Scribd is the worlds largest social reading and publishing site. This book features a host of problems, the most significant of which have come to be called diophantine equations. Mathematics in the time of the pharos free ebook download as pdf file. For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more. Find two square numbers whose di erence is a given number, say 60. Nonetheless, restating them algebraically can aid in understanding them. The eighth problem of the second book of diophantus s arithmetica is to divide a square into a sum of two squares. In diophantus there is another problem, v, 5, on the same subject2.
Book ii, iii, iv, and v contain indeterminate problems, and book vi contains. This book tells the story of abels problem and his proof. Diophantus of alexandria had a great impact in the world of mathematics. The first ten propositions of book ii can be easily interpreted in modern algebraic notation. The eighth problem of the second book of diophantuss arithmetica is to divide a square into a sum of two squares.
In the margin of his copy of diophantus s book on polygonal numbers, he wrote that he had discovered this proposition and called it beautiful and wonderful. Intersection of the line cb and the circle gives a rational point x 0,y 0. Of course, in doing so the geometric flavor of the propositions is lost. Abels step ii implies that r 5 is a 2 3 4 rational function of the roots. To divide a given square into a sum of two squares. In it he introduced algebraic manipulations on equations including a symbol for one unknown probably following other authors in alexandria. Book x presumably greek book vi deals with rightangled triangles with rational sides and subject to various further conditions. It consists of a short paragraph from theons commentary on the first book of. Thus the problem has been reduced to a linear equation, which.
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. These numbers are 0,1,2, none of which give you 3 modulo 4, a. 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. Diophantus of alexandria, arithmetica and diophantine equations. On the contrary, it is impossible to divide either a cube into two cubes. Immediately preceding book i, diophantus gives the following definitions to solve these simple problems. Diophantus of alexandria arithmetica book i joseph. Alexandrian algebra according to diophantus mathematics. The problems in book i of the arithmetica are determinate ie, having a unique solution or a. Murty,esmonde problems in algebraic number theory summation. A similar problem involves decomposing a given integer into the sum of three squares. Diophantuss arithmetica1 is a list of about 128 algebraic problems with so lutions.
1246 40 529 772 88 1166 509 1538 349 866 119 1432 153 562 296 866 1504 1564 828 875 146 766 891 1554 471 629 1147 1352 1543 825 1515 470 932 296 1118 1052 347 219 864 462 1145 588 1075