magnitudes, are intended also to be applicable to numbers. The first 8 propositions of the second book, for instance, are for geometrical purposes proved by inspection. No one can doubt them who looks at the figures. But as arithmetical propositions they are not self-evident if stated with any arithmetical symbolism. In such a form, the first 10 propositions (the 9th and 10th are not treated in the same way as the first 8) are as follows1: (1) ab+ ac+ad+......a (b+c+d+......). (2) (a + b)2 = (a + b) a + (a + b) b. (3) (a + b) a = ab + a2. (4) (a+b)2= a2 + b2 + 2ab. 2 2 The eleventh proposition is the geometrical way of solving the quadratic equation a (a-b)=b2 and the fourteenth solves the quadratic a2 = bc. From this statement, in algebraical form, of the chief contents of the 2nd Book, it will at once be seen what an advantage Greek mathematicians found in a geometrical symbolism. These propositions are all true for incommensurable, as well as commensurable, magnitudes, irrational as well as rational, numbers. no symbolism at all for surds. 1 It will be observed that Theon's method of finding a square root, cited above, is founded on Eucl. II. 4. So also Diophantus (infra, p. 104) uses Euclid II. as an arithmetical book. 2 This is the famous problem of 'the golden section,' which is used again in Euclid Iv. 10 for the purpose of con But in numbers the Greeks had structing a regular pentagon. Euclid's : that there was no exact numerical equivalent, for instance, for the root of 2 but they knew also that the diagonal of a square side 2 1'. Hence lines, which were merely convenient symbols for other numbers, became the indispensable symbols for surds. Thus, Euclid's 10th book, which deals with incommensurables, is in form purely geometrical, though its contents are of purely arithmetical utility: and every arithmetical proposition, in the proof or application of which a surd might possibly occur, was necessarily exhibited in a geometrical form. It is not, therefore, surprising that a linear symbolism became habitual to the Greek mathematicians and that their attention was wholly diverted from the customary arithmetical signs of the unlearned. 3 48. It is in the 7th book of the Elements that Euclid first turns to the consideration of numbers only. It begins with 21 definitions which serve for the 7th 8th and 9th books. The most important of these are the following: (1) Unity (uovás) is that by virtue of which everything is called 'one' (ềv Xéyetai)*. (3) and (4) A less number, which is a measure of a 1 This fact, according to an old scholiast (said to be Proclus) on the 10th book of Euclid, remained for a long time the profoundest secret of the Pythagorean school. The man who divulged it was drowned. See Cantor, Vorles. pp. 155, 156, quoting Knoche, Untersuch. über die Schol. des Proklus etc., Herford, 1865, pp. 17-28, esp. p. 23. 2 The use of lines of course avoided the necessity of calculation. A rectangle represented a product: its side a quotient. Thus, for instance, Euclid (x. 21), wishing to show that a rational number divided by a rational gave a rational quotient, states that 'if a rational rectangle be constructed on a rational line, its side is also rational.' 3 In the 7th 8th and 9th books, no geometrical figures are given, as indeed none are necessary. In the 7th book according to our MSS. numbers are generally represented by dots (in Peyrard's edition by lines), in the 8th book particular numbers are given by way of illustration: in the 9th book both dots and particular numbers occur. Euclid probably used lines only, except where a number was to be represented as odd or even, in which case perhaps he used dots. At any rate, he does not, any more than in the geometrical books, use division, and his treatment of the propositions is purely synthetic, as elsewhere. The arithmetical books of Euclid are included in Williamson's translation, Oxford 1781-1788. 4 In the 2nd definition μovás means 'the unit.' 'Number' is there defined as ‘τὸ ἐκ μονάδων συγκείμενον πλῆθος. greater, is a pépos (part) of it: but if not a measure, it is μépn (parts) of the other1. (6) and (7) 'Odd' and 'even' numbers (Tepiσool and aρtioi). (11) Prime'numbers (πρῶτος ὁ μονάδι μόνον μετρούμενος). (12) Numbers 'prime to one another (πρῶτος πρὸς ἀλλήλους). (13) Composite numbers (σúveTOL). (16) Products of two numbers are 'plane' (èπíπedoɩ) and each factor is a ‘side' (πλeʊpá). (17) Products of three numbers (λevpaí) are 'solid' (στερεοί). (18) 'Square' numbers (TETρáywvos ó loákis loos). (19) ‘Cubes (κύβος ὁ ἰσάκις ἴσος ἰσάκις). (20) Numbers are 'proportional' (áváλoyov eiσí) when the 1st is the same multiple, part or parts of the 2nd as the 3rd of the 4th. (21) Plane and solid numbers are 'similar' when their sides are proportional. (22) A 'perfect' (Téλelos) number is that which is the sum of all its factors (uépn). It will be seen that this nomenclature is purely Pythagorean. The class of 'prime' (πρôτо) numbers is not indeed mentioned by any earlier writer now known, but it can hardly be doubted that they were defined by the Pythagoreans, as a sub-class of odd numbers. The book deals with the following matters: Prop. I. If of two given unequal numbers the less be subtracted from the greater as often as possible and the remainder from the less and the next remainder from the preceding remainder and so on, and no remainder is a measure of the preceding remainder until 1 is reached, the two given numbers are prime to one another. This (which is proved by reductio ad absurdum) leads to Propp. II., III. To find the greatest common measure of two or more numbers. (The procedure is identical with ours.) Propp. IV.-XXII. These deal with submultiples and fractions 1 μέρος ἐστὶν ἀριθμὸς ἀριθμοῦ, ὁ ἐλάσσων τοῦ μείζονος, ὅταν καταμετρῇ τοῦ μείζονα. μέρη δὲ, ὅταν μὴ καταμετρῇ. This word μέρη is the plural of μέρος, and is a very inconvenient expression. and apply to numbers the doctrines of proportion which had been previously proved for magnitudes in the 5th book1. Propp. XXIII.-XXX. Of numbers prime to one another. E. g. XXIX. If two numbers are prime to one another, all their powers are prime to one another. Propp. XXXI.-XXXIV. Of prime numbers in composition. E. g. XXXIV. Every number is prime or is divisible by a prime. Propp. xxxv.-XLI. Miscellanea: e. g. XXXV. To find the lowest numbers which are in the same ratio with any given numbers. XXXVI. To find the L. C. M. of two, and XXXVIII. of three, numbers. XLI. To find the lowest number which is divisible into given parts. 49. The 8th book deals, in the first half, chiefly with numbers in continued proportion (ἀριθμοὶ ἑξῆς ἀνάλογον) e. g. III. If any numbers are in a continued proportion and are the least which have the same ratio to one another, the extreme terms will be prime to one another. VII. If the 1st term is a divisor of the last, so is it of the 2nd. But a few other propositions are inserted, e. g. V. Plane numbers are to one another in the ratio which is compounded of their sides. XI. There is one mean proportional between two squares and XII. two between two cubes. The last half of the book (Propp. XIV. to XXVII.) is entirely devoted to the mutual relations of squares, cubes and plane numbers, e. g. XXII. If three numbers are in continued proportion and the first is a square, so is the third. XXIII. If four numbers are in continued proportion, and the first is a cube so is the fourth. 50. The 9th book continues the same subject for a few propositions: e.g. III. If a cube be multiplied by itself the product is a cube. Then follow (VIII.-XV.) some more propositions on numbers in continued proportion, or geometrical series: e. g. IX. If in a series, commencing from unity, the 2nd term is a square, so are the following terms. And if the 2nd 1 E.g. IV. Every number is either 2 μέρος οι μέρη of every higher number. v. vI. If A is the same μépos (or μépn) of B as C of D, A+ C is the same of B + D. If A: B: C:D, then AD=BC XIX. and conversely. 2 i. e. are the least which can form a continued proportion of the same number of terms, bearing the same ratio to one another, as in the given case. term is a cube, the following terms are cubes. A few propositions on prime numbers (XVI.-XX.) are then given of which the most important is XX. The number of primes is greater than any given number. The discussion of odd and even numbers is then introduced (XXI.—XXXIV.), the propositions being of such a character as XXIV. If an even number be subtracted from an even number, the remainder is even. Then suddenly, appears the following proposition, XXXV. "If any numbers be in continued proportion, and the first term be deducted from the 2nd and also from the last, the remainder of the 2nd will be to the 1st as the remainder of the last to the sum of all the preceding terms1." Stated in another form, this proposition is: If a, ar, ar2, ar3... ar" be a geometrical series, then ar a: a :: (ar" − a) : a + ar + ar2... + ar”−1. It is an easy step further to conclude that and thus to sum the series, but Euclid does not take this step. The proposition, as it stands, is apparently introduced solely for the purpose of proving the next (xXXVI.), the last in the book. This is, in effect, that in a geometrical series of the powers of 2 from 1 onwards, the sum of the first n terms (if a prime number) multiplied by the nth term is a perfect number2. In the proof, which is too long to be here inserted, the sum of n terms is assumed to be known by simple addition. 1 Euclid takes only four numbers. His proof, put shortly, is as follows: Y a η В к እ € Take yn = 50=α, ¿ê=By, Fλ=d. 8 and componendo 50 : 0k :: 50+5K+§λ :: Ok + kλ +λe. By substitution (taking the terms backwards) e0: a + By + d :: ßn: a. i.e. es-a: a + By + d :: By - a: a. Q. E. D. 2 ̓Εὰν ἀπὸ μονάδος ὁποσοιοῦν ἀριθμοὶ ἑξῆς ἐκτεθῶσιν ἐν τῇ διπλασίονι ἀναλογίᾳ From this also it is evident that the proposition is untrue unless p is a prime |