#### Diophantus

（*Dio/fantos), of Alexandria, the only Greek writer on Algebra. His period is wholly unknown, which is not to be wondered at if we consider that he stands quite alone as to the subject which he treated. But, looking at the improbability of all mention of such a writer being omitted by Proclus and Pappus, we feel strongly inclined to place him towards the end of the fifth century of our era at the earliest. If the Diophantus, on whose astronomical work (according to Suidas) Hypatia wrote a commentary, and whose arithmetic Theon mentions in his commentary on the Almagest, be the subject of our article, he must have lived before the fifth century: but it would be by no means safe to assume this identity. Abulpharagius, according to Montucla, places him at A. D. 365. The first writer who mentions him, (if it be not Theon) is John, patriarch of Jerusalem, in his life of Johannes Damascenus, written in the eighth century. It matters not much where we place him, as far as Greek literature is concerned: the question will only become of importance when we have the means of investigating whether or not he derived his algebra, or any of it, from an Indian source. Colebrooke, as to this matter, is content that Diophantus should be placed in the fourth century. (See the__Penny Cyclopaedia,__art.

*Viya Ganita.*）

It is singular that, though his date is uncertain to a couple of centuries at least, we have some reason to suppose that he married at the age of 33, and that in five years a son was born of this marriage, who died at the age of 42, four years before his father: so that Diophantus lived to 84. Bachet, his editor, found a problem proposed in verse, in an unpublished Greek anthology, like some of those which Diophantus himself proposed in verse, and composed in the manner of an epitaph. The unknown quantity is the age to which Diophantus lived, and the simple equation of condition to which it leads gives, when solved, the preceding information. But it is just as likely as not that the maker of the epigram invented the dates.

#### Works

When the manuscripts of Diophantus came to light in the 16th century, it was said that there were thirteen books of the__Arithmetica__: but no more than six have ever been produced with that title; besides which we have one book,

__De Multangulis Numeris__, on polygonal numbers. These books contain a system of reasoning on numbers by the aid of general symbols, and with some use of symbols of operation; so that, though the demonstrations are very much conducted in words at length, and arranged so as to remind us of Euclid, there is no question that the work is algebraical: not a treatise

*on algebra,*but an algebraical treatise on the relations of integer numbers, and on the solution of equations of more than one variable in integers. Hence such questions obtained the name of Diophantine, and the modern works on that pecuculiar branch of numerical analysis which is called the theory of numbers, such as those of Gauss and Legendre, would have been said, a century ago, to be full of

*Diophantine analysis.*As there are many classical students who will not see a copy of Diophantus in their lives, it may be desirable to give one simple proposition from that writer in modern words and symbols, annexing the algebraical phrases from the original.

Book i. qu. 30. Having given the sum of two numbers (20) and their product (96), required the numbers. Observe that the square of the half sum should be greater than the product. Let the difference of the numbers be 2ς (σσοὶ Β`); then the sum being 20 (κ᾽) and the half sum 10 (ὶ) the greater number will be *s*+10 (τετάχθω οὖν ὁ μείζων σοῦ ἑνὸς καὶ μο̂ ὶ) and the less will be 10--ς (μο̂ ὶ λείψει σοῦ ἑνὸς, which he would often write μο̂ ὶ ψ σὸς ὰ).
But the product is 96 (γ̀σ᾽) which is also 100--ς^{2} (ρ᾽ λείψει δυνάμεως μιᾶς, or ρ᾽ ψ δῦ ὰ). Hence ς=2 (γίνεται ὁ σὸς μο̂ Β᾽) &c.

A young algebraist of our day might hardly be inclined to give the name of algebraical notation to the preceding, though he might admit that there was algebraical reasoning. But if he had consulted the Hindu or Mahommedan writers, or Cardan, Tartaglia, Stevinus, and the other European algebraists, who preceded Vieta, he would see that he must either give the name to the notation above exemplified, or refuse it to everything which preceded the seventeenth century. Diophantus declines his letters, just as we now speak of m th or (m+ 1 ) th; and μο̂ is an abbreviation of υονάς or μονάδος, as the case may be.

The question whether Diophantus was an original inventor, or whether he had received a hint from India, the only country we know of which could then have given one, is of great difficulty. We cannot enter into it at length: the very great similarity of the Diophantine and Hindu algebra (as far as the former goes) makes it almost certain that the two must have had a common origin, or have come one from the other; though it is clear that Diophantus, if a borrower, has completely recast the subject by the introduction of Euclid's form of demonstration. On this point we refer to the article of the Penny Cyclopaedia already cited.

#### Editions

There are many paraphrases, so-called translations, and abbreviations of Diophantus, but very few editions.*Joseph Auria prepared an edition (Gr. Lat.) of the whole, with the Scholia of the monk Maximus Planudes on the first two books ; but it was never printed.*

*The first edition is that of Xylander, Basle, 1575, folio, in Latin only, with the Scholia and notes.*

*The first Greek edition, with Latin, (and original notes, the Scholia being rejected as useless,) is that of Bachet de Meziriac, Paris, 1621, folio.*Fermat left materials for the second and best edition (Gr. Lat.), in which is preserved all that was good in Bachet, and in particular his Latin version, and most valuable comments and additions of his own (it being peculiarly his subject).

*These materials were collected by J. de Billy, and published by Fermat's son, Toulouse, 1670, folio.*

#### Translation

An English lady, the late Miss Abigail Baruch Lousada, whose successful cultivation of mathematics and close attention to this writer for many years was well known to scientific persons, left a complete translation of Diophantus, with notes: it has not yet been published, and we trust, will not be lost.[A. De M.]