedia.org/wiki/Arithmetica_Universalis-0.md
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title: "Arithmetica Universalis"
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Arithmetica Universalis ("Universal Arithmetic") is a mathematics text by Isaac Newton. Written in Latin, it was edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Cambridge. The Arithmetica was based on Newton's lecture notes.
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== Publication history ==
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Whiston's original edition was published in 1707. It was translated into English by Joseph Raphson, who published it in 1720 as the Universal Arithmetick. John Machin published a second Latin edition in 1722.
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== Lack of credit for the writer ==
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None of these editions credit Newton as author; Newton was unhappy with the publication of the Arithmetica, and so refused to have his name appear. In fact, when Whiston's edition was published, Newton was so upset he considered purchasing all of the copies so he could destroy them.
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== Content ==
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The Arithmetica touches on algebraic notation, arithmetic, the relationship between geometry and algebra, and the solution of equations. Newton also applied Descartes' rule of signs to imaginary roots. He also offered, without proof, a rule to determine the number of imaginary roots of polynomial equations. A rigorous proof of Newton's counting formula for equations up to and including the fifth degree was published by James Joseph Sylvester in 1864.
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== References ==
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The Arithmetica Universalis from the Grace K. Babson Collection, including links to PDFs of English and Latin versions of the Arithmetica
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Centre College Library information on Newton's works
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== External links ==
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Arithmetica Universalis (1707), first edition
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Universal Arithmetick (1720), English translation by Joseph Raphson
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Arithmetica Universalis (1722), second edition
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title: "Ars Magna (Cardano book)"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Ars_Magna_(Cardano_book)"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:22.876678+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Ars Magna (The Great Art, 1545) is an important Latin-language book on algebra written by Gerolamo Cardano. It was first published in 1545 under the title Artis Magnae, Sive de Regulis Algebraicis, Lib. unus (The Great Art, or The Rules of Algebra, Book one). There was a second edition in Cardano's lifetime, published in 1570. It is considered one of the three greatest scientific treatises of the early Renaissance, together with Copernicus' De revolutionibus orbium coelestium and Vesalius' De humani corporis fabrica. The first editions of these three books were published within a two-year span (1543–1545).
|
||||
|
||||
|
||||
== History ==
|
||||
In 1535, Niccolò Fontana Tartaglia became famous for having solved cubics of the form x3 + ax = b (with a,b > 0). However, he chose to keep his method secret. In 1539, Cardano, then a lecturer in mathematics at the Piatti Foundation in Milan, published his first mathematical book, Pratica Arithmeticæ et mensurandi singularis (The Practice of Arithmetic and Simple Mensuration). That same year, he asked Tartaglia to explain to him his method for solving cubic equations. After some reluctance, Tartaglia did so, but he asked Cardano not to share the information until he published it. Cardano submerged himself in mathematics during the next several years working on how to extend Tartaglia's formula to other types of cubics. Furthermore, his student Lodovico Ferrari found a way of solving quartic equations, but Ferrari's method depended upon Tartaglia's, since it involved the use of an auxiliary cubic equation. Then Cardano became aware of the fact that Scipione del Ferro had discovered Tartaglia's formula before Tartaglia himself, a discovery that prompted him to publish these results.
|
||||
|
||||
|
||||
== Contents ==
|
||||
The book, which is divided into forty chapters, contains the first published algebraic solution to cubic and quartic equations. Cardano acknowledges that Tartaglia gave him the formula for solving a type of cubic equations and that the same formula had been discovered by Scipione del Ferro. He also acknowledges that it was Ferrari who found a way of solving quartic equations.
|
||||
Since at the time negative numbers were not generally acknowledged, knowing how to solve cubics of the form x3 + ax = b did not mean knowing how to solve cubics of the form x3 = ax + b (with a,b > 0), for instance. Besides, Cardano also explains how to reduce equations of the form x3 + ax2 + bx + c = 0 to cubic equations without a quadratic term, but, again, he has to consider several cases. In all, Cardano was driven to the study of thirteen different types of cubic equations (chapters XI–XXIII).
|
||||
In Ars Magna the concept of multiple root appears for the first time (chapter I). The first example that Cardano provides of a polynomial equation with multiple roots is x3 = 12x + 16, of which −2 is a double root.
|
||||
Ars Magna also contains the first occurrence of complex numbers (chapter XXXVII). The problem mentioned by Cardano which leads to square roots of negative numbers is: find two numbers whose sum is equal to 10 and whose product is equal to 40. The answer is 5 + √−15 and 5 − √−15. Cardano called this "sophistic," because he saw no physical meaning to it, but boldly wrote "nevertheless we will operate" and formally calculated that their product does indeed equal 40. Cardano then says that this answer is "as subtle as it is useless".
|
||||
It is a common misconception that Cardano introduced complex numbers in solving cubic equations. Since (in modern notation) Cardano's formula for a root of the polynomial x3 + px + q is
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
−
|
||||
|
||||
|
||||
q
|
||||
2
|
||||
|
||||
|
||||
+
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
q
|
||||
|
||||
2
|
||||
|
||||
|
||||
4
|
||||
|
||||
|
||||
+
|
||||
|
||||
|
||||
|
||||
p
|
||||
|
||||
3
|
||||
|
||||
|
||||
27
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
3
|
||||
|
||||
|
||||
|
||||
+
|
||||
|
||||
|
||||
|
||||
−
|
||||
|
||||
|
||||
q
|
||||
2
|
||||
|
||||
|
||||
−
|
||||
|
||||
|
||||
|
||||
|
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|
||||
q
|
||||
|
||||
2
|
||||
|
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|
||||
4
|
||||
|
||||
|
||||
+
|
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|
||||
|
||||
|
||||
p
|
||||
|
||||
3
|
||||
|
||||
|
||||
27
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
3
|
||||
|
||||
|
||||
|
||||
,
|
||||
|
||||
|
||||
{\displaystyle {\sqrt[{3}]{-{\frac {q}{2}}+{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}}+{\sqrt[{3}]{-{\frac {q}{2}}-{\sqrt {{\frac {q^{2}}{4}}+{\frac {p^{3}}{27}}}}}},}
|
||||
|
||||
|
||||
square roots of negative numbers appear naturally in this context. However, q2/4 + p3/27 never happens to be negative in the specific cases in which Cardano applies the formula.
|
||||
|
||||
|
||||
== Notes ==
|
||||
|
||||
|
||||
== Bibliography ==
|
||||
Calinger, Ronald (1999), A contextual history of Mathematics, Prentice-Hall, ISBN 0-02-318285-7
|
||||
Cardano, Gerolamo (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3 {{citation}}: ISBN / Date incompatibility (help)
|
||||
Gindikin, Simon (1988), Tales of physicists and mathematicians, Birkhäuser, ISBN 3-7643-3317-0
|
||||
|
||||
|
||||
== External links ==
|
||||
PDF of Ars Magna (in Latin)
|
||||
Cardano's biography
|
||||
@ -0,0 +1,48 @@
|
||||
---
|
||||
title: "Art Gallery Theorems and Algorithms"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Art_Gallery_Theorems_and_Algorithms"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:24.102157+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Art Gallery Theorems and Algorithms is a mathematical monograph on topics related to the art gallery problem, on finding positions for guards within a polygonal museum floorplan so that all points of the museum are visible to at least one guard, and on related problems in computational geometry concerning polygons. It was written by Joseph O'Rourke, and published in 1987 in the International Series of Monographs on Computer Science of the Oxford University Press. Only 1000 copies were produced before the book went out of print, so to keep this material accessible O'Rourke has made a pdf version of the book available online.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The art gallery problem, posed by Victor Klee in 1973, asks for the number of points at which to place guards inside a polygon (representing the floor plan of a museum) so that each point within the polygon is visible to at least one guard. Václav Chvátal provided the first proof that the answer is at most
|
||||
|
||||
|
||||
|
||||
⌊
|
||||
n
|
||||
|
||||
/
|
||||
|
||||
3
|
||||
⌋
|
||||
|
||||
|
||||
{\displaystyle \lfloor n/3\rfloor }
|
||||
|
||||
for a polygon with
|
||||
|
||||
|
||||
|
||||
n
|
||||
|
||||
|
||||
{\displaystyle n}
|
||||
|
||||
corners, but a simplified proof by Steve Fisk based on graph coloring and polygon triangulation is more widely known. This is the opening material of the book, which goes on to covers topics including visibility, decompositions of polygons, coverings of polygons, triangulations and triangulation algorithms, and higher-dimensional generalizations, including the result that some polyhedra such as the Schönhardt polyhedron do not have triangulations without additional vertices. More generally, the book has as a theme "the interplay between discrete and computational geometry".
|
||||
It has 10 chapters, whose topics include the original art gallery theorem and Fisk's triangulation-based proof; rectilinear polygons; guards that can patrol a line segment rather than a single point; special classes of polygons including star-shaped polygons, spiral polygons, and monotone polygons; non-simple polygons; prison yard problems, in which the guards must view the exterior, or both the interior and exterior, of a polygon; visibility graphs; visibility algorithms; the computational complexity of minimizing the number of guards; and three-dimensional generalizations.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
The book only requires an undergraduate-level knowledge of graph theory and algorithms. However, it lacks exercises, and is organized more as a monograph than as a textbook. Despite warning that it omits some details that would be important to implementors of the algorithms that it describes, and does not describe algorithms that perform well on random inputs despite poor worst-case complexity, reviewer Wm. Randolph Franklin recommends it "for the library of every geometer".
|
||||
Reviewer Herbert Edelsbrunner writes that "This book is the most comprehensive collection of results on polygons currently available and thus earns its place as a standard text in computational geometry. It is very well written and a pleasure to read." However, reviewer Patrick J. Ryan complains that some of the book's proofs are inelegant, and reviewer David Avis, writing in 1990, noted that already by that time there were "many new developments" making the book outdated. Nevertheless, Avis writes that "the book succeeds on a number of levels", as an introductory text for undergraduates or for researchers in other areas, and as an invitation to solve the "many unsolved questions" remaining in this area.
|
||||
|
||||
|
||||
== References ==
|
||||
22
data/en.wikipedia.org/wiki/Automorphic_Forms_on_GL(2)-0.md
Normal file
22
data/en.wikipedia.org/wiki/Automorphic_Forms_on_GL(2)-0.md
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@ -0,0 +1,22 @@
|
||||
---
|
||||
title: "Automorphic Forms on GL(2)"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Automorphic_Forms_on_GL(2)"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:25.244788+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Automorphic Forms on GL(2) is a mathematics book by H. Jacquet and Robert Langlands (1970) where they rewrite Erich Hecke's theory of modular forms in terms of the representation theory of GL(2) over local fields and adele rings of global fields and prove the Jacquet–Langlands correspondence. A second volume by Jacquet (1972) gives an interpretation of some results by Rankin and Selberg in terms of the representation theory of GL(2) × GL(2).
|
||||
|
||||
|
||||
== References ==
|
||||
Godement, R. (1970), Notes on Jacquet–Langlands' theory, Institute for Advanced Study
|
||||
Jacquet, H; Langlands, R. P. (1970), Automorphic Forms on GL (2), Lecture Notes in Mathematics, vol. 114, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3540049036, MR 0401654, S2CID 122773458
|
||||
Jacquet, Hervé (1972), Automorphic Forms on GL(2) Part II, Lecture Notes in Mathematics, vol. 278, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0058503, ISBN 978-3-540-05931-8, MR 0562503
|
||||
Langlands, R. (1971), "Automorphic forms on GL(2)" (PDF), Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 2, Paris: Gauthier-Villars, pp. 327–329, MR 0562505
|
||||
|
||||
|
||||
== External links ==
|
||||
Buzzard, Kevin (2012), What is in the book Automorphic forms on GL(2) by Jacquet and Langlands? (PDF)
|
||||
19
data/en.wikipedia.org/wiki/Basic_Number_Theory-0.md
Normal file
19
data/en.wikipedia.org/wiki/Basic_Number_Theory-0.md
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|
||||
---
|
||||
title: "Basic Number Theory"
|
||||
chunk: 1/2
|
||||
source: "https://en.wikipedia.org/wiki/Basic_Number_Theory"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:27.589697+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Basic Number Theory is an influential book by André Weil, an exposition of algebraic number theory and class field theory with particular emphasis on valuation-theoretic methods. Based in part on a course taught at Princeton University in 1961–62, it appeared as Volume 144 in Springer's Grundlehren der mathematischen Wissenschaften series. The approach handles all 'A-fields' or global fields, meaning finite algebraic extensions of the field of rational numbers and of the field of rational functions of one variable with a finite field of constants. The theory is developed in a uniform way, starting with topological fields, properties of Haar measure on locally compact fields, the main theorems of adelic and idelic number theory, and class field theory via the theory of simple algebras over local and global fields. The word `basic’ in the title is closer in meaning to `foundational’ rather than `elementary’, and is perhaps best interpreted as meaning that the material developed is foundational for the development of the theories of automorphic forms, representation theory of algebraic groups, and more advanced topics in algebraic number theory. The style is austere, with a narrow concentration on a logically coherent development of the theory required, and essentially no examples.
|
||||
|
||||
== Mathematical context and purpose ==
|
||||
In the foreword, the author explains that instead of the “futile and impossible task” of improving on Hecke's classical treatment of algebraic number theory, he “rather tried to draw the conclusions from the developments of the last thirty years, whereby locally compact groups, measure and integration have been seen to play an increasingly important role in classical number theory”. Weil goes on to explain a viewpoint that grew from work of Hensel, Hasse, Chevalley, Artin, Iwasawa, Tate, and Tamagawa in which the real numbers may be seen as but one of infinitely many different completions of the rationals, with no logical reason to favour it over the various p-adic completions. In this setting, the adeles (or valuation vectors) give a natural locally compact ring in which all the valuations are brought together in a single coherent way in which they “cooperate for a common purpose”. Removing the real numbers from a pedestal and placing them alongside the p-adic numbers leads naturally – “it goes without saying” to the development of the theory of function fields over finite fields in a “fully simultaneous treatment with number-fields”. In a striking choice of wording for a foreword written in the United States in 1967, the author chooses to drive this particular viewpoint home by explaining that the two classes of global fields “must be granted a fully simultaneous treatment […] instead of the segregated status, and at best the separate but equal facilities, which hitherto have been their lot. That, far from losing by such treatment, both races stand to gain by it, is one fact which will, I hope, clearly emerge from this book.”
|
||||
After World War II, a series of developments in class field theory diminished the significance of the cyclic algebras (and, more generally, the crossed product algebras) which are defined in terms of the number field in proofs of class field theory. Instead cohomological formalism became a more significant part of local and global class field theory, particularly in work of Hochschild and Nakayama, Weil, Artin, and Tate during the period 1950–1952.
|
||||
Alongside the desire to consider algebraic number fields alongside function fields over finite fields, the work of Chevalley is particularly emphasised. In order to derive the theorems of global class field theory from those of local class field theory, Chevalley introduced what he called the élément idéal, later called idèle, at Hasse's suggestion. The idèle group of a number field was first introduced by Chevalley in order to describe global class field theory for infinite extensions, but several years later he used it in a new way to derive global class field theory from local class field theory. Weil mentioned this (unpublished) work as a significant influence on some of the choices of treatment he uses.
|
||||
|
||||
== Reception ==
|
||||
The 1st edition was reviewed by George Whaples for Mathematical Reviews and Helmut Koch for Zentralblatt. Later editions were reviewed by Fernando Q. Gouvêa for the Mathematical Association of America and by Koch for Zentralblatt; in his review of the second edition Koch makes the remark "Shafarevich showed me the first edition in autumn 1967 in Moscow and said that this book will be from now on the book about class field theory". The coherence of the treatment, and some of its distinctive features, were highlighted by several reviewers, with Koch going on to say "This book is written in the spirit of the early forties and just this makes it a valuable source of information for everyone who is working about problems related to number and function fields."
|
||||
66
data/en.wikipedia.org/wiki/Basic_Number_Theory-1.md
Normal file
66
data/en.wikipedia.org/wiki/Basic_Number_Theory-1.md
Normal file
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|
||||
---
|
||||
title: "Basic Number Theory"
|
||||
chunk: 2/2
|
||||
source: "https://en.wikipedia.org/wiki/Basic_Number_Theory"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:27.589697+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
== Contents ==
|
||||
Roughly speaking, the first half of the book is modern in its consistent use of adelic and idèlic methods and the simultaneous treatment of algebraic number fields and rational function fields over finite fields. The second half is arguably pre-modern in its development of simple algebras and class field theory without the language of cohomology, and without the language of Galois cohomology in particular. The author acknowledges this as a trade-off, explaining that “to develop such an approach systematically would have meant loading a great deal of unnecessary machinery on a ship which seemed well equipped for this particular voyage; instead of making it more seaworthy, it might have sunk it.” The treatment of class field theory uses analytic methods on both commutative fields and simple algebras. These methods show their power in giving the first unified proof that if K/k is a finite normal extension of A-fields, then any automorphism of K over k is induced by the Frobenius automorphism for infinitely many places of K. This approach also allows for a significantly simpler and more logical proof of algebraic statements, for example the result that a simple algebra over an A-field splits (globally) if and only if it splits everywhere locally. The systematic use of simple algebras also simplifies the treatment of local class field theory. For instance, it is more straightforward to prove that a simple algebra over a local field has an unramified splitting field than to prove the corresponding statement for 2-cohomology classes.
|
||||
|
||||
=== Chapter I ===
|
||||
The book begins with Witt’s formulation of Wedderburn’s proof that a finite division ring is commutative ('Wedderburn's little theorem'). Properties of Haar measure are used to prove that `local fields’ (commutative fields locally compact under a non-discrete topology) are completions of A-fields. In particular – a concept developed later – they are precisely the fields whose local class field theory is needed for the global theory. The non-discrete non-commutative locally compact fields are then division algebras of finite dimension over a local field.
|
||||
|
||||
=== Chapter II ===
|
||||
Finite-dimensional vector spaces over local fields and division algebras under the topology uniquely determined by the field's topology are studied, and lattices are defined topologically, an analogue of Minkowski's theorem is proved in this context, and the main theorems about character groups of these vector spaces, which in the commutative one-dimensional case reduces to `self duality’ for local fields, are shown.
|
||||
|
||||
=== Chapter III ===
|
||||
Tensor products are used to study extensions of the places of an A-field to places of a finite separable extension of the field, with the more complicated inseparable case postponed to later.
|
||||
|
||||
=== Chapter IV ===
|
||||
This chapter introduces the topological adele ring and idèle group of an A-field, and proves the `main theorems’ as follows:
|
||||
|
||||
both the adele ring and the idèle group are locally compact;
|
||||
the A-field, when embedded diagonally, is a discrete and co-compact subring of its adele ring;
|
||||
the adele ring is self dual, meaning that it is topologically isomorphic to its Pontryagin dual, with similar properties for finite-dimensional vector spaces and algebras over local fields.
|
||||
The chapter ends with a generalized unit theorem for A-fields, describing the units in valuation terms.
|
||||
|
||||
=== Chapter V ===
|
||||
This chapter departs slightly from the simultaneous treatment of number fields and function fields. In the number field setting, lattices (that is, fractional ideals) are defined, and the Haar measure volume of a fundamental domain for a lattice is found. This is used to study the discriminant of an extension.
|
||||
|
||||
=== Chapter VI ===
|
||||
This chapter is focused on the function field case; the Riemann-Roch theorem is stated and proved in measure-theoretic language, with the canonical class defined as the class of divisors of non-trivial characters of the adele ring which are trivial on the embedded field.
|
||||
|
||||
=== Chapter VII ===
|
||||
The zeta and L-functions (and similar analytic objects) for an A-field are expressed in terms of integrals over the idèle group. Decomposing these integrals into products over all valuations and using Fourier transforms gives rise to meromorphic continuations and functional equations. This gives, for example, analytic continuation of the Dedekind zeta-function to the whole plane, along with its functional equation. The treatment here goes back ultimately to a suggestion of Artin, and was developed in Tate's thesis.
|
||||
|
||||
=== Chapter VIII ===
|
||||
Formulas for local and global different and discriminants, ramification theory, and the formula for the genus of an algebraic extension of a function field are developed.
|
||||
|
||||
=== Chapter IX ===
|
||||
A brief treatment of simple algebras is given, including explicit rules for cyclic factor sets.
|
||||
|
||||
=== Chapters X and XI ===
|
||||
The zeta-function of a simple algebra over an A-field is defined, and used to prove further results on the norm group and groupoid of maximal ideals in a simple algebra over an A-field.
|
||||
|
||||
=== Chapter XII ===
|
||||
The reciprocity law of local class field theory over a local field in the context of a pairing of the multiplicative group of a field and the character group of the absolute Galois group of the algebraic closure of the field is proved. Ramification theory for abelian extensions is developed.
|
||||
|
||||
=== Chapter XIII ===
|
||||
The global class field theory for A-fields is developed using the pairings of Chapter XII, replacing multiplicative groups of local fields with idèle class groups of A-fields. The pairing is constructed as a product over places of local Hasse invariants.
|
||||
|
||||
=== Third edition ===
|
||||
Some references are added, some minor corrections made, some comments added, and five appendices are included, containing the following material:
|
||||
|
||||
A character version of the (local) transfer theorem and its extension to the global transfer theorem.
|
||||
Šafarevič's theorem on the structure of Galois groups of local fields using the theory of Weil groups.
|
||||
Theorems of Tate and Sen on the Herbrand distribution.
|
||||
Examples of L-functions with Grössencharacter.
|
||||
|
||||
== Editions ==
|
||||
Weil, André (1974). Basic Number Theory. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-642-61945-8. ISBN 978-3-540-58655-5.
|
||||
|
||||
== References ==
|
||||
@ -0,0 +1,41 @@
|
||||
---
|
||||
title: "Black Mathematicians and Their Works"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Black_Mathematicians_and_Their_Works"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:31.112492+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Black Mathematicians and Their Works is an edited volume of works in and about mathematics, by African-American mathematicians. It was edited by Virginia Newell, Joella Gipson, L. Waldo Rich, and Beauregard Stubblefield, with a foreword by Wade Ellis, and published in 1980 by Dorrance & Company.
|
||||
The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.
|
||||
|
||||
|
||||
== Contents ==
|
||||
The book celebrates the achievements of black mathematicians and also records their struggle against racism. It includes reprints of 23 papers of mathematics research and three more on mathematics education, by black mathematicians. It provides brief biographies and photographs of 62 black mathematicians, all long-established at the time of publication (having doctorates prior to 1973). It also reproduces several letters by Lee Lorch documenting racist behavior in mathematical societies, such as exclusion from conferences and their associated social gatherings. An appendix lists universities that have worked with black mathematicians, by the number of doctorates conferred and the number of faculty hired.
|
||||
As well as two of the editors (Gipson and Stubblefield), the authors whose works are reproduced in the book include
|
||||
Albert Turner Bharucha-Reid,
|
||||
David Blackwell,
|
||||
Lillian K. Bradley,
|
||||
Marjorie Lee Browne,
|
||||
Edward M. Carroll,
|
||||
William Schieffelin Claytor,
|
||||
Vivienne Malone-Mayes,
|
||||
Clarence F. Stephens,
|
||||
Walter Richard Talbot, and
|
||||
J. Ernest Wilkins Jr.
|
||||
|
||||
|
||||
== Reception ==
|
||||
Black Mathematicians and Their Works was the first book to collect the works of black mathematicians, and 40 years after its publication it remained the only such book. By demonstrating the successes of black mathematicians, it aimed to counter the then-current opinion that black people could not do mathematics, and provide encouragement to young black future mathematicians.
|
||||
Edray Herber Goins has named this book as his "mathematical comfort food", writing:
|
||||
|
||||
Whenever I question whether black folk are making progress in these United States, I think of the articles in this volume, and those pioneers who continued to do math in the face of blatant racism.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Black Mathematicians and Their Works on the Internet Archive
|
||||
@ -0,0 +1,41 @@
|
||||
---
|
||||
title: "Book on Numbers and Computation"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Book_on_Numbers_and_Computation"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:34.610392+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Book on Numbers and Computation (Chinese: 筭數書; pinyin: Suàn shù shū), or the Writings on Reckoning, is one of the earliest known Chinese mathematical treatises. It was written during the early Western Han dynasty, sometime between 202 BC and 186 BC. It was preserved among the Zhangjiashan Han bamboo texts and contains similar mathematical problems and principles found in the later Eastern Han period text of The Nine Chapters on the Mathematical Art.
|
||||
|
||||
|
||||
== Discovery ==
|
||||
The text was found in tomb M247 of the burial grounds near Zhangjiashan, Jiangling County, in Hubei province, excavated in December–January 1983–1984. This tomb belonged to an anonymous civil servant in early West Han dynasty. In the tomb were 1200 bamboo strips written in ink. Originally the strips were bound together with string, but the string had rotted away and it took Chinese scholars 17 years to piece together the strips. As well as the mathematical work the strips covered government statutes, law reports and therapeutic gymnastics.
|
||||
On the back of the sixth strip, the top has a black square mark, followed by the three characters 筭數書, which serve as the title of the rolled up book.
|
||||
|
||||
|
||||
== Content ==
|
||||
The Suàn shù shū consists of 200 strips of bamboo written in ink, 180 strips are intact, the others have rotted. They consist of 69 mathematical problems from a variety of sources, two names Mr Wáng and Mr Yáng were found, probably two of the writers. Each problem has a question, an answer, followed by a method. The problems cover elementary arithmetic; fractions; inverse proportion; factorization of numbers; geometric progressions, in particular interest rate calculations and handling of errors; conversion between different units; the false position method for finding roots and the extraction of approximate square roots; calculation of the volume of various 3-dimensional shapes; relative dimensions of a square and its inscribed circle; calculation of unknown side of rectangle, given area and one side. All the calculations about circumference and area of circle are approximate, equivalent to taking π = 3. Calculations of pi were made more accurate with the work of Liu Xin (c. 46 BC – 23 AD), Zhang Heng (78–139 AD), Liu Hui (fl. 3rd century AD), and Zu Chongzhi (429–500).
|
||||
Prior to discovery the oldest Chinese mathematical text were the Zhoubi Suanjing and The Nine Chapters on the Mathematical Art which dates from around 100 CE. Many topics are covered in both texts, however, error correction problems only appear in the Suàn shù shū, and the last two chapter of the nine chapters have no corresponding material in the Suàn shù shū.
