that Diophantus, in search of a convenient symbol for his unknown quantity, would select the only letter of the Greek alphabet which was not already appropriated as a numeral'. But he made the acute observation that, as the symbol occurred in many places (of course in Bachet's text) for apropós used in the ordinary untechnical sense, and was therefore, as it appeared, not exclusively used to designate the unknown quantity, the technical apieμós, it must after all be more of the nature of an abbreviation than an algebraical symbol like our x. It is true that this uncertainty in the use of the sign in the MSS. is put an end to by Tannery, who uses it for the technical apieμós alone and writes the untechnical apieμós in full; but, even if Diophantus' practice was as strict as this, I do not think this argues any difference in the nature of the abbreviation. There is also a doubt whether the final sigma, s, was developed as distinct from the form σ so early as the date of the MSS. of Diophantus, or rather so early as the first copy of his work, if the author himself really gave the explanation of the sign as found in our text of his second definition. These considerations suggested to me that the sign was not the final sigma at all, but must be explained in some other way. I had to look for confirmation of this to the precise shape of the sign as found in extant MSS. The only MS. which I had the opportunity of inspecting personally was the MS. of the first ten problems of Diophantus in the Bodleian; but here I found strong confirmation of my view in the fact that the sign appeared as ', quite different in shape from, and much larger than, the final sigma at the end of words in the same MS. (There is in the Oxford MS. the same irregularity as was pointed out by Nesselmann in the use of the sign sometimes for the technical, and sometimes for the untechnical, api@pós3.) But I found evidence that the sign appeared elsewhere in somewhat different forms. Thus Rodet in the Journal Asiatique of January, 1878, quoted certain passages from Diophantus for the purpose of comparison with the algebra of Muhammad b. Mūsā al-Khuwārazmi. Rodet says he ccpied these passages exactly from Bachet's MS.; but, while he generally gives the sign as the final sigma, he has in one case чч for apipoi. In this last case 1 Nesselmann, pp. 290-1. 2 ibid. pp. 300-1. 3 An extreme case is ἔταξα τὸ τοῦ δευτέρου, ἀριθμοῦ ἑνός, where the sign (contrary to what would be expected) means the untechnical ȧpioμós, and the technical is written in full. Also in the definition ὁ δὲ μηδὲν τούτων τῶν ἰδιωμάτων κτησάμενος... ἀριθμὸς καλείται the word dpiμós is itself denoted by the symbol, showing that the word and the symbol are absolutely convertible. Bachet himself reads 55". But the same form чy which Rodet gives is actually found in three places in Bachet's own edition. (1) In his note to IV. 3 he gives a reading from his own MS. which he has corrected in his own text and in which the signs чā and ч4 occur, evidently meaning apieμòs ā and ȧpilμoin, though the sign should have been that for åρiðμоσтóv (= 1/x). (2) In the text of IV. 13 there is a sentence (marked by Bachet as interpolated) which contains the expression 445, where the context again shows that 44 is for apioμoí. (3) At the beginning of v. 9 there is a difficulty in the text, and Bachet notes that his MS. has μnte ó dimλaσíwv avтoû ч where a Vatican MS. reads apieμóv (Xylander notes that his MS. had in this place μήτε ὁ διπλασίων αὐτοῦ ἀρ μό ā ...). It is thus clear that the MS. (Paris. 2379) which Bachet used sometimes has the sign for åpeμós in a form which is at least sufficiently like ч to be taken for it. Tannery states that the form of the sign found in the Madrid MS. (A) is ч, while B, has it in a form (S) nearly approaching Bachet's reproduction of it. It appeared also that the use of the sign, or something like it, was not confined to MSS. of Diophantus; on reference to Gardthausen, Griechische Palaeographie, I found under the head "hieroglyphisch-conventionell" an abbreviation, for apiμós, -o, which is given as occurring in the Bodleian MS. of Euclid (D'Orville 301) of the 9th century. Similarly Lehmann' notes as a sign for apoμós found in that MS. a curved line similar to that which was used as an abbreviation for xaí. He adds that the