The author thanks benjamin braun, for whose history of mathematics course this paper was originally written, and an anonymous referee for their guidance and suggestions. In diophantus arithmetica, how would the string of. The symbolic and mathematical influence of diophantuss. Answer to solve problems, which are from the arithmetica of diophantus. The general assertion concerning fn was proved by the german mathematician david hilbert in 1909.
Selected problems of second degree from arithmetica. Indeterminate problems, which are number theory because the solutions are required to be rational numbers the only kind recognized by diophantus, begin in book 2. Heath d 2 furthermore, wilbur knorr concluded diophantus dates to be. Diophantus solution is quite clear and can be followed easily. Some problems of diophantus franz lemmermeyer december 21, 2003 it is believed that diophantus worked around 250 ad. 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. Diophantuss book is for the truly dedicated scholars and hobbyists who may still be searching for a proof for f. Thus, it is clear that diophantus did not invent algebra but rather collected, expanded, and generalized the work of the earlier algebraists. He solved only equations that used whole numbers and their powers. Diophantus of alexandria department of mathematics. This book features a host of problems, the most significant of which have come to be called diophantine equations. Heath argues that diophantus is contemporary to anatolius, who was the bishop of laodicea around 280ce. If this information is correct, by solving this algebraic problem, diophantus married at 33 and had a son who died at 43, four years before diophantus himself died at 84. Diophantus takes the square to be 16 and solves the problem as follows.
Archimedes has 51780 v3 265153 which is pretty accurate, but he does not tell how he got it. And if diophantus states a necessary condition for dividing a number into two or three squares as in the previous case of v. 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. The construction of each problem in arithmetica follows this pattern.
I describe diophantuss arithmetica and abu kamils book on algebra, the tw o main 2 such is the case with roshdi rasheds accounts of arabic alg ebra. The number he gives his readers is 100 and the given difference is 40. Diophantus was the first greek mathematician who recognized fractions as numbers, thus allowed positive rational numbers for the coefficients and solutions. The problem in the very first problem in the very first book of arithmetica diophantus asks his readers to divide a given number into two numbers that have a given difference. From reading the problem i intend to discuss, found in book two of arithmetica, it can. In these books, diophantus solves indeterminate equations. Let the first summand be, and thus the second the latter is to be a square. In 1912 the german mathematicians arthur wieferich and aubrey kempner proved that f3 9. Arithmetica work by diophantus arith,etica a history of mathematics second ed.
The purpose of any diophantine equation is to solve for all the unknowns in the problem. To find a number such that when two given numbers 100, 20. The symbolic and mathematical influence of diophantuss arithmetica. Book x presumably greek book vi deals with rightangled triangles with rational sides and subject to various further conditions. In it he introduced algebraic manipulations on equations including. The following is problem 7 of the first book of arithmetica. 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. For example to find a square between 54 and 2 he multiplies both by 64, spots the square 100 between 80 and 128, so obtaining the solution 2516 to. When diophantus was dealing with 2 or more unknowns, he would try to write all the unknowns in terms of only one of them. It is therefore difficult for a modern, after studying 100 diophantic equations, to solve the. However, the necessity of his necessary condition must be explored. Babylonians yes v 2 pretty much ignored by the classical greeks because of the difficulty with irrational numbers. An example of this is found in problem 16, book i of the arithmetica, and it. The solution diophantus writes we use modern notation.
Introduction the works of the mathematician diophantus have often struck readers as idiosyncratic. The general assertion concerning f n was proved by the german mathematician david hilbert in diophantus of alexandria c. Diophantus of alexandria arithmetica book i joseph. The thirteen books of the almagest are the most influential and significant. Diophantus of alexandria, arithmetica and diophantine equations. Diophantus was a hellenistic greek or possibly egyptian, jewish or even chaldean mathematician who lived in alexandria during the 3rd century ce.
A similar problem involves decomposing a given integer into the sum of three squares. For example, book ii, problem 8, seeks to express a given square number as the sum of two square numbers here read more. Forty two problems of first degree from diophantus arithmetica a thesis by tinka davis bachelor of science, so. Another type of problem which diophantus studies, this time in book iv, is to find powers between given limits. For example, in problem 14, book i of the arithmetica, he chose a given ratio as well as a second value for x, thus creating a rather simple problem to solve gow 120. The eighth problem of the second book of diophantuss arithmetica is to divide a square into a sum of two squares. Diophantus arithmetica is a work in books, but only survived in the original greek. The six books of the arithmetica present a collection of both determinate and in. The following is a statement of arithmetica book ii, problem 28 and its solution. Diophantus illustrates this problem using the number 16 as an example. Problem 24 of book iv of arithmetica is particularly prophetic, although it is the only example of this kind in the entire work. We produced a computer package to solve a problem called the sunit equation, with the hope that number theorists of all stripes can more easily attack a wide variety of unsolved problems in mathematics.
Books iv to vii of diophantus arithmetica springerlink. Problem find two square numbers such that the sum of the product of the two numbers with either number is also a square number. Of the original thirteen books of the arithmetica, only six have survived, although some diophantine problems from arithmetica have also been found in later arabic sources. This gives rise to a linear equation in diophantus age x much simpler than anything. Arithmetica is the major work of diophantus and the most prominent work on algebra in greek mathematics. All the equations in the book were latter given a name of diophantine equations and the method for solving them is known as diophantine analysis. It seems more like a book about diophantus s arithmetica, not the translation of the actual book. Go to abbreviations for forms go to rules for manipulations of forms go to prob.
To find a number such that when two given numbers 100,20. 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 first known work to use algebra in a modern style is the arithmetica of the greek mathematician diophantus of alexandria. The second one expands the square of the modulus of zw ztimes the complex conjugate of w. Pythagorean numerology and diophantus arithmetica a note on hippolytus elenchos i 2 eugene afonasin novosibirsk state university keenness backed by teaching is a swift road to knowledge. Diophantuss arithmetica1 is a list of about 128 algebraic problems with so lutions. In his text arithmetica, the mathematician diophantus looked at algebraic equations whose solutions are. Diophantus project gutenberg selfpublishing ebooks. It seems more like a book about diophantuss arithmetica, not the translation of the actual book. For simplicity, modern notation is used, but the method is due to diophantus. Of the original thirteen books of which arithmetica consisted only six have survived, though there are some who believe that four arabic books discovered in 1968 are also by diophantus.
Of course, these are our modern symbolic representations of the papyrus rhind problems. Diophantus lived in alexandria in times of roman domination ca 250 a. On the other hand, diophantus is quoted around 350ce by theon of alexandria, heath, 2 giving us a possible interval of about five hundred years. Problem 2 to split a given number 60 in two parts having a given ratio 3. Problem 3 to split a given number 80 in two parts, the larger of which has a given ratio 3. Diophantus arithmetica diophantus was the author of three books, one is called the arithmetica that deals with solving algebraic equations, while the other two books are. It was diophantus who began using letters as symbols for operations in algebra. To divide a given square into a sum of two squares. Alexandrian algebra according to diophantus mathematics. Pythagorean numerology and diophantus arithmetica a. Joseph muscat 2015 2 2 problems problem 1 to split a given number 100 in two parts having a given di erence 40.