|
||||
The text has been translated to English by Christopher Cullen, director of the Needham Research Institute.
|
||||
|
||||
|
||||
== Notes ==
|
||||
|
||||
|
||||
== References ==
|
||||
Christopher Cullen: The Suan shu shu Writings on reckoning, Needham Research Institute
|
||||
Cullen, Christopher (2007). "The Suàn shù shū, "Writings on reckoning": Rewriting the history of early Chinese mathematics in the light of an excavated manuscript". Historia Mathematica. 34: 10–44. doi:10.1016/j.hm.2005.11.006.
|
||||
Dauben, Joseph W. (2004). "The Suan Shu Shu (A Book on Numbers and Computation), A Preliminary Investigation" in Form, Zahl, Ordnung, 151–168. München: Franz Steiner Verlag. ISBN 3-515-08525-4.
|
||||
Dauben, Joseph W. (2007). "Chinese Mathematics" in The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, 187–384. Edited by Victor J. Katz. Princeton: Princeton University Press. ISBN 0-691-11485-4.
|
||||
Guilin Liu, Lisheng Feng, Airong Jiang, and Xiaohui Zheng. (2003). The Development of E-mathematics Resources at Tsinghua University Library (THUL)," in Electronic Information and Communication in Mathematics, 1–13. Edited by Fengshen Bai and Bernd Wegner. Berlin: Springer. ISBN 3-540-40689-1.
|
||||
Stephanie Pain, Histories: China's oldest mathematical puzzles, New Scientist, 30 July 2006.
|
||||
Péng Hào, Zhāngjiāshān Hànjiǎn "Suàn shù shū" zhùshì (The Hàn dynasty book on wooden strips "Suàn shù shū" found at Zhāngjiāshān with a commentary and explanation) Beijing, Science Press, (2001).
|
||||
Wu Wenjun ed, Zhong Guo Shu Xue Shi Da Xi(The Grand Series of History of Chinese Mathematics) vol 1, chapter 2, "Suan Shu Shu". ISBN 7-303-04555-4
|
||||
Rémi Anicotte (2019). Le Livre sur les calculs effectués avec des bâtonnets – Un manuscrit du -IIe siècle excavé à Zhangjiashan, Paris: Presses de l'Inalco. http://www.inalco.fr/publication/livre-calculs-effectues-batonnets-manuscrit-iie-siecle-excave-zhangjiashan
|
||||
|
||||
|
||||
== External links ==
|
||||
Christopher Cullen, Suan shu shu Writings on reckoning Needham Research Institute
|
||||
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|
||||
---
|
||||
title: "Braids, Links, and Mapping Class Groups"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Braids,_Links,_and_Mapping_Class_Groups"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:35.756825+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Braids, Links, and Mapping Class Groups is a mathematical monograph on braid groups and their applications in low-dimensional topology. It was written by Joan Birman, and published in 1974 by the Princeton University Press and University of Tokyo Press, as volume 82 of the book series Annals of Mathematics Studies.
|
||||
Although braid groups had been introduced in 1891 by Adolf Hurwitz and formalized in 1925 by Emil Artin, this was the first book devoted to them. It has been described as a "seminal work", one that "laid the foundations for several new subfields in topology".
|
||||
|
||||
|
||||
== Topics ==
|
||||
Braids, Links, and Mapping Class Groups is organized into five chapters and an appendix. The first introductory chapter defines braid groups, configuration spaces, and the use of configuration spaces to define braid groups on arbitrary two-dimensional manifolds. It provides a solution to the word problem for braids, the question of determining whether two different-looking braid presentations really describe the same group element. It also describes the braid groups as automorphism groups of free groups and of multiply-punctured disks.
|
||||
The next three chapters present connections of braid groups to three different areas of mathematics. Chapter 2 concerns applications to knot theory, via Alexander's theorem that every knot or link can be formed by closing off a braid, and provides the first complete proof of the Markov theorem on equivalence of links formed in this way. It also includes material on the conjugacy problem, important in this area because conjugate braids close off to form the same link, and on the "algebraic link problem" (not to be confused with algebraic links) in which one must determine whether two links can be related to each other by finitely many moves of a certain type, equivalent to the homeomorphism of link complements. Chapter 3 concerns representation theory, and includes Fox derivatives and Fox's free differential calculus, the Magnus representation of free groups and the Gassner and Burau representations of braid groups. Chapter 4 concerns the mapping class groups of 2-manifolds, Dehn twists and the Lickorish twist theorem, and plats, braids closed off in a different way than in Alexander's theorem.
|
||||
Chapter 5 is titled "plats and links". It moves from 2-dimensional topology to 3-dimensional topology, and is more speculative, concerning connections between braid groups, 3-manifolds, and the classification of links. It includes also an analog of Alexander's theorem for plats, where the number of strands of the resulting plat turns out to be determined by the bridge number of a given link. The appendix provides a list of 34 open problems. By the time Wilbur Whitten wrote his review, in June 1975, a handful of these had already been solved.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
This is a book for advanced mathematics students and professionals, who are expected to already be familiar with algebraic topology and presentations of groups by generators and relators. Although it is not a textbook, it could possibly be used for graduate seminars.
|
||||
Reviewer Lee Neuwirth calls the book "most readable", "a nice mix of known results on the subject and new material". Whitten describes it as "thorough, skillfully written" and "a pleasure to read". Wilhelm Magnus finds it "remarkable" that while covering the subject with full mathematical rigor, Birman has preserved the intuitive appeal of some of its earliest works.
|
||||
|
||||
|
||||
== References ==
|
||||
@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Calendrical_Calculations"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:35:53.364318+00:00"
|
||||
date_saved: "2026-05-05T08:43:38.156804+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
28
data/en.wikipedia.org/wiki/Canon_arithmeticus-0.md
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28
data/en.wikipedia.org/wiki/Canon_arithmeticus-0.md
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|
||||
---
|
||||
title: "Canon arithmeticus"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Canon_arithmeticus"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:39.276631+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Canon arithmeticus is a set of mathematical tables of indices and powers with respect to primitive roots for prime powers less than 1000, originally published by Carl Gustav Jacob Jacobi (1839), with introductory text in Latin. The tables were at one time used for arithmetical calculations modulo prime powers, though like many mathematical tables, they have now been replaced by digital computers. Jacobi also reproduced Burkhardt's table of the periods of decimal fractions of 1/p and Ostrogradsky's tables of primitive roots of primes less than 200 and gave tables of indices of some odd numbers modulo powers of 2 with respect to the base 3 (Dickson 2005, p. 185–186).
|
||||
Although the second edition of 1956 has Jacobi's name on the title, it has little in common with the first edition apart from the topic: the tables were completely recalculated, usually with a different choice of primitive root, by Wilhelm Patz. Jacobi's original tables use 10 or −10 or a number with a small power of this form as the primitive root whenever possible, while the second edition uses the smallest possible positive primitive root (Fletcher 1958).
|
||||
The term "canon arithmeticus" is occasionally used to mean any table of indices and powers of primitive roots.
|
||||
|
||||
|
||||
== References ==
|
||||
Dickson, Leonard Eugene (2005) [1919], History of the theory of numbers, vol. I: Divisibility and primality, New York: Dover Publications, ISBN 978-0-486-44232-7, MR 0245499
|
||||
Fletcher, A. (1958), "Canon Arithmeticus by C. G. J. Jacobi; H. Brandt", The Mathematical Gazette, Review, 42 (339), The Mathematical Association: 76–77, doi:10.2307/3608400, ISSN 0025-5572, JSTOR 3608400, S2CID 246262528
|
||||
Jacobi, Carl Gustav Jacob (1839), Canon arithmeticus, sive tabulae quibus exhibentur pro singulis numeris primis vel primorum potestatibus infra 1000 numeri ad datos indices et indices ad datos numeros pertinentes (in Latin), Berlin: Typis Academicis, Berolini, MR 0081559
|
||||
Jacobi, Carl Gustav Jacob (1956) [1839], Brandt, Heinrich; Patz, Wilhelm (eds.), Canon arithmeticus, Mathematische Lehrbücher und Monographien: Mathematische Monographien (in German), vol. 2, Berlin: Akademie-Verlag, MR 0081559
|
||||
|
||||
|
||||
== See also ==
|
||||
A. W. Faber Model 366, a discrete slide rule incorporating similar concepts to the Canon arithmeticus
|
||||
|
||||
|
||||
== External links ==
|
||||
The 1839 edition of the Canon arithmeticus at the Internet Archive
|
||||
27
data/en.wikipedia.org/wiki/Cardinal_and_Ordinal_Numbers-0.md
Normal file
27
data/en.wikipedia.org/wiki/Cardinal_and_Ordinal_Numbers-0.md
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|
||||
---
|
||||
title: "Cardinal and Ordinal Numbers"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Cardinal_and_Ordinal_Numbers"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:40.431703+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Cardinal and Ordinal Numbers is a book on transfinite numbers, by Polish mathematician Wacław Sierpiński. It was published in 1958 by Państwowe Wydawnictwo Naukowe, as volume 34 of the series Monografie Matematyczne of the Institute of Mathematics of the Polish Academy of Sciences. Sierpiński wrote on the same topic earlier, in his 1928 book Leçons sur les nombres transfinis, but his 1958 book on the topic was completely rewritten and significantly longer. A second edition of Cardinal and Ordinal Numbers was published in 1965.
|
||||
|
||||
|
||||
== Topics ==
|
||||
After five introductory chapters on naive set theory and set-theoretic notation, and a sixth chapter on the axiom of choice, the book has four chapters on cardinal numbers, their arithmetic, and series and products of cardinal numbers, comprising approximately 50 pages. Following this, four longer chapters (totalling roughly 180 pages) cover orderings of sets, order types, well-orders, ordinal numbers, ordinal arithmetic, and the Burali-Forti paradox according to which the collection of all ordinal numbers cannot be a set. Three final chapters concern aleph numbers and the continuum hypothesis, statements equivalent to the axiom of choice, and consequences of the axiom of choice.
|
||||
The second edition makes only minor changes to the first except for adding footnotes concerning two later developments in the area: the proof by Paul Cohen of the independence of the continuum hypothesis, and the construction by Robert M. Solovay of the Solovay model in which all sets of real numbers are Lebesgue measurable.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
Sierpiński was known for his significant contributions to the theory of transfinite numbers;, reviewer Reuben Goodstein calls his book "a goldmine of results", and similarly Leonard Gillman writes that it is highly valuable "as a compendium of interesting mathematical information, presented with care and clarity". Both Gillman and John C. Oxtoby call the writing style "leisurely" and "unhurried", and although Gillman criticizes the translation from an earlier Polish-language manuscript into English as unpolished, and points to several errors in the bibliography, he does not find the writing in the text of the book to be problematic.
|
||||
In the 1970 text General Topology by Stephen Willard, Willard lists this book as one of five "standard references" on set theory.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== Further reading ==
|
||||
28
data/en.wikipedia.org/wiki/Chases_and_Escapes-0.md
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28
data/en.wikipedia.org/wiki/Chases_and_Escapes-0.md
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|
||||
---
|
||||
title: "Chases and Escapes"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Chases_and_Escapes"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:41.600966+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Chases and Escapes: The Mathematics of Pursuit and Evasion is a mathematics book on continuous pursuit–evasion problems. It was written by Paul J. Nahin, and published by the Princeton University Press in 2007. It was reissued as a paperback reprint in 2012. The Basic Library List Committee of the Mathematical Association of America has rated this book as essential for inclusion in undergraduate mathematics libraries.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The book has four chapters, covering the solutions to 21 continuous pursuit–evasion problems, with an additional 10 "challenge problems" left for readers to solve, with solutions given in an appendix. The problems are presented as entertaining stories that "breathe life into the mathematics and invite wider engagement", and their solutions use varied methods, including the computer calculation of numerical solutions for differential equations whose solutions have no closed form.
|
||||
Most of the material was previously known, but is collected here for the first time. The book also provides background material on the history of the problems it describes, although this is not its main focus.
|
||||
Even before beginning its main content, the preface of the book begins with an example of pure evasion from known pursuit, the path used by the Enola Gay to escape the blast of the nuclear bomb it dropped on Hiroshima. The first chapter of the book concerns the opposite situation of "pure pursuit" without evasion, including the initial work in this area by Pierre Bouguer in 1732. Bouger studied a problem of pirates chasing a merchant ship, in which the merchant ship (unaware of the pirates) travels on a straight line while the pirate ship always travels towards the current position of the merchant ship. The resulting pursuit curve is called a radiodrome, and this chapter studies several similar problems and stories involving a linearly moving target, including variations where the pursuer may aim ahead of the target and the tractrix curve generated by a pursuer that follows the target at constant distance.
|
||||
Chapter 2 considers targets moving to evade their pursuers, beginning with an example of circular evasive motion described in terms of a dog chasing a duck in a pond, with the dog beginning at the center and the duck moving circularly around the bank. Other variants considered in this chapter include cases where the target is hidden from view, and moving on an unknown trajectory. Chapter 3 considers "cyclic pursuit" problems in which multiple agents pursue each other, as in the mice problem.
|
||||
The fourth and final chapter is entitled "Seven classic evasion problems". It begins with a problem from Martin Gardner's Mathematical Games, the reverse of the dog-and-duck problem, in which a person on a raft in a circular lake tries to reach the shore before a pursuer on land reaches the same point. It also includes hide-and-seek problems and their formulation using game theory, and the work of Richard Rado and Abram Samoilovitch Besicovitch on a man and lion of equal speed trapped in a circular arena, with the lion trying to catch the man, first popularized in A Mathematician's Miscellany by J. E. Littlewood.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
The book assumes an undergraduate-level understanding of calculus and differential equations. It also uses some game theory but its coverage of the necessary material in this area is self-contained. It is not a textbook, but could be used to provide motivating examples for courses in calculus and differential equations, or as the basis of an undergraduate research project to a student who has completed this material.
|
||||
As well, the book may be of interest to any reader with the requisite background who enjoys mathematics.
|
||||
Game theorist Gerald A. Heuer writes that "The treatment in general is very good, and readers are likely to appreciate the author's friendly and lively writing style." On the other hand, Mark Colyvan, a philosopher, would have preferred to see heavier coverage of the game-theoretic aspects of the subject, and notes that the mathematical idealizations used here can lead to inaccurate conclusions for real-world problems. Despite these quibbles, Colyvan writes that "this book provides an excellent vehicle to pursue the mathematics in question, and the mathematics in question is most certainly worth pursuing". Reviewer Bill Satzer calls the book "highly readable", and reviewer Justin Mullins writes that author Paul Nahin "guides us masterfully through the maths".
|
||||
|
||||
|
||||
== References ==
|
||||
41
data/en.wikipedia.org/wiki/Chicka_Chicka_1,_2,_3-0.md
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41
data/en.wikipedia.org/wiki/Chicka_Chicka_1,_2,_3-0.md
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@ -0,0 +1,41 @@
|
||||
---
|
||||
title: "Chicka Chicka 1, 2, 3"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Chicka_Chicka_1,_2,_3"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:43.953353+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Chicka Chicka 1, 2, 3 is the title of a children's picture book written by Bill Martin, Jr. and Michael Sampson, and illustrated by Lois Ehlert in 2004. It was published by Simon & Schuster. It is a sequel to 1989's Chicka Chicka Boom Boom.
|
||||
|
||||
|
||||
== Plot ==
|
||||
Anthropomorphic numbers from 1 to 20 consecutively (including 5 wearing a top hat), then 30 to 90 by tens (including 70 with long hair), and finally 99, climb up an apple tree. While watching them climb, the number 0 tries to find a place available for him in the tree.
|
||||
However, 0 soon realizes there is no more room left for him, until a colony of bumblebees make everyone in the tree (except 10 who hides) fall out, counting backwards. While this happens, a few of the numbers are revealed to have suffered certain injuries from the fall.
|
||||
The number 0 finally finds his place in the tree and goes to the top, joining with 10 and forming the large number 100 to scare the bees away. Then all the other numbers return and climb back up the apple tree, cheering for 10 and 0's bravery.
|
||||
|
||||
|
||||
== Development ==
|
||||
The publisher, Simon & Schuster, originally asked Bill Martin, Jr. to write a sequel to his book Chicka Chicka Boom Boom. But when he and co-author Michael Sampson turned the manuscript in, it was rejected. That manuscript was published by Henry Holt as the title Rock It, Sock It, Number Line. Five years later, Martin and Sampson wrote a second counting book, and it became Chicka Chicka 1, 2, 3.
|
||||
|
||||
|
||||
== Reception ==
|
||||
The book quickly became a best-seller, and is used by teachers throughout the United States to teach counting and place value to young children.
|
||||
|
||||
|
||||
== Awards ==
|
||||
The book has won numerous awards from a variety of publications, libraries, and parenting groups, including Best Book of 2004 by Parenting Magazine.
|
||||
|
||||
|
||||
== Adaptations ==
|
||||
At the same year when the book was published, Weston Woods Studios made an animated musical short film adaptation of the story. As with the original Chicka Chicka Boom Boom cartoon, its music was composed and performed by Crystal Taliefero.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Michael Sampson personal website
|
||||
Bill Martin, Jr. personal website
|
||||
14
data/en.wikipedia.org/wiki/Clavis_mathematicae-0.md
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14
data/en.wikipedia.org/wiki/Clavis_mathematicae-0.md
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|
||||
---
|
||||
title: "Clavis mathematicae"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Clavis_mathematicae"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:47.337039+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Clavis mathematicae (English: The Key of Mathematics) is a mathematics book written by William Oughtred, originally published in 1631 in Latin. It was an attempt to communicate the contemporary mathematical practices, and the European history of mathematics, into a concise and digestible form. The book contains an addition in its 1647 English edition, "Easy Way of Delineating Sun-Dials by Geometry", which had been written by Oughtred earlier in life. The original edition brought the autodidactic Oughtred acclaim amongst mathematicians, but the English-language reissue brought him celebrity, especially amongst tradesman who made use of the arithmetic in their labors. The book is also notable for using the symbol "x" for multiplication, a method invented by Oughtred.
|
||||
|
||||
|
||||
== References ==
|
||||
38
data/en.wikipedia.org/wiki/Color_and_Symmetry-0.md
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38
data/en.wikipedia.org/wiki/Color_and_Symmetry-0.md
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@ -0,0 +1,38 @@
|
||||
---
|
||||
title: "Color and Symmetry"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Color_and_Symmetry"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:48.503230+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Color and Symmetry is a book by Arthur L. Loeb published by Wiley Interscience in 1971. The author adopts an unconventional algorithmic approach to generating the line and plane groups based on the concept of "rotocenter" (the invariant point of a rotation). He then introduces the concept of two or more colors to derive all of the plane dichromatic symmetry groups and some of the polychromatic symmetry groups.
|
||||
|
||||
|
||||
== Structure and topics ==
|
||||
The book is divided into three parts. In the first part, chapters 1–7, the author introduces his "algorismic" (algorithmic) method based on "rotocenters" and "rotosimplexes" (a set of congruent rotocenters). He then derives the 7 frieze groups and the 17 wallpaper groups.
|
||||
In the second part, chapters 8–10, the dichromatic (black-and-white, two-colored) patterns are introduced and the 17 dichromatic line groups and the 46 black-and-white dichromatic plane groups are derived.
|
||||
In the third part, chapters 11–22, polychromatic patterns (3 or more colors), polychromatic line groups, and polychromatic plane groups are derived and illustrated. Loeb's synthetic approach does not enable a comparison of colour symmetry concepts and definitions by other authors, and it is therefore not surprising that the number of polychromatic patterns he identifies are different from that published elsewhere.
|
||||
|
||||
|
||||
== Audience ==
|
||||
Unusually, the author does not state the target audience for his book; his publisher, in their dust jacket blurb, say "Color and Symmetry will be of primary interest on the one hand to crystallographers, chemists, material scientists, and mathematicians. On the other hand, this volume will serve the interests of those active in the fields of design, visual and environmental studies and architecture."
|
||||
Only a school-level mathematical background is required to follow the author's logical development of his argument. Group theory is not used in the book, which is beneficial to readers without this specific mathematical background, but it makes some of the material more long-winded than it would be if it had been developed using standard group theory.
|
||||
Michael Holt in his review for Leonardo said: "In this erudite and handsomely presented monograph, then, designers should find a rich source of explicit rules for pattern-making and mathematicians and crystallographers a welcome and novel slant on symmetry operations with colours."
|
||||
|
||||
|
||||
== Reception ==
|
||||
The book had a generally positive reception from contemporary reviewers. W.E. Klee in a review for Acta Crystallographica wrote: "Color and Symmetry will surely stimulate new interest in colour symmetries and will be of special interest to crystallographers. People active in design may also profit from this book." D.M. Brink in a review for Physics Bulletin published by the Institute of Physics said: "The book will be useful to workers with a technical interest in periodic structures and also to more general readers who are fascinated by symmetrical patterns. The illustrations encourage the reader to understand the mathematical structure underlying the patterns."
|
||||
J.D.H. Donney in a review for Physics Today said: "This book should prove useful to physicists, chemists, crystallographers
|
||||
(of course), but also to decorators and designers, from textiles to ceramics. It will be enjoyed, not only by mathematicians, but by all lovers of orderliness, logic and beauty." David Harker in a review for Science said: "It may well be that this work will become a classic essay on planar color symmetry."
|
||||
|
||||
|
||||
== Criticism ==
|
||||
The author's idiosyncratic approach was not adopted by researchers in the field, and later assessments of Loeb's contribution to color symmetry were more critical of his work than earlier reviewers had been. Marjorie Senechal said that Loeb's work on polychromatic patterns, whilst not wrong, imposed artificial restrictions which meant that some valid colored patterns with three or more colors were excluded from his lists.
|
||||
R.L.E. Schwarzenberger in 1980 said: "The study of colour symmetry has been bedevilled by a lack of precise definitions when the number of colours is greater than two ... it is unfortunate that this paper was apparently ignored by Shubnikov and Loeb whose books give incomplete and unsystematic listings." In a 1984 review paper Schwarzenberger remarks: "... these authors [including Loeb] confine themselves to a restricted class of colour group ... for N > 2 the effect is to dramatically limit the number of colour groups considered."
|
||||
Branko Grünbaum and G.C. Shephard in their book Tilings and patterns gave an assessment of previous work in the field. Commenting on Color and Symmetry they said:"Loeb gives an original, interesting and satisfactory account of the 2-color groups ... unfortunately when discussing multicolor patterns, Loeb restricts the admissible color changes so severely that he obtains a total of only 54 periodic k-color configurations with k ≥ 3." Later authors determined that the total number of k-color configurations with 3 ≤ k ≤ 12 is 751.
|
||||
|
||||
|
||||
== References ==
|
||||
35
data/en.wikipedia.org/wiki/Colored_Symmetry-0.md
Normal file
35
data/en.wikipedia.org/wiki/Colored_Symmetry-0.md
Normal file
@ -0,0 +1,35 @@
|
||||
---
|
||||
title: "Colored Symmetry"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Colored_Symmetry"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:49.656087+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Colored Symmetry is a book by A.V. Shubnikov and N.V. Belov and published by Pergamon Press in 1964. The book contains translations of materials originally written in Russian and published between 1951 and 1958. The book was notable because it gave English-language speakers access to new work in the fields of dichromatic and polychromatic symmetry.
|
||||
|
||||
|
||||
== Structure and topics ==
|
||||
The book is divided into two parts. The first part is a translation into English of A.V. Shubnikov's book Symmetry and antisymmetry of finite figures (Russian: Симметрия и антисимметрия конечных фигур) originally published in 1951. As the editor says in his preface, this book rekindled interest in the field of antisymmetry after a break of 20 years. The book defines symmetry elements, operations and groups; it then introduces the concept of antisymmetry, and derives the full set of dichromatic three-dimensional point groups. A paper entitled Antisymmetry of textures is appended to part 1; it analyzes the antisymmetry of groups containing infinity-fold axes.
|
||||
The second part, entitled Infinite groups of colored symmetry, consists of translations of six papers by N.V. Belov and his co-workers in the new field of polychromatic symmetry. These papers cover the derivation of the 42 magnetic Bravais lattices and the 1651 magnetic space groups, the 46 dichromatic plane groups, mosaics for the 46 dichromatic plane groups, one-dimensional infinite crystallographic groups, polychromatic plane groups, and three-dimensional mosaics with colored symmetry.
|
||||
|
||||
|
||||
== Audience ==
|
||||
The book is written for crystallographers, mathematicians and physics researchers who are interested in the application of color symmetry to crystal structure analysis and physics experiments involving magnetic or ferroelectric materials.
|
||||
|
||||
|
||||
== Reception ==
|
||||
The book had a mixed reception from reviewers. Allen Nussbaum in American Scientist praised the editor for constructing a consistent story from the original works, but criticised the papers in part two for being difficult to read. G.S. Pawley in a review for Science Progress gave credit to the editor for adding the international notation next to the authors' "retrograde personal notation". However, he criticised claims that the book is a "valuable reference book" as being "optimistic". Martin Buerger in an extensive review for Science also offered both praise and criticism. He stated that previous work in the field by William Barlow and H.J. Woods is not given sufficient credit by the authors and is largely missing from the, otherwise full, bibliography. He praised Shubnikov's book (part 1) as being "very clearly written, well illustrated, and easy to understand", but criticised Belov's papers in part 2 because they "lack a central unifying theme." R.J. Davis in a brief review in Mineralogical Magazine said "this book is therefore unique in English and forms an essential introduction to modern developments in symmetry theory."
|
||||
|
||||
|
||||
== Influence ==
|
||||
In later reviews of the literature by R.L.E. Schwarzenberger and by Branko Grünbaum and G.C. Shephard in their book Tilings and patterns the work of the Russian color symmetry school led by A.V. Shubnikov and N.V. Belov was put into its proper historical context. Schwarzenberger, and Grünbaum and Shephard, give credit to Shubnikov and Belov for relaunching the field of color symmetry after the work of Heinrich Heesch and H.J. Woods in the 1930s was largely ignored. However, they criticise Shubnikov and Belov for taking a crystallographic rather than a group-theoretic approach, and for using their own confusing notation rather than adopting the international standard Hermann–Mauguin notation for crystallographic symmetry elements.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
"Colored Symmetry". The Macmillan Company. 1964. at the Internet Archive
|
||||
0
data/en.wikipedia.org/wiki/Complexities
Normal file
0
data/en.wikipedia.org/wiki/Complexities
Normal file
@ -0,0 +1,36 @@
|
||||
---
|
||||
title: "Complexity and Real Computation"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Complexity_and_Real_Computation"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:51.906904+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Complexity and Real Computation is a book on the computational complexity theory of real computation. It studies algorithms whose inputs and outputs are real numbers, using the Blum–Shub–Smale machine as its model of computation. For instance, this theory is capable of addressing a question posed in 1991 by Roger Penrose in The Emperor's New Mind: "is the Mandelbrot set computable?"
|
||||
The book was written by Lenore Blum, Felipe Cucker, Michael Shub and Stephen Smale, with a foreword by Richard M. Karp, and published by Springer-Verlag in 1998 (doi:10.1007/978-1-4612-0701-6, ISBN 0-387-98281-7).
|
||||
|
||||
|
||||
== Purpose ==
|
||||
Stephen Vavasis observes that this book fills a significant gap in the literature: although theoretical computer scientists working on discrete algorithms had been studying models of computation and their implications for the complexity of algorithms since the 1970s, researchers in numerical algorithms had for the most part failed to define their model of computation, leaving their results on a shaky foundation. Beyond the goal of making this aspect of the topic more well-founded, the book also has the aims of presenting new results in the complexity theory of real-number computation, and of collecting previously-known results in this theory.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The introduction of the book reprints the paper "Complexity and real computation: a manifesto", previously published by the same authors. This manifesto explains why classical discrete models of computation such as the Turing machine are inadequate for the study of numerical problems in areas such as scientific computing and computational geometry, motivating the newer model studied in the book. Following this, the book is divided into three parts.
|
||||
Part I of the book sets up models of computation over any ring, with unit cost per ring operation. It provides analogues of recursion theory and of the P versus NP problem in each case, and proves the existence of NP-complete problems analogously to the proof of the Cook–Levin theorem in the classical model, which can be seen as the special case of this theory for modulo-2 arithmetic. The ring of integers is studied as a particular example, as are the algebraically closed fields of characteristic zero, which are shown from the point of view of NP-completeness within their computational models to all be equivalent to the complex numbers. (Eric Bach points out that this equivalence can be seen as a form of the Lefschetz principle.)
|
||||
Part II focuses on numerical approximation algorithms, on the use of Newton's method for these algorithms, and on author Stephen Smale's alpha theory for numerical certification of the accuracy of the results of these computations. Other topics considered in this section include finding the roots of polynomials and the intersection points of algebraic curves, the condition number of systems of equations, and the time complexity of linear programming with rational coefficients.
|
||||
Part III provides analogues of structural complexity theory and descriptive complexity theory for real-number computation, including many separations of complexity classes that are provable in this theory even though the analogous separations in classical complexity theory remain unproven. A key tool in this area is the use of the number of connected components of a semialgebraic set to provide a lower bound on the time complexity of an associated computational problem.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
The book is aimed at the level of a graduate student or researcher in these topics, and in places it assumes background knowledge of classical computational complexity theory, differential geometry, topology, and dynamical systems.
|
||||
Reviewer Klaus Meer writes that the book is "very well written", "perfect to use on a graduate level", and well-represents both the state of the art in this area and the strong connections that can be made between fields as diverse as algebraic number theory, algebraic geometry, mathematical logic, and numerical analysis.
|
||||
As a minor criticism, aimed more at the Blum–Shub–Smale model than the book, Stephen Vavasis observes that (unlike with Turing machines) seemingly-minor details in the model, such as the ability to compute the floor and ceiling functions, can make big differences in what is computable and how efficiently it can be computed. However, Vavasis writes, "this difficulty is probably inherent in the topic". Relatedly, Eric Bach complains that assigning unit cost to all arithmetic operations can give a misleading idea of a problem's complexity in practical computation, and Vavasis also notes that, as of the publication date of his review, this work had seemingly had little effect on practical research in scientific computing. Despite these issues, he recommends the book as a convenient and clearly-written compendium of the theory of numerical computing.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Complexity and Real Computation at the Internet Archive
|
||||
@ -0,0 +1,26 @@
|
||||
---
|
||||
title: "Computability in Analysis and Physics"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Computability_in_Analysis_and_Physics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:53.082106+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Computability in Analysis and Physics is a monograph on computable analysis by Marian Pour-El and J. Ian Richards. It was published by Springer-Verlag in their Perspectives in Mathematical Logic series in 1989, and reprinted by the Association for Symbolic Logic and Cambridge University Press in their Perspectives in Logic series in 2016.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The book concerns computable analysis, a branch of mathematical analysis founded by Alan Turing and concerned with the computability of constructions in analysis. This area is connected to, but distinct from, constructive analysis, reverse mathematics, and numerical analysis. The early development of the field was summarized in a book by Oliver Aberth, Computable Analysis (1980), and Computability in Analysis and Physics provides an update, incorporating substantial developments in this area by its authors. In contrast to the Russian school of computable analysis led by Andrey Markov Jr., it views computability as a distinguishing property of mathematical objects among others, rather than developing a theory that concerns only computable objects.
|
||||
After an initial section of the book, introducing computable analysis and leading up to an example of John Myhill of a computable continuously differentiable function whose derivative is not computable, the remaining two parts of the book concerns the authors' results. These include the results that, for a computable self-adjoint operator, the eigenvalues are individually computable, but their sequence is (in general) not; the existence of a computable self-adjoint operator for which 0 is an eigenvalue of multiplicity one with no computable eigenvectors; and the equivalence of computability and boundedness for operators. The authors' main tools include the notions of a computability structure, a pair of a Banach space and an axiomatically characterized set of its sequences, and of an effective generating set, a member of the set of sequences whose linear span is dense in the space.
|
||||
The authors are motivated in part by the computability of solutions to differential equations. They provide an example of computable and continuous initial conditions for the wave equation (with however a non-computable gradient) that lead to a continuous but not computable solution at a later time. However, they show that this phenomenon cannot occur for the heat equation or for Laplace's equation.
|
||||
The book also includes a collection of open problems, likely to inspire its readers to more research in this area.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
The book is self-contained, and targeted at researchers in mathematical analysis and computability; reviewers Douglas Bridges and Robin Gandy disagree over which of these two groups it is better aimed at. Although co-author Marian Pour-El came from a background in mathematical logic, and the two series in which the book was published both have logic in their title, readers are not expected to be familiar with logic.
|
||||
Despite complaining about the formality of the presentation and that the authors did not aim to include all recent developments in computable analysis, reviewer Rod Downey writes that this book "is clearly a must for anybody whose research is in this area", and Gandy calls it "an interesting, readable and very well written book".
|
||||
|
||||
|
||||
== References ==
|
||||
@ -0,0 +1,21 @@
|
||||
---
|
||||
title: "Concepts of Modern Mathematics"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Concepts_of_Modern_Mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:54.236463+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Concepts of Modern Mathematics is a book by mathematician and science popularizer Ian Stewart about then-recent developments in mathematics. It was originally published by Penguin Books in 1975, updated in 1981, and reprinted by Dover publications in 1995 and 2015.
|
||||
|
||||
|
||||
== Overview ==
|
||||
The book arose out of an extramural class that Ian Stewart taught at the University of Warwick about "Modern mathematics". In the 1995 Dover edition Stewart wrote that the aim of the class was:
|
||||
|
||||
to explain why the underlying abstract point of view had gained currency among research mathematicians, and to examine how it opened up entirely new realms of mathematical thought.
|
||||
The book is aimed at non-mathematicians. However, there are frequent equations and diagrams and the level of presentation is more technical than some of Stewart's other popular books such as Flatterland. Topics covered include analytic geometry, set theory, abstract algebra, group theory, topology, and probability.
|
||||
|
||||
|
||||
== References ==
|
||||
41
data/en.wikipedia.org/wiki/Convex_Polyhedra_(book)-0.md
Normal file
41
data/en.wikipedia.org/wiki/Convex_Polyhedra_(book)-0.md
Normal file
@ -0,0 +1,41 @@
|
||||
---
|
||||
title: "Convex Polyhedra (book)"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Convex_Polyhedra_(book)"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:56.619640+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Convex Polyhedra is a book on the mathematics of convex polyhedra, written by Soviet mathematician Aleksandr Danilovich Aleksandrov, and originally published in Russian in 1950, under the title Выпуклые многогранники. It was translated into German by Wilhelm Süss as Konvexe Polyeder in 1958. An updated edition, translated into English by Nurlan S. Dairbekov, Semën Samsonovich Kutateladze and Alexei B. Sossinsky, with added material by Victor Zalgaller, L. A. Shor, and Yu. A. Volkov, was published as Convex Polyhedra by Springer-Verlag in 2005.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The main focus of the book is on the specification of geometric data that will determine uniquely the shape of a three-dimensional convex polyhedron, up to some class of geometric transformations such as congruence or similarity. It considers both bounded polyhedra (convex hulls of finite sets of points) and unbounded polyhedra (intersections of finitely many half-spaces).
|
||||
The 1950 Russian edition of the book included 11 chapters. The first chapter covers the basic topological properties of polyhedra, including their topological equivalence to spheres (in the bounded case) and Euler's polyhedral formula. After a lemma of Augustin Cauchy on the impossibility of labeling the edges of a polyhedron by positive and negative signs so that each vertex has at least four sign changes, the remainder of chapter 2 outlines the content of the remaining book. Chapters 3 and 4 prove Alexandrov's uniqueness theorem, characterizing the surface geometry of polyhedra as being exactly the metric spaces that are topologically spherical locally like the Euclidean plane except at a finite set of points of positive angular defect, obeying Descartes' theorem on total angular defect that the total angular defect should be
|
||||
|
||||
|
||||
|
||||
4
|
||||
π
|
||||
|
||||
|
||||
{\displaystyle 4\pi }
|
||||
|
||||
. Chapter 5 considers the metric spaces defined in the same way that are topologically a disk rather than a sphere, and studies the flexible polyhedral surfaces that result.
|
||||
Chapters 6 through 8 of the book are related to a theorem of Hermann Minkowski that a convex polyhedron is uniquely determined by the areas and directions of its faces, with a new proof based on invariance of domain. A generalization of this theorem implies that the same is true for the perimeters and directions of the faces. Chapter 9 concerns the reconstruction of three-dimensional polyhedra from a two-dimensional perspective view, by constraining the vertices of the polyhedron to lie on rays through the point of view. The original Russian edition of the book concludes with two chapters, 10 and 11, related to Cauchy's theorem that polyhedra with flat faces form rigid structures, and describing the differences between the rigidity and infinitesimal rigidity of polyhedra, as developed analogously to Cauchy's rigidity theorem by Max Dehn.
|
||||
The 2005 English edition adds comments and bibliographic information regarding many problems that were posed as open in the 1950 edition but subsequently solved. It also includes in a chapter of supplementary material the translations of three related articles by Volkov and Shor, including a simplified proof of Pogorelov's theorems generalizing Alexandrov's uniqueness theorem to non-polyhedral convex surfaces.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
Robert Connelly writes that, for a work describing significant developments in the theory of convex polyhedra that was however hard to access in the west, the English translation of Convex Polyhedra was long overdue. He calls the material on Alexandrov's uniqueness theorem "the star result in the book", and he writes that the book "had a great influence on countless Russian mathematicians". Nevertheless, he complains about the book's small number of exercises, and about an inconsistent level presentation that fails to distinguish important and basic results from specialized technicalities.
|
||||
Although intended for a broad mathematical audience, Convex Polyhedra assumes a significant level of background knowledge in material including topology, differential geometry, and linear algebra.
|
||||
Reviewer Vasyl Gorkaviy recommends Convex Polyhedra to students and professional mathematicians as an introduction to the mathematics of convex polyhedra. He also writes that, over 50 years after its original publication, "it still remains of great interest for specialists", after being updated to include many new developments and to list new open problems in the area.
|
||||
|
||||
|
||||
== See also ==
|
||||
List of books about polyhedra
|
||||
|
||||
|
||||
== References ==
|
||||
@ -0,0 +1,30 @@
|
||||
---
|
||||
title: "Crocheting Adventures with Hyperbolic Planes"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Crocheting_Adventures_with_Hyperbolic_Planes"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:58.864644+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Crocheting Adventures with Hyperbolic Planes is a book on crochet and hyperbolic geometry by Daina Taimiņa. It was published in 2009 by A K Peters, with a 2018 second edition by CRC Press.
|
||||
|
||||
|
||||
== Topics ==
|
||||
|
||||
The book is on the use of crochet to make physical surfaces with the geometry of the hyperbolic plane. The full hyperbolic plane cannot be embedded smoothly into three-dimensional space, but pieces of it can. Past researchers had made models of these surfaces out of paper, but Taimiņa's work is the first work to do so using textile arts. She had previously described these models in a research paper and used them as illustrations for an undergraduate geometry textbook, but this book describes more of the background for the project, makes it more widely accessible, and provides instructions for others to follow in making these models.
|
||||
The book has nine chapters. The first chapter introduces the notion of the curvature of a surface, provides instructions for an introductory project in crocheting a patch of the hyperbolic plane, and provides an initial warning about the exponential growth in the area of this plane as a function of its radius, which will cause larger crochet projects to take a very long time to complete. Chapter two covers more concepts in the geometry of the hyperbolic plane, connecting them to crocheted models of the plane.
|
||||
The next three chapters take a step back to look at the broader history of the topics discussed in the book: geometry and its connection to human arts and architecture in chapter 3, crochet in chapter 4, and non-Euclidean geometry in chapter 5. Chapters 6, 7, and 8 cover specific geometric objects with negatively-curved surfaces, including the pseudosphere, helicoid, and catenoid, investigate mathematical toys, and use these crocheted models "to explore otherwise hard to visualize objects". A final chapter covers the applications of hyperbolic geometry and its ongoing research interest.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
The book is written for a general audience. However, although suggesting that it has a place on mathematical coffee tables, reviewer Keith Leatham wonders who its real readers are likely to be. Reviewer Hinke Osinga, however, feels that the book can be of interest to readers interested in either crochet or mathematics, rather than (as Leatham suggests) requiring both interests. She writes "I highly recommend this book, perhaps not only as a beautiful coffee-table book with the subtle message that mathematics is fun, but also because crochet is a perfect tool for testing and exploring deep mathematical theories."
|
||||
|
||||
|
||||
== Recognition ==
|
||||
Crocheting Adventures with Hyperbolic Planes won the 2009 Bookseller/Diagram Prize for Oddest Title of the Year.
|
||||
It also won the 2012 Euler Book Prize of the Mathematical Association of America.
|
||||
|
||||
|
||||
== References ==
|
||||
@ -0,0 +1,148 @@
|
||||
---
|
||||
title: "Davenport–Schinzel Sequences and Their Geometric Applications"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Davenport–Schinzel_Sequences_and_Their_Geometric_Applications"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:01.151538+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Davenport–Schinzel Sequences and Their Geometric Applications is a book in discrete geometry. It was written by Micha Sharir and Pankaj K. Agarwal, and published by Cambridge University Press in 1995, with a paperback reprint in 2010.
|
||||
|
||||
|
||||
== Topics ==
|
||||
Davenport–Schinzel sequences are named after Harold Davenport and Andrzej Schinzel, who applied them to certain problems in the theory of differential equations. They are finite sequences of symbols from a given alphabet, constrained by forbidding pairs of symbols from appearing in alternation more than a given number of times (regardless of what other symbols might separate them). In a Davenport–Schinzel sequence of order
|
||||
|
||||
|
||||
|
||||
k
|
||||
|
||||
|
||||
{\displaystyle k}
|
||||
|
||||
, the longest allowed alternations have length
|
||||
|
||||
|
||||
|
||||
k
|
||||
|
||||
|
||||
{\displaystyle k}
|
||||
|
||||
. For instance, a Davenport–Schinzel sequence of order three could have two symbols
|
||||
|
||||
|
||||
|
||||
x
|
||||
|
||||
|
||||
{\displaystyle x}
|
||||
|
||||
and
|
||||
|
||||
|
||||
|
||||
y
|
||||
|
||||
|
||||
{\displaystyle y}
|
||||
|
||||
that appear either in the order
|
||||
|
||||
|
||||
|
||||
x
|
||||
…
|
||||
y
|
||||
…
|
||||
x
|
||||
|
||||
|
||||
{\displaystyle x\dots y\dots x}
|
||||
|
||||
or
|
||||
|
||||
|
||||
|
||||
y
|
||||
…
|
||||
x
|
||||
…
|
||||
y
|
||||
|
||||
|
||||
{\displaystyle y\dots x\dots y}
|
||||
|
||||
, but longer alternations like
|
||||
|
||||
|
||||
|
||||
x
|
||||
…
|
||||
y
|
||||
…
|
||||
x
|
||||
…
|
||||
y
|
||||
|
||||
|
||||
{\displaystyle x\dots y\dots x\dots y}
|
||||
|
||||
would be forbidden. The length of such a sequence, for a given choice of
|
||||
|
||||
|
||||
|
||||
k
|
||||
|
||||
|
||||
{\displaystyle k}
|
||||
|
||||
, can be only slightly longer than its number of distinct symbols. This phenomenon has been used to prove corresponding near-linear bounds on various problems in discrete geometry, for instance showing that the unbounded face of an arrangement of line segments can have complexity that is only slightly superlinear. The book is about this family of results, both on bounding the lengths of Davenport–Schinzel sequences and on their applications to discrete geometry.
|
||||
The first three chapters of the book provide bounds on the lengths of Davenport–Schinzel sequences whose superlinearity is described in terms of the inverse Ackermann function
|
||||
|
||||
|
||||
|
||||
α
|
||||
(
|
||||
n
|
||||
)
|
||||
|
||||
|
||||
{\displaystyle \alpha (n)}
|
||||
|
||||
. For instance, the length of a Davenport–Schinzel sequence of order three, with
|
||||
|
||||
|
||||
|
||||
n
|
||||
|
||||
|
||||
{\displaystyle n}
|
||||
|
||||
symbols, can be at most
|
||||
|
||||
|
||||
|
||||
Θ
|
||||
(
|
||||
n
|
||||
α
|
||||
(
|
||||
n
|
||||
)
|
||||
)
|
||||
|
||||
|
||||
{\displaystyle \Theta (n\alpha (n))}
|
||||
|
||||
, as the second chapter shows; the third concerns higher orders. The fourth chapter applies this theory to line segments,
|
||||
and includes a proof that the bounds proven using these tools are tight: there exist systems of line segments whose arrangement complexity matches the bounds on Davenport–Schinzel sequence length.
|
||||
The remaining chapters concern more advanced applications of these methods. Three chapters concern arrangements of curves in the plane, algorithms for arrangements, and higher-dimensional arrangements, following which the final chapter (comprising a large fraction of the book) concerns applications of these combinatorial bounds to problems including Voronoi diagrams and nearest neighbor search, the construction of transversal lines through systems of objects, visibility problems, and robot motion planning. The topic remains an active area of research and the book poses many open questions.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
Although primarily aimed at researchers, this book (and especially its earlier chapters) could also be used as the textbook for a graduate course in its material. Reviewer Peter Hajnal calls it "very important to any specialist in computational geometry" and "highly recommended to anybody who is interested in this new topic at the border of combinatorics, geometry, and algorithm theory".
|
||||
|
||||
|
||||
== References ==
|
||||
@ -0,0 +1,53 @@
|
||||
---
|
||||
title: "De Beghinselen Der Weeghconst"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/De_Beghinselen_Der_Weeghconst"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:29.997146+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
De Beghinselen der Weeghconst (lit. "The Principles of the Art of Weighing") is a book about statics written by the Flemish physicist Simon Stevin in Dutch. It was published in 1586 in a single volume with De Weeghdaet (lit. "The Act of Weighing"), De Beghinselen des Waterwichts ("The Principles of Hydrostatics") and an Anhang (an appendix). In 1605, there was another edition.
|
||||
|
||||
|
||||
== Importance ==
|
||||
The importance of the book was summarized by the Encyclopædia Britannica:
|
||||
|
||||
In De Beghinselen der Weeghconst (1586; “Statics and Hydrostatics”) Stevin published the theorem of the triangle of forces. The knowledge of this triangle of forces, equivalent to the parallelogram diagram of forces, gave a new impetus to the study of statics, which had previously been founded on the theory of the lever. He also discovered that the downward pressure of a liquid is independent of the shape of its vessel and depends only on its height and base.
|
||||
|
||||
|
||||
== Contents ==
|
||||
The first part consists of two books, together account for 95 pages, here divided into 10 pieces.
|
||||
|
||||
|
||||
=== Book I ===
|
||||
Start: panegyrics, Mission to Rudolf II, Uytspraeck Vande Weerdicheyt of Duytsche Tael, Cortbegryp
|
||||
Bepalinghen and Begheerten (definitions and assumptions)
|
||||
|
||||
Proposal 1 t / m 4: hefboomwet
|
||||
Proposal 5 t / m 12: a balance with weights pilaer
|
||||
Proposition 13 t / m 18: follow-up, with hefwicht, two supports
|
||||
Proposition 19: balance on an inclined plane, with cloot Crans
|
||||
Proposal 20 t / m 28: pilaer with scheefwichten, hanging, body
|
||||
|
||||
|
||||
=== Book II ===
|
||||
Proposal 1 t / m 6: center of gravity boards – triangle, rectilinear flat
|
||||
Proposal 7 t / m 13: trapezium, divide, cut fire
|
||||
Proposition 14 t / m 24: center of gravity of bodies – pillar, pyramid, burner
|
||||
The Weeghdaet
|
||||
The Beghinselen des Waterwichts
|
||||
Anhang
|
||||
Byvough
|
||||
|
||||
|
||||
== See also ==
|
||||
Simon Stevin
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== Further reading ==
|
||||
Stevin, Simon (1955). Dijksterhuis, E.J (ed.). The principal works of Simon Stevin, Mechanics, edition Vol. I, volume (PDF) (in English and Dutch). C.V. Swets and Zeitlinger.
|
||||
14
data/en.wikipedia.org/wiki/De_arte_supputandi-0.md
Normal file
14
data/en.wikipedia.org/wiki/De_arte_supputandi-0.md
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@ -0,0 +1,14 @@
|
||||
---
|
||||
title: "De arte supputandi"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/De_arte_supputandi"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:02.309998+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
De arte supputandi libri quattuor was the first printed work on arithmetic published in England. Published in 1522, it was written by Cuthbert Tunstall, Bishop of London, and based on Luca Pacioli's Summa de arithmetica, geometria, proportioni et proportionalità. It is dedicated to Sir Thomas More.
|
||||
|
||||
|
||||
== References ==
|
||||
@ -0,0 +1,28 @@
|
||||
---
|
||||
title: "De quinque corporibus regularibus"
|
||||
chunk: 1/2
|
||||
source: "https://en.wikipedia.org/wiki/De_quinque_corporibus_regularibus"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:04.742568+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
De quinque corporibus regularibus (sometimes called Libellus de quinque corporibus regularibus) is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is a manuscript, in the Latin language; its title means [the little book] on the five regular solids. It is one of three books known to have been written by della Francesca.
|
||||
Along with the Platonic solids, De quinque corporibus regularibus includes descriptions of five of the thirteen Archimedean solids, and of several other irregular polyhedra coming from architectural applications. It was the first of what would become many books connecting mathematics to art through the construction and perspective drawing of polyhedra, including Luca Pacioli's 1509 Divina proportione (which incorporated without credit an Italian translation of della Francesca's work).
|
||||
Lost for many years, De quinque corporibus regularibus was rediscovered in the 19th century in the Vatican Library and the Vatican copy has since been republished in facsimile.
|
||||
|
||||
== Background ==
|
||||
|
||||
The five Platonic solids (the regular tetrahedron, cube, octahedron, dodecahedron, and icosahedron) were known to della Francesca through two classical sources: Timaeus, in which Plato theorizes that four of them correspond to the classical elements making up the world (with the fifth, the dodecahedron, corresponding to the heavens), and the Elements of Euclid, in which the Platonic solids are constructed as mathematical objects. Two apocryphal books of the Elements concerning the metric properties of the Platonic solids, sometimes called pseudo-Euclid, were also commonly considered to be part of the Elements in the time of della Francesca. It is the material from the Elements and pseudo-Euclid, rather than from Timaeus, that forms della Francesca's main inspiration.
|
||||
The thirteen Archimedean solids, convex polyhedra in which the vertices but not the faces are symmetric to each other, were classified by Archimedes in a book that has long been lost. Archimedes' classification was later briefly described by Pappus of Alexandria in terms of how many faces of each kind these polyhedra have. Della Francesca had previously studied and copied the works of Archimedes, and includes citations to Archimedes in De quinque corporibus regularibus. But although he describes six of the Archimedean solids in his books (five in De quinque corporibus regularibus), this appears to be an independent rediscovery; he does not credit Archimedes for these shapes and there is no evidence that he knew of Archimedes' work on them. Similarly, although both Archimedes and Della Francesca found formulas for the volume of a cloister vault (see below), their work on this appears to be independent, as Archimedes' volume formula remained unknown until the early 20th century.
|
||||
De quinque corporibus regularibus is one of three books known to have been written by della Francesca. The other two, De prospectiva pingendi and Trattato d'abaco, concern perspective drawing and arithmetic in the tradition of Fibonacci's Liber Abaci, respectively. The other mathematical book, Trattato d'abaco, was part of a long line of abbacist works, teaching arithmetic, accounting, and basic geometrical calculations through many practical exercises, beginning with the work of Fibonacci in his book Liber Abaci (1202). Although the early parts of De quinque corporibus regularibus also borrow from this line of work, and overlap extensively with Trattato d'abaco, Fibonacci and his followers had previously applied their calculation methods only in two-dimensional geometry. The later parts of De quinque corporibus regularibus are more original in their application of arithmetic to the geometry of three-dimensional shapes.
|
||||
|
||||
== Contents ==
|
||||
|
||||
After its dedication, the title page of De quinque corporibus regularibus begins Petri pictoris Burgensis De quinque corporibus regularibus. The first three words mean "Of Peter the painter, from Borgo", and refer to the book's author, Piero della Francesca (from Borgo Santo Sepolcro); the title proper begins after that. A decorative initial begins the text of the book.
