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Diophantos writes 30y+1= (5y + 1), whence y = 2 and

36.

=

And 33 +36= a square 121

2p

[As before, if we assume 30y2+1 = (py +1)2, y = 30 — p2,

and since

1

y

must be a small proper fraction, 30 - p2 should <2p

or p2+2p+1> 31, and 5 is the smallest possible value of p. For this reason Diophantos chooses it.]

We have now (says Diophantos) to make the sides of the required squares as near as may be to .

Now

10=9+1=32+ (3)2 + (3)3,

and 3,, are the sides of three squares whose sum = 10. Bringing (3, 3, 4) and u to a common denominator, we have

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If then we took 3-35, +33, +31 for the sides, the sum of their squares would be 3 (1) or 363, which is > 10. Diophantos accordingly assumes as the sides of the three required squares 3-35x, +37x, +31x,

where a must therefore be not exactly, but near it.

Solving (3-35x)2 + (§ +37x)2 + (§ + 31x)2 = 10,

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The two instances here given, though only instances, serve perfectly to illustrate the method of Diophantos. To have put them generally with the use of algebraical symbols, instead of

concrete numbers, would have rendered necessary the introduction of a large number of such symbols, and the number of conditions (e.g. that such and such an expression shall be a square) which it would have been necessary to express would have nullified all the advantages of this general treatment.

As it only lies within my scope to explain what we actually find in Diophantos' work, I shall not here introduce certain investigations embodied by Poselger in his article "Beiträge zur Unbestimmten Analysis," published in the Abhandlungen der Königlichen Akademie der Wissenschaften zu Berlin Aus dem Jahre 1832, Berlin, 1834. One section of this paper Poselger entitles "Annäherungs-methoden nach Diophantus," and obtains in it, upon Diophantos' principles', a method of approximation to the value of a surd which will furnish the same results as the method by means of continued fractions, except that the approximation by what he calls the "Diophantine method" is quicker than the method of continued fractions, so that it may serve to expedite the latter'.

1 "Wenn wir den Weg des Diophantos verfolgen."

2 "Die Diophantische Methode kann also dazu dienen, die Convergenz der Partialbrüche des Kettenbruchs zu beschleunigen."

CHAPTER VI.

§1. THE PORISMS OF DIOPHANTOS.

WE have already spoken (in the Historical Introduction) of the Porisms of Diophantos as having probably formed a distinct part of the work of our author. We also discussed the question as to whether the Porisms now lost formed an integral portion of the Arithmetics or whether it was a completely separate treatise. What remains for us to do under the head of Diophantos' Porisms is to collect such references to them and such enunciations of definite porisms as are directly given by Diophantos. If we confine our list of Porisms to those given under that name by Diophantos, it does not therefore follow that many other theorems enunciated, assumed or implied in the extant work, but not distinctly called Porisms, may not with equal propriety be supposed to have been actually propounded in the Porisms. For distinctness, however, and in order to make our assumptions perfectly safe, it will be better to separate what are actually called porisms from other theorems implied and assumed in Diophantos' problems.

First then with regard to the actual Porisms. I shall not attempt to discuss here the nature of the proposition which was called a porism, for such a discussion would be irrelevant to my purpose. The Porisms themselves too have been well enumerated and explained by Nesselmann in his tenth chapter; he has also given, with few omissions, the chief of the other theorems assumed by Diophantos as known. Of necessity, therefore, in this section and the next I shall have to cover very much the same ground, and shall accordingly be as brief as may be.

Porism 1. The first porism enunciated by Diophantos occurs in v. 3. He says "We have from the Porisms that if each of two numbers and their product when severally added to the same number produce squares, the numbers are the squares of two consecutive numbers." This theorem is not correctly enunciated, for two consecutive squares are not the only two numbers which will satisfy the condition. For suppose

x+a=m2, y+a = n2, xy+a= p2.

Now by help of the first two equations we find

xy + a = m3n2 — a (m2 + n2 − 1) + a2,

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and this is equal to p3. In order that

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may be a square certain conditions must be satisfied. One sufficient condition is

or

m2 + n2 - 1 = 2mn,

m− n = ± 1,

and this is Diophantos' condition.

But we may also regard

m3n2 — a (m2 + n2 − 1) + a2 = p2

as an indeterminate equation in m of which we know one solution, namely

m = n± 1.

Other solutions are then found by substituting z + (n + 1) for m, whence we have the equation

(n2 − a) z2 + 2 {n2 (n + 1) − a (n + 1)} z + (n2 − a) (n + 1)2

or

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(n2 − a) z2 + 2 (n2 − a) (n + 1) z + {n (n + 1) − a}2 = p2, which is easy to solve in Diophantos' manner, the absolute term being a square.

But in the problem v. 3 three numbers are required such that each of them, and the product of each pair, severally added

1 καὶ ἐπεὶ ἔχομεν ἐν τοῖς πορίσμασιν, ὅτι ἐὰν δύο ἀριθμοὶ ἑκάτερός τε καὶ ὁ ὑπ' αὐτῶν μετὰ τοῦ αὐτοῦ δοθέντος ποιῇ τετράγωνον, γεγόνασιν ἀπὸ δύο τετραγώνων τῶν κατὰ τὸ ἑξῆς.

to a given number produce squares. Thus, if the third number bez, three more conditions must be added, namely, z+a, zx+a, yz + a should be squares. The two last conditions are satisfied, if m + 1 = n, by putting

when

z=2(x+y)−1 = 4m2 + 4m +1 −4a,

xz + a = {m (2m+ 1) − 2a}2,

yz + a = {m (2m+3) — (2a − 1)}3,

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and this means of satisfying the conditions may have affected the formulating of the Porism.

v. 4 gives another case of the Porism with a for + a.

Porism 2. In v. 5 Diophantos says, "We have in the Porisms that in addition to any two consecutive squares we can find another number which, being double of the sum of both and increased by 2, makes up three numbers, the product of any pair of which plus the sum of that pair or the third number produces a square,” i.e.

m2, m2 + 2m+1, 4 (m2 +m+1),

are three numbers which satisfy the conditions.

The same porism is assumed and made use of in the following problem, v. 6.

Porism 3 occurs in v. 19. Unfortunately the text of the enunciation is corrupt, but there can be no doubt that the correct statement of the porism is "The difference of two cubes can be transformed into the sum of two cubes." Diophantos contents himself with the mere enunciation and does not proceed to effect the actual transformation. Thus we do not know his method, or how far he was able to prove the porism as a perfectly general theorem. The theorems upon the transformation of sums and differences of cubes were investigated by Vieta, Bachet and Fermat.

1 καὶ ἔχομεν πάλιν ἐν τοῖς πορίσμασιν ὅτι πᾶσι δύο τετραγώνοις τοῖς κατὰ τὸ ἑξῆς προσευρίσκεται ἕτερος ἀριθμὸς ὃς ὢν διπλασίων συναμφοτέρου καὶ δυάδι μείζων, τρεῖς ἀριθμοὺς ποιεῖ ὧν ὁ ὑπὸ ὁποιωνοῦν ἐάντε προσλάβῃ συναμφότερον, ἐάντε λοιπὸν ποιεῖ τετράγωνον.

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