a conjunction here and there, and so on. Neither is it the Europeans posterior to the middle of the seventeenth century who were the first to use Symbolic forms of Algebra. In this they were anticipated many centuries by the Indians.

As examples of these three stages Nesselmann gives three instances quoting word for word the solution of a quadratic equation by Mohammed ibn Mūsā as an example of the first stage, and the solution of a problem from Diophantos to illustrate the second. Thus:

First Stage. Example from Mohammed ibn Musa (ed. Rosen, p. 5). “A square and ten of its roots are equal to nine and thirty dirhems, that is, if you add ten roots to one square, the sum is equal to nine and thirty. The solution is as follows: halve the number of roots, that is in this case five; then multiply this by itself, and the result is five and twenty. Add this to the nine and thirty, which gives sixty-four; take the square root, or eight, and subtract from it half the number of roots, namely five, and there remain three: this is the root of the square which was required and the square itself is nine1."

Here we observe that not even are symbols used for numbers, so that this example is even more "rhetorical" than the work of Iamblichos who does use the Greek symbols for his numbers.

2

Second stage. As an example of Diophantos I give a translation word for word of II. 8. So as to make the symbols correspond exactly I use S (Square) for dû (dúvaμis), N (Number) for 5, U for Units (povádes).

"To divide the proposed square into two squares. Let it be proposed then to divide 16 into two squares. And let the first

1 Thus Mohammed ibn Musa states in words the solution

2 I have used the full words whenever Diophantos does so, and to avoid confusion have written Square and Number in the technical sense with a capital letter, and italicised them.

be supposed to be One Square. Thus 16 minus One Square must be equal to a square. I form the square from any number of N's minus as many U's as there are in the side of 16 U's. Suppose this to be 2 N's minus 4 U's. Thus the square itself will be 4 Squares, 16 U. minus 16 N.'s. These are equal to 16 Units minus One Square. Add to each the negative term (λeîus, deficiency) and take equals from equals. Thus 5 Squares are equal to 16 Numbers; and the Number is 16 fifths. One [square] will be 256 twenty-fifths, and the other 144 twenty-fifths, and the sum of the two makes up 400 twenty-fifths, or 16 Units, and each [of the two found] is a square.

Of the third stage any exemplification is unnecessary.

§ 6. To the form of Diophantos' notation is due the fact that he is unable to introduce into his questions more than one unknown quantity. This limitation has made his procedure often very different from our modern work. In the first place he performs eliminations, which we should leave to be done in the course of the work, before he prepares to work out the problem, by expressing everything which occurs in such a way as to contain only one unknown. This is the case in the great majority of questions of the first Book, which are cases of the solution of determinate simultaneous equations of the first order with two, three, or four variables; all these Diophantos expresses in terms of one unknown, and then proceeds to find it from a simple equation. In cases where the relations between these variables are complicated, Diophantos shows extraordinary acuteness in the selection of an unknown quantity. Secondly, however, this limitation affects much of Diophantos' work injuriously, for while he handles problems which are by nature indeterminate and would lead with our notation to an indeterminate equation containing two or three unknowns, he is compelled by limitation of notation to assign to one or other of these arbitrarily-chosen numbers which have the effect of making the problem a determinate one. However it is but fair to say that Diophantos in assigning an arbitrary value to a quantity is careful to tell us so, saying "for such and such a quantity we put any number whatever, say such and such

a one." Thus it can hardly be said that there is (in general) any loss of universality. We may say, then, that in general Diophantos is obliged to express all his unknowns in terms, or as functions, of one variable. There is something excessively interesting in the clever devices by which he contrives so to express them in terms of his single unknown, s, as that by that very expression of them all conditions of the problem are satisfied except one, which serves to complete the solution by determining the value of s. Another consequence of Diophantos' 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 &c. of the functions of s which express the quantities to be found, in this case the required unknown which is used for the solution of the new subsidiary problem is denoted by the same symbol s; hence we have often 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. 29 and 30, where for the proper working-out of the problem two unknowns are imperatively necessary. We should of course use x and y; but Diophantos calls the first s as usual; the second, for want of a term, he agrees to call "one unit," i.e. 1. 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 "1"-the second variable-and so finds it. This is the most curious case I have met with, and the way in which Diophantos after having worked with this "1" along with other numerals is yet able to pounce 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 using x and 1 for the variables as originally taken by Diophantos without falling into confusion. Perhaps, however, it may not be impossible that Diophantos in working out the problems before writing them down as we have them may have given the "1" which stood for a variable some mark by which

H. D.

6

he could recognise it and distinguish it from other numbers. For the problems themselves see Appendix.

It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in which, in addition to surds and imaginary quantities, he includes negative quantities. Of a negative quantity per se, i.e. without some positive quantity to subtract it from, Diophantos 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 adúvaros, impossible. So we find him describing the equation 4 = 4.x + 20 as aтоTоs because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos 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, Diophantos 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. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.

CHAPTER V.

DIOPHANTOS' METHODS OF SOLUTION.

§ 1. Before I give an account in detail of the different methods which Diophantos employs for the solution of his problems, so far as they can be classified, I must take exception to some remarks which Hankel has made in his account of Diophantos (Zur Geschichte der Mathematik in Alterthum und Mittelalter, Leipzig, 1874, pp. 164-5). This account does not only possess literary merit: it is the work of a man who has read Diophantos. His remarks therefore possess exceptional value as those of a man particularly well qualified to speak on matters relating to the history of mathematics, and also from the contrast to the mass of writers who have thought themselves capable of pronouncing upon Diophantos and his merits, while they show unmistakeably that they have not studied his work. Hankel, who has read Diophantos with appreciation, says in the place referred to, "The reader will now be desirous to become acquainted with the classes of indeterminate problems which Diophantos treats of, and his methods of solution. As regards the first point, we must observe that in the 130 (or so) indeterminate questions, of which Diophantos treats in his great work, there are over 50 different classes of questions, which are arranged one after the other without any recognisable classification, except that the solution of earlier questions facilitates that of the later. The first Book only contains determinate algebraic equations; Books II. to V. contain for the most part indeterminate questions, in which expressions which involve in the first or second degree two or more variables are to be made squares or cubes. Lastly, Book VI. is concerned