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ARITHMET´ICA

ARITHMET´ICA (ἀριθμητική, sc. τέχνη or ἐπιστήυη, Plat. Gory. 450 D, &c.: in Latin, Vitr. 1.1, Plin. Nat. 35.10, 36, § 76: arithmetica as n. pl. in Cic. Att. 14.1. 2, fin.: numeri is used, more idiomatically, with the same meaning, Cic. Fin. 5.29, 87) means generally the theory of numbers as opposed to the practical art of calculation. The distinction is especially insisted on by Plato (Gorg. 451 B, C; Euthyd. 290 B, C; cf. Charm. 174 B), who refers λογιστική to an inferior order of thought. But the line of division cannot be sharply drawn, for the discoveries of ἀριθμητική (e. g. the method of finding a least common multiple) are often appropriated by. λογιστική, and the processes of the latter are indispensable to many of the inquiries of the former. Hence Plato at a later time distinguishes popular ἀρ.π and λογ. together from the philosophical species of both (Phileb. 56 D sqq.), and the same distinction seems to underlie Rep. 525 A sqq., where again ἀρ. and λογ. are named together as ἀγωγὰ πρὸς ἀληθείαν, and the study of λογ. is further commended as leading the soul finally περὶ αὐτῶν τῶν ἀριθμῶν διαλέγεσθαι. From the same passage it appears that Plato's main objection to popular λογ. was that it had no logical unit, but dealt indifferently with all manner of tangible objects--apples, cows, &c.--at the bidding of any huckster or merchant. From this it would seem that any arithmetical operation in which numbers alone were considered would fall under ἀρ., while such propositions as that 600 obols make a mina would belong to λογ. (cf. Geminus in Procl. Comm. Eucl. ed. [p. 1.188]Friedlein, p. 38). But the opposition between the two terms came soon to cover an opposition, not of matter only, but of method. For philosophical purposes, numbers were generally represented by dots or lines arranged in geometrical figures (cf. Plat. Theaet. 147, 148; and Euclid, bks. vii. viii. ix.), and operations with the customary symbols ά, β́, &c., were referred to λογ., and were seldom used, as indeed they were seldom required, in pure mathematics. Hence it was that no Greek mathematician ever seriously attempted to improve the ordinary Greek numerical symbolism, and thereby to simplify the cumbrous processes of elementary calculation (cf. Cantor, Vorles. über Gesch. der Mathem. p. 133).

The introduction of the study of numbers into Greece is universally attributed to Pythagoras, of whom Aristoxenus (ap. Stob. Ecl. Phys. 1.16, 100.2, ad init.) says, in words which illustrate the passages of Plato above cited, that he first raised arithmetic above τῆς τῶν ἐμπόρων χρείας, πάντα τὰ πράγματα ἀπεικάζων τοῖς ἀριθμοῖς. (On Egyptian and Chaldean arithmetic, and the possible indebtedness of Pythagoras to these foreign sources, comp. Cantor, op. cit. pp. 1-93.) The studious secrecy of the earliest Pythagorean school, and the almost complete loss of later Pythagorean writings, must always cause great doubt as to the development of αριθμητική: but the frequent references of Plato and Aristotle, when compared with Euclid vii.-ix., and with such late writers as Nicomachus of Gerasa, Theon Smyrnaeus, and Iamblichus, show that very little addition was from time to time made in the subject-matter of the science, although the original nomenclature received, in some cases, new applications. The chief subjects of άρ. were from the first, and remained always, the classification of numbers, the theory of proportion, and the summation of series. Some attempt at a theory of permutations and combinations may possibly have been made (cf. Plut. Quaest. Conv. 8.9, 13), but the few problems and answers recorded by historians do not suggest an accurate knowledge of this branch of mathematics. The solution of equations will be mentioned later.

The unpractical nature of ἀρ. may be well seen from the books of Euclid above cited, in which only five propositions are of real importance in calculation. The mode of finding a G. C. M. is exhibited in 7.2, 3, and of an L. C. M. in 7.36, 38. In 9.35, a proposition is stated from which the sum of a geometrical series may easily be found. But the great bulk of the books is devoted to the investigation of prime and other numbers, or to propositions in proportion, such as 8.22: “If three numbers are proportional, and the first is a square, the third is also a square.” (Cf. Nesselmann, Algebra der Griechen, pp. 158-183.)

It is not proposed in this article to give more than a brief statement of those arithmetical terms which occur most frequently in Greek philosophical literature, and a few references to passages where some problems are worked out. The first classification of numbers is that into ἄρτιοι and περισσοί, “odd” and “even.” This division no doubt was older than Pythagoras, for a game, common in Plato's time, was founded on it (Lysis, 206 E). Of περισσοί, some are “prime,” πρῶτοι, μονάδι μόνῃ μεπρούμενοι (Eucl. vii. def. 11). Further classifications are then founded on the geometrical representation mentioned above. Plane numbers (ἐπιπεδοι) are the products of two factors (πλευραί), and solid numbers (στερεοί) of three (Eucl. vii. deff. 16 and 18). Of plane numbers some are triangular (τρίγωνοι), falling under the formula n (n + 1)/2 (Theon Smyrn. Math. Plat. