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CHAPTER IV

DIOPHANTUS' METHODS OF SOLUTION

BEFORE I give an account in detail of the different methods which Diophantus employs for the solution of his problems, so far as they can be classified, it is worth while to quote some remarks which Hankel has made in his account of Diophantus'. Hankel, writing with his usual brilliancy, says in the place referred to, "The reader will now be desirous to become acquainted with the classes of indeterminate problems which Diophantus treats of, and with his methods of solution. As regards the first point, we must observe that included in the 130 (or so) indeterminate problems, of which Diophantus treats in his great work, there are over 50 different classes of problems, strung together on no recognisable principle of grouping, except that the solution of the earlier problems facilitates that of the later. The first Book is confined to determinate algebraic equations; Books II. to v. contain for the most part indeterminate problems, in which expressions involving in the first or second degree two or more variables are to be made squares or cubes. Lastly, Book VI. is concerned with right-angled triangles regarded purely arithmetically, in which some linear or quadratic function of the sides is to be made a square or a cube. That is all that we can pronounce about this varied series of problems without exhibiting singly each of the fifty classes. Almost more different in kind than the problems are their solutions, and we are completely unable to give an even tolerably exhaustive review of the different turns which his procedure takes. Of more general comprehensive methods there is in our author no trace discoverable: every question requires a quite special method, which often will not serve even for the most closely allied problems. It is on that

1 Zur Geschichte der Mathematik in Alterthum und Mittelalter, Leipzig, 1874, pp. 164-5.

account difficult for a modern mathematician even after studying 100 Diophantine solutions to solve the 101st problem; and if we have made the attempt, and after some vain endeavours read Diophantus' own solution, we shall be astonished to see how suddenly he leaves the broad high-road, dashes into a side-path and with a quick turn reaches the goal, often enough a goal with reaching which we should not be content; we expected to have to climb a toilsome path, but to be rewarded at the end by an extensive view; instead of which our guide leads by narrow, strange, but smooth ways to a small eminence; he has finished! He lacks the calm and concentrated energy for a deep plunge into a single important problem; and in this way the reader also hurries with inward unrest from problem to problem, as in a game of riddles, without being able to enjoy the individual one. Diophantus dazzles more than he delights. He is in a wonderful measure shrewd, clever, quick-sighted, indefatigable, but does not penetrate thoroughly or deeply into the root of the matter. As his problems seem framed in obedience to no obvious scientific necessity, but often only for the sake of the solution, the solution itself also lacks completeness and deeper signification. He is a brilliant performer in the art of indeterminate analysis invented by him, but the science has nevertheless been indebted, at least directly, to this brilliant genius for few methods, because he was deficient in the speculative thought which sees in the True more than the Correct. That is the general impression which I have derived from a thorough and repeated study of Diophantus' arithmetic."

It might be inferred from these remarks of Hankel that Diophantus' object was less to teach methods than to obtain a multitude of mere results. On the other hand Nesselmann observes that Diophantus, while using (as he must) specific numbers for numbers which are "given" or have to be arbitrarily assumed, always makes it clear how by varying our initial assumptions we can obtain any number of particular solutions of the problem, showing "that his whole attention is directed to the explanation of the method, to which end numerical examples only serve as means"; this is proved by his frequently stopping short, when the method has been made sufficiently clear, and the remainder of the work is mere straightforward calculation. The truth seems to be that there is as much in the shape of general

1 Algebra der Griechen, pp. 308–9.

methods to be found in Diophantus as his notation and the nature of the subject admitted of. On this point I can quote no better authority than Euler, who says1: "Diophantus himself, it is true, gives only the most special solutions of all the questions which he treats, and he is generally content with indicating numbers which furnish one single solution. But it must not be supposed that his method was restricted to these very special solutions. In his time the use of letters to denote undetermined numbers was not yet established, and consequently the more general solutions which we are now enabled to give by means of such notation could not be expected from him. Nevertheless, the actual methods which he uses for solving any of his problems are as general as those which are in use today; nay, we are obliged to admit that there is hardly any method yet invented in this kind of analysis of which there are not sufficiently distinct traces to be discovered in Diophantus."

In his 8th chapter, entitled "Diophantus' treatment of equations"," Nesselmann gives an account of Diophantus' solutions of (1) Determinate, (2) Indeterminate equations, classified according to their kind. In chapter 9, entitled "Diophantus' methods of solution"," he classifies these "methods" as follows: (1) "The adroit assumption of unknowns," (2) "Method of reckoning backwards and auxiliary questions," (3) "Use of the symbol for the unknown in different significations," (4) "Method of Limits," (5) “Solution by mere reflection," (6) "Solution in general expressions," (7) "Arbitrary determinations and assumptions," (8) "Use of the rightangled triangle."