|
||||
The first of the book's four parts concerns problems in plane geometry, primarily concerning the measurement of polygons, such as calculating their area, perimeter, or side length, given a different one of these quantities. The second part concerns the circumscribed spheres of the Platonic solids, and asks similar questions on lengths, areas, or volumes of these solids relative to the measurements of the sphere that surrounds them. It also includes the (very likely novel) derivation for the height of an irregular tetrahedron, given its side lengths,
|
||||
equivalent (using the standard formula relating height and volume of tetrahedra) to a form of Heron's formula for tetrahedra.
|
||||
The third part includes additional exercises on circumscribed spheres, and then considers pairs of Platonic solids inscribed one within another, again focusing on their relative measurements.
|
||||
This part is inspired most directly by the 15th (apocryphal) book of the Elements,
|
||||
which constructs certain inscribed pairs of polyhedral figures (for instance, a regular tetrahedron inscribed within a cube and sharing its four vertices with the four of the cube). De quinque corporibus regularibus aims to arithmetize these constructions, making it possible to calculate the measurements for one polyhedron given measurements of the other.
|
||||
@ -0,0 +1,37 @@
|
||||
---
|
||||
title: "De quinque corporibus regularibus"
|
||||
chunk: 2/2
|
||||
source: "https://en.wikipedia.org/wiki/De_quinque_corporibus_regularibus"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:04.742568+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The fourth and final part of the book concerns other shapes than the Platonic solids. These include six Archimedean solids: the truncated tetrahedron (which appears also in an exercise in his Trattato d'abaco), and the truncations of the other four Platonic solids. The cuboctahedron, another Archimedean solid, is described in the Trattato but not in De quinque corporibus regularibus; since De quinque corporibus regularibus appears to be a later work than the Trattato, this omission appears to be deliberate, and a sign that della Francesca was not aiming for a complete listing of these polyhedra. The fourth part of De quinque corporibus regularibus also includes domed shapes like the domes of the Pantheon, Rome or the (at the time newly constructed) Santa Maria presso San Satiro in Milan formed from a ring of triangles surrounded by concentric rings of irregular quadrilaterals, and other shapes arising in architectural applications. The result that Peterson (1997) calls della Francesca's "most sophisticated" is the derivation of the volume of a Steinmetz solid (the intersection of two cylinders, the shape of a cloister vault), which della Francesca had illustrated in his book on perspective. Despite its curves, this shape has a simple but non-obvious formula for its volume, 2/3 of the volume of its enclosing cube. This result was known to both Archimedes and, in ancient China, Zu Chongzhi, but della Francesca was unaware of either prior discovery.
|
||||
De quinque corporibus regularibus is illustrated in a variety of styles by della Francesca, not all of which are in correct mathematical perspective. It includes many exercises, roughly half of which overlap with the geometric parts of della Francesca's Trattato d'abaco, translated from the Italian of the Trattato to the Latin of the De quinque corporibus regularibus.
|
||||
|
||||
== Dissemination ==
|
||||
Della Francesca dedicated De quinque corporibus regularibus to Guidobaldo da Montefeltro, the Duke of Urbino. Although the book is not dated, this dedication narrows the date of its completion to the range from 1482 when Guidobaldo, ten years old, became duke, until 1492 when Della Francesca died. However, della Francesca likely wrote his book first in Italian, before translating it into Latin either himself or with the assistance of a friend, Matteo dal Borgo, so its original draft may have been from before Guidobaldo's accession. In any case, the book was added to the library of the duke. It was kept there together with della Francesca's book on perspective, which he had dedicated to the previous duke.
|
||||
In what has been called "probably the first full-blown case of plagiarism in the history of mathematics", Luca Pacioli copied exercises from Trattato d'abaco into his 1494 book Summa de arithmetica, and then, in his 1509 book Divina proportione, incorporated a translation of the entire book De quinque corporibus regularibus into Italian, without crediting della Francesca for any of this material. It is through Pacioli that much of della Francesca's work became widely known. Although Giorgio Vasari denounced Pacioli for plagiarism in his 1568 book, Lives of the Most Excellent Painters, Sculptors, and Architects, he did not provide sufficient detail to verify these claims. Della Francesca's original work became lost until, in 1851 and again in 1880, it was rediscovered in the Urbino collection of the Vatican Library by Scottish antiquary James Dennistoun and German art historian Max Jordan, respectively, allowing the accuracy of Vasari's accusations to be verified.
|
||||
Subsequent works to study the regular solids and their perspectives in similar ways, based on the work of della Francesca and its transmission by Pacioli, include Albrecht Dürer's Underweysung der Messung (1525), which focuses on techniques for both the perspective drawing of regular and irregular polyhedra as well as for their construction as physical models, and Wenzel Jamnitzer's Perspectiva corporum regularium (1568), which presents images of many polyhedra derived from the regular polyhedra, but without mathematical analysis.
|
||||
Although a book with the same title was recorded to exist in the 16th century in the private library of John Dee, the Vatican copy of De quinque corporibus regularibus (Vatican Codex Urbinas 632) is the only extant copy known. An 1895 catalog of the Vatican collection lists it between volumes of Euclid and Archimedes. Reproductions of it have been published by the Accademia dei Lincei in 1916, and by Giunti in 1995.
|
||||
|
||||
== See also ==
|
||||
List of books about polyhedra
|
||||
|
||||
== Notes ==
|
||||
|
||||
== References ==
|
||||
|
||||
Banker, James R. (March 2005), "A manuscript of the works of Archimedes in the hand of Piero della Francesca", The Burlington Magazine, 147 (1224): 165–169, JSTOR 20073883, S2CID 190211171
|
||||
Davis, Margaret Daly (1977), Piero Della Francesca's Mathematical Treatises: The Trattato D'abaco and Libellus de Quinque Corporibus Regularibus (in English and Italian), Longo Editore
|
||||
Dee, John (2006), Halliwell-Phillipps, J. O. (ed.), The Private Diary of Dr. John Dee, and the Catalog of His Library of Manuscripts, Project Gutenberg, p. 77
|
||||
Dennistoun, James (1851), Memoirs of the Dukes of Urbino, illustrating the arms, arts, and literature of Italy, from 1440 to 1630, Longman, Brown, Green, and Longmans, pp. 195–197
|
||||
Field, J. V. (1997), "Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler", Archive for History of Exact Sciences, 50 (3–4): 241–289, doi:10.1007/BF00374595, JSTOR 41134110, MR 1457069, S2CID 118516740
|
||||
Jordan, M. (1880), "Der vermisste Traktat des Piero della Francesca über die fünf regelmässigen Körper", Jahrbuch der Königlich Preussischen Kunstsammlungen (in German), 1 (2–4): 112–119, JSTOR 4301707
|
||||
Montebelli, Vico (2015), "Luca Pacioli and perspective (part I)", Lettera Matematica, 3 (3): 135–141, doi:10.1007/s40329-015-0090-4, MR 3402538, S2CID 193533200
|
||||
Mungello, David E. (1985), Curious Land: Jesuit Accommodation and the Origins of Sinology, Studia Leibnitiana Supplementa, vol. 25, Franz Steiner Verlag
|
||||
Peterson, Mark A. (1997), "The geometry of Piero della Francesca", The Mathematical Intelligencer, 19 (3): 33–40, doi:10.1007/BF03025346, MR 1475147, S2CID 120720532
|
||||
Stornajalo, Cosimo (1895), Codices urbinates graeci Bibliothecae Vaticanae, descripti praeside Alfonso cardinali Capecelatro, Vatican Library, p. 97
|
||||
Swetz, Frank J. (February 1995), "The volume of a sphere: A Chinese derivation", The Mathematics Teacher, 88 (2): 142–145, doi:10.5951/MT.88.2.0142, JSTOR 27969235
|
||||
26
data/en.wikipedia.org/wiki/Descriptive_Complexity-0.md
Normal file
26
data/en.wikipedia.org/wiki/Descriptive_Complexity-0.md
Normal file
@ -0,0 +1,26 @@
|
||||
---
|
||||
title: "Descriptive Complexity"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Descriptive_Complexity"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:05.899385+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Descriptive Complexity is a book in mathematical logic and computational complexity theory by Neil Immerman. It concerns descriptive complexity theory, an area in which the expressibility of mathematical properties using different types of logic is shown to be equivalent to their computability in different types of resource-bounded models of computation. It was published in 1999 by Springer-Verlag in their book series Graduate Texts in Computer Science.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The book has 15 chapters, roughly grouped into five chapters on first-order logic, three on second-order logic, and seven independent chapters on advanced topics.
|
||||
The first two chapters provide background material in first-order logic (including first-order arithmetic, the BIT predicate, and the notion of a first-order query) and complexity theory (including formal languages, resource-bounded complexity classes, and complete problems). Chapter three begins the connection between logic and complexity, with a proof that the first-order-recognizable languages can be recognized in logarithmic space, and the construction of complete languages for logarithmic space, nondeterministic logarithmic space, and polynomial time. The fourth chapter concerns inductive definitions, fixed-point operators, and the characterization of polynomial time in terms of first-order logic with the least fixed-point operator. The part of the book on first-order topics ends with a chapter on logical characterizations of resource bounds for parallel random-access machines and circuit complexity.
|
||||
Chapter six introduces Ehrenfeucht–Fraïssé games, a key tool for proving logical inexpressibility, and chapter seven introduces second-order logic. It includes Fagin's theorem characterizing nondeterministic polynomial time in terms of existential second-order logic, the Cook–Levin theorem on the existence of NP-complete problems, and extensions of these results to the polynomial hierarchy. Chapter eight uses games to prove the inexpressibility of certain languages in second-order logic.
|
||||
Chapter nine concerns the complementation of languages and the transitive closure operator, including the Immerman–Szelepcsényi theorem that nondeterministic logarithmic space is closed under complementation. Chapter ten provides complete problems and a second-order logical characterization of polynomial space. Chapter eleven concerns uniformity in circuit complexity (the distinction between the existence of circuits for solving a problem, and their algorithmic constructibility), and chapter twelve concerns the role of ordering and counting predicates in logical characterizations of complexity classes. Chapter thirteen uses the switching lemma for lower bounds, and chapter fourteen concerns applications to databases and model checking. A final chapter outlines topics still in need of research in this area.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
The book is primarily aimed as a reference to researchers in this area, but it could also be used as the basis of a graduate course, and comes equipped with exercises for this purpose. Although it claims to be self-contained, reviewer W. Klonowski writes that its readers need to already understand both classical complexity and the basics of mathematical logic.
|
||||
Reviewer Anuj Dawar writes that some of the early promise of descriptive complexity has been dampened by its inability to bring logical tools to bear on the core problems of complexity theory, and by the need to add computation-like additions to logical languages in order to use them to characterize computation. Nevertheless, he writes, the book is useful as a way of introducing researchers to this line of research, and to a less heavily explored way of approaching computational complexity.
|
||||
|
||||
|
||||
== References ==
|
||||
63
data/en.wikipedia.org/wiki/Disquisitiones_Arithmeticae-0.md
Normal file
63
data/en.wikipedia.org/wiki/Disquisitiones_Arithmeticae-0.md
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@ -0,0 +1,63 @@
|
||||
---
|
||||
title: "Disquisitiones Arithmeticae"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Disquisitiones_Arithmeticae"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:09.351017+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Disquisitiones Arithmeticae (Latin for Arithmetical Investigations) is a textbook on number theory written in Latin by Carl Friedrich Gauss in 1798, when Gauss was 21, and published in 1801, when he was 24. It had a revolutionary impact on number theory by making the field truly rigorous and systematic and paved the path for modern number theory. In this book, Gauss brought together and reconciled results in number theory obtained by such eminent mathematicians as Fermat, Euler, Lagrange, and Legendre, while adding profound and original results of his own.
|
||||
|
||||
|
||||
== Scope ==
|
||||
The Disquisitiones covers both elementary number theory and parts of the area of mathematics now called algebraic number theory. Gauss did not explicitly recognize the concept of a group, which is central to modern algebra, so he did not use this term. His own title for his subject was Higher Arithmetic. In his Preface to the Disquisitiones, Gauss describes the scope of the book as follows:
|
||||
|
||||
The inquiries which this volume will investigate pertain to that part of Mathematics which concerns itself with integers.
|
||||
Gauss also writes, "When confronting many difficult problems, derivations have been suppressed for the sake of brevity when readers refer to this work." ("Quod, in pluribus quaestionibus difficilibus, demonstrationibus syntheticis usus sum, analysinque per quam erutae sunt suppressi, imprimis brevitatis studio tribuendum est, cui quantum fieri poterat consulere oportebat")
|
||||
|
||||
|
||||
== Contents ==
|
||||
|
||||
The book is divided into seven sections:
|
||||
|
||||
Congruent Numbers in General
|
||||
Congruences of the First Degree
|
||||
Residues of Powers
|
||||
Congruences of the Second Degree
|
||||
Forms and Indeterminate Equations of the Second Degree
|
||||
Various Applications of the Preceding Discussions
|
||||
Equations Defining Sections of a Circle
|
||||
|
||||
These sections are subdivided into 366 numbered items, which state a theorem with proof or otherwise develop a remark or thought.
|
||||
Sections I to III are essentially a review of previous results, including Fermat's little theorem, Wilson's theorem and the existence of primitive roots. Although few of the results in these sections are original, Gauss was the first mathematician to bring this material together in a systematic way. He also realized the importance of the property of unique factorization (assured by the fundamental theorem of arithmetic, first studied by Euclid), which he restates and proves using modern tools.
|
||||
From Section IV onward, much of the work is original. Section IV develops a proof of quadratic reciprocity; Section V, which takes up over half of the book, is a comprehensive analysis of binary and ternary quadratic forms. Section VI includes two different primality tests. Finally, Section VII is an analysis of cyclotomic polynomials, which concludes by giving the criteria that determine which regular polygons are constructible, i.e., can be constructed with a compass and unmarked straightedge alone.
|
||||
Gauss started to write an eighth section on higher-order congruences, but did not complete it, and it was published separately after his death with the title Disquisitiones generales de congruentiis (Latin: 'General Investigations on Congruences'). In it Gauss discussed congruences of arbitrary degree, attacking the problem of general congruences from a standpoint closely related to that taken later by Dedekind, Galois, and Emil Artin. The treatise paved the way for the theory of function fields over a finite field of constants. Ideas unique to that treatise are clear recognition of the importance of the
|
||||
Frobenius morphism, and a version of Hensel's lemma.
|
||||
The Disquisitiones was one of the last mathematical works written in scholarly Latin. An English translation was not published until 1965, by Jesuit scholar Arthur A. Clarke. Clarke was the first dean at the Lincoln Center campus of Fordham College.
|
||||
|
||||
|
||||
== Importance ==
|
||||
Before the Disquisitiones was published, number theory consisted of a collection of isolated theorems and conjectures. Gauss brought the work of his predecessors together with his own original work into a systematic framework, filled in gaps, corrected unsound proofs, and extended the subject in numerous ways.
|
||||
The logical structure of the Disquisitiones (theorem statement followed by proof, followed by corollaries) set a standard for later texts. While recognising the primary importance of logical proof, Gauss also illustrates many theorems with numerical examples.
|
||||
The Disquisitiones was the starting point for other 19th-century European mathematicians, including Ernst Kummer, Peter Gustav Lejeune Dirichlet and Richard Dedekind. Many of Gauss's annotations are in effect announcements of further research of his own, some of which remained unpublished. They must have appeared particularly cryptic to his contemporaries; they can now be read as containing the germs of the theories of L-functions and complex multiplication, in particular.
|
||||
The Disquisitiones continued to exert influence in the 20th century. For example, in section V, article 303, Gauss summarized his calculations of class numbers of proper primitive binary quadratic forms, and conjectured that he had found all of them with class numbers 1, 2, and 3. This was later interpreted as the determination of imaginary quadratic number fields with even discriminant and class number 1, 2, and 3, and extended to the case of odd discriminant. Sometimes called the class number problem, this more general question was eventually confirmed in 1986 (the specific question Gauss asked was confirmed by Landau in 1902 for class number one). In section VII, article 358, Gauss proved what can be interpreted as the first nontrivial case of the Riemann hypothesis for curves over finite fields (the Hasse–Weil theorem).
|
||||
|
||||
|
||||
== Bibliography ==
|
||||
Gauss, Carl Friedrich (1801), Disquisitiones Arithmeticae (in Latin), Leipzig: Gerh. Fleischer
|
||||
Gauss, Carl Friedrich (1807) [1801], Recherches Arithmétiques (in French), translated by Poullet-Delisle, A.-C.-M., Paris: Courcier
|
||||
Gauss, Carl Friedrich (1889) [1801], Carl Friedrich Gauss' Untersuchungen über höhere Arithmetik (in German), translated by Maser, H., Berlin: Springer; Reprinted 1965, New York: Chelsea, ISBN 0-8284-0191-8
|
||||
Gauss, Carl Friedrich (1966) [1801], Groth, Paul; Bressi, Todd W. (eds.), Disquisitiones Arithmeticae (PDF), translated by Clarke, Arthur A., New Haven: Yale, doi:10.12987/9780300194258, ISBN 978-0-300-09473-2; Corrected ed. 1986, New York: Springer, doi:10.1007/978-1-4939-7560-0, ISBN 978-0-387-96254-2
|
||||
Dunnington, G. Waldo (1935), "Gauss, His Disquisitiones Arithmeticae, and His Contemporaries in the Institut de France", National Mathematics Magazine, 9 (7): 187–192, doi:10.2307/3028190, JSTOR 3028190
|
||||
Goldstein, Catherine; Schappacher, Norbert; Schwermer, Joachim (2010), The Shaping of Arithmetic after C.F. Gauss's Disquisitiones Arithmeticae, Springer, ISBN 978-3-642-05802-8
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Media related to Disquisitiones Arithmeticae at Wikimedia Commons
|
||||
Latin Wikisource has original text related to this article: Disquisitiones arithmeticae (Latin original) (first ed. 1801) (ed. 1870)
|
||||
French Wikisource has original text related to this article: Recherches arithmétiques (French translation) (ed. 1807)
|
||||
68
data/en.wikipedia.org/wiki/Divina_proportione-0.md
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|
||||
---
|
||||
title: "Divina proportione"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Divina_proportione"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:03.564557+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Divina proportione (15th century Italian for Divine proportion), later also called De divina proportione (converting the Italian title into a Latin one) is a book on mathematics written by Luca Pacioli and illustrated by Leonardo da Vinci, completed by February 9th, 1498 in Milan and first printed in 1509. Its subject was mathematical proportions (the title refers to the golden ratio) and their applications to geometry, to visual art through perspective, and to architecture. The clarity of the written material and Leonardo's excellent diagrams helped the book to achieve an impact beyond mathematical circles, popularizing contemporary geometric concepts and images.
|
||||
Some of its content was plagiarised from an earlier book by Piero della Francesca, De quinque corporibus regularibus.
|
||||
|
||||
|
||||
== Contents of the book ==
|
||||
|
||||
The book consists of three separate manuscripts, which Pacioli worked on between 1496 and 1498. He credits Fibonacci as the main source for the mathematics he presents.
|
||||
|
||||
|
||||
=== Compendio divina proportione ===
|
||||
The first part, Compendio divina proportione (Compendium on the Divine Proportion), studies the golden ratio from a mathematical perspective (following the relevant work of Euclid), giving mystical and religious meanings to this ratio, in seventy-one chapters. Pacioli points out that golden rectangles can be inscribed by an icosahedron, and in the fifth chapter, gives five reasons why the golden ratio should be referred to as the "Divine Proportion":
|
||||
|
||||
Its value represents divine simplicity.
|
||||
Its definition invokes three lengths, symbolizing the Holy Trinity.
|
||||
Its irrationality represents God's incomprehensibility.
|
||||
Its self-similarity recalls God's omnipresence and invariability.
|
||||
Its relation to the dodecahedron, which represents the quintessence
|
||||
It also contains a discourse on the regular and semiregular polyhedra, as well as a discussion of the use of geometric perspective by painters such as Piero della Francesca, Melozzo da Forlì and Marco Palmezzano.
|
||||
|
||||
|
||||
=== Trattato dell'architettura ===
|
||||
The second part, Trattato dell'architettura (Treatise on Architecture), discusses the ideas of Vitruvius (from his De architectura) on the application of mathematics to architecture in twenty chapters. The text compares the proportions of the human body to those of artificial structures, with examples from classical Greco-Roman architecture.
|
||||
|
||||
|
||||
=== Libellus in tres partiales divisus ===
|
||||
The third part, Libellus in tres partiales divisus (Book divided into three parts), is a translation into Italian of Piero della Francesca's Latin book De quinque corporibus regularibus [On [the] Five Regular Solids]. It does not credit della Francesca for this material, and in 1550 Giorgio Vasari wrote a biography of della Francesca, in which he accused Pacioli of plagiarism and claimed that he stole della Francesca's work on perspective, on arithmetic and on geometry. Because della Francesca's book had been lost, these accusations remained unsubstantiated until the 19th century, when a copy of della Francesca's book was found in the Vatican Library and a comparison confirmed that Pacioli had copied it.
|
||||
|
||||
|
||||
=== Illustrations ===
|
||||
After these three parts are appended two sections of illustrations, the first showing twenty-three capital letters drawn with a ruler and compass by Pacioli and the second with some sixty illustrations in woodcut after drawings by Leonardo da Vinci. Leonardo drew the illustrations of the regular solids while he lived with and took mathematics lessons from Pacioli. Leonardo's drawings are probably the first illustrations of skeletonic solids which allowed an easy distinction between front and back.
|
||||
Another collaboration between Pacioli and Leonardo existed: Pacioli planned a book of mathematics and proverbs called De Viribus Quantitatis (The powers of numbers) which Leonardo was to illustrate, but Pacioli died before he could publish it.
|
||||
|
||||
|
||||
== History ==
|
||||
Pacioli produced three manuscripts of the treatise by different scribes. He gave the first copy with a dedication to the Duke of Milan, Ludovico il Moro; this manuscript is now preserved in Switzerland at the Bibliothèque de Genève in Geneva. A second copy was donated to Galeazzo da Sanseverino and now rests at the Biblioteca Ambrosiana in Milan. On 1 June 1509 the first printed edition was published in Venice by Paganino Paganini; it has since been reprinted several times.
|
||||
|
||||
The book was displayed as part of an exhibition in Milan between October 2005 and October 2006 together with the Codex Atlanticus. The "M" logo used by the Metropolitan Museum of Art in New York was adapted from one in Divina proportione.
|
||||
|
||||
|
||||
== See also ==
|
||||
|
||||
List of works by Leonardo da Vinci
|
||||
Frederik Macody Lund
|
||||
Samuel Colman
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
=== Works cited ===
|
||||
|
||||
|
||||
== External links ==
|
||||
Full text of original edition
|
||||
Full text of 1509 edition
|
||||
Title page of a reprint in Vienna, 1889
|
||||
A video featuring a 1509 edition on display at Stevens Institute of Technology
|
||||
Full text of original edition (1498) in English
|
||||
0
data/en.wikipedia.org/wiki/Divine_Proportions
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0
data/en.wikipedia.org/wiki/Divine_Proportions
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data/en.wikipedia.org/wiki/Do_Not_Erase
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0
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@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Does_God_Play_Dice?"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T06:17:34.317128+00:00"
|
||||
date_saved: "2026-05-05T08:44:12.875523+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
74
data/en.wikipedia.org/wiki/Elements_of_Dynamic-0.md
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74
data/en.wikipedia.org/wiki/Elements_of_Dynamic-0.md
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@ -0,0 +1,74 @@
|
||||
---
|
||||
title: "Elements of Dynamic"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Elements_of_Dynamic"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:16.835680+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Elements of Dynamic is a book published by William Kingdon Clifford in 1878. In 1887 it was supplemented by a fourth part and an appendix. The subtitle is "An introduction to motion and rest in solid and fluid bodies". It was reviewed positively, has remained a standard reference since its appearance, and is now available online as a Historical Math Monograph from Cornell University.
|
||||
On page 95 Clifford deconstructed the quaternion product of William Rowan Hamilton into two separate products of two vectors: vector product and scalar product. This separation of the quaternion product into two was followed by J. W. Gibbs in his development of vector analysis, first in a pamphlet acknowledging Clifford's Kinematic, and later in a textbook published by Yale University, called Vector Analysis. He apparently remained unaware of Clifford's seminal paper on the unification of these two products into what he termed a geometric algebra.
|
||||
Elements of Dynamic was the debut of the term cross-ratio for a four-argument function frequently used in geometry.
|
||||
Clifford uses the term twist to discuss (pages 126 to 131) the screw theory that had recently been introduced by Robert Stawell Ball.
|
||||
The operation on vectors by matrix multiplication was given with
|
||||
|
||||
|
||||
|
||||
ϕ
|
||||
=
|
||||
|
||||
|
||||
(
|
||||
|
||||
|
||||
|
||||
a
|
||||
|
||||
|
||||
h
|
||||
|
||||
|
||||
|
||||
|
||||
|
||||
h
|
||||
′
|
||||
|
||||
|
||||
|
||||
b
|
||||
|
||||
|
||||
|
||||
)
|
||||
|
||||
|
||||
|
||||
|
||||
{\displaystyle \phi ={\begin{pmatrix}a&h\\h'&b\end{pmatrix}}}
|
||||
|
||||
on page 163.
|
||||
|
||||
|
||||
== Reviews ==
|
||||
A review in the Philosophical Magazine explained for prospective readers that kinematics is the "study of the theory of pure motion". Noting the nature of "progressive training" required for mathematics, the reviewer wondered "For what class of readers is the book designed?"
|
||||
|
||||
Richard A. Proctor noted in The Contemporary Review (33:65) that there are "few errors in the work, and even misprints are few and far between for a treatise of this kind." He did not approve of Clifford's coining of "odd new words as squirts, sinks, twists, and whirls." Proctor quoted the last sentence of the book: "Every continuous motion of an infinite body may be built up of squirts and vortices."
|
||||
In a "Sketch of Professor Clifford" in June 1879 the journal Popular Science said "It will probably not take high rank as a university text-book, for which it was intended, but is much admired by mathematicians for the elegance, freshness, and originality displayed in the treatment of mathematical problems."
|
||||
After Clifford had died, and Book IV and Appendix were published in 1887, the literary magazine
|
||||
Athenaeum said "we have here Clifford pure and simple." It explained that he "had entirely shaken off the concept of force as an explanatory cause." It also expressed "the oft-told regret that Clifford did not live to reshape the teaching of elementary dynamics in this country, and we wait somewhat impatiently for his successor in this labour, who seems long in appearing."
|
||||
In 1901 Alexander Macfarlane spoke at Lehigh University on Clifford. Reviewing Elements of Dynamic he said
|
||||
|
||||
The work is unique for the clear ideas given of the science; ideas and principles are more prominent than symbols and formulae. He takes such familiar words as spin, twist, squirt, whirl, and gives them exact meaning. The book is an example of what he meant by scientific insight,...
|
||||
In 2004 Gowan Dawson reviewed the situation of the book's publication. On the basis of a letter from Lucy Clifford to Alexander MacMillan, the publisher, Dawson wrote
|
||||
|
||||
Clifford, by the time of his death, had published just a single monograph, The Elements of Dynamic, and that had been rushed through the presses in an incomplete form only during the last months of his life. Clifford's standing as both a leading mathematical specialist and an iconoclastic scientific publicist had instead been forged largely in the pages of the Victorian periodical press...
