Diophantus(Διόφαντος), 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 Bachet, his editor, found a problem proposed in verse, in an unpublished Greek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 Β̀); then the sum being 20 (κʼ) and the half sum 10 (ὶ) the greater
number will be s+10 (τετάχθω οὖν ὁ μείζων
σοῦ ἑνὸς καὶ μο͂ ὶ) and the less will be 10--ς (μο͂ ὶ λείψει σοῦ ἑνὸς, which he would
often write μο͂ ὶ ψ σὸς ὰ). But the product is 96
(γ̀σʼ) which is also 100--ς2 (ρʼ λείψει δυνάμεως
μιᾶς, or ρʼ ψ δῦ ὰ). Hence ς=2 (γίνεται ὁ σὸς μο͂
Βʼ) &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 μο͂ 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.](author.A.DE.M)]