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Not merely is nothing known of the life of DIOPHANTUS, except that he lived in Alexandria, but his very century is doubtful. Theon, the father of Hypatia, mentioned him, and Hypatia herself is said to have written a commentary on his work. It appears not improbable that he was a contemporary of Pappus. Others place him in the fifth century.
The works known to have been written by him are the Arithmetics, in thirteen books; a work on Porisms, and another on Polygonal Numbers. Of the thirteen books on Arithmetics, six only have come down to us, unless, indeed, which is not improbable, the other works formed part of this treatise. The Porisms are not extant; of the Polygonal Numbers we have a fragment. The missing portions of the Arithmetics are probably those intermediate between the first and the remaining books. The works of Diophantus were much studied in the Arabian schools; they were translated by Abul Wafa in the 10th century, and were probably known to the greatest of Arabian algebraists, Mohammed Ibn Musa, in the previous century. How far their influence extended to India is matter for conjecture. In Western Europe Diophantus was not known till the 15th century, and was not seriously studied till the sixteenth.
The Arithmetics of Diophantus differ from any other mathematical work of the Greeks, in presenting to us Alegbra dissociated from Geometry, and applied to the abstract study of Number. It is not, however, so much a text-book on algebra as a collection of algebraic problems, arranged to some extent, though not entirely, in the order of complexity. The first book premises a few definitions and explanations of notation, and then presents a series of problems leading to determinate equations of the first degree. The method for handling these is given; the solution of quadratic equations is promised, and was probably given in one of the missing books; it is presupposed in the problems that follow. The greater part of the work as we now have it is occupied with indeterminate analysis of the second degree. Only one unknown quantity is used; to make a given function of this unknown quantity "equal to a square," is the form in which the problem is presented: corresponding to what in modern style would take the shape Ax²+Bx+C=y². Extraordinary dexterity is shown in handling the problems proposed. He is seen to be in possession of many results in the theory of numbers, as that a square number could be divided into two squares in any number of ways; and generally of the properties of numbers which are the sum of two squares. Of numbers regarded as the sum of three, and even of four, squares he had evidently made some, though doubtless an imperfect, study.
Diophantus uses an elaborate system of notation, including certain symbols of operation. The unknown quantity was expressed by the word αριθμόs, Number, and by a sign commonly called Sigma, but perhaps an abridgment of the first two letters αρ. Absolute numbers are indicated by a prefix of μόναδες or units. Fractions are noted by placing the deonominator above the line to the right of the numerator. There is a symbol for subtraction, and also for equality, taken from the initial letters of the corresponding Greek words.
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| This biography is
reprinted from The New Calendar of Great Men. Ed. Frederic
Harrison. London: Macmillan and Co., 1920. |
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