Diophantus of alexandria arithmetica book i joseph. The problem of finding a general solution was finally solved by euclid 3rd century bc, with. One of the most famous problems that diophantus treated was writing a square as the sum of two squares book ii, problem 8. Solve problems, which are from the arithmetica of dioph. Heath briefly goes through the histories of the various translations. This book features a host of problems, the most significant of which have come to be called diophantine equations. From aristarchus to diophantus, clarendon press, oxford 1921. To divide a given square into a sum of two squares.
Let one of the required squares be x2 then 16 x2 16x2 must be equal to a square. Problem find two square numbers such that the sum of the product of the two numbers with either number is also a square number. 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. Another unresolved question about diophantus is that concerning the. Find two numbers such that the square of either added to the sum of both gives a square. He had his first beard in the next 112 of his life. One of his greatest contributions is the book of arithmetica. In other words, for the given numbers a and b, to find x and y such that x y a and x3 y3 b. The history of algebra is very intriguing because of the many cultures that contributed to its origins. Instead of the square root of 81 144 required by the formula, he takes the square root of 144 81.
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. A problem of diophantus and dicksons conjecture andrej dujella abstract. 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. Book iv problem 21 to nd four numbers such that the product of any two added to one gives a square. Some of the things i found though were that he studied at the university of alexandria in egypt. Generalized solution in which the sides of triangle oab form a rational triple if line cb has a rational gradient t. Problem to nd a number whose di erences from two given numbers 9,21 are both squares. 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.
In it he introduced algebraic manipulations on equations including a symbol for one unknown probably following other authors in alexandria. Historia mathematica 28 2001, 167206, and the introduction to the first volume of the new english translation of archimedes. Diophantine equations i putnam practice october 27, 2004 in his book arithmetica diophantus discussed the problem of. Diophantus of alexandria, arithmetica and diophantine. World heritage encyclopedia, the aggregation of the largest online encyclopedias available, and the. Diophantus was another mathematician who lived at alexandria and about. For simplicity, modern notation is used, but the method is due to diophantus. To read this file, the font must be set to uniicode, i. The thirteen books of the almagest are the most influential and significant trigonometric work of all. Of particular note is problem 8, since it is to this problem which fermat appended his famous last theorem. At the end of the following 17 of his life diophantus got married. We may generalize diophantus s solution to solve the problem for any given square, which we will represent algebraically as a 2. Find three numbers such that when any two of them are added, the sum is one of three given numbers.
He did numerous things for the world of mathematics. 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. Includes bibliographical references addeddate 20061222 03. For example, book ii, problem 8, seeks to express a given. From the same required number to subtract two given numbers so as to make. Other readers will always be interested in your opinion of the books youve read. Theres just an abstract from the books, mostly an abbreviated description of the problems and their solutions which doesnt seem to be a 1.
Answer to solve problems, which are from the arithmetica of diophantus. Book iii problem 9 to nd three squares at equal intervals. Diophantus of alexandria a study in the history of greek. A free powerpoint ppt presentation displayed as a flash slide show on id. Diophantus was the first greek mathematician who recognized fractions as numbers, thus allowed positive rational numbers for the coefficients and solutions. Diophantus, alexandria, mathematics, greek, fermats theorem publisher cambridge. 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. At the close of the introduction, diophantus speaks of the thirteen books into which he. Although there were many ancient civilizations that studied algebra, there are two men that are best know for bringing algebra to our modern day. A contribution of diophantus to mathematics the following is a statement of arithmetica book ii, problem 28 and its solution. 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. Books iv to vii of diophantus arithmetica springerlink.
Diophantus wrote a seminal series of books called the arithmetica. Diophantus selected a particular instance of a perfect square to set this equal to, one that was particularly useful in. I found very little known facts about diophantuss life. Some parts of greek mathematics the prime example is book ii of the elements are, in the original. For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more. 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 following is problem 7 of the first book of arithmetica. Fourteen centuries later fermat was reading diophantus book and asked the famous question. The problems of book i are not characteristic, being mostly simple problems used to illustrate algebraic reckoning. Problem for arithmetica book ii, problem 8 divide a given square number, say 16, into the sum of two squares. 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. Diophantus died 4 years after the death of his son. It seems more like a book about diophantuss arithmetica, not the translation of the actual book. 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.
The following is a statement of arithmetica book ii, problem 28 and its solution. The first part of the book presents the elementary facts of algebraic geometry essential to understanding the rest of it. For a new translation in french of the ten known books of the. Sesiano, jacques, books iv to vii of diophantus arithmetica in the arabic translation attributed to qusta ibn luqa new york, 1982, 8. 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. Diophantus project gutenberg selfpublishing ebooks. Book ii problem 8 to split a given square 16 in two squares. 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. Diophantuss arithmetica1 is a list of about 128 algebraic problems with so lutions. Derive the necessary condition on a and b that ensures a rational solution. 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. Solve problems, which are from the arithmetica of diophantus.
1152 201 1031 1539 689 1474 1578 633 1401 985 130 273 904 1295 1318 1371 1153 594 1411 321 1027 1031 939 678 206 659 655 361 193 1329 871 188 808 788 948 744 1165 1461 63 1047 701 1018 387 1347