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equations containing two unknowns, and of the same sort as Diophantus I. 1-6; or, of course, they can be reduced to a simple equation with one unknown by means of an easy elimination. One other (XIV. 51) gives simultaneous equations in three unknowns

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and one (XIV. 49) gives four equations in four unknowns

x+y= 40, x + 2 = 45, x + u = 36, x + y + s + u = 60.

With these may be compared Diophantus I. 16-21. (3) Six more are problems of the usual type about the filling of vessels by pipes: e.g. (XIV. 130) one pipe fills the vessel in one day, a second in two, and a third in three; how long will all three running together take to fill it? Another about brickmakers (XIV. 136) is of the

same sort.

The Anthology contains (4) two indeterminate equations of the first degree which can be solved in positive integers in an infinite number of ways (XIV. 48 and 144); the first is a distribution of apples satisfying the equation x-3y=y, where y is not less than 2, and the original number of apples is 3; the second leads to the following three equations between four unknown quantities: x+y=x1+1/1,

x = 231,

x1 = 31,

the general solution of which is x = 4k, y = k, x, = 3k, y1 = 2k. These very equations, made however determinate by assuming that x+y=x1+y1 = 100, are solved in Diophantus I. 12.

We mentioned above the problem in the Anthology (XIV. 49) leading to the following four simultaneous linear equations with four unknown quantities,

x+y=a,

x+2=b,

x+u= c,

x+y+s+u= d.

The general solution of any number of simultaneous linear equations of this type with the same number of unknown quantities was given by Thymaridas, apparently of Paros, and an early Pythagorean. He gave a rule, ěpodos, or method of attack (as

Iamblichus1, our informant, calls it) which must have been widely known, inasmuch as it was called by the name of the érávenμa, "flower" or "bloom," of Thymaridas. The rule is stated in general terms, but, though no symbols are used, the content is pure algebra. Thymaridas, too, distinguishes between what he calls αόριστον, the undefined or unknown quantity, and the ὡρισμένον, the definite or known, therein anticipating the very phrase of Diophantus, πλῆθος μονάδων αόριστον, “ an undefined number of units," by which he describes his apieμós or x. Thymaridas' rule, though obscurely expressed, states in effect that, if there are n equations between n unknown quantities x, x, x...n-1 of the following form,

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Iamblichus goes on to show that other types of equations can be reduced to this, so that the rule does not leave us in the lurch (oỷ πaρéλкei) in those cases either. Thus we can reduce to Thymaridas' form the indeterminate problem represented by the following three linear equations between four unknown quantities: x+y=a (x+u),

x + 8 = b (u + 1),

x + u = c(y + z).

From the first equation we obtain

x+y+z+u= (a + 1) (≈ + u),

from which it follows that, if x, y, z, u are all to be integers, x+y+z+u must contain a +1 as a factor.

contain b+I and c+ I as factors.

Similarly it must

Suppose now that x+y+z+ u = (a + 1 ) (b + 1) (c + 1). Therefore, by means of the first equation, we get

(x+y) (1

(x + y) ( 1 + 1 ) = (a + 1) (b + 1) (c + 1),

1 Iamblichus, In Nicomachi arithmeticam introductionem (ed. Pistelli), pp. 62, 18-68, 26.

or

Similarly

x+y=a(b+1) (c + 1).

x + z = b (c + 1)(a + 1),

x + u = c(a + 1)(b + 1),

and the equations are in the form to which Thymaridas' rule is applicable.

Hence, by that rule,

x=

a (b + 1) (c + 1) + ... − (a + 1) (b + 1) (c + 1) ̧

2

In order to ensure that x may always be integral, it is only necessary to assume

x+y+z+u= 2 (a + 1) (b + 1) (c + 1).

The factor 2 is of course determined by the number of unknowns. If there are n unknowns, the factor to be put in place of 2 is n - 2.

Iamblichus has the particular case corresponding to a = 2, b=3,c=4. He goes on to apply the method to the equations

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for the case where k/l=3, m/n = §, p/q = {.

