DIOPHANTUS
, a celebrated mathematician of Alexandria, who has been reputed to be the inventor of Algebra; at least his is the earliest work extant on that science. It is not certain when Diophantus lived. Some have placed him before Christ, and some after, in the reigns of Nero and the Antonines; but all with equal uncertainty. It seems he is the same Diophantus who wrote the Canon Astronomicus, which Suidas says was commented on by the celebrated Hypatia, daughter of Theon of Alexandria. His reputation must have been very high among the ancients, since they ranked him with Pythagoras and Euclid in mathematical learning. Bachet, in his notes upon the 5th book De Arithmeticis, has collected, from Diophantus's epitaph in the Anthologia, the following circumstances of his life; namely, that he was married when he was 33 years old, and had a son born 5 years after; that this son died when he was 42 years of age, and that his father did not survive him above 4 years; from which it appears, that Diophantus was 84 years old when he died.
DIOPHANTUS wrote 13 books of Arithmetic, or Algebra, which Regiomontanus in his preface to Alfraganus, tells us, are still preserved in manuscript in the Vatican library. Indeed Diophantus himself tells us | that his work consisted of 13 books, viz, at the end of his address to Dionysius, placed at the beginning of the work; and from hence Regiomontanus might be led to say the 13 books were in that library. No more than 6 whole books, with part of a seventh, have ever been published; and I am of opinion there are no more in being; indeed Bombelli, in the preface to his Algebra, written 1572, says there were but 6 of the books then in the library, and that he and another were about a translation of them.
Those 6 books, with the imperfect 7th, were first published at Basil by Xylander in 1575, but in a Latin version only, with the Greek scholia of Maximus Planudes upon the two first books, and observations of his own. The same books were afterwards published in Greek and Latin at Paris in 1621, by Bachet, an ingenious and learned Frenchman, who made a new Latin version of the work, and enriched it with very learned commentaries. Bachet did not entirely neglect the notes of Xylander in his edition, but he treated the scholiast Planudes with the utmost contempt. He seems to intimate, in what he says upon the 28th question of the 2d book, that the 6 books which we have of Diophantus, may be nothing more than a collection made by some novice, of such propositions as he judged proper, out of the whole 13: but Fabricius thinks there is no just ground for such a supposition.
DIOPHANTINE Problems, are certain questions relating to square and cubic numbers, and to rightangled triangles, &c; the nature of which were first and chiesly treated of by Diophantes, in his Arithmetic, or rather Algebra.
In these questions, it is chiefly intended to find commensurable numbers to answer indeterminate problems; which often bring out an infinite number of incommensurable quantities. For example, let it be proposed to find a right-angled triangle, whose three sides x, y, z are expressed by rational numbers; from the nature of the figure it is known that , where z denotes the hypothenuse. Now it is plain that x and y may also be so taken, that z shall be irrational; for if x = 1, and y = 2, then is z = √5.
Now the art of resolving such problems, consists in ordering the unknown quantity or quantities, in such a manner, that the square or higher power may vanish out of the equation, and then by means of the unknown quantity in its first dimension, the equation may be resolved without having recourse to incommensurables. For ex. in the equation above, , suppose , then is , out of which equation x2 vanishes, and then it is , which gives . Hence, assuming y and u equal to any numbers at pleasure, the three sides of the triangle will be y, (y2 - u2)/2u, and (y2 + u2)/2u, which are all rational whenever y and u are rational. For ex. if y = 3, and u = 1, then , and x + u or . It is evident that this problem admits of infinite numbers of solutions, as y or u may be assumed infinitely various. See Algebra, and Diophantus.
Abundant information on this sort of problems may be found in the writings of a great many authors, particularly Fermat, Bachet, Ozanam, Kersey, Saunderson, Euler, &c.