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
Elements of Dynamic, books I, II, III (1878) London: MacMillan & Co; on-line presentation by Cornell University Historical Mathematical Monographs.
|
||||
Books I, II, III (1878) at the Internet Archive
|
||||
Book IV (1887) at the Internet Archive
|
||||
@ -0,0 +1,28 @@
|
||||
---
|
||||
title: "Encyclopedic Dictionary of Mathematics"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Encyclopedic_Dictionary_of_Mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:19.213869+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Encyclopedic Dictionary of Mathematics is a translation of the Japanese Iwanami Sūgaku Jiten (岩波数学辞典). The editor of the first and second editions was Shokichi Iyanaga; the editor of the third edition was Kiyosi Itô; the fourth edition was edited by the Mathematical Society of Japan.
|
||||
|
||||
|
||||
== Editions ==
|
||||
Iyanaga, Shōkichi, ed. (1954), Iwanami Sūgaku Jiten (in Japanese) (1st ed.), Tokyo: Iwanami Shoten, MR 0062668
|
||||
Iyanaga, Shōkichi, ed. (1960), Iwanami mathematical dictionary, Edited by the Mathematical Society of Japan. Revised and enlarged (in Japanese), Tokyo: Iwanami Shoten, MR 0130788
|
||||
Iyanaga, Shōkichi, ed. (1968), Iwanami Sūgaku Jiten (in Japanese) (2nd ed.), Iwanami Shoten Publishers, Tokyo, MR 0241228
|
||||
Iyanaga, Shōkichi, ed. (1977), Encyclopedic Dictionary of Mathematics, Volumes I, II, hardback, translated from the 2nd Japanese edition (1st ed.), MIT Press, pp. 1750, ISBN 978-0-262-09016-2, MR 0490100
|
||||
Iyanaga, Shōkichi; Kawada, Yukiyosi, eds. (1980) [1977], Encyclopedic Dictionary of Mathematics, Volumes I, II, Translated from the 2nd Japanese edition, paperback version of the 1977 edition (1st ed.), MIT Press, ISBN 978-0-262-59010-5, MR 0591028
|
||||
Sūgaku Jiten (in Japanese) (3rd ed.), Tokyo: Iwanami Shoten, 1985, ISBN 978-4-00-080016-7, MR 0924841
|
||||
Itô, Kiyosi, ed. (1987), Encyclopedic Dictionary of Mathematics. Vol. I--IV, Hardback, Translated from the Japanese 3rd edition (2nd ed.), MIT Press, p. 2148, ISBN 978-0-262-09026-1, MR 0901762, archived from the original on 2011-06-29
|
||||
Itô, Kiyosi, ed. (1993), Encyclopedic Dictionary of Mathematics. Vol. I--II, Paperback, Translated from the Japanese 3rd edition (2nd ed.), MIT Press, p. 2148, ISBN 0-262-59020-4; paperback version of the 1987 edition
|
||||
Iwanami Sūgaku Jiten (in Japanese) (Fourth ed.), Tokyo: Iwanami Shoten, 2007, ISBN 978-4-00-080309-0, MR 2383190
|
||||
|
||||
|
||||
== References ==
|
||||
Dieudonné, Jean (1979), "Reviews: Encyclopedic Dictionary of Mathematics", The American Mathematical Monthly, 86 (3): 232–233, doi:10.2307/2321544, ISSN 0002-9890, JSTOR 2321544, MR 1538996
|
||||
Halmos, Paul (1981), "About books: review of Encyclopedic Dictionary of Mathematics", The Mathematical Intelligencer, 3 (3): 138–140, doi:10.1007/BF03022868, ISSN 0343-6993, S2CID 189887339
|
||||
@ -0,0 +1,29 @@
|
||||
---
|
||||
title: "Equivalents of the Axiom of Choice"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Equivalents_of_the_Axiom_of_Choice"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:21.545527+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Equivalents of the Axiom of Choice is a book in mathematics, collecting statements in mathematics that are true if and only if the axiom of choice holds. It was written by Herman Rubin and Jean E. Rubin, and published in 1963 by North-Holland as volume 34 of their Studies in Logic and the Foundations of Mathematics series. An updated edition, Equivalents of the Axiom of Choice, II, was published as volume 116 of the same series in 1985.
|
||||
|
||||
|
||||
== Topics ==
|
||||
At the time of the book's original publication, it was unknown whether the axiom of choice followed from the other axioms of Zermelo–Fraenkel set theory (ZF), or was independent of them, although it was known to be consistent with them from the work of Kurt Gödel. This book codified the project of classifying theorems of mathematics according to whether the axiom of choice was necessary in their proofs, or whether they could be proven without it. At approximately the same time as the book's publication, Paul Cohen proved that the negation of the axiom of choice is also consistent, implying that the axiom of choice, and all of its equivalent statements in this book, are indeed independent of ZF.
|
||||
The first edition of the book includes over 150 statements in mathematics that are equivalent to the axiom of choice, including some that are novel to the book. This edition is divided into two parts, the first involving notions expressed using sets and the second involving classes instead of sets. Within the first part, the topics are grouped into statements related to the well-ordering principle, the axiom of choice itself, trichotomy (the ability to compare cardinal numbers), and Zorn's lemma and related maximality principles. This section also includes three more chapters, on statements in abstract algebra, statements for cardinal numbers, and a final collection of miscellaneous statements. The second section has four chapters, on topics parallel to four of the first section's chapters.
|
||||
The book includes the history of each statement, and many proofs of their equivalence. Rather than ZF, it uses Von Neumann–Bernays–Gödel set theory for its proofs, mainly in a form called NBG0 that allows urelements (contrary to the axiom of extensionality) and also does not include the axiom of regularity.
|
||||
The second edition adds many additional equivalent statements, more than twice as many as the first edition, with an additional list of over 80 statements that are related to the axiom of choice but not known to be equivalent to it. It includes two added sections, one on equivalent statements that need the axioms of extensionality and regularity in their proofs of equivalence, and another on statements in topology, mathematical analysis, and mathematical logic. It also includes more recent developments on the independence of the axiom of choice, and an improved account of the history of Zorn's lemma.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
This book is written as a reference for professional mathematicians, especially those working in set theory. Reviewer Chen Chung Chang writes that it "will be useful both to the specialist in the field and to the general working mathematician", and that its presentation of results is "clear and lucid". By the time of the second edition, reviewers J. M. Plotkin and David Pincus both called this "the standard reference" in this area.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Equivalents of the Axiom of Choice, II at the Internet Archive
|
||||
49
data/en.wikipedia.org/wiki/Euclid's_Optics-0.md
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49
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|
||||
---
|
||||
title: "Euclid's Optics"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Euclid's_Optics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:23.896975+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Optics (Ancient Greek: Ὀπτικά) is a work on the geometry of vision written by the Greek mathematician Euclid around 300 BC. The earliest surviving manuscript of Optics is in Greek and dates from the 10th century AD.
|
||||
The work deals almost entirely with the geometry of vision, with little reference to either the physical or psychological aspects of sight. No Western scientist had previously given such mathematical attention to vision. Euclid's Optics influenced the work of later Greek, Islamic, and Western European Renaissance scientists and artists, and is further credited with laying the foundations of classical optics.
|
||||
|
||||
|
||||
== Historical significance ==
|
||||
|
||||
Writers before Euclid had developed theories of vision. However, their works were mostly philosophical in nature and lacked the mathematics that Euclid introduced in his Optics. Efforts by the Greeks prior to Euclid were concerned primarily with the physical dimension of vision. Whereas Plato and Empedocles thought of the visual ray as "luminous and ethereal emanation", Euclid’s treatment of vision in a mathematical way was part of the larger Hellenistic trend to quantify a whole range of scientific fields.
|
||||
Because Optics contributed a new dimension to the study of vision, it influenced later scientists. In particular, Ptolemy used Euclid's mathematical treatment of vision and his idea of a visual cone in combination with physical theories in Ptolemy's Optics, which has been called "one of the most important works on optics written before Newton". Renaissance artists such as Brunelleschi, Alberti, and Dürer used Euclid's Optics in their own work on linear perspective.
|
||||
|
||||
|
||||
== Structure and method ==
|
||||
Similar to Euclid's much more famous work on geometry, Elements, Optics begins with a small number of definitions and postulates, which are then used to prove, by deductive reasoning, a body of geometric propositions about vision.
|
||||
The postulates in Optics are:
|
||||
|
||||
Let it be assumed
|
||||
That rectilinear rays proceeding from the eye diverge indefinitely;
|
||||
That the figure contained by a set of visual rays is a cone of which the vertex is at the eye and the base at the surface of the objects seen;
|
||||
That those things are seen upon which visual rays fall and those things are not seen upon which visual rays do not fall;
|
||||
That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;
|
||||
That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;
|
||||
That, similarly, things seen by rays further to the right appear further to the right, and things seen by rays further to the left appear further to the left;
|
||||
That things seen under more angles are seen more clearly.
|
||||
The geometric treatment of the subject follows the same methodology as the Elements.
|
||||
|
||||
|
||||
== Content ==
|
||||
According to Euclid, the eye sees objects that are within its visual cone. The visual cone is made up of straight lines, or visual rays, extending outward from the eye. These visual rays are discrete, but we perceive a continuous image because our eyes, and thus our visual rays, move very quickly. Because visual rays are discrete, however, it is possible for small objects to lie unseen between them. This accounts for the difficulty in searching for a dropped needle. Although the needle may be within one's field of view, until the eye's visual rays fall upon the needle, it will not be seen. Discrete visual rays also explain the sharp or blurred appearance of objects. According to postulate 7, the closer an object, the more visual rays fall upon it and the more detailed or sharp it appears. This is an early attempt to describe the phenomenon of optical resolution.
|
||||
Much of the work considers perspective, how an object appears in space relative to the eye. For example, in proposition 8, Euclid argues that the perceived size of an object is not related to its distance from the eye by a simple proportion.
|
||||
An English translation was published in the Journal of the Optical Society of America.
|
||||
|
||||
|
||||
== Notes ==
|
||||
|
||||
|
||||
== References ==
|
||||
Smith, M.A. (1996). Ptolemy's Theory of Visual Perception: An English Translation of the Optics with Introduction and Commentary. Philadelphia: The American Philosophical Society.
|
||||
Smith, M.A. (2014). From Sight to Light. The Passage from Ancient to Modern Optics. Chicago & London: The University of Chicago Press. Bibcode:2014fslp.book.....S.
|
||||
English translation of Euclid's Optics
|
||||
Latin text of Euclid's Optics from Euclidis Opera Omnia, ed. J.L. Heiberg, vol. VII
|
||||
22
data/en.wikipedia.org/wiki/Euclid_and_His_Modern_Rivals-0.md
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22
data/en.wikipedia.org/wiki/Euclid_and_His_Modern_Rivals-0.md
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|
||||
---
|
||||
title: "Euclid and His Modern Rivals"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Euclid_and_His_Modern_Rivals"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:22.731487+00:00"
|
||||
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|
||||
---
|
||||
|
||||
Euclid and His Modern Rivals is a mathematical book published in 1879 by the English mathematician Charles Lutwidge Dodgson (1832–1898), better known under his literary pseudonym "Lewis Carroll". It considers the pedagogic merit of thirteen contemporary geometry textbooks, demonstrating how each in turn is either inferior to or functionally identical to Euclid's Elements.
|
||||
In it Dodgson supports using Euclid's geometry textbook The Elements as the geometry textbook in schools against more modern geometry textbooks that were replacing it, advocated by the Association for the Improvement of Geometrical Teaching, satirized in the book as the "Association for the Improvement of Things in General". Euclid's ghost returns in the play to defend his book against its modern rivals and tries to demonstrate how all of them are inferior to his book.
|
||||
Despite its scholarly subject and content, the work takes the form of a whimsical dialogue, principally between a mathematician named Minos (taken from Minos, judge of the underworld in Greek mythology) and a "devil's advocate" named Professor Niemand (German for 'nobody') who represents the "Modern Rivals" of the title.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
Robin Wilson (2008). Lewis Carroll in Numberland. Allen Lane. pp. 91–95. ISBN 978-0-7139-9757-6.
|
||||
|
||||
|
||||
== External links ==
|
||||
Euclid and His Modern Rivals, scan of the 1885 second edition.
|
||||
32
data/en.wikipedia.org/wiki/Euclides_Danicus-0.md
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32
data/en.wikipedia.org/wiki/Euclides_Danicus-0.md
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|
||||
---
|
||||
title: "Euclides Danicus"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Euclides_Danicus"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:25.024623+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Euclides Danicus (lit. 'Danish Euclid') is one of three books of mathematics written by Georg Mohr. It was published in 1672 simultaneously in Copenhagen and Amsterdam, in Danish and Dutch respectively. It contains the first proof of the Mohr–Mascheroni theorem, which states that every geometric construction that can be performed using a compass and straightedge can also be done with compass alone.
|
||||
|
||||
|
||||
== Contents ==
|
||||
The book is divided into two parts. In the first part, Mohr shows how to perform all of the constructions of Euclid's Elements using a compass alone. In the second part, he includes some other specific constructions, including some related to the mathematics of the sundial.
|
||||
|
||||
|
||||
== 1928 rediscovery of the book ==
|
||||
Euclides Danicus languished in obscurity, possibly caused by its choice of language, until its rediscovery in 1928 in a bookshop in Copenhagen. Until then, the Mohr–Mascheroni theorem had been credited to Lorenzo Mascheroni, who published a proof in 1797, independently of Mohr's work. Soon after the rediscovery of Mohr's book, publications about it by Florian Cajori and Nathan Altshiller Court made its existence much more widely known. The Danish version was republished in facsimile in 1928 by the Royal Danish Academy of Sciences and Letters, with a foreword by Johannes Hjelmslev, and a German translation was published in 1929.
|
||||
|
||||
|
||||
== Surviving copies ==
|
||||
Only eight copies of the original publication of the book are known to survive. In 2005, one of these original copies was sold at auction, to Fry's Electronics, for what Gerald L. Alexanderson calls a "ridiculously low price": US$13,000.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
Wikicommons has a copy of the original: https://commons.wikimedia.org/wiki/File:Georg_Mohr%27s_Euclides_Danicus.pdf
|
||||
|
||||
|
||||
== Further reading ==
|
||||
Zühlke, P. (1956), "Auf den Spuren des Euclides Danicus", Mathematisch-Physikalische Semesterberichte, 5: 118–119, MR 0082435.
|
||||
@ -0,0 +1,34 @@
|
||||
---
|
||||
title: "Evolution and the Theory of Games"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Evolution_and_the_Theory_of_Games"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:26.180617+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Evolution and the Theory of Games is a book by the British evolutionary biologist John Maynard Smith on evolutionary game theory. The book was initially published in December 1982 by Cambridge University Press.
|
||||
|
||||
|
||||
== Overview ==
|
||||
In the book, John Maynard Smith summarises work on evolutionary game theory that had developed in the 1970s, to which he made several important contributions.
|
||||
The main contribution of the book is in introducing the concept of Evolutionarily Stable Strategy (ESS). ESS states that for a set of behaviours to be conserved over evolutionary time, they must be the most beneficial avenue of action when common, so that no alternative behaviour can invade. Supposing, for instance, that in a population of frogs, males fight to the death over breeding ponds. This would be an ESS if any one cowardly frog that does not fight to the death always fares worse (in terms of evolutionary survival fitness). A more likely scenario is one in which fighting to the death is not an ESS because a frog might arise that will stop fighting if it realises that it is going to lose. This frog would thus reap the benefits of fighting, but not the ultimate cost. Hence, fighting to the death would easily be invaded by a mutation that causes this "informed fighting." Much complexity can develop from this.
|
||||
The structure of the book is as follows: Following an introductory chapter, the basic model is outlined. Here the Hawk-Dove model is introduced its assumptions reviewed and then an extended model, playing the field, is outlined. Chapters then include The war of attrition, Games with genetic models and Learning the ESS. There are then two chapters on Mixed Strategies and three on Asymmetric games. These are followed by chapters entitled; Life history strategies and the size game, Honesty, bargaining and commitment and The evolution of cooperation. The main body of the book concludes with a Postscript. After the Postscript are 11 Appendices dealing with some more technical background and illustrative material. The technical appendices cover the following topics: Matrix notation for game theory / A game with two pure strategies always has an ESS / The Bishop-Cannings theorem / Dynamics and stability / Retaliation / Games between relatives / The war of attrition with random rewards / The ESS when the strategy set is defined by one or more continuous variables / To find the ESS from a set of recurrence relations / Asymmetric games with cyclic dynamics / The reiterated Prisoner's Dilemma
|
||||
|
||||
|
||||
== Reception ==
|
||||
In their review evolutionary game theory, its past and likely influence in the future, Traulsen and Glynatsi (2023) find that Maynard Smith's book ‘Evolution and the theory of games’ ranks first in terms of citations within the field. This they argue demonstrates the substantial influence that he has had on the field (Traulsen and Glynatsi, 2023, page 2.
|
||||
Bennett (1983) reviewed the book in the European Journal of Operational Research as did Mart Gross (1984) in The Quarterly Review of Biology.
|
||||
|
||||
|
||||
== See also ==
|
||||
|
||||
Evolutionary biology
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Cambridge University Press
|
||||
26
data/en.wikipedia.org/wiki/Hey's_Mineral_Index-0.md
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26
data/en.wikipedia.org/wiki/Hey's_Mineral_Index-0.md
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|
||||
---
|
||||
title: "Hey's Mineral Index"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Hey's_Mineral_Index"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:47.010914+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Hey's Mineral Index is a standard reference work in mineralogy.
|
||||
It is an alphabetical index of known mineral species and varieties, and includes synonyms. For species and major varieties more detail is provided. It includes a classification of minerals based on their chemistry into 32 top-level groups, and breaks these groups down further.
|
||||
|
||||
|
||||
== Editions ==
|
||||
First edition, An Index of Mineral Species & Varieties Arranged Chemically, With An Alphabetical Index of Accepted Mineral Names and Synonyms by Max Hutchinson Hey, published by order of the British Library, was published in 1950.
|
||||
Second edition by Max Hutchinson Hey was published in 1962.
|
||||
Third edition by Andrew M. Clark was published in 1993.
|
||||
Fourth edition by Andrew M. Clark due for publication in 2004 but was never released.
|
||||
|
||||
|
||||
== See also ==
|
||||
Nickel–Strunz classification
|
||||
|
||||
|
||||
== References ==
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
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|
||||
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|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
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|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
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|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
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|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
date_saved: "2026-05-05T08:43:04.376499+00:00"
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
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|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
date_saved: "2026-05-05T08:43:04.376499+00:00"
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
date_saved: "2026-05-05T08:43:04.376499+00:00"
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
date_saved: "2026-05-05T08:43:04.376499+00:00"
|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/List_of_publications_in_mathematics"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:24:10.821694+00:00"
|
||||
date_saved: "2026-05-05T08:43:04.376499+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
30
data/en.wikipedia.org/wiki/On_a_Piece_of_Chalk-0.md
Normal file
30
data/en.wikipedia.org/wiki/On_a_Piece_of_Chalk-0.md
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@ -0,0 +1,30 @@
|
||||
---
|
||||
title: "On a Piece of Chalk"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/On_a_Piece_of_Chalk"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:49.295780+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
On a Piece of Chalk was a lecture given by Thomas Henry Huxley on 26 August 1868 to the working men of Norwich during a meeting of the British Association for the Advancement of Science. It was published as an essay in Macmillan's Magazine in London later that year. The piece reconstructs the geological history of Britain from a simple piece of chalk and demonstrates science as "organized common sense".
|
||||
On a Piece of Chalk was republished by Scribner in 1967 with an introduction by Loren Eiseley and illustrations by Rudolf Freund.
|
||||
|
||||
|
||||
== Reception ==
|
||||
In 1967, Dael Wolfle of the AAAS gave a favorable review for On a Piece of Chalk, writing:
|
||||
|
||||
That the lessons of paleontology are now so much more widely appreciated than they were when Huxley drew them from a piece of carpenter's chalk is in good measure a tribute to Huxley's genius. We have much more factual knowledge than he had, but we have no better exemplar of the art of explaining in compelling and understandable terms what science is about, nor a more vigorous example of the scientist's obligation to practice that art.
|
||||
In April 2015, physicist and Nobel laureate Steven Weinberg included On a Piece of Chalk in a personal list of "the 13 best science books for the general reader".
|
||||
|
||||
|
||||
== See also ==
|
||||
White Cliffs of Dover
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Full text of the essay, hosted by Clark University.
|
||||
19
data/en.wikipedia.org/wiki/Oryctographia_Carniolica-0.md
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19
data/en.wikipedia.org/wiki/Oryctographia_Carniolica-0.md
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|
||||
---
|
||||
title: "Oryctographia Carniolica"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Oryctographia_Carniolica"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:50.458586+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Oryctographia Carniolica (Carniolan Mineralogy; with the subtitle oder Physikalische Erdbeschreibung des Herzogthums Krain, Istrien, und zum Theil der benachbarten Länder 'or a Physical Geography of the Duchy of Carniola, Istria, and in Part the Neighboring Lands') is a four-volume work by Belsazar Hacquet, published in Leipzig in 1778, 1781, 1784, and 1789. It discusses the physical properties of the Duchy of Carniola, Istria, and parts of the neighboring lands. It also includes an in-depth description of the Idrija mercury mine.
|
||||
The front page of the work presents the first known depiction of Triglav, the highest mountain in Slovenia. The copper engraving was produced by C. Conti after a drawing by Franz Xaver Baraga.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Media related to Oryctographia Carniolica at Wikimedia Commons
|
||||
54
data/en.wikipedia.org/wiki/Otherlands_(book)-0.md
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54
data/en.wikipedia.org/wiki/Otherlands_(book)-0.md
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|
||||
---
|
||||
title: "Otherlands (book)"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Otherlands_(book)"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:51.633748+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Otherlands: A Journey Through Earth's Extinct Worlds is a nonfiction book about palaeontology written by Thomas Halliday, a British palaeontologist. He goes as far back in time as approximately 555 million years ago. The book was simultaneously published in English by Allen Lane (UK and Commonwealth), Random House (USA and Philippines), and Penguin Canada (Canada) in February 2022.
|
||||
|
||||
|
||||
== Summary ==
|
||||
Halliday uncovers for the lay public the vast changes in fauna, flora, topography, and climate over the past 555 million years. He observes cyclical changes, including cycles of ice ages and climate warming, and periods of mass extinction followed by periods of mass flourishing. However at each renewal, life takes on completely different forms that are adapted to the new environmental conditions. Thus, life goes on but species do not. Looking forward on a paleontology time scale, humankind will inevitably go extinct.
|
||||
|
||||
|
||||
== Period covered ==
|
||||
Halliday starts the first chapter by going back in time about 20,000 years ago. And, through the next 15 chapters he goes further back in time to approximately 555 million years ago. Each chapter covers a specific geological time period and part of the world. Such time and place most often represent key turning points in the paleontological history of life on Earth.
|
||||
The table below provides more detailed information about the specific locations and periods covered.
|
||||
|
||||
|
||||
== "Otherlands" & Thomas Halliday on climate change ==
|
||||
Halliday indicates that humankind bears a huge responsibility in the trajectory of our contemporary climate change. Today's atmosphere has a similar composition as during the Oligocene (an epoch ranging from 34 million to 23 million years ago during the Palaeogene period shown within the table above). However, by the end of this century, the Intergovernmental Panel for Climate Change (IPCC) projects that the level of CO2 in the atmosphere could reach levels of CO2 similar to the Eocene (the preceding epoch to the Oligocene ranging from 56 million to 34 million years ago). Temperature ranges during the Eocene were a lot higher than contemporary ones. And, the only way to reduce this prospective increase in temperature is by decreasing CO2 emissions and flatten the upward trajectory of atmospheric CO2 concentration.
|
||||
By studying the distant past, Halliday can envision prospective climate change scenarios. Depending on how much CO2 is emitted, the Earth could very well be heading towards Eocene-temperature levels far faster than any underlying long term paleontology-cycle would suggest.
|
||||
|
||||
|
||||
== Author's background ==
|
||||
Halliday is a paleontologist and evolutionary biologist. He has held research positions at University College London and the University of Birmingham, and has been part of paleontology field crews in Argentina and India. He holds a Leverhulme Early Career Fellowship at the University of Birmingham, and is a scientific associate of the Natural History Museum. His research combines theoretical and real data to investigate long-term patterns in the fossil record, particularly in mammals. He was the winner of the Linnean Society's John C. Marsden Medal in 2016 and the Hugh Miller Writing Competition in 2018.
|
||||
"Otherlands" is Halliday's first book.
|
||||
Halliday has published numerous scientific papers often related to the explanatory narrative within "Otherlands." The book itself includes 45 pages of scientific references (between 20 and 30 references per chapter).
