number." Thus it can hardly be said that there is (as a rule) any loss of generality. We may say, then, that in general Diophantus is obliged to express all his unknowns in terms, or as functions, of one variable. He compels our admiration by the clever devices by which he contrives so to express them in terms of his single unknown, s, as to satisfy by that very expression of them all conditions of the problem except one, which then enables us to complete the solution by determining the value of s. Another consequence of Diophantus' want of other symbols besides s to express more variables than one is that, when (as often happens) it is necessary in the course of a problem to work out a subsidiary problem in order to obtain the coefficients etc. in the functions of s which express the numbers to be found, the unknown quantity which it is the object of the new subsidiary problem to find is also in its turn denoted by the same symbols; hence we often have in the same problem the same variables used with two different meanings. This is an obvious inconvenience and might lead to confusion in the mind of a careless reader. Again we find two cases, II. 28 and 29, where for the proper working-out of the problem two unknowns are imperatively necessary. We should of course use x and y; but Diophantus calls the first s as usual; the second, for want of a term, he agrees to call "one unit," i.e. I. Then, later, having completed the part of the solution necessary to find s, he substitutes its value, and uses s over again to denote what he had originally called "I"-the second variable-and so finds it. This is the most curious case of all, and the way in which Diophantus, after having worked with this "I" along with other numerals, is yet able to put his finger upon the particular place where it has passed to, so as to substitutes for it, is very remarkable. This could only be possible in particular cases such as those which I have mentioned; but, even here, it seems scarcely possible now to work out the problem by using x and I for the variables as originally taken by Diophantus without falling into confusion. Perhaps, however, in working out the problems before writing them down as we have them Diophantus may have given the "I" which stood for a variable some mark by which he could recognise it and distinguish it from other numbers.

Diophantus will have in his solutions no numbers whatever except "rational" numbers; and in pursuance of this restriction he excludes not only surds and imaginary quantities, but also negative quantities. Of a negative quantity per se, i.e. without some positive quantity to subtract it from, Diophantus had apparently no conception. Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution. is in these cases ἀδύνατος, impossible. So we find him (v. 2) describing the equation 4 = 4x + 20 as ἄτοπος, absurd, because it would give x=-4. Diophantus makes it his object throughout to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, the conditions which must be satisfied in order to secure a result rational in his sense of the word. In the great majority of cases, when Diophantus arrives in the course of a solution at an equation which would give an irrational result, he retraces his steps and finds out how his equation has arisen, and how he may, by altering the previous work, substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantus has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 30 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.

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 Diophantus1. 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

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 indetcininate 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 observes1 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 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 says': "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

Novi Commentarii Academiae Petropolitanae, 1756-7, Vol. VI. (1761), p. 155=Commentationes arithmeticae collectae (ed. Fuss), 1849, 1. p. 193. • "Diophant's Behandlung der Gleichungen."

• "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."