Enough has been said to show that Diophantus was not the inventor of Algebra. Nor was he the first to solve indeterminate problems of the second degree.

Take, first, the problem of dividing a square number into two squares (Diophantus II. 8), or of finding a right-angled triangle with sides in rational numbers. This problem was, as we learn from Proclus', attributed to Pythagoras, who was credited with the discovery of a general formula for finding such triangles which may be shown thus:

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where n is an odd number. Plato again is credited, according to the same authority, with another formula of the same sort,

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Both these formulae are readily connected with the geometrical proposition in Eucl. II. 5, the algebraical equivalent of which may be stated as

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The content of Euclid Book II. is beyond doubt Pythagorean, and this way of stating the proposition quoted could not have escaped the Pythagoreans. If we put 1 for b and the square of any odd number for a, we have the Pythagorean formula; and, if we put a = 2n2, b = 2, we obtain Plato's formula. Euclid finds a more general formula in Book x. (Lemma following X. 28). Starting with numbers u = c + b and v = c - b, so that

uv = c — b2,

Euclid points out that, in order that uv may be a square, and v must be "similar plane numbers" or numbers of the form mnp3, mnq. Substituting we have

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But the problem of finding right-angled triangles in rational numbers was not the only indeterminate problem of the second degree solved by the Pythagoreans. They solved the equation

2x2 — y2 = ± 1

in such a way as to prove that there are an infinite number of solutions of that equation in integral numbers. The Pythagoreans used for this purpose the system of "side-" and "diagonal-" numbers', afterwards fully described by Theon of Smyrna We begin with unity as both the first “side” and the first "diagonal";

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We then form (a2, d1⁄2), (a,, d ̧), etc., in accordance with the following law,

and so on.

a1 = a1 + d1, d2 = 2a1 + d1;

a1 = a2+d2, d1 = 2a,+ d1;

Theon states, with reference to these numbers, the general proposition that

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and observes that (1) the signs alternate as successive d's and a's

1 See Proclus, In Platonis rempublicam commentarii (Teubner, Leipzig), Vol. II. c. 27. p. 27, 11-18.

* Theon of Smyrna, ed. Hiller, pp. 43, 44.

are taken, d-2a, being equal to 1, d2a, equal to + I, d2a, equal to I and so on, (2) the sum of the squares of all the d's will be double of the sum of the squares of all the a's. For the purpose of (2) the number of successive terms in each series, if finite, must of course be even. The algebraical proof is easy.

dn2 - 2an2 = (2an−1 + dn−1)2 — 2 (an−1 + dn−1)3

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and so on, while d3- 2a2 = 1. Proclus tells us that the property was proved by means of the theorems of Eucl. II. 9, 10, which are indeed equivalent to

(2x+y)2 -2(x+y)2 = 2x2 — y2.

Diophantus does not particularly mention the indeterminate equation 2x2-1, still less does he mention "side-" and "diagonal-" numbers. But from the Lemma to VI. 15 (quoted above, p. 69) it is clear that he knew how to find any number of solutions when one is known. Thus, seeing that x=1, y = I is one solution, he would put

whence

2 (1 + x)3 − 1 = a square

=(px-1) say,

x = (4+2p)/(p3 — 2).

Take the value = 2, and we have x=4, or x+1=5; and 2.5214972. Putting +5 in place of x, we find a still higher value, and so on.

In a recent paper Heiberg has published and translated, and Zeuthen has commented on, still further Greek examples of indeterminate analysis1. They come from the Constantinople MS. (probably of 12th c.) from which Schöne edited the Metrica of Heron. The first two of the thirteen problems had been published before (though in a less complete form); the others are new.

The first bids us find two rectangles such that the perimeter of the second is three times that of the first, and the area of the first is three times that of the second (the first of the two conditions is, by some accident, omitted in the text). The number 3

1 Bibliotheca Mathematica, VIII, 1907-8, pp. 118-134.

2 Hultsch's Heron, Geeponica, 78, 79. The two problems are discussed by Cantor, Agrimensoren, p. 62, and Tannery, Mém. de la soc. des sc. de Bordeaux, IV, 1882.

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