|
||||
|
||||
|
||||
== Book reception ==
|
||||
This book was well received with positive reviews from nonfiction book critics. It was shortlisted for the 2022 Wainwright Prize, and was highly commended.
|
||||
"An extraordinary history of our almost-alien Earth"
|
||||
"A stirring, eye-opening journey into deep time, from the Ice Age to the first appearance of microbial life 550 million years ago, by a brilliant young paleobiologist."
|
||||
"A bracing pleasure for Earth-science buffs and readers interested in diving into deep history."
|
||||
"Our planet has been many different worlds over its 4.5-billion-year history. Imagining what they were like is hard—with our limited lifespan, deep time eludes us by its very nature. Otherlands, the debut of Scottish palaeontologist Thomas Halliday, presents you with a series of past worlds. Though this is a non-fiction book thoroughly grounded in fact, it is the quality of the narrative that stands out. Beyond imaginative metaphors to describe extinct lifeforms, some of his reflections on deep time, taxonomy, and evolution are simply spine-tingling."
|
||||
"In this remarkable book, the award-winning scientist Thomas Halliday takes us on a tour of the landscapes, flora and fauna of the distant past."
|
||||
"A fascinating journey through Earth's history."
|
||||
"An extraordinary history of our almost-alien Earth."
|
||||
"As a paleontologist, Halliday is at home with an amazing range of technical terms, casually rattling off thorny ones like Anthropornis nordenskjoeldi or palaeoscolecids. Fortunately, you can safely skim over these tongue twisters and focus on the big picture – 16 chapters reaching back from a mere 20,000 years in the past, when humans moved into the Americas, to the Ediacaran Period, 550 million years ago. Each chapter presents its own point in time in memorable fashion, homing in on everyday experiences to make that particular “otherland” come alive."
|
||||
|
||||
|
||||
== See also ==
|
||||
List of popular science books on evolution
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Thomas Halliday's website
|
||||
26
data/en.wikipedia.org/wiki/Protogaea-0.md
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||||
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|
||||
title: "Protogaea"
|
||||
chunk: 1/1
|
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source: "https://en.wikipedia.org/wiki/Protogaea"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:53.977497+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Protogaea is a work by Gottfried Leibniz on geology and natural history. Unpublished in his lifetime, but made known by Johann Georg von Eckhart in 1719, it was conceived as a preface to his incomplete history of the House of Brunswick.
|
||||
|
||||
|
||||
== Life ==
|
||||
Protogaea is a history of the Earth written in conjectural terms; it was composed by Leibniz in the period from 1691 to 1693. A summary in Latin was published in 1693 in the Leipzig Acta Eruditorum. The text was first published in full at Göttingen in 1749, shortly after Benoît de Maillet's more far-reaching ideas on the origin of the Earth, circulated in manuscript, had been printed.
|
||||
|
||||
|
||||
== Views ==
|
||||
Protogaea built on, and criticized, the natural philosophy of René Descartes, as expressed in his Principia Philosophiae. Leibniz in the work adopted the Cartesian theory of the Earth as a sun crusted over with sunspots. He relied on the authority of Agostino Scilla writing about fossils to discredit the speculations of Athanasius Kircher and Johann Joachim Becher; he had met Scilla in Rome a few years earlier. He took up suggestions of Nicolaus Steno that argued for the forms of fossils being prior to their inclusion in rocks, for stratification, and for the gradual solidification of the Earth.
|
||||
|
||||
|
||||
== Notes ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Protogaea (1749, German), full digital facsimile from Linda Hall Library
|
||||
@ -0,0 +1,15 @@
|
||||
---
|
||||
title: "Quaternary Geology and Geomorphology"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Quaternary_Geology_and_Geomorphology"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:55.189390+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Quaternary Geology and Geomorphology(《第四纪地质学与地貌学》)is a teaching and researching textbook for higher teaching institutions in China. It was first published by Geology Press in August 2009, and the chief editors of the book are TianMingZhong, ChengJie, and so on.
|
||||
Quaternary Geology and Geomorphology contains the most advanced knowledge about Quaternary geology until the final editing. It exhibits the internal connection along dynamic, geomorphology sediment and environment. Besides, the book systematically instructs quaternary geology and geomorphology. Authors also provided some popular research methods about quaternary geology.
|
||||
|
||||
|
||||
== References ==
|
||||
@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Sylva,_or_A_Discourse_of_Forest-Trees_and_the_Propagation_of_Timber"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T06:19:50.284520+00:00"
|
||||
date_saved: "2026-05-05T08:42:57.589336+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
@ -0,0 +1,19 @@
|
||||
---
|
||||
title: "T. rex and the Crater of Doom"
|
||||
chunk: 1/2
|
||||
source: "https://en.wikipedia.org/wiki/T._rex_and_the_Crater_of_Doom"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:58.785888+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
T. rex and the Crater of Doom is a nonfiction book by UC Berkeley professor Walter Alvarez that was published by Princeton University Press in 1997. The book discusses the research and evidence that led to the creation of the Alvarez hypothesis, which explains how an impact event was the main cause that resulted in the Cretaceous–Paleogene extinction event.
|
||||
|
||||
== Content ==
|
||||
The book begins by discussing Alvarez's research in the 1970s, before ever investigating the cause of the Cretaceous–Paleogene extinction event, when he was researching plate tectonics involving the Apennine Mountains with William "Bill" Lowrie. The method of this research was to use the evidence of the Earth's magnetic field to show that the plate upon which the rocks of the mountains rested had rotated over millions of years. While investigating limestone deposits in Gubbio, they discovered that some of the rocks were not aligned with the magnetic north pole, but in the opposite direction, implying the Earth undergoes geomagnetic reversal over time. This, with the plentiful fossilized material of extinct foraminifera, allowed them to date the time differences between each reversal and catalog the species of microscopic life found in each era. In doing so, they discovered that a certain time period had resulted in very few foraminifera fossils being formed, a boundary of little life now known as the Cretaceous–Paleogene boundary or the KT boundary. Just above this boundary, there was a thick layer that had no evidence of fossils at all, pointing to an almost complete extinction of microscopic sea life. This discovery was in direct opposition to the theory of gradualism, the leading belief of the period that evolutionary change occurred slowly across large time periods, rather than in bursts of short, distinct events, a theory known as catastrophism.
|
||||
Suspecting that the layer without fossilized remains was evidence of a catastrophic event, Alvarez set out to determine how quickly the layer of clay had been deposited, which would either prove or disprove his hypothesis. His father, Luis Alvarez, suggested that the amount of iridium, an element deposited from cosmic dust at a fixed rate, might provide evidence for his claim. If the amount of iridium in the layer was higher than would be expected, that would imply an asteroid or comet impact had caused impactor dust to fall in high amounts all around the world, building up a higher concentration of iridium. The amount of iridium when tested was found to be 9 parts per billion (ppb), rather than the 0.1 ppb that would have accumulated naturally in the layer. The next step was to determine whether this high concentration of iridium was unique to Gubbio or whether it could be found worldwide, as would be expected for a catastrophic impact event. While locations with clear rock layers of the KT boundary were rare, Alvarez was able to confirm his findings with the Stevns Klint deposits in Zealand.
|
||||
Alvarez's hypothesis at the time conflicted over whether an impact was the cause or whether the iridium had been deposited by a supernova from a nearby star, which could have also killed most life on Earth due to gamma ray bursts and cosmic radiation. In order to confirm or rule out this alternative hypothesis, Alvarez worked with Frank Asaro and Helen Michel to determine if the clay layer also contained plutonium-244, a distinctive isotope that a supernova would also have deposited if it had been the cause. While their initial testing came out as positive for the isotope, it turned out to be a false positive under further scrutiny and testing. This caused Alvarez to abandon the supernova possibility and focus singularly on an impact event being the cause. However, Alvarez was uncertain on how such an impact could have wiped out species all around the world. After investigating the effects of the 1883 eruption of Krakatoa, he determined that a large enough impact could force enough ash and dust into the atmosphere to block out the sun, leading to a global mass extinction.
|
||||
By 1980, evidence of the KT boundary and high iridium levels had been independently reported on at dozens of other sites, moving Alvarez's hypothesis toward a global hunt for the impact crater, in competition with several other scientists such as Jan Smit. Many teams continued to dispute the impact hypothesis, instead theorizing that a volcanic eruption could have been the cause of the mass extinction. An eruption in an area known as the Deccan Traps was dated to the same time period of the boundary, making the eruption hypothesis stronger. The search for an impact crater caused Alvarez to turn to evidence of a tsunami, which a large impact would have likely caused if it had occurred in the middle of the ocean. By the late 1980s, he found evidence at the Brazos River that a tsunami has swept across the Gulf of Mexico millions of years earlier. The discovery was made thanks to a graduate student named Alan Hildebrand that notified Alvarez of the evidence of a crater on the Yucatan Peninsula, which had never been published in the scientific literature by the Mexican petroleum geologists that had found it. The age of the crater needed to be determined if it was going to be a candidate for the KT boundary impact, but access to the region was limited due to the crater having been buried over time and the core samples obtained by the geologists having been lost. The only option Alvarez had left was to find undisturbed sediment left over from the impact still on the surface rock layer somewhere in northeastern Mexico. After several weeks of searching, his team found evidence in a riverbed named Arroyo el Mimbral with the exact signature of the impact that was expected. Several years later, in 1991, the core samples were re-discovered and confirmed the findings from Alvarez's expedition.
|
||||
|
||||
== Style and tone ==
|
||||
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|
||||
---
|
||||
title: "T. rex and the Crater of Doom"
|
||||
chunk: 2/2
|
||||
source: "https://en.wikipedia.org/wiki/T._rex_and_the_Crater_of_Doom"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:58.785888+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
In a review for Geological Magazine, Simon Conway Morris noted that the controversy and debate over the impact hypothesis had led to comments against it that are "querulous, petulant, otiose, and sometimes simply poisonous", but that Alvarez's book "acknowledges the differences that have arisen, but never does he descend to insult and injury", leading to a "gracious and generous book". Timothy Ferris, writing for The New York Times, stated that Alvarez "gets the facts across in a lighthearted, almost playful manner", but still manages to present "solid science" that presents a "clear and efficient exposition that conveys plenty of cogent detail while keeping an eye on the subtle interplay of thought, action and personality that makes scientific research such arresting human behavior."
|
||||
|
||||
== Critical reception ==
|
||||
Clark R. Chapman, writing for Nature, stated that Alvarez's "slim" book can be read "in a single sitting and I recommend it highly – if only as a jumping-off point to other perspectives on this dramatic scientific revolution." In a review for Scientific American of T. rex and the Crater of Doom and the opposing theories presented in The Great Dinosaur Extinction Controversy, Michael Benton advocated reading the book "for an excellent account of the pro-impact position and for insight into how scientists pose questions and seek to resolve them by sometimes roundabout means". For The Quarterly Review of Biology, paleontologist Mark Norell reviewed the book, criticizing it for not presenting more connective evidence between the impact and the saurian extinction event, stating, "the evidence marshaled by Alvarez is conclusive. Just over 65 million years ago an impact happened. But is this responsible for "the crime"? The jury is still out." William Glen, in the journal Isis, explained how the book presented in "simple, compelling language a fascinating autobiographical chronicle of cutting-edge scientific research that includes much not yet known to history" and that it was "indispensable for anyone interested in the science, the history, or life in science." Los Angeles Times writer Dave A. Russell said that the book was "very well written and so engrossing that a reader with little or no background in the earth's geologic history will enjoy an easy and vastly entertaining summary of how we came to our present understanding of the past." Douglas Palmer in New Scientist described the historical story as a reading similar to "Arthurian legend, full of temptations which lead the hero astray and distract him and his followers from the true path" and that "this personal account of the search for a geological Excalibur makes fascinating reading."
|
||||
|
||||
== References ==
|
||||
|
||||
== External links ==
|
||||
T. rex and the Crater of Doom on the Princeton University Press, Publisher website
|
||||
34
data/en.wikipedia.org/wiki/The_Analyst-0.md
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|
||||
---
|
||||
title: "The Analyst"
|
||||
chunk: 1/2
|
||||
source: "https://en.wikipedia.org/wiki/The_Analyst"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:13.533929+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Analyst: A Discourse Addressed to an Infidel Mathematician: Wherein It Is Examined Whether the Object, Principles, and Inferences of the Modern Analysis Are More Distinctly Conceived, or More Evidently Deduced, Than Religious Mysteries and Points of Faith, is a book by George Berkeley. It was first published in 1734, first by J. Tonson (London), then by S. Fuller (Dublin). The "infidel mathematician" is believed to have been Edmond Halley, though others have speculated that Isaac Newton was intended.
|
||||
The book contains a direct attack on the foundations of calculus, specifically on Isaac Newton's notion of fluxions and on Leibniz's notion of infinitesimal change.
|
||||
|
||||
== Background and purpose ==
|
||||
|
||||
From his earliest days as a writer, Berkeley had taken up his satirical pen to attack what were then called 'free-thinkers' (secularists, sceptics, agnostics, atheists, etc.—in short, anyone who doubted the truths of received Christian religion or called for a diminution of religion in public life). In 1732, in the latest installment in this effort, Berkeley published his Alciphron, a series of dialogues directed at different types of 'free-thinkers'. One of the archetypes Berkeley addressed was the secular scientist, who discarded Christian mysteries as unnecessary superstitions, and declared his confidence in the certainty of human reason and science. Against his arguments, Berkeley mounted a subtle defense of the validity and usefulness of these elements of the Christian faith.
|
||||
Alciphron was widely read and caused a bit of a stir. But it was an offhand comment mocking Berkeley's arguments by the 'free-thinking' royal astronomer Sir Edmund Halley that prompted Berkeley to pick up his pen again and try a new tack. The result was The Analyst, conceived as a satire attacking the foundations of mathematics with the same vigour and style as 'free-thinkers' routinely attacked religious truths.
|
||||
Berkeley sought to take apart the then foundations of calculus, claimed to uncover numerous gaps in proof, attacked the use of infinitesimals, the diagonal of the unit square, the very existence of numbers, etc. The general point was not so much to mock mathematics or mathematicians, but rather to show that mathematicians, like the Christians they criticized, relied upon unknowable mysteries in the foundations of their reasoning. Moreover, the existence of these "superstitions" was not fatal to mathematical reasoning; indeed, it was an aid. So too with the Christian faithful and their mysteries. Berkeley concluded that the certainty of mathematics is no greater than the certainty of religion.
|
||||
|
||||
== Content ==
|
||||
The Analyst was a direct attack on the foundations of calculus, specifically on Newton's notion of fluxions and on Leibniz's notion of infinitesimal change. In section 16, Berkeley criticises
|
||||
|
||||
...the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment . . . Since if this second Supposition had been made before the common Division by o, all had vanished at once, and you must have got nothing by your Supposition. Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nxn-1. But, notwithstanding all this address to cover it, the fallacy is still the same.
|
||||
It is a frequently quoted passage, particularly when he wrote:
|
||||
|
||||
And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?
|
||||
Berkeley did not dispute the results of calculus; he acknowledged the results were true. The thrust of his criticism was that Calculus was not more logically rigorous than religion. He instead questioned whether mathematicians "submit to Authority, take things upon Trust" just as followers of religious tenets did. According to Burton, Berkeley introduced an ingenious theory of compensating errors that were meant to explain the correctness of the results of calculus. Berkeley contended that the practitioners of calculus introduced several errors which cancelled, leaving the correct answer. In his own words, "by virtue of a twofold mistake you arrive, though not at science, yet truth."
|
||||
|
||||
== Analysis ==
|
||||
The idea that Newton was the intended recipient of the discourse is put into doubt by a passage that appears toward the end of the book:
|
||||
"Query 58: Whether it be really an effect of Thinking, that the same Men admire the great author for his Fluxions, and deride him for his Religion?"
|
||||
Here Berkeley ridicules those who celebrate Newton (the inventor of "fluxions", roughly equivalent to the differentials of later versions of the differential calculus) as a genius while deriding his well-known religiosity. Since Berkeley is here explicitly calling attention to Newton's religious faith, that seems to indicate he did not mean his readers to identify the "infidel (i.e., lacking faith) mathematician" with Newton.
|
||||
Mathematics historian Judith Grabiner comments, "Berkeley's criticisms of the rigor of the calculus were witty, unkind, and — with respect to the mathematical practices he was criticizing — essentially correct". While his critiques of the mathematical practices were sound, his essay has been criticised on logical and philosophical grounds.
|
||||
For example, David Sherry argues that Berkeley's criticism of infinitesimal calculus consists of a logical criticism and a metaphysical criticism. The logical criticism is that of a fallacia suppositionis, which means gaining points in an argument by means of one assumption and, while keeping those points, concluding the argument with a contradictory assumption. The metaphysical criticism is a challenge to the existence itself of concepts such as fluxions, moments, and infinitesimals, and is rooted in Berkeley's empiricist philosophy which tolerates no expression without a referent. Andersen (2011) showed that Berkeley's doctrine of the compensation of errors contains a logical circularity. Namely, Berkeley's determination of the derivative of the quadratic function relies on Apollonius's determination of the tangent of the parabola.
|
||||
53
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|
||||
---
|
||||
title: "The Analyst"
|
||||
chunk: 2/2
|
||||
source: "https://en.wikipedia.org/wiki/The_Analyst"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:13.533929+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
== Influence ==
|
||||
Two years after this publication, Thomas Bayes published anonymously "An Introduction to the Doctrine of Fluxions, and a Defence of the Mathematicians Against the Objections of the Author of the Analyst" (1736), in which he defended the logical foundation of Isaac Newton's calculus against the criticism outlined in The Analyst. Colin Maclaurin's two-volume Treatise of Fluxions published in 1742 also began as a response to Berkeley attacks, intended to show that Newton's calculus was rigorous by reducing it to the methods of Greek geometry.
|
||||
Despite these attempts, calculus continued to be developed using non-rigorous methods until around 1830 when Augustin Cauchy, and later Bernhard Riemann and Karl Weierstrass, redefined the derivative and integral using a rigorous definition of the concept of limit. The idea of using limits as a foundation for calculus had been suggested by d'Alembert, but d'Alembert's definition was not rigorous by modern standards. The concept of limits had already appeared in the work of Newton, but was not stated with sufficient clarity to hold up to the criticism of Berkeley.
|
||||
In 1966, Abraham Robinson introduced Non-standard Analysis, which provided a rigorous foundation for working with infinitely small quantities. This provided another way of putting calculus on a mathematically rigorous foundation, the way it was done before the (ε, δ)-definition of limit had been fully developed.
|
||||
|
||||
=== Ghosts of departed quantities ===
|
||||
Towards the end of The Analyst, Berkeley addresses possible justifications for the foundations of calculus that mathematicians may put forward. In response to the idea fluxions could be defined using ultimate ratios of vanishing quantities, Berkeley wrote:
|
||||
|
||||
It must, indeed, be acknowledged, that [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?
|
||||
|
||||
Edwards describes this as the most memorable point of the book. Katz and Sherry argue that the expression was intended to address both infinitesimals and Newton's theory of fluxions.
|
||||
Today the phrase "ghosts of departed quantities" is also used when discussing Berkeley's attacks on other possible foundations of Calculus. In particular it is used when discussing infinitesimals, but it is also used when discussing differentials, and adequality.
|
||||
|
||||
== Text and commentary ==
|
||||
The full text of The Analyst can be read on Wikisource, as well as on David R. Wilkins' website, which includes some commentary and links to responses by Berkeley's contemporaries.
|
||||
The Analyst is also reproduced, with commentary, in recent works:
|
||||
|
||||
William Ewald's From Kant to Hilbert: A Source Book in the Foundations of Mathematics.
|
||||
Ewald concludes that Berkeley's objections to the calculus of his day were mostly well taken at the time.
|
||||
|
||||
D. M. Jesseph's overview in the 2005 "Landmark Writings in Western Mathematics".
|
||||
|
||||
== References ==
|
||||
|
||||
== Sources ==
|
||||
Kirsti, Andersen (2011), "One of Berkeley's arguments on compensating errors in the calculus.", Historia Mathematica, 38 (2): 219–318, doi:10.1016/j.hm.2010.07.001
|
||||
Arkeryd, Leif (Dec 2005), "Nonstandard Analysis", The American Mathematical Monthly, 112 (10): 926–928, doi:10.2307/30037635, JSTOR 30037635
|
||||
Błaszczyk, Piotr; Katz, Mikhail; Sherry, David (2012), "Ten misconceptions from the history of analysis and their debunking", Foundations of Science, 18: 43–74, arXiv:1202.4153, doi:10.1007/s10699-012-9285-8, S2CID 254508527
|
||||
Boyer, C; Merzbach, U (1991), A History of Mathematics (2 ed.)
|
||||
Burton, David (1997), The History of Mathematics: An Introduction, McGraw-Hill
|
||||
Edwards, C. H. (1994), The Historical Development of the Calculus, Springer
|
||||
Grabiner, Judith (May 1997), "Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions", The American Mathematical Monthly, 104 (5): 393–410, doi:10.2307/2974733, JSTOR 2974733
|
||||
Grabiner, Judith V. (Dec 2004), "Newton, Maclaurin, and the Authority of Mathematics", The American Mathematical Monthly, 111 (10): 841–852, doi:10.2307/4145093, JSTOR 4145093
|
||||
Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, 78 (3): 571–625, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y, S2CID 254471766
|
||||
Kleiner, I.; Movshovitz-Hadar, N. (Dec 1994), "The Role of Paradoxes in the Evolution of Mathematics", The American Mathematical Monthly, 101 (10): 963–974, doi:10.2307/2975163, JSTOR 2975163
|
||||
Leader, Solomon (May 1986), "What is a Differential? A New Answer from the Generalized Riemann Integral", The American Mathematical Monthly, 93 (5): 348–356, doi:10.2307/2323591, JSTOR 2323591
|
||||
Pourciau, Bruce (2001), "Newton and the notion of limit", Historia Math., 28 (1): 393–30, doi:10.1006/hmat.2000.2301
|
||||
Robert, Alain (1988), Nonstandard analysis, New York: Wiley, ISBN 978-0-471-91703-8
|
||||
Sherry, D. (1987), "The wake of Berkeley's Analyst: Rigor mathematicae?", Studies in Historical Philosophy and Science, 18 (4): 455–480, Bibcode:1987SHPSA..18..455S, doi:10.1016/0039-3681(87)90003-3
|
||||
Wren, F. L.; Garrett, J. A. (May 1933), "The Development of the Fundamental Concepts of Infinitesimal Analysis", The American Mathematical Monthly, 40 (5): 269–281, doi:10.2307/2302202, JSTOR 2302202
|
||||
|
||||
== External links ==
|
||||
Works related to The Analyst: a Discourse addressed to an Infidel Mathematician at Wikisource
|
||||
51
data/en.wikipedia.org/wiki/The_Annotated_Turing-0.md
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51
data/en.wikipedia.org/wiki/The_Annotated_Turing-0.md
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|
||||
---
|
||||
title: "The Annotated Turing"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Annotated_Turing"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:17.061886+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine is a book by Charles Petzold, published in 2008 by John Wiley & Sons, Inc.
|
||||
Petzold annotates Alan Turing's paper "On Computable Numbers, with an Application to the Entscheidungsproblem". The book takes readers sentence by sentence through Turing's paper, providing explanations, further examples, corrections, and biographical information.
|
||||
|
||||
|
||||
== Table of contents ==
|
||||
Part I. Foundations
|
||||
Chapter 1: This Tomb Holds Diophantus
|
||||
Chapter 2: The Irrational and the Transcendental
|
||||
Chapter 3: Centuries of Progress
|
||||
Part II. Computable Numbers
|
||||
Chapter 4: The Education of Alan Turing
|
||||
Chapter 5: Machines at Work
|
||||
Chapter 6: Addition and Multiplication
|
||||
Chapter 7: Also Known as Subroutines
|
||||
Chapter 8: Everything is a Number
|
||||
Chapter 9: The Universal Machine
|
||||
Chapter 10: Computers and Computability
|
||||
Chapter 11: Of Machines and Men
|
||||
Part III. Das Entscheidungsproblem
|
||||
Chapter 12: Logic and Computability
|
||||
Chapter 13: Computable Functions
|
||||
Chapter 14: The Major Proof
|
||||
Chapter 15: The Lambda Calculus
|
||||
Chapter 16: Conceiving the Continuum
|
||||
Part IV. And Beyond
|
||||
Chapter 17: Is Everything a Turing Machine?
|
||||
Chapter 18: The Long Sleep of Diophantus
|
||||
|
||||
|
||||
== See also ==
|
||||
Alan Turing: The Enigma (1983)
|
||||
Prof: Alan Turing Decoded (2015)
|
||||
The Turing Guide (2017)
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Book website
|
||||
Q&A with Mr Charles Petzold 2-2013 vNextOC from YouTube
|
||||
@ -0,0 +1,31 @@
|
||||
---
|
||||
title: "The Banach–Tarski Paradox (book)"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Banach–Tarski_Paradox_(book)"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:26.367041+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Banach–Tarski Paradox is a book in mathematics on the Banach–Tarski paradox, the fact that a unit ball can be partitioned into a finite number of subsets and reassembled to form two unit balls. It was written by Stan Wagon and published in 1985 by the Cambridge University Press as volume 24 of their Encyclopedia of Mathematics and its Applications book series. A second printing in 1986 added two pages as an addendum, and a 1993 paperback printing added a new preface.
|
||||
In 2016 the Cambridge University Press published a second edition, adding Grzegorz Tomkowicz as a co-author, as volume 163 of the same series. The Basic Library List Committee of the Mathematical Association of America has recommended its inclusion in undergraduate mathematics libraries.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The Banach–Tarski paradox, proved by Stefan Banach and Alfred Tarski in 1924, states that it is possible to partition a three-dimensional unit ball into finitely many pieces and reassemble them into two unit balls, a single ball of larger or smaller area, or any other bounded set with a non-empty interior. Although it is a mathematical theorem, it is called a paradox because it is so counter-intuitive; in the preface to the book, Jan Mycielski calls it the most surprising result in mathematics. It is closely related to measure theory and the non-existence of a measure on all subsets of three-dimensional space, invariant under all congruences of space, and to the theory of paradoxical sets in free groups and the representation of these groups by three-dimensional rotations, used in the proof of the paradox. The topic of the book is the Banach–Tarski paradox, its proof, and the many related results that have since become known.
|
||||
The book is divided into two parts, the first on the existence of paradoxical decompositions and the second on conditions that prevent their existence. After two chapters of background material, the first part proves the Banach–Tarski paradox itself, considers higher-dimensional spaces and non-Euclidean geometry, studies the number of pieces necessary for a paradoxical decomposition, and finds analogous results to the Banach–Tarski paradox for one- and two-dimensional sets. The second part includes a related theorem of Tarski that congruence-invariant finitely-additive measures prevent the existence of paradoxical decompositions, a theorem that Lebesgue measure is the only such measure on the Lebesgue measurable sets, material on amenable groups, connections to the axiom of choice and the Hahn–Banach theorem. Three appendices describe Euclidean groups, Jordan measure, and a collection of open problems.
|
||||
The second edition adds material on several recent results in this area, in many cases inspired by the first edition of the book. Trevor Wilson proved the existence of a continuous motion from the one-ball assembly to the two-ball assembly, keeping the sets of the partition disjoint at all times; this question had been posed by de Groot in the first edition of the book. Miklós Laczkovich solved Tarski's circle-squaring problem, asking for a dissection of a disk to a square of the same area, in 1990. And Edward Marczewski had asked in 1930 whether the Banach–Tarski paradox could be achieved using only Baire sets; a positive answer was found in 1994 by Randall Dougherty and Matthew Foreman.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
The book is written at a level accessible to mathematics graduate students, but provides a survey of research in this area that should also be useful to more advanced researchers. The beginning parts of the book, including its proof of the Banach–Tarski paradox, should also be readable by undergraduate mathematicians.
|
||||
Reviewer Włodzimierz Bzyl writes that "this beautiful book is written with care and is certainly worth reading". Reviewer John J. Watkins writes that the first edition of the book "became the classic text on paradoxical mathematics" and that the second edition "exceeds any possible expectation I might have had for expanding a book I already deeply treasured".
|
||||
|
||||
|
||||
== See also ==
|
||||
List of paradoxes
|
||||
History of mathematics
|
||||
|
||||
|
||||
== References ==
|
||||
@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Beauty_of_Fractals"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T06:17:00.877624+00:00"
|
||||
date_saved: "2026-05-05T08:43:28.808626+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
@ -0,0 +1,21 @@
|
||||
---
|
||||
title: "The Book of Numbers (math book)"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Book_of_Numbers_(math_book)"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:32.292880+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Book of Numbers is a 1996 mathematics book by John H. Conway and Richard K. Guy. It discusses individual numbers, and types of number, that have proved conceptually significant. Topics include the origin of the nursery rhyme "Hickory Dickory Dock", figurate numbers, the Fibonacci sequence, transcendental numbers, the Metonic cycle, combinatorics, the complex plane, nimbers, and surreal numbers.
|
||||
The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries.
|
||||
|
||||
|
||||
== Reception ==
|
||||
Andrew Bremner called the book "a delight" and opined that readers of Martin Gardner would appreciate it. A. Robert Pargeter found it "fascinating" both for systematic reading and for browsing, and he recommended that school and college libraries carry it.
|
||||
Sarah Gourlie called the book "organized and enlightening", while observing that some topics were considerably more demanding than others. Likewise, reviewing the book for the Mathematical Association of America, Allen Stenger noted that while the book only presumed knowledge of high-school algebra and trigonometry, it also in places demanded a "high level of mathematical reasoning". Stenger expected that many readers would be unable to follow all of the explanations unaided. A retrospective by Ezra Brown also commented on the "more than a little sophistication" required to follow some of Conway and Guy's discussions, while finding that the authors' joy "comes through on every page".
|
||||
The MacTutor History of Mathematics Archive quotes a review of this book in its biography of Conway, saying that "the publishers should have been required to post a warning label on the front cover indicating that this book contains extremely addictive material."
|
||||
|
||||
|
||||
== References ==
|
||||
54
data/en.wikipedia.org/wiki/The_Book_of_Squares-0.md
Normal file
54
data/en.wikipedia.org/wiki/The_Book_of_Squares-0.md
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@ -0,0 +1,54 @@
|
||||
---
|
||||
title: "The Book of Squares"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Book_of_Squares"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:33.488575+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Book of Squares, (Liber Quadratorum in the original Latin) is a book on algebra by Leonardo Fibonacci, published in 1225. It was dedicated to Frederick II, Holy Roman Emperor.
|
||||
|
||||
After being brought to Pisa by Master Dominick to the feet of your celestial majesty, most glorious prince, Lord F.,
|
||||
The Liber quadratorum has been passed down by a single 15th-century manuscript, the so-called ms. E 75 Sup. of the Biblioteca Ambrosiana (Milan, Italy), ff. 19r–39v. During the 19th century, the work was published for the first time in a printed edition by Baldassarre Boncompagni Ludovisi, prince of Piombino.
|
||||
Appearing in the book is Fibonacci's identity, establishing that the set of all sums of two squares is closed under multiplication. The book anticipated the works of later mathematicians such as Fermat and Euler. The book examines several topics in number theory, among them an inductive method for finding Pythagorean triples based on the sequence of odd integers, the fact that the sum of the first
|
||||
|
||||
|
||||
|
||||
n
|
||||
|
||||
|
||||
{\displaystyle n}
|
||||
|
||||
odd integers is
|
||||
|
||||
|
||||
|
||||
|
||||
n
|
||||
|
||||
2
|
||||
|
||||
|
||||
|
||||
|
||||
{\displaystyle n^{2}}
|
||||
|
||||
, and the solution to the congruum problem.
|
||||
|
||||
|
||||
== Notes ==
|
||||
|
||||
|
||||
== Further reading ==
|
||||
B. Boncompagni Ludovisi, Opuscoli di Leonardo Pisano secondo un codice della Biblioteca Ambrosiana di Milano contrassegnato E.75. Parte Superiore, in Id., Scritti di Leonardo Pisano matematico del secolo decimoterzo, vol. II, Roma 1862, pp. 253–283
|
||||
P. Ver Eecke, Léonard de Pise. Le livre des nombres carrés. Traduit pour la première fois du Latin Médiéval en Français, Paris, Blanchard-Desclée – Bruges 1952.
|
||||
G. Arrighi, La fortuna di Leonardo Pisano alla corte di Federico II, in Dante e la cultura sveva. Atti del Convegno di Studi, Melfi, 2–5 novembre 1969, Firenze 1970, pp. 17–31.
|
||||
E. Picutti, Il Libro dei quadrati di Leonardo Pisano e i problemi di analisi indeterminata nel Codice Palatino 557 della Biblioteca Nazionale di Firenze, in «Physis. Rivista Internazionale di Storia della Scienza» XXI, 1979, pp. 195–339.
|
||||
L.E. Sigler, Leonardo Pisano Fibonacci, the book of squares. An annotated translation into modern English, Boston 1987.
|
||||
M. Moyon, Algèbre & Practica geometriæ en Occident médiéval latin: Abū Bakr, Fibonacci et Jean de Murs, in Pluralité de l’algèbre à la Renaissance, a cura di S. Rommevaux, M. Spiesser, M.R. Massa Esteve, Paris 2012, pp. 33–65.
|
||||
|
||||
|
||||
== External links ==
|
||||
Fibonacci and Square Numbers at "Fibonacci and Square Numbers – Introduction | Mathematical Association of America". maa.org. Retrieved 2023-09-10.
|
||||
37
data/en.wikipedia.org/wiki/The_Calculating_Machines-0.md
Normal file
37
data/en.wikipedia.org/wiki/The_Calculating_Machines-0.md
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@ -0,0 +1,37 @@
|
||||
---
|
||||
title: "The Calculating Machines"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Calculating_Machines"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:36.966881+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Die Rechenmaschinen, by Ernst Martin, and its English translation, The Calculating Machines (Die Rechenmaschinen): Their History and Development, are books on mechanical desktop calculators from prior to World War II.
|
||||
|
||||
|
||||
== Publication history ==
|
||||
Die Rechenmaschinen, the original book by Martin, was published in 1925, and revised in 1937. Both editions are very rare. Little is known about Martin beyond these books.
|
||||
The 1925 edition was edited and translated into English by Peggy A. Kidwell and Michael R. Williams, and published in 1992 by the MIT Press as the final 16th volume of its The Charles Babbage Institute Reprint Series for the History of Computing (ISBN 0-262-13278-8). Kidwell and Williams chose this edition, rather than the revised edition, because of "the rarity of the books and the poor condition of the illustrations in extant copies". Indeed, they were only able to locate three copies of Martin's book.
|
||||
The book and its translation includes many illustrations, and
|
||||
the translation preserves some idiosyncrasies of the original work, including a set of advertisements for calculating machines at the end of the book.
|
||||
|
||||
|
||||
== Topics ==
|
||||
After an introduction grouping calculating machines into seven types,
|
||||
the book describes over 200 machines, comprising "almost every desk-top calculator available before World War II", ordered chronologically. It also contains biographical information about some of the people who contributed to the design of these machines, including Blaise Pascal, Gottfried Wilhelm Leibniz, and Giovanni Poleni.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
At the time Martin wrote the book, "mechanical calculating machines were a symbol of high-tech sophistication in the workplace"; reviewer Jonathan Samuel Golan suggests that it was aimed at collectors rather than historians, while the editors of the Bulletin of Science, Technology & Society suggest that instead its purpose was to inform the public. Nowadays, reviewer A. D. Booth suggests that readers of the book are likely to be people who once used these machines, looking back at them with nostalgia, while the Bulletin editors point to new use by collectors, and Golan instead suggests that it can be used to study the history of a bygone technology.
|
||||
In terms of its content, Booth complains that the contributions of Samuel Morland are overlooked, and that Morland's calculator was at least the equal of Pascal's in priority and quality. Similarly, Doron Swade notes the omission of the work of Wilhelm Schickard, earlier than that of both Morland and Pascal, but excuses this lapse by noting that Schickard's work was forgotten and only rediscovered after Martin's book was published. Golan writes that the descriptions of older calculating machines are "cursory" and secondhand, while the later ones seem to be copied from advertisements.
|
||||
Booth praises the quality of the translation, and calls the newly reprinted edition "an invaluable window on the past". Similarly, Golan calls it "a valuable document, providing a fascinating portrait of the state of the art". Swade is more cautious, pointing to the book's clear biases, but still noting its value as "reference material for collectors and curators as well as historians".
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
The Calculating Machines on the Internet Archive
|
||||
The Calculating Machines, online PDF edition with permission of the MIT Press
|
||||
24
data/en.wikipedia.org/wiki/The_Classical_Groups-0.md
Normal file
24
data/en.wikipedia.org/wiki/The_Classical_Groups-0.md
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@ -0,0 +1,24 @@
|
||||
---
|
||||
title: "The Classical Groups"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Classical_Groups"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:46.222532+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Classical Groups: Their Invariants and Representations is a mathematics book by Hermann Weyl published in 1939. The book describes classical invariant theory in terms of representation theory. It is largely responsible for the revival of interest in invariant theory, which had been almost killed off by David Hilbert's solution of its main problems in the 1890s.
|
||||
The second edition was published in 1946. Weyl gave an informal talk about the topic of his book in 1939.
|
||||
|
||||
|
||||
== Reception ==
|
||||
Roger Howe called the book "wonderful and terrible".
|
||||
|
||||
|
||||
== References ==
|
||||
Howe, Roger (1988), "The classical groups and invariants of binary forms", in Wells, R. O. Jr. (ed.), The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., vol. 48, Providence, R.I.: American Mathematical Society, pp. 133–166, ISBN 978-0-8218-1482-6, MR 0974333
|
||||
Howe, Roger (1989), "Remarks on classical invariant theory.", Transactions of the American Mathematical Society, 313 (2), American Mathematical Society: 539–570, doi:10.2307/2001418, ISSN 0002-9947, JSTOR 2001418, MR 0986027
|
||||
Jacobson, Nathan (1940), "Book Review: The Classical Groups", Bulletin of the American Mathematical Society, 46 (7): 592–595, doi:10.1090/S0002-9904-1940-07236-2, ISSN 0002-9904, MR 1564136
|
||||
Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255 {{citation}}: ISBN / Date incompatibility (help)
|
||||
Weyl, Hermann (1939a), "Invariants", Duke Mathematical Journal, 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030
|
||||
@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Construction_and_Principal_Uses_of_Mathematical_Instruments"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:37:05.140614+00:00"
|
||||
date_saved: "2026-05-05T08:43:55.455451+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
24
data/en.wikipedia.org/wiki/The_Cube_Made_Interesting-0.md
Normal file
24
data/en.wikipedia.org/wiki/The_Cube_Made_Interesting-0.md
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@ -0,0 +1,24 @@
|
||||
---
|
||||
title: "The Cube Made Interesting"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Cube_Made_Interesting"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:59.993538+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Cube Made Interesting is a geometry book aimed at high school mathematics students, on the geometry of the cube. It was originally written in Polish by Aniela Ehrenfeucht (née Miklaszewska, 1905–2000), titled Ciekawy Sześcian [the interesting cube], and published by Polish Scientific Publishers PWN in 1960. Wacław Zawadowski translated it into English, and the translation was published in 1964 by the Pergamon Press and Macmillan Inc., in their "Popular Lectures in Mathematics" series.
|
||||
|
||||
|
||||
== Topics ==
|
||||
The book begins with Euler's polyhedral formula, and includes material on the symmetries of their cube and their realization as geometric rotations, and on the shape of plane sections through cubes. It describes the 30 combinatorially distinct colorings of the six faces of the cube by six different colors, and discusses arrangements of colored cubes that match pairs of equal-colored faces. The final chapter of the book concerns Prince Rupert's cube, the problem of fitting a cube through a hole drilled through a smaller cube without breaking the smaller cube into pieces.
|
||||
An unusual feature of the book is its heavy illustration with red-blue anaglyphs; provided with the book are red-blue glasses allowing readers to see these images as three-dimensional shapes.
|
||||
|
||||
|
||||
== Audience and reception ==
|
||||
This book is based on talks given by Ehrenfeucht to students and teachers, and is aimed at a secondary-school audience. Reviewer A. A. Kosinski writes that it "would contribute profitably to the development of the geometric imagination of a student", and Martyn Cundy writes that "the claim of the title is certainly justified".
|
||||
However, H. S. M. Coxeter notes that some of the terminology has become incorrect or nonstandard in the translation, suggesting that copyediting by someone more familiar with English mathematical terminology would have helped avoid these problems. Cundy complains that the material on Prince Rupert's cube does not provide its optimal solution, and suggests that the color printing and inclusion of 3D viewing glasses made it unnecessarily expensive.
|
||||
|
||||
|
||||
== References ==
|
||||
@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Design_of_Experiments"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T06:17:30.777398+00:00"
|
||||
date_saved: "2026-05-05T08:44:07.114263+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Emperor's_New_Mind"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T06:17:41.471061+00:00"
|
||||
date_saved: "2026-05-05T08:44:18.035212+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
@ -0,0 +1,26 @@
|
||||
---
|
||||
title: "The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Equidistribution_of_Lattice_Shapes_of_Rings_of_Integers_of_Cubic,_Quartic,_and_Quintic_Number_Fields"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:44:20.399254+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields: An Artist's Rendering is a mathematics book by Piper Harron (also known as Piper H and Piper Harris), based on her Princeton University doctoral thesis of the same title. It has been described as "feminist", "unique", "honest", "generous", and "refreshing".
|
||||
|
||||
|
||||
== Thesis and reception ==
|
||||
Harron was advised by Fields Medalist Manjul Bhargava, and her thesis deals with the properties of number fields, specifically the shape of their rings of integers. Harron and Bhargava showed that, viewed as a lattice in real vector space, the ring of integers of a random number field does not have any special symmetries. Rather than simply presenting the proof, Harron intended for the thesis and book to explain both the mathematics and the process (and struggle) that was required to reach this result.
|
||||
The writing is accessible and informal, and the book features sections targeting three different audiences: laypeople, people with general mathematical knowledge, and experts in number theory. Harron intentionally departs from the typical academic format as she is writing for a community of mathematicians who "do not feel that they are encouraged to be themselves". Unusually for a mathematics thesis, Harron intersperses her rigorous analysis and proofs with cartoons, poetry, pop-culture references, and humorous diagrams. Science writer Evelyn Lamb, in Scientific American, expresses admiration for Harron for explaining the process behind the mathematics in a way that is accessible to non-mathematicians, especially "because as a woman of color, she could pay a higher price for doing it." Mathematician Philip Ording calls her approach to communicating mathematical abstractions "generous".
|
||||
Her thesis went viral in late 2015, especially within the mathematical community, in part because of the prologue which begins by stating that "respected research math is dominated by men of a certain attitude". Harron had left academia for several years, later saying that she found the atmosphere oppressive and herself miserable and verging on failure. She returned determined that, even if she did not do math the "right way", she "could still contribute to the community". Her prologue states that the community lacks diversity and discourages diversity of thought. "It is not my place to make the system comfortable with itself", she concludes.
|
||||
A concise proof was published in Compositio Mathematica in 2016.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
The Equidistribution of Lattice Shapes of Rings of Integers of Cubic, Quartic, and Quintic Number Fields (Harron's PhD thesis)
|
||||
The Liberated Mathematician
|
||||
@ -0,0 +1,55 @@
|
||||
---
|
||||
title: "The Map that Changed the World"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Map_that_Changed_the_World"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:48.161693+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Map that Changed the World is a 2001 book by Simon Winchester about English geologist William Smith and his great achievement, the first geological map of England, Wales and southern Scotland.
|
||||
Smith's was the first national-scale geological map, and by far the most accurate of its time. His pivotal insights were that each local sequence of rock strata was a subsequence of a single universal sequence of strata and that these rock strata could be distinguished and traced for great distances by means of embedded fossilized organisms.
|
||||
Winchester's book narrates the intellectual context of the time, the development of Smith's ideas and how they contributed to the theory of evolution and more generally to a dawning realisation of the true age of the Earth. The book describes the social, economic or industrial context for Smith's insights and work, such as the importance of coal mining and the transport of coal by means of canals, both of which were a stimulus to the study of geology and the means whereby Smith supported his research. Landowners wished to know if coal might be found on their holdings. Canal planning and construction depended on understanding the rock and soil along its route.
|
||||
Related topics, such as the founding of the Geological Society of London, are included. Smith's map was published by John Cary, a leading map publisher. Winchester describes the practice of publishing at the time as well as the system of debtor's prisons through his account of the sojourn of Smith in the King's Bench Prison.
|
||||
|
||||
|
||||
== Book format details ==
|
||||
The Map that Changed the World was published in 2001 by HarperCollins, while Winchester retains the copyright. The first edition is illustrated by Soun Vannithone. It includes an extensive index, glossary of geological terms, recommended reading and (lengthy) acknowledgements, as well as many stippled images (of consistent style). The last numbered page is page 329. There are 16 chapters, and single clay paper sheet in the middle containing colour plates of Smith's famous map and a modern geological map for comparison. (Smith's map is less complete, but essentially in agreement with the modern map). An image of Smith's first table of strata, and first (circular) geological map are also included. Just after the contents section, there is a 5-page section giving extensive details on the illustrations (such as the names of the chapter heading fossils). Each chapter begins with an inset image of a fossil, and a large first Capital. The dust-cover of the book can be removed and unfolded to reveal a larger print of the map in question.
|
||||
|
||||
|
||||
== The contents ==
|
||||
|
||||
One: Escape on the Northbound Stage
|
||||
Two: A Land Awakening from Sleep
|
||||
Three: The Mystery of the Chedworth Bun
|
||||
Four: The Duke and the Baronet's Widow
|
||||
Five: A Light in the Underworld
|
||||
Six: The Slicing of Somerset
|
||||
Seven: The View from York Minster
|
||||
Eight: Notes from the Swan
|
||||
Nine: The Dictator in the Drawing Room
|
||||
Ten: The Great Map Conceived
|
||||
Eleven: A Jurassic Interlude
|
||||
Twelve: The Map That Changed the World
|
||||
Thirteen: An Ungentlemanly Act
|
||||
Fourteen: The Sale of the Century
|
||||
Fifteen: The Wrath of Leviathan
|
||||
Sixteen: The Lost and Found Man
|
||||
Seventeen: All Honor to the Doctor.
|
||||
|
||||
|
||||
=== Escape on a Northbound Stage ===
|
||||
A plausible but whimsical description of the day on which William Smith was let out of debtor's prison. It inducts the reader into the interpretation of the time and place to be held consistently throughout the book. Smith is described physically, as heavy-set balding and plain-looking, and emotionally as quitting London in disgust. He is leaving London with nothing other than his wife, nephew, and such possessions as they can carry. It is implied that these circumstances are the result of unjustified discrimination from the scientific elite. The chapter ends with a brief note that 12 years later the injustice was in some measure redressed.
|
||||
|
||||
|
||||
=== A Land Awakening from Sleep ===
|
||||
A description of the social circumstances of the time of the birth of Smith. It begins by emphasising that the date of 4004 BC, for the beginning of the world, computed from the genealogy tables of the Bible, was firmly accepted by most; the idea that the world was any older was considered implausible. Explanations based on Noah's flood were acceptable in scientific circles. But, in the year 1769, as Smith was born, James Watt was patenting a steam engine, cloth manufacture was improving, the postal service was viable. New technology and information was rapidly becoming available or even common-place. "William Smith appeared on the stage at a profoundly interesting moment: he was about to make it more so."
|
||||
These claims by Winchester are inaccurate. Geologists had begun to recognize that the earth was old in the late 1600s. Archbishop Ussher's 4004 BC date for the Creation of the Earth, along with similar estimates by Isaac Newton and other academics of the 17th century, was merely a historical footnote in academia by Smith's lifetime. In 1787, Scottish geologist James Hutton argued that the Earth's age was immeasurable. Smith was in no way challenging the church or risking jail – American paleontologist Stephen Jay Gould refuted such claims in his review of The Map That Changed the World. In fact, Smith's single-minded focus on recognizing layers made him rather late to realize that the Earth was old. Additionally, Smith's focus on recognizing layers based on the fossils in them was not unique. Though he was especially thorough, similar work (especially on the mainland of Europe) around the same time independently cemented the principle that fossils change over time and can be reliably used to identify layers. Martin Rudwick's Earth's Deep History: How It Was Discovered and Why It Matters is perhaps the most accessible account of the development of geology in this time interval.
|
||||
|
||||
|
||||
== Footnotes ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Presentation by Winchester on The Map that Changed the World, September 7, 2001, C-SPAN
|
||||
49
data/en.wikipedia.org/wiki/The_Math(s)_Fix-0.md
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49
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|
||||
---
|
||||
title: "The Math(s) Fix"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Math(s)_Fix"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:05.533110+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Math(s) Fix: An Education Blueprint for the AI Age is a 2020 book by Conrad Wolfram, a British technologist, entrepreneur, and co-founder of Wolfram Research Europe. The book argues that mathematics education worldwide is fundamentally broken and proposes a comprehensive restructuring centred on real-world problem-solving and computational thinking, rather than manual calculation.
|
||||
|
||||
|
||||
== Synopsis ==
|
||||
The book's core claim is that school mathematics curricula around the world teach the wrong thing: they overwhelmingly focus on training students to perform hand calculations that computers can now perform far more accurately and quickly. Wolfram contends this conflation of *mathematics* with *calculating by hand* has persisted long past its usefulness and is actively harmful, crowding out the genuinely important mathematical skills of:
|
||||
|
||||
Problem Definition - translating a real-world situation into a mathematical question
|
||||
Abstraction - identifying the relevant structure of a problem
|
||||
Computation - executing a solution (which in the modern world almost always means using a computer)
|
||||
Interpretation - applying and verifying the answer back in context
|
||||
Wolfram calls these the four steps of the "computational thinking" process, and argues they are what mathematicians, scientists, engineers, and data analysts actually do - yet schools spend nearly all their time on a narrow slice of step three (manual arithmetic and symbolic manipulation) while largely ignoring the other three. His argument is that computers have largely automated calculating, so the valuable human skill is now the surrounding thinking - knowing what to compute, why, and how to interpret the result. On this view, a student who cannot engage critically with a statistical claim, a data visualisation, or an algorithmic decision is not merely mathematically underprepared but functionally illiterate in an increasingly quantitative world.
|
||||
|
||||
|
||||
== Reception ==
|
||||
The Math(s) Fix was broadly well-received among progressive educators and those working at the intersection of technology and education policy. Supporters praised its clarity of argument and the comprehensiveness of its reform vision. In her review, Rachelle Dené Poth said that “I was immediately drawn in when I started reading this book and found that I couldn’t put it down”. Kara.Reviews writes that “The Math(s) Fix should be read by anyone with a strong interest in education policy, reform, or decision-making at any level.”.
|
||||
|
||||
|
||||
== See also ==
|
||||
|
||||
Computational thinking
|
||||
Computational literacy
|
||||
Mathematical modeling and simulations
|
||||
Mathematical notation
|
||||
Mathematical visualization
|
||||
STEM
|
||||
Artificial Intelligence in education
|
||||
Stephen Wolfram
|
||||
Wolfram Language
|
||||
Wolfram Mathematica
|
||||
Wolfram Research
|
||||
|
||||
|
||||
== External links ==
|
||||
computerbasedmath.org/
|
||||
Official page on Wolfram.com
|
||||
Conrad Wolfram TED Talk
|
||||
|
||||
|
||||
== References ==
|
||||
@ -0,0 +1,35 @@
|
||||
---
|
||||
title: "The Physics of Blown Sand and Desert Dunes"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/The_Physics_of_Blown_Sand_and_Desert_Dunes"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:52.778659+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
The Physics of Blown Sand and Desert Dunes is a scientific book written by Ralph A. Bagnold. The book laid the foundations of the scientific investigation of the transport of sand by wind. It also discusses the formation and movement of sand dunes in the Libyan Desert. During his expeditions into the Libyan Desert, Bagnold had been fascinated by the shapes of the sand dunes, and after returning to England he built a wind tunnel and conducted the experiments which are the basis of the book.
|
||||
Bagnold finished writing the book in 1939, and it was first published on 26 June 1941. A reprinted version, with minor revisions by Bagnold, was published by Chapman and Hall in 1953, and reprinted again in 1971. The book was reissued by Dover Publications in 2005.
|
||||
The book explores the movement of sand in desert environments, with a particular emphasis on how wind affects the formation and movement of dunes and ripples. Bagnold's interest in this subject was spurred by his extensive desert expeditions, during which he observed various sand storms. One pivotal observation was that the movement of sand, unlike that of dust, predominantly occurs near the ground, within a height of one metre, and was less influenced by large-scale eddy currents in the air.
|
||||
The book emphasises the feasibility of replicating these natural phenomena under controlled conditions in a laboratory. By using a wind tunnel, Bagnold sought to gain a deeper understanding of the physics governing the interaction between airstreams and sand grains, and vice versa. His aim was to ensure that findings from controlled experiments mirrored real-world conditions, with verifications of these laboratory results conducted through field observations in the Libyan Desert in the late 1930s.
|
||||
Bagnold delineates his research into two distinct stages. The first, which constitutes the primary focus of the book, investigates the dynamics of sand movement across mostly flat terrains. This includes understanding how sand is lifted, transported, and accumulated on a plane surface. Bagnold's wind tunnel experiments from the mid-1930s form the core of his analysis, though the book also dedicates chapters to the morphology of naturally occurring sand formations.
|
||||
The second stage, which Bagnold indicates is yet to be fully explored, delves into aeolian transport and the aerodynamics of airstreams as they navigate the curved surfaces of sand accumulations, hinting at the complexities of studying such natural systems. Apart from examining sand and its dynamics in the context of wind, Bagnold also makes comparisons between sand and dry granular snow, noting the parallels in their movement. Differences and similarities between sand movement in air versus water are also highlighted.
|
||||
Acknowledging the pioneering nature of his work, Bagnold expressed an awareness of the potential limitations and omissions in his study. He also emphasised the balance he attempted to strike between providing a rigorous scientific treatment, incorporating mathematical models and diagrams, while ensuring the content remained accessible to experts across diverse fields, from hydraulic engineering, to geophysics and geomorphology.
|
||||
The book is still a main reference in the field, and was used by NASA for studying sand dunes and the development of sand-driving mechanisms on Mars.
|
||||
|
||||
|
||||
== See also ==
|
||||
Aeolian processes
|
||||
Bagnold formula
|
||||
Bagnold number
|
||||
Barchan
|
||||
Bibliography of Aeolian Research
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
A short film containing an interview with R.A. Bagnold
|
||||
British Army Officers 1939–1945
|
||||
The Bibliography of Aeolian Research
|
||||
@ -4,7 +4,7 @@ chunk: 1/6
|
||||
source: "https://en.wikipedia.org/wiki/The_Structure_and_Distribution_of_Coral_Reefs"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:31:40.659577+00:00"
|
||||
date_saved: "2026-05-05T08:42:56.403568+00:00"
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/The_Structure_and_Distribution_of_Coral_Reefs"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:31:40.659577+00:00"
|
||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/The_Structure_and_Distribution_of_Coral_Reefs"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:31:40.659577+00:00"
|
||||
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/The_Structure_and_Distribution_of_Coral_Reefs"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:31:40.659577+00:00"
|
||||
date_saved: "2026-05-05T08:42:56.403568+00:00"
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/The_Structure_and_Distribution_of_Coral_Reefs"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:31:40.659577+00:00"
|
||||
date_saved: "2026-05-05T08:42:56.403568+00:00"
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||||
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|
||||
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|
||||
|
||||
|
||||
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|
||||
source: "https://en.wikipedia.org/wiki/The_Structure_and_Distribution_of_Coral_Reefs"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:31:40.659577+00:00"
|
||||
date_saved: "2026-05-05T08:42:56.403568+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
31
data/en.wikipedia.org/wiki/Theory_of_the_Earth-0.md
Normal file
31
data/en.wikipedia.org/wiki/Theory_of_the_Earth-0.md
Normal file
@ -0,0 +1,31 @@
|
||||
---
|
||||
title: "Theory of the Earth"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Theory_of_the_Earth"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:42:59.946789+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Theory of the Earth is a publication by James Hutton which laid the foundations for geology. In it he showed that the Earth is the product of natural forces. What could be seen happening today, over long periods of time, could produce what we see in the rocks. It also hypothesized that the age of the Earth was much older than what biblical literalists claim. This idea, uniformitarianism, was used by Charles Lyell in his work, and Lyell's textbook was an important influence on Charles Darwin. The work was first published in 1788 by the Royal Society of Edinburgh, and later in 1795 as two book volumes.
|
||||
Hutton recognized that rocks record the evidence of the past action of processes which still operate today. He also anticipated natural selection, as follows: "Those which depart most from the best adapted constitution, will be the most liable to perish, while, on the other hand, those organised bodies, which most approach to the best constitution for the present circumstances, will be best adapted to continue, in preserving themselves and multiplying the individuals of their race".
|
||||
|
||||
|
||||
== History ==
|
||||
Hutton's prose hindered his theories. They were not taken seriously until 1802, when Edinburgh University mathematics professor John Playfair restated Hutton's geological ideas in clearer, much simpler English. However, he left out Hutton's thoughts on evolution. Charles Lyell in the 1830s popularised the idea of an infinitely repeating cycle (of the erosion of rocks and the building up of sediment). Lyell believed in gradual change, and thought even Hutton gave too much credit to catastrophic changes.
|
||||
Hutton's work was published in different forms and stages:
|
||||
|
||||
1788. Theory of the Earth; or an investigation of the laws observable in the composition, dissolution, and restoration of land upon the Globe. Transactions of the Royal Society of Edinburgh, vol. 1, Part 2, pp. 209–304.
|
||||
1795. Theory of the Earth; with proofs and illustrations. 2 vols, Edinburgh: Creech.
|
||||
1899. Theory of the Earth; with proofs and illustrations, vol III. Edited by Sir Archibald Geikie. Geological Society, Burlington House, London.
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
|
||||
eBooks provided by Project Gutenberg:
|
||||
Theory of the Earth (1795, vol. 1)
|
||||
Theory of the Earth (1795, vol. 2)
|
||||
@ -4,7 +4,7 @@ chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Traces_of_Catastrophe"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:34:54.225437+00:00"
|
||||
date_saved: "2026-05-05T08:43:01.112862+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
|
||||
0
data/en.wikipedia.org/wiki/Volcano
Normal file
0
data/en.wikipedia.org/wiki/Volcano
Normal file
33
data/en.wikipedia.org/wiki/Álgebra_de_Baldor-0.md
Normal file
33
data/en.wikipedia.org/wiki/Álgebra_de_Baldor-0.md
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|
||||
---
|
||||
title: "Álgebra de Baldor"
|
||||
chunk: 1/1
|
||||
source: "https://en.wikipedia.org/wiki/Álgebra_de_Baldor"
|
||||
category: "reference"
|
||||
tags: "science, encyclopedia"
|
||||
date_saved: "2026-05-05T08:43:10.080065+00:00"
|
||||
instance: "kb-cron"
|
||||
---
|
||||
|
||||
Álgebra, commonly known as Álgebra de Baldor (Spanish: Baldor's Algebra), is a book by the Cuban mathematician, lawyer, and professor Aurelio Baldor. The first edition was published on 19 June 1941. Baldor’s Algebra contains a total of 5,790 exercises, averaging 19 exercises per test. It is considered one of the most comprehensive works on algebra.
|
||||
|
||||
|
||||
== Publication history ==
|
||||
Rhe first recorded edition of Baldor’s Algebra was published in 1941 by Editorial Cultural in Havana, Cuba. In 1948, Aurelio Baldor sold the rights to the Mexican publisher Publicaciones Culturales in order to invest the proceeds in his educational institute. This publisher continued to publish the book from Mexico. Upon his arrival in Mexico, already in exile, the book was published by Editorial Cultural Mexicana. According to this website, the book was also published in Venezuela, Colombia, and Spain. Publications continue in Mexico, following reorganizations and name changes of the original publishing companies, by Grupo Editorial Patria.
|
||||
|
||||
|
||||
== Description ==
|
||||
In the early editions of Algebra, the cover was red, and until the 2005 edition the illustrations were originally created by the Cuban artist D.G. Terminel. The cover featured the Persian Muslim mathematician Al-Khwarizmi and, in the background, a depiction of his native Baghdad, which covers part of the front and back covers. The front flap is illustrated with a portrait of the Greek mathematician and physicist Archimedes, as well as the siege of his city, Syracuse. The back flap featured a portrait of the Scottish mathematician John Napier, one of the developers of logarithms.
|
||||
For the 2007 edition, published by Grupo Editorial Patria, the book underwent a complete redesign, with the graphic elements updated by graphic designer Juan Bernardo Rosado Ortiz, illustrator José Luis Mendoza Monroy, and layout artist Carlos Sánchez. This edition is the first to include a CD-ROM to supplement the printed material.
|
||||
|
||||
|
||||
== Content ==
|
||||
The book consists of a preface, 39 chapters, and an appendix. The chapters, in order, are: Addition, Subtraction, Grouping Signs, Multiplication, Division, Notable Products and Quotients, The Remainder Theorem (also called the Residual Theorem), First-degree Integer Equations with One Unknown, Problems Involving First-degree Integer Equations with One Unknown, Factorial Decomposition, Greatest Common Divisor, Least Common Multiple, Algebraic Fractions—Simplifying Fractions, Operations with Algebraic Fractions, First-Degree Fractional Numerical Equations with One Unknown, First-Degree Algebraic Equations with One Unknown, Problems Involving First-Degree Fractional Equations—Movable-Point Problems, Formulas, Inequalities, Functions, Graphical representation of functions, Graphs—practical applications, Indeterminate equations, First-degree simultaneous equations with two unknowns, First-degree simultaneous equations with three or more unknowns, Problems solved using simultaneous equations, Elementary study of coordinate theory, Powers, Roots, Theory of exponents, Radicals, Imaginary numbers, Second-degree equations with one unknown, Problems solved using second-degree equations—The lamp problem, Theory of second-degree equations—Study of the quadratic trinomial, Binomial and trinomial equations, Sequences, Logarithms, Compound Interest, Amortization, and Impositions.
|
||||
The appendix contains tables for calculating compound interest and decreasing compound interest, a chart of the basic forms of factorial decomposition, and a table of powers and roots. Finally, it includes the answers to the more than 1,500 exercises found in some standard textbooks.
|
||||
Each chapter begins with an illustrated heading. The introductory sections are headed by a drawing alluding to prehistory and pre-Columbian civilizations, which signifies the origin of the concept of number. Chapter 1 is headed by an illustration alluding to mathematics in ancient Egypt. The brief accompanying text mentions the Rhind Papyrus. The next illustration deals with calculation in Chaldea and Assyria. Chapter 3 deals with Thales of Miletus. The following are, in that order: Pythagoras, Plato, Euclid, Archimedes, Ptolemy, Diophantus, Hypatia. Then there is an illustration on algebra in India, alongside its three main figures: Aryabhata, Brahmagupta, and Bhaskara. The next one is about the three greatest figures of the so-called Baghdad School: Al-Khwarizmi, Al-Battani, and Omar Khayyam. Next is an illustration discussing mathematics in the Hispano-Arabic universities and the contributions made by its best-known figures: Juan of Spain, Johannes de Sacrobosco, and Adelard of Bath. From there, the evolution of mathematics from the Late Middle Ages to the 20th century is presented, mentioning other mathematicians such as: Leonardo de Pisa, Raimundo Lulio, Nicolo Tartaglia, Gerolamo Cardano, François Viète, John Neper, René Descartes, Pierre Fermat, Blas Pascal, Isaac Newton, Gottfried Leibnitz, Brook Taylor, Leonardo Euler, Jean D'Alembert, Joseph-Louis Lagrange, Gaspard Monge, Pierre-Simon Laplace, Carl Friedrich Gauss, Augustin Louis Cauchy, Nikolai Lobachevsky, Niels Henrik Abel, Carl Gustav Jacobi, Évariste Galois, Karl Weierstrass, Henri Poincaré, Max Planck, and lastly Albert Einstein.
|
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|
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|
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== References ==
|
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|
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== External links ==
|
||||
Website
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The Éléments de géométrie algébrique (EGA; from French: "Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné) is a rigorous treatise on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut des Hautes Études Scientifiques. In it, Grothendieck established systematic foundations of algebraic geometry, building upon the concept of schemes, which he defined. The work is now considered the foundation and basic reference of modern algebraic geometry.
|
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|
||||
|
||||
== Editions ==
|
||||
Initially thirteen chapters were planned, but only the first four (making a total of approximately 1500 pages) were published. Much of the material which would have been found in the following chapters can be found, in a less polished form, in the Séminaire de géométrie algébrique (known as SGA). Indeed, as explained by Grothendieck in the preface of the published version of SGA, by 1970 it had become clear that incorporating all of the planned material in EGA would require significant changes in the earlier chapters already published, and that therefore the prospects of completing EGA in the near term were limited. An obvious example is provided by derived categories, which became an indispensable tool in the later SGA volumes, but was not yet used in EGA III as the theory was not yet developed at the time. Considerable effort was therefore spent to bring the published SGA volumes to a high degree of completeness and rigour. Before work on the treatise was abandoned, there were plans in 1966–67 to expand the group of authors to include Grothendieck's students Pierre Deligne and Michel Raynaud, as evidenced by published correspondence between Grothendieck and David Mumford. Grothendieck's letter of 4 November 1966 to Mumford also indicates that the second-edition revised structure was in place by that time, with Chapter VIII already intended to cover the Picard scheme. In that letter he estimated that at the pace of writing up to that point, the following four chapters (V to VIII) would have taken eight years to complete, indicating an intended length comparable to the first four chapters, which had been in preparation for about eight years at the time.
|
||||
Grothendieck nevertheless wrote a revised version of EGA I which was published by Springer-Verlag. It updates the terminology, replacing "prescheme" by "scheme" and "scheme" by "separated scheme", and heavily emphasizes the use of representable functors. The new preface of the second edition also includes a slightly revised plan of the complete treatise, now divided into twelve chapters.
|
||||
Grothendieck's EGA V which deals with Bertini type theorems is to some extent available from the Grothendieck Circle website. Monografie Matematyczne in Poland has accepted this volume for publication, but the editing process is quite slow (as of 2010).
|
||||
James Milne has preserved some of the original Grothendieck notes and a translation of them into English. They may be available from his websites connected with the University of Michigan in Ann Arbor.
|
||||
|
||||
|
||||
== Chapters ==
|
||||
The following table lays out the original and revised plan of the treatise and indicates where (in SGA or elsewhere) the topics intended for the later, unpublished chapters were treated by Grothendieck and his collaborators.
|
||||
|
||||
In addition to the actual chapters, an extensive "Chapter 0" on various preliminaries was divided between the volumes in which the treatise appeared. Topics treated range from category theory, sheaf theory and general topology to commutative algebra and homological algebra. The longest part of Chapter 0, attached to Chapter IV, is more than 200 pages.
|
||||
Grothendieck never gave permission for the 2nd edition of EGA I to be republished, so copies are rare but found in many libraries. The work on EGA was finally disrupted by Grothendieck's departure first from IHÉS in 1970 and soon afterwards from the mathematical establishment altogether. Grothendieck's incomplete notes on EGA V can be found at Grothendieck Circle.
|
||||
In historical terms, the development of the EGA approach set the seal on the application of sheaf theory to algebraic geometry, set in motion by Serre's basic paper FAC. It also contained the first complete exposition of the algebraic approach to differential calculus, via principal parts. The foundational unification it proposed (see for example unifying theories in mathematics) has stood the test of time.
|
||||
EGA has been scanned by NUMDAM and is available at their website under "Publications mathématiques de l'IHÉS", volumes 4 (EGAI), 8 (EGAII), 11 (EGAIII.1re), 17 (EGAIII.2e), 20 (EGAIV.1re), 24 (EGAIV.2e), 28 (EGAIV.3e) and 32 (EGAIV.4e).
|
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||||
== Bibliographic information ==
|
||||
Grothendieck, Alexandre; Dieudonné, Jean (1971). Éléments de géométrie algébrique: I. Le langage des schémas. Grundlehren der Mathematischen Wissenschaften (in French). Vol. 166 (2nd ed.). Berlin; New York: Springer-Verlag. ISBN 978-3-540-05113-8.
|
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Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4: 5–228. doi:10.1007/bf02684778. MR 0217083.
|
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Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8: 5–222. doi:10.1007/bf02699291. MR 0217084.
|
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Grothendieck, Alexandre; Dieudonné, Jean (1961). "Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie". Publications Mathématiques de l'IHÉS. 11: 5–167. doi:10.1007/bf02684274. MR 0217085.
|
||||
Grothendieck, Alexandre; Dieudonné, Jean (1963). "Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Seconde partie". Publications Mathématiques de l'IHÉS. 17: 5–91. doi:10.1007/bf02684890. MR 0163911.
|
||||
Grothendieck, Alexandre; Dieudonné, Jean (1964). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie". Publications Mathématiques de l'IHÉS. 20: 5–259. doi:10.1007/bf02684747. MR 0173675.
|
||||
Grothendieck, Alexandre; Dieudonné, Jean (1965). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie". Publications Mathématiques de l'IHÉS. 24: 5–231. doi:10.1007/bf02684322. MR 0199181.
|
||||
Grothendieck, Alexandre; Dieudonné, Jean (1966). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie". Publications Mathématiques de l'IHÉS. 28: 5–255. doi:10.1007/bf02684343. MR 0217086.
|
||||
Grothendieck, Alexandre; Dieudonné, Jean (1967). "Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie". Publications Mathématiques de l'IHÉS. 32: 5–361. doi:10.1007/bf02732123. MR 0238860.
|
||||
|
||||
|
||||
== See also ==
|
||||
Fondements de la Géometrie Algébrique (FGA)
|
||||
Séminaire de Géométrie Algébrique du Bois Marie (SGA)
|
||||
|
||||
|
||||
== References ==
|
||||
|
||||
|
||||
== External links ==
|
||||
Scanned copies and partial English translations: Mathematical Texts (published) Archived 2012-11-04 at the Wayback Machine
|
||||
Detailed table of contents: EGA
|
||||
SGA, EGA, FGA by Mateo Carmona
|
||||
The Grothendieck circle maintains copies of EGA, SGA, and other of Grothendieck's writings
|
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Éléments de mathématique (English: Elements of Mathematics) is a series of mathematics books written by the pseudonymous French collective Nicolas Bourbaki. Begun in 1939, the series has been published in several volumes, and remains in progress. The series is noted as a large-scale, self-contained, formal treatment of mathematics.
|
||||
The members of the Bourbaki group originally intended the work as a textbook on analysis, with the working title Traité d'analyse (Treatise on Analysis). While planning the structure of the work they became more ambitious, expanding its scope to cover several branches of modern mathematics. Once the plan of the work was expanded to treat other fields in depth, the title Éléments de mathématique was adopted. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras.
|
||||
The unusual singular "mathématique" (mathematic) of the title is deliberate, meant to convey the authors' belief in the unity of mathematics. A companion volume, Éléments d'histoire des mathématiques (Elements of the History of Mathematics), collects and reproduces several of the historical notes that previously appeared in the work.
|
||||
|
||||
== History ==
|
||||
In late 1934, a group of mathematicians including André Weil resolved to collectively write a textbook on mathematical analysis. They intended their work as a modern replacement for Édouard Goursat's Course in Mathematical Analysis (1902) —and also to fill a void in instructional material caused by the death of a generation of mathematics students in World War I. The group adopted the collective pseudonym Nicolas Bourbaki, after the French general Charles-Denis Bourbaki. During the late 1930s and early 1940s, the Bourbaki group expanded the plan of their work beyond analysis, and began publishing texts under the title Éléments de mathématique.
|
||||
Volumes of the Éléments have appeared periodically since the publication of the first Fascicule ("Installment") in 1939 by Éditions Hermann, with several being published during the 1950s and 1960s, Bourbaki's most productive period and time of greatest influence. Several years have sometimes passed before the publication of a new volume, and various factors have contributed to a slow pace of publication. The group's working style is slow and rigorous, and a final product is not deemed acceptable unless it is unanimously approved by the group. Further, World War II interrupted Bourbaki's activities during its early years. In the 1970s a legal dispute arose with Hermann, the group's original publisher, concerning copyright and royalty payments. The Bourbaki group won the involved lawsuit, retaining copyright over the work authored under the pseudonym, but at a price: the legal battle had dominated the group's attention during the 1970s, preventing them from doing productive mathematical work under the Bourbaki name. Following the lawsuit and during the 1980s, publication of new volumes was resumed via Éditions Masson. From the 1980s through the 2000s Bourbaki published very infrequently, with the result that in 1998 Le Monde pronounced the collective "dead". However, in 2012 Bourbaki resumed publication of the Éléments with a revised and expanded edition of the eighth chapter of Algebra, followed by the first of new books on algebraic topology (covering also material that had originally been planned as the eleventh chapter of the group's book on general topology), a significantly expanded book on spectral theory in two volumes, and in 2026 two new chapter on topological vector spaces. Furthermore, two entirely new books (on category theory and modular forms) are stated to be under preparation.
|
||||
Springer Verlag became Bourbaki's current publisher during the 21st century, reprinting the Éléments while also publishing new volumes. Some early versions of the Éléments can be viewed at an online archive, and the mathematical historian Liliane Beaulieu has documented the sequence of publication.
|
||||
The Éléments have had a complex publication history. From the 1940s through the 1960s, Bourbaki published the Éléments in booklet form as small installments of individual chapters, known in the French as fascicules. Despite having settled on a logical sequence for the work (see below), Bourbaki did not publish the Éléments in the order of its logical structure. Rather, the group planned the arc of the work in broad strokes and published disparate chapters wherever they could agree on a final product, with the understanding that (logically) later chapters published (chronologically) first would ultimately have to be grounded in the later publication of logically earlier chapters. The first installment of the Éléments to be published was the Summary of Results for the Theory of Sets in 1939; the first proper chapter of content on set theory—with proofs and theorems—did not appear until 1954. Independently of the work's logical structure, the early fascicules were assigned chronological numberings by the publisher Hermann for historical reference. Gradually, the small fascicules were collected and reprinted in larger volumes, forming the basis of the modern edition of the work.
|
||||
The large majority of the Éléments has been translated into an English edition, although this translation is incomplete. Currently the complete French edition of the work consists of 12 books printed in 29 volumes, with 73 chapters. The English edition completely reproduces seven books and partially reproduces two, with three unavailable; it comprises 14 volumes, reproducing 58 of the original's 73 chapters. However, the English General Topology is not based on latest revised French edition (of 1971 and 1974) and misses some material added there (for example on quaternions and rotation groups in Chapter VIII).
|
||||
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date_saved: "2026-05-05T08:44:15.713125+00:00"
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|
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|
||||
== Structure ==
|
||||
Éléments de mathématique is divided into books, volumes, and chapters. A book refers to a broad area of investigation or branch of mathematics (Algebra, Integration); a given book is sometimes published in multiple volumes (physical books) or else in a single volume. The work is further subdivided into chapters with some volumes consisting of a single chapter.
|
||||
Typically of mathematics textbooks, the Éléments' chapters present definitions, mathematical notation, proofs of theorems and exercises, forming the core mathematical content of the work. The chapters are supplemented by historical notes and summaries of results. The former usually appear after a given chapter to contextualize the development of its topics, and the latter are occasionally used sections in which a book's major results are collected and stated without proof. Eléments d'histoire des mathématiques is a compilation volume of several of the historical note sections previously published in the Éléments proper, through the book on Lie groups and Lie algebras.
|
||||
When Bourbaki's founders originally planned the Treatise on Analysis, they conceived of an introductory and foundational section of the text, which would describe all prerequisite concepts from scratch. This proposed area of the text was referred to as the "Abstract Packet" (Paquet Abstrait). During the early planning stages the founders greatly expanded the scope of the abstract packet, with the result that it would require several volumes for its expression rather than a section or chapter in a single volume. This portion of the Éléments was gradually realized as its first three books, dealing with set theory, abstract algebra, and general topology.
|
||||
Today, the Éléments divide into two parts. Bourbaki structured the first part of the work into six sequentially numbered books: I. Theory of Sets, II. Algebra, III. General Topology, IV. Functions of a Real Variable, V. Topological Vector Spaces, and VI. Integration. The first six books are given the unifying subtitle Les structures fondamentales de l’analyse (Fundamental Structures of Analysis), fulfilling Bourbaki's original intent to write a rigorous treatise on analysis, together with a thorough presentation of set theory, algebra and general topology.
|
||||
Throughout the Fundamental Structures of Analysis, any statements or proofs presented within a given chapter assume as given the results established in previous chapters, or previously in the same chapter. In detail, the logical structure within the first six books is as follows, with each section taking as given all preceding material:
|
||||
|
||||
I: Theory of Sets
|
||||
II (1): Algebra, chapters 1-3
|
||||
III (1): General Topology, chapters 1-3
|
||||
II (2): Algebra, from chapter 4 onwards
|
||||
III (2): General topology, from chapter 4 onwards
|
||||
IV: Functions of a Real Variable
|
||||
V: Topological Vector Spaces
|
||||
VI: Integration
|
||||
Thus the six books are also "logically ordered", with the caveat that some material presented in the later chapters of Algebra, the second book, invokes results from the early chapters of General Topology, the third book.
|
||||
Following the Fundamental Structures of Analysis, the second part of the Éléments consists of books treating more modern research topics: Lie Groups and Lie Algebras, Commutative Algebra, Spectral Theory, Differential and Analytic Manifolds, and Algebraic Topology. Whereas the Éléments' first six books followed a strict, sequential logical structure, each book in the second part is dependent on the results established in the first six books, but not on those of the second part's other books. The second part of the work also lacks a unifying subtitle comparable to the Fundamental Structures of Analysis.
|
||||
|
||||
== Volumes ==
|
||||
The Éléments are published in French and English volumes, detailed below.
|
||||
|
||||
== See also ==
|
||||
Euclid's Elements
|
||||
Principia Mathematica
|
||||
|
||||
== Notes ==
|
||||
|
||||
== References ==
|
||||
|
||||
== Further reading ==
|
||||
Leo Corry Writing the ultimate mathematical textbook: Nicholas Bourbaki's Éléments de mathématique
|
||||
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