Math 488H and Math 575H: Topics in the History of Mathematics

Information Page, Spring 2016

For a great account of the developement of numbers see "The Universal History of Numbers" by Georges Ifrah.

Ishango bone, ants-that-count, fish-that-count, animals-that-count and more animals-that-count.

Here you can see Babylonian digits. Here is a table of multiplication by 4.

Here are some pages of the book "Episodes from the early history of mathematic" by A. Aaboe.

Here is a nice article by J Friberg about Babylonian mathematics, and here are some pages of the book "A remarkable collection of Babylonian mathematical texts" by J. Friberg.

Here, here, and here is the tablet with square root of 2. Here is an article about Babylonian way of aproximation of square roots.

Here and here is a discussion of the tablet Plimpton 322. Here, here , and here are nice articles about interpratations of Plimpton 322.

Here is another interesting paper by J. Friberg about some amazing Babylonian tablets.

Here is a nice discyssio of Babylonian approximation to the area of a circle.

Here and here are some very recent and very interesting discoveries about Babylonian mathematics.

Here is a nice overview of the Baylonian mathematics.

Here is a short account of the history of ancient Egypt.

Here is a short discussion of Egyptian hieroglyphic numerals, here is a picture of a stone with some numerals, here and here are both the hieroglyphic and hieratic numerals.

Here is a nice discussion of Egyptian fractions, here is an interesting article about modern results on Egyptian fractions, and here is the 2/n table of from the Rhind (Ahmes) papyrus.

Here is a discussion of the Egyptian papyri.

Here is a short discussion of the Ahmes papyrus, and here is a discussion of the Moscow papyrus.

Here is a discussion of problem 10 from the Moscow papyrus.

Here is an article about mathematics of Liu Hui, who among other things computes volumes of some pyramids and of frustum. Frustum

Here is a short history of Antient Greece.

Here and here is a discussion of Greek number systems.

A short biography of Thales and Pythagoras.

Here is an interesting view on Pythagoras and his brotherhood. Here is a discussion of the Wallace-Gerwien-Bolyai Theorem. Here is a discussion of an analogous problem in three dimensions (often called Hilbert's third problem).

A short biography of Hippocrates, and Archytas.

Here is a discussion of the lunes and here is another discussion of the lunes.

Here is a short history of the problem of angle trisection and here and here is a discussion of the problem of doubling a cube.

Here is a short discussion of the problems of antiquity.

Here and here you can see the quadratrix.

A short biography of Plato, Theaetetus, Eudoxus, and Eudoxus again.

A short biography of Euclid and Euclid again.

Here is a discussion of the model of planetary motion dues to Eudoxus.

Here is a nice version of Euclids "Elements", and here is a short note about Euclid and his "Elements".

You should be familiar at least with the following items from the "Elements":

Here is a short history of perfect numbers.

Here and here is information about the Platonic solids.

Here and here are some basic facts about continued fractions.

Here is a nice note about continued fractions and its connections to Greek geometry.

Here is a very interesting take on continued fractions and Greek mathematics (it is a challenging read though). See in particular the Appendix.

A short biography of Archimedes.

Here are the works of Archimedes.

Here is a nice description of Archimedes' approximation of pi.

Here is a discussion of how Archimedes arrived at the formula for a volume of a ball (his "Method").

Here you can read about the history of the Archimedes palimpsest.

Here you can see how the Archimedes text in the pailmpsest was recently deciphered, and here is a longer discussion of the history of the palimpsest.

Here is an interesting lecture on Archimedes and his "Method".

Here you can read about tomb of Archimedes.

Here is a nice translation of the "Quadrature of the Parabola" by Archimedes.

Here is a discussion of the spiral of Archimedes.

Here is a an interesting description of Stomachion and the recent developments about it. Here and here is more about Stomachion.

Here is a discussion of the angle trisection method of Archimedes.

Here is a short biography of Apollonius.

Here is a nice essay on conic sections.

Here are some basic properties of conics.

Here is a translation of "Conics" by Apollonius.

Here is a copy of "On burning mirrors" by Diocles. Here is a short biography of Diocles.

Here you can play with Apollonius circles, and here you can find a proof.

Here is the idea for angle trisection using a conchoid (note that it is incorrectly atributed to Hippocrates; it was definitely known to Nicomedes).

Here is a discussion of conchoid and its use to construct two mean proportionals. Here you can play with conchoids. Here is a conchoidograph and here and here is a more classical version of the tool.

A short biography of Aristarchus, Nicomedes, Eratosthenes, Hipparchus, and Ptolemy.

Here is interesting modern take on Almagest and astronomy in general.

Here and here are short movies about how Eratosthenes measured the circumference of the Earth.

Here you can see how Hipparchus measured the distance to the moon.

Here is a discussion of Ptolemy's theorem.

A short biography of Heron and Menelaus.

Here is an applet to play with spherical geometry.

Here is a discussion of Menealus's Theorem, with several proofs and additional topics.

Here is Newton's approach to the duplication of a cube (at the bottom of the page, construction using a marked ruler). A homework problem asks you to justify this method.

Here, and here are are 2 different proofs of Heron's formula.

A short biography of Diophantus. Here is discussion of problem 8 in book 2 of Diophantus's "Aritmetica". Here is a page from an edition of "Arithmetica" with Fermat's commentaries.

Here is a translation of Diophantus's "Aritmetica" with extensive introduction (you need djvu viewer to open it), here are other formats available (pdf, kindle, etc.).

Here ( old) is a very nice paper about Diophantus and his influence on the development of algebra and number theory.

A short biography of Pappus, Theon, Hypatia, and Proclus.

Here is the Pappus' hexagon theorem. Here is a discussion of the Pappus-Guldin theorem. --

Biographies of first Indian mathematicians Aryabhata I, Brahmagupta, Bhaskara I

A short biography of Al-Khwarizmi, Al-Karaji, and Omar Khayyam.

A short biography of Fibonacci, Luca Pacioli, Scipione del Ferro, Nicolo Fontana (Tartaglia), Girolamo Cardano, Lodovico Ferrari, and Rafael Bombelli.

Here is a discussion of the controversy between Trataglia and Cardano.

Here is a short discussion of the history of equations of degree 2,3,4.

page 17

A short biography of François Viète and Thomas Harriot.

Adrian von Roomen's problem.

A short biography of René Descartes, Pierre de Fermat, and Blaise Pascal.

A short biography of John Wallis, Isaac Newton and Gottfried Wilhelm von Leibniz.

Here is a nice article about how Wallis discovered some of his results.

Here you can see how Newton discovered his binomial theorem.

Here is a short account of the main contributions of Newton and Leibniz to calculus.

Here you can see Newton's notebook with his discoveries between 1664-1665 (he was 21).

Here is a nice talk by William Dunham about Newton.

A short biography of Jacob (Jacques) Bernoulli, Johann Bernoulli, de L'Hopital, and Leonhard Euler.

Here is an archive of all the works of Euler. In particular, here is a translation of the original articles in which Euler computes the sum of squeres of reciprocals of the natural numbers.

Here is a discussion of Fermat numbers.

Assignments

Solutions to each problem must be written clearly, carefully, using complete sentences. All details should be provided with careful explanations.