second power by MM or, in words, deurépa pvpiás. The denominator 187474560 in v. 8 would thus be written μopíov devrépas μυριάδος α καὶ μυριάδων πρώτων ηψμζ καὶ Μ δφξ, and the fraction 131299224/1629586560 would be written devrépa μvpiàs ā πpútai (μυριάδες) γρκθ Μ θσκί μορίου δευτέρων μυριάδων 15 πρώτων (μυριάδων) βήνη Μ εφέ. But there is another kind of fraction, besides the purely numerical one, which is continually occurring in the Arithmetica, such fractions namely as involve the unknown quantity in some form or other in their denominators. The simplest case is that in which the denominator is merely a power of the unknown, S. Concerning fractions of this kind Diophantus says (Def. 3): "As fractions named after numbers have similar names to those of the numbers themselves (thus a third is named from three, a fourth from four), so the fractions homonymous with the numbers just defined are called after them; thus from ȧpilμós we name the fraction apieμooтóv [ie. 1/x from x], Tò Suvaμоoтóv from δύναμις, τὸ κυβοστόν from κύβος, το δυναμοδυναμοστόν from δυναμοδύναμις, το δυναμοκυβοστόν from δυναμόκυβος, and τὸ κυβοκυβοστόν from κυβόκυβος. And every such fraction shall have, above the sign for the homonymous number, a line to indicate the species." Thus we find, for example, IV. 3, sñ corresponding to 8/x and, IV. 15, s* Xe for 35/x. Cf. 4"σv for 250/x. Where the denominator is a compound expression involving the unknown and its powers, Diophantus uses the expedient which he often adopts with numerical fractions when the numerators and denominators are large numbers, namely the insertion of ev popi or μoplov between the expressions for the numerator and denominator. Thus in VI. 12 we have Δ' Ε Μ βφκ ἐν μορίῳ Δ' Δ ΜΑΛΔ and in VI. 14 = (60x2+2520)/(x+900-6ar*), Δ' τε Λ Μ Ν ἐν μορίῳ Δ Δ Μ Α Λ Δ' β = = (15* – 36)/(z + 36 – 127*). For toos, equal, connecting the two sides of an equation, the sign in the archetype seems to have been ; but copyists intro 1 Hultsch, loc. cit. duced a sign which was sometimes confused with the sign 4 for apilμós; this was no doubt the same abbreviation ч as that shown (with terminations of cases added above) in the list given at the end of Codex Parisinus 2360 (Archimedes) of contractions found in the "very ancient" MS. from which it was copied and which was at one time the property of Georgius Valla1. Diophantus evidently put down his equations in the ordinary course of writing, ie. they were written straight on, as are the steps in the propositions of Euclid, and not put in separate lines for each step in the process of simplification. In the scholia of Maximus Planudes however we find conspectuses of the problems with steps in separate lines which, except for the slightly more cumbrous notation, make the work scarcely more difficult to follow than it is in our notation. Though in the MSS. we have the abbreviation to denote equality, Bachet makes no use of any symbol for the purpose in his Latin translation. He uses throughout the full Latin word. It is interesting however to observe that in the notes to his earlier translation (1575) Xylander had already used a symbol to denote equality, namely ||, two short vertical parallel lines. Thus we find, for example (p. 76), IQ + 12 || 1Q + 6N+9, . which we should express by + 12 = x2+6x+9. Now that we have described in detail Diophantus' method of expressing algebraical quantities and relations, it is clear that it is essentially different in its character from the modern notation. While in modern times signs and symbols have been developed 1 Heiberg, Quaestiones Archimedeae, p. 115. One instance will suffice. On the left Planudes has abbreviations for the words showing the nature of the steps or the operations they involve, e.g. EKO. = EKOcois (settingout), τετρ. = τετραγωνισμός (squaring), σύνθ. = σύνθεσις (adding), ἀφ. = ἀφαίρεσις (subtraction), μep. = μeproμós (division), üw.= ̃wapķis (resulting fact). which have no intrinsic relationship to the things which they represent, but depend for their use upon convention, the case is quite different in Diophantus, where algebraic notation takes the form of mere abbreviation of words which are considered as pronounced or implied. In order to show in what place, in respect of systems of algebraic notation, Diophantus stands, Nesselmann observes that we can, as regards the form of exposition of algebraic operations and equations, distinguish three historical stages of development, well marked and easily discernible. (1) The first stage Nesselmann represents by the name Rhetorical Algebra or "reckoning by complete words." The characteristic of this stage is the absolute want of all symbols, the whole of the calculation being carried on by means of complete words, and forming in fact continuous prose. As representatives of this first stage Nesselmann mentions Iamblichus (of whose algebraical work he quotes a specimen in his fifth chapter) "and all Arabian and Persian algebraists who are at present known." In their works we find no vestige of algebraic symbols; the same may be said of the oldest Italian algebraists and their followers, and among them Regiomontanus. (2) The second stage Nesselmann proposes to call the Syncopated Algebra. This stage is essentially rhetorical, and therein like the first in its treatment of questions; but we now find for often-recurring operations and quantities certain abbreviational symbols. To this stage belong Diophantus and, after him, all the later Europeans until about the middle of the seventeenth century (with the exception of Vieta, who was the first to establish, under the name of Logistica speciosa, as distinct from Logistica numerosa, a regular system of reckoning with letters denoting magnitudes and not numbers only). (3) To the third stage Nesselmann gives the name Symbolic Algebra, which uses a complete system of notation by signs having no visible connexion with the words or things which they represent, a complete language of symbols, which supplants entirely the rhetorical system, it being possible to work out a solution without using a single word of the ordinary written language, with the exception (for clearness' sake) of a connecting word or two here and there, and so on1. Neither 1 It may be convenient to note here the beginnings of some of our ordinary algebraical symbols. The signs + and first appeared in print in Johann Widman's arithmetic (1489), where however they are scarcely used as regular symbols of operation; next they are found in the Rechenbuch of Henricus Grammateus (Schreiber), written in 1518 but perhaps not published till 1521, and then regularly in Stifel's Arithmetica integra (1544) H. D. 4 is it the Europeans from the middle of the seventeenth century onwards who were the first to use symbolic forms of Algebra. In this they were anticipated by the Indians. Nesselmann illustrates these three stages by three examples, quoting word for word the solution of a quadratic equation by Muḥammad b. Mūsā as an example of the first stage, and the solution of a problem from Diophantus as representing the second. First Stage. Example from Muḥammad b. Mūsā (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. Take half 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 Iamblichus who does use the Greek symbols for his numbers. as well as in his edition of Rudolff's Coss (1553). Vieta (1540-1603) has, in addition, = for ~. Robert Recorde (1510-1558) had already in his Algebra (The Whetstone of Witte, 1557) used (but with much longer lines) to denote equality ("bicause noe.2. thynges, can be moare equalle"). Harriot (1560-1621) denoted multiplication by a dot, and also by mere juxtaposition of letters; Stifel (1487-1567) had however already expressed the product of two magnitudes by the juxtaposition of the two letters representing them. Oughtred (1574-1660) used the sign x for multiplication. Harriot also introduced the signs > and < for greater and less respectively. for division is found in Rahn's Algebra (1659). Descartes introduced in his Geometry (1637) our method of writing powers, as «3, a1 etc. (except a3, for which he wrote aa); but this notation was practically anticipated by Pierre Hérigone (Cours mathématique, 1634), who wrote aa, az, 44, etc., and the idea is even to be found in the Rechenbuch of Grammateus above mentioned, where the successive powers of the unknown are denoted by pri, se, ter, etc. The use of x for the unknown quantity began with Descartes, who first used &, then y, and then x for this purpose, showing that he intentionally chose his unknowns from the last letters of the alphabet. ✔✅ for the square root is traceable to Rudolff, with whom it had only two strokes, the first (down) stroke being short, and the other relatively long. 1 Thus Muḥammad b. Mūsā states in words the following solution. Second Stage. As an example of Diophantus I give a translation word for word of II. 8. So as to make the symbols correspond exactly I use S (Square) for 4o (dúvaμış), N (Number) for s, U . (Units) for M (μονάδες). "To divide the proposed square into two squares. Let it be proposed then to divide 16 into two squares. And let the first be supposed to be 1S; therefore the second will be 16 U-1S. Thus 16 U-1S must be equal to a square. I form the square from any number of N's minus as many 's as there are in the side of 16 U's. Suppose this to be 2N-4U. Thus the square itself will be 4S 16U-16N. These are equal to 16U-1S. Add to each the negative term (ý λeîfis, the deficiency) and take likes from likes. Thus 5S are equal to 16N, and the N is 16 fifths. One [square] will be, and the other, and the sum of the two makes up 400, or 16U, and each of the two is a square.” Of the third stage any exemplification is unnecessary. To the form of Diophantus' notation is due the fact that he is unable to introduce into his solutions more than one unknown quantity. This limitation has made his procedure often very different from our modern work. In the first place we can begin with any number of unknown quantities denoted by different symbols, and eliminate all of them but one by gradual steps in the course of the work; Diophantus on the other hand has to perform all his eliminations beforehand, as a preliminary to the actual work, by expressing every quantity which occurs in the problem in terms of only one unknown. This is the case in the great majority of questions of the first Book, which involve the solution of determinate simultaneous equations of the first degree with two, three, or four variables; all these Diophantus expresses in terms of one unknown, and then proceeds to find it from a simple equation. Secondly, however, this limitation affects much of Diophantus' work injuriously; for, when 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 assume for one or other of these some particular number arbitrarily chosen, the effect of the assumption being to make the problem a determinate one. However, it is but fair to say that Diophantus, 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 |