At the end of chapter 8 Nesselmann observes that it is not his solutions of equations that we have to wonder at, but the art, amounting to virtuosity, which enabled Diophantus to avoid such equations as he could not technically solve. We look (says Nesselmann) with astonishment at his operations, when he reduces the most difficult problems by some surprising turn to a quite simple

1 Novi Commentarii Academiae Petropolitanae, 1756–7, Vol. VI. (1761), p. 155= Commentationes arithmeticae collectae (ed. Fuss), 1849, I. p. 193.

2 "Diophant's Behandlung der Gleichungen."

3 "Diophant's Auflösungsmethoden."

(1) "Die geschickte Annahme der Unbekannten," (2) "Methode der Zurückrechnung und Nebenaufgabe," (3) "Gebrauch des Symbols für die Unbekannte in verschiedenen Bedeutungen," (4) "Methode der Grenzen," (5) " Auflösung durch blosse Reflexion," (6) "Auflösung in allgemeinen Ausdrücken," (7) "Willkührliche Bestimmungen und Annahmen," (8) "Gebrauch des rechtwinkligen Dreiecks."

equation. Then, when in the 9th chapter Nesselmann passes to the "methods," he prefaces it by saying: "To give a complete picture of Diophantus' methods in all their variety would mean nothing else than copying his book outright. The individual characteristics of almost every problem give him occasion to try upon it a peculiar procedure or found upon it an artifice which cannot be applied to any other problem.... Meanwhile, though it may be impossible to exhibit all his methods in any short space, yet I will try to describe some operations which occur more often or are particularly remarkable for their elegance, and (where possible) to bring out the underlying scientific principle by a general exposition and by a suitable grouping of similar cases under common aspects or characters." Now the possibility of giving a satisfactory account of the methods of Diophantus must depend largely upon the meaning we attach to the word "method." Nesselmann's arrangement seems to me to be faulty inasmuch as (1) he has treated Diophantus' solutions of equations, which certainly proceeded on fixed rules, and therefore by "method," separately from what he calls "methods of solution," thereby making it appear as though he did not look upon the "treatment of equations" as "methods"; (2) the classification of the "Methods of solution" seems unsatisfactory. Some of the latter can hardly be said to be methods of solution at all; thus the third, "Use of the symbol for the unknown in different significations," might be more justly described as a "hindrance to the solution"; it is an inconvenience to which Diophantus was subjected owing to the want of notation. Indeed, on the assumption of the eight "methods," as Nesselmann describes them, it is really not surprising that no complete account of them could be given without copying the whole book. To take the first, "the adroit assumption of unknowns." Supposing that a number of essentially different problems are proposed, the differences make a different choice of an unknown in each case absolutely necessary. That being so, how could a rule be given for all cases? The best that can be done is to give a number of typical instances. Precisely the same remark applies to "methods" (2), (5), (6), (7). The case of (4), "Method of Limits," is different; here we have a "method" in the true sense of the term, ie. in the sense of an instrument for solution. And accordingly in this case the method can be exhibited, as I hope to show later on; (8) also deserves. to some extent the name of a "method."

In one particular case, Diophantus formally states a method or rule; this is his rule for solving what he calls a "double-equation," and will be found in II. II, where such an equation appears for the first time. Apart from this, we do not find in Diophantus' work statements of method put generally as book-work to be applied to examples. Thus we do not find the separate rules and limitations for the solution of different kinds of equations systematically arranged, but we have to seek them out laboriously from the whole of his work, gathering scattered indications here and there, and to formulate them in the best way that we can.

I shall now attempt to give a short account of those methods running through Diophantus which admit of general statement. For the reasons which I have stated, my arrangement will be different from that of Nesselmann; I shall omit some of the heads in his classification of "methods of solution"; and, in accordance with his remark that these "methods" can only be adequately described by a transcription of the entire work, I shall leave them to be gathered from a perusal of my reproduction of Diophantus' book.

I shall begin my account with

I. DIOPHANTUS' TREATMENT OF EQUATIONS.

This subject falls naturally into two divisions: (A) Determinate equations of different degrees, (B) Indeterminate equations.

(A) Determinate equations.

Diophantus was able without difficulty to solve determinate equations of the first and second degrees; of a cubic equation we find in his Arithmetica only one example, and that is a very special case. The solution of simple equations we may pass over; we have then to consider Diophantus' methods of solution in the case of (1) Pure equations, (2) Adfected, or mixed, quadratics.

(1) Pure determinate equations.

By pure equations I mean those equations which contain only one power of the unknown, whatever the degree. The solution is effected in the same way whatever the exponent of the term in the

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