ending is placed above it and the sign is doubled for the plural. Lehmann's facsimile is like the form given by Gardthausen, but has the angle a little more rounded. The form чy above mentioned is also given by Lehmann, with the remark that it seems to be only a modification of the other. Again, from the critical notes to Heiberg's texts of the Arenarius of Archimedes it is clear that the sign for appós occurred several times in the MSS. in a form approximating to that of the final sigma, and that there was the usual confusion caused by the similarity of the signs for apiμos and kai. In Hultsch's edition of Heron, similarly, the critical notes to the Geodaesia show that one MS. had an abbreviation for 1 Lehmann, Die tachygraphischen Abkürzungen der griechischen Handschriften, 1880, p. 107: "Von Sigeln, welchen ich auch anderwärts begegnet bin, sind zu nennen åp‹Ðμós, das in der Oxforder Euclidhandschrift mit einer der Note xai ähnlichen Schlangenlinie bezeichnet wird.” * Cf. Heiberg, Quaestiones Archimedear, pp. 172, 174, 187, 188, 191, 191; Archimedis opera omnia, 11., pp. 268 sqq. apioμós in various forms with the case-endings superposed; sometimes they resembled the letter, sometimes p, sometimes O and once 1. Lastly, the sign for apieμós resembling the final sigma evidently appeared in a MS. of Theon of Smyrna1. All these facts strongly support the assumption that the sign was a mere tachygraphic abbreviation and not an algebraical symbol like our x, though discharging much the same function. The next question is, what is its origin? The facts (1) that the sign has the breathing prefixed in the Bodleian MS., which writes. 'for apibuós, and (2) that in one place Xylander's MS. read ap for the full word, suggested to me the question whether it could be a contraction of the first two letters of pioμós; and, on consideration, this seemed to me quite possible when I found a contraction for ap given by Gardthausen, namely P. It is easy to see that a simplification of this in different ways would readily produce signs like the different forms shown above. This then was the hypothesis which I put forward twenty-five years ago, and which I still hold to be the easiest and best explanation. Two alternatives are possible. (1) Diophantus may not have made the contraction himself. In that case I suppose the sign to be a cursive contraction made by scribes; and I conceive it to have come about through the intermediate form S. The loss of the downward stroke, or of the loop, would produce a close approximation to the forms which we know. (2) Diophantus may have used a sign approximately, if not exactly, like that which we find in the MSS. For it is from a papyrus of 154 A.D., in writing of the class which Gardthausen calls the "Majuskelcursive," that the contraction P for the two letters is taken. The great advantage of my hypothesis is that it makes the sign for appós exactly parallel to those for the powers of the unknown, e.g., 4 for dúvaμs and K' for xúßos, and to that for the unit povas which is denoted by M, with the sole difference that the letters coalesce into one instead of being written separately. Tannery's views on the subject are, I think, not very consistent, and certainly they do not commend themselves to me. He seems to suggest that the sign is the ancient letter Koppa, perhaps slightly modified; he first says that the sign in Diophantus is peculiar to him and that, although the word apibμós is very often Heron, ed. Hultsch, pp. 146, 148, 149, 150. * Theon of Smyrna, ed. Hiller, p. 56, critical notes. represented in mathematical MSS. by an abbreviation, it has much oftener the form or something similar, closely resembling the ancient Koppa. In the next sentence he seems to say that "on the contrary the Diophantine abbreviation is an inverted digamma"; yet lower down he says that the copyist of a (copied from the archetype a) got the form 4 by simplifying the more complicated Koppa. And, just before the last remark, he has stated that in the archetype a the form must have been 5 or very like it, as is shown by the confusion with the sign for xai. (If this is so, it can hardly have been peculiar to Diophantus, seeing that the same confusion occurs fairly often in the MSS. of other authors, as above shown.) I think the last consideration (the confusion with xaí) is very much against the Koppa-hypothesis; and, in any case, it seems to me very unlikely that a sign would be used by Diophantus for the unknown which was already appropriated to the number 90. And I confess I am unable to see in the sign any resemblance to an inverted digamma. Hultsch' regards it as not impossible that Diophantus may have adopted one of the signs used by the Egyptians for their unknown quantity hau, which, if turned round from left to right, would give; but here again I see no particular resemblance. Prof. D'Arcy Thompson' has a suggestion that the sign might be the first letter of awpós, a heap. But, apart from the fact that the final sigma (s) is not that first letter, there is no trace whatever in Diophantus of such a use of the word σwpós; and, when Pachymeres speaks of a number being owpeía μovádwv, he means no more than the los μovádwv which he is explaining his words have no connexion with the Egyptian hau. Notwithstanding that the sign is not the final sigma, I shall not hesitate to use for it in the sequel, for convenience of printing. Tannery prints it rather differently as s. We pass to the notation which Diophantus used to express the different powers of the unknown quantity, corresponding to x2, x3, and so on. He calls the square of the unknown quantity dúvapis, and denotes it by the abbreviation 4". The word Sivaμis, literally "power," is constantly used in Greek mathematics for 1 Art. Diophantus in Pauly-Wissowa's Real-Encyclopädie der classischen Altertumswissenschaften. 2 Transactions of the Royal Society of Edinburgh, Vol. xxxv111. (1896), pp. 607-9. 3 Dioph. II. p. 78, 4. Cf. Iamblichus, ed. Pistelli, p. 7, 7; 34, 3; 81, 14, where σwpela is similarly used to elucidate πλῆθος. square'. With Diophantus, however, it is not any square, but only the square of the unknown; where he speaks of any particular square number, it is TeTρáywvos dápioμós. The higher powers of the unknown quantity which Diophantus makes use of he calls xúßos, δυναμοδύναμις, δυναμόκυβος, κυβόκυβος, corresponding respectively to r3, x2, x3, xo. Beyond the sixth power he does not go, having no occasion for higher powers in the solutions of his problems. For these powers he uses the abbreviations K, 44, 4K, KK respectively. There is a difference between Diophantus' use of the word Súvapis and of the complete words for the third and higher powers, namely that the latter are not always restricted like dúvaμis to powers of the unknown, but may denote powers of ordinary known numbers as well. This is no doubt owing to the fact that, while there are two words dúvaμıs and tetpárywvos which both signify "square,” there is only one word for a third power, namely Kúßos. It is important, however, to observe that the abbreviations KY, 4o4, ΔΚΥ, ΚΥΚ, are, like δύναμις and 4', only used to denote powers of the unknown. The coefficients of the different powers of the unknown, like that of the unknown itself, are expressed by the addition of the Greek letters denoting numerals, e.g., 4K 5 corresponds to 263. Thus in Diophantus' system of notation the signs 4" and the rest represent not merely the exponent of a power like the 2 in 2, but the whole expression. There is no obvious connexion between the symbol 4" and the symbols of which it is the square, as there is between 2 and x, and in this lies the great inconvenience of the notation. But upon this notation no advance was made by Xylander, or even by Bachet and Fermat. They wrote N (which was short for Numerus) for the s of Diophantus, Q (Quadratus) for 4, C (Cubus) for K3, so that we find, for example, IQ+5N = 24, corresponding to x2+5x=24. Other symbols were however used even before the publication of Xylander's Diophantus, e.g. in Bombelli's Algebra. Bombelli denotes the unknown and its powers by the symbols 1, 2, 3, and so on. But it is certain that up to this time (1572) the common symbols had been R (Radix or Res), Z (Zensus, i.e. square), C (Cubus). Apparently the first important step towards 2, 3, etc., was taken by Vieta (1540— 1 In Plato we have divas used for a square number (Timaeus, 31) and also (Theaetetus, 147 D) for a square root of a number which is not a complete square, i.e. for a surd; but the commonest use is in geometry, in the form duváμec, “in square," e.g. “AB is dvráμe double of BC" means ་་ AB 2BC." = |