Diophantus of alexandria, arithmetica and diophantine. In other words, for the given numbers a and b, to find x and y such that x y a and x3 y3 b. By the egyptian method psellus means simply the arithmetic techniques expounded by diophantus, based on the special nomenclature of the arithmos and its powers sc. Diophantus and pappus ca 300 represent a shortlived revival of greek mathematics in a society that did not value math as the greeks had done 500750 years earlier. He had his first beard in the next 112 of his life. A contribution of diophantus to mathematics the following is a statement of arithmetica book ii, problem 28 and its solution. Diophantine equations i putnam practice october 27, 2004 in his book arithmetica diophantus discussed the problem of.
For example, book ii, problem 8, seeks to express a given. Diophantus project gutenberg selfpublishing ebooks. Theres just an abstract from the books, mostly an abbreviated description of the problems and their solutions which doesnt seem to be a 1. The distinctive features of diophantus s problems appear in the later books. Other readers will always be interested in your opinion of the books youve read. Diophantus wrote a seminal series of books called the arithmetica.
I found very little known facts about diophantuss life. Diophantus, alexandria, mathematics, greek, fermats theorem publisher cambridge. For simplicity, modern notation is used, but the method is due to diophantus. This book features a host of problems, the most significant of. Generalized solution in which the sides of triangle oab form a rational triple if line cb has a rational gradient t. Book ii problem 8 to split a given square 16 in two squares. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them. Derive the necessary condition on a and b that ensures a rational solution. One of his greatest contributions is the book of arithmetica. Problem find two square numbers such that the sum of the product of the two numbers with either number is also a square number. Find two numbers such that the square of either added to the sum of both gives a square. Problem to nd a number whose di erences from two given numbers 9,21 are both squares.
Another unresolved question about diophantus is that concerning the. This problem became important when fermat, in his copy of diophantus arithmetica edited by bachet, noted that he had this wonderful proof that cubes cant. The first part of the book presents the elementary facts of algebraic geometry essential to understanding the rest of it. Sesiano, jacques, books iv to vii of diophantus arithmetica in the arabic translation attributed to qusta ibn luqa new york, 1982, 8. To divide a given square into a sum of two squares. Of particular note is problem 8, since it is to this problem which fermat appended his famous last theorem. Diophantus was another mathematician who lived at alexandria and about. Diophantus lived in alexandria in times of roman domination ca 250 a. Diophantus selected a particular instance of a perfect square to set this equal to, one that was particularly useful in. The problem of finding a general solution was finally solved by euclid 3rd century bc, with.
Solve problems, which are from the arithmetica of dioph. At the close of the introduction, diophantus speaks of the thirteen books into which he. This edition of books iv to vii of diophantus arithmetica, which are extant only in a recently discovered arabic translation, is the outgrowth of a doctoral dissertation submitted to the brown university department of the history of mathematics in may 1975. For a new translation in french of the ten known books of the. The denominations in anatolius are the same as in diophantus for the first, the fourth and the sixth powers, they differ though for the fifth power, i. Instead of the square root of 81 144 required by the formula, he takes the square root of 144 81. Diophantus of alexandria a study in the history of greek. Problem for arithmetica book ii, problem 8 divide a given square number, say 16, into the sum of two squares. From aristarchus to diophantus, clarendon press, oxford 1921. In this work, sesiano presents an english translation of and commentary on four of the thirteen books of the arithmetica, which had been considered lost for well over six hundred years. Heath briefly goes through the histories of the various translations.
Includes bibliographical references addeddate 20061222 03. The history of algebra is very intriguing because of the many cultures that contributed to its origins. The following is problem 7 of the first book of arithmetica. Some parts of greek mathematics the prime example is book ii of the elements are, in the original. The text used is the edition of tannery 1893, but i have also consulted the translation of ver eecke 1959 and the paraphrase of heath 1910. Diophantus died 4 years after the death of his son. To read this file, the font must be set to uniicode, i. The following is a statement of arithmetica book ii, problem 28 and its solution.
The problems one of the most famous problems that diophantus treated was writing a square as the sum of two squares book ii, problem 8. The thirteen books of the almagest are the most influential and significant trigonometric work of all. Book iii problem 9 to nd three squares at equal intervals. The system is described in the first part of the text tannery, 189395, ii, 37. In short, diophantus quite uncommonly provides his readers with all the tools necessary to follow and understand step by step the whole procedure of solution of problem 2.
Book iv problem 21 to nd four numbers such that the product of any two added to one gives a square. Find three numbers such that when any two of them are added, the sum is one of three given numbers. A problem of diophantus and dicksons conjecture andrej dujella abstract. Answer to solve problems, which are from the arithmetica of diophantus. At the end of the following 17 of his life diophantus got married. Books iv to vii of diophantus arithmetica springerlink.
A free powerpoint ppt presentation displayed as a flash slide show on id. The second half of the book considers the evolution of the theory of diophantine equations from the renaissance to the middle of the 20th century. The translation and meaning of the adjective sparked a longlasting controversy that has become a nonnegligible aspect of the debate about the possibility of interpreting diophantus approach and, more generally, greek mathematics in algebraic terms. He did numerous things for the world of mathematics. Fourteen centuries later fermat was reading diophantus book and asked the famous question. Let one of the required squares be x2 then 16 x2 16x2 must be equal to a square. Diophantuss book text book is wonderful if one wants to learn about greek mathematics by puzzling through and by attempting to follow how he solved a lot of complex, complicated algebra problems. One of these poems relates to the life, and the age at death, of a thirdcentury mathematician named diophantus, who lived in or around alexandria, egypt but was probably of greek heritage. For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more. From the same required number to subtract two given numbers so as to make. Diophantus was the first greek mathematician who recognized fractions as numbers, thus allowed positive rational numbers for the coefficients and solutions. Historia mathematica 28 2001, 167206, and the introduction to the first volume of the new english translation of archimedes.
This book features a host of problems, the most significant of which have come to be called diophantine equations. The problems of book i are not characteristic, being mostly simple problems used to illustrate algebraic reckoning. We may generalize diophantus s solution to solve the problem for any given square, which we will represent algebraically as a 2. Solve problems, which are from the arithmetica of diophantus. It seems more like a book about diophantuss arithmetica, not the translation of the actual book. Diophantus of alexandria arithmetica book i joseph. Although there were many ancient civilizations that studied algebra, there are two men that are best know for bringing algebra to our modern day. In it he introduced algebraic manipulations on equations including a symbol for one unknown probably following other authors in alexandria. World heritage encyclopedia, the aggregation of the largest online encyclopedias available, and the. His writing, the arithmetica, originally in books six survive in greek, another four in medieval arabic translation, sets out hundreds of arithmetic problems with their solutions. Some of the things i found though were that he studied at the university of alexandria in egypt. One of the most famous problems that diophantus treated was writing a square as the sum of two squares book ii, problem 8. We find an echo of euclids search for pythagorean triples in the work of diophantus 3rd century ad who in his arithmetica book ii, problem 8, seeks to divide a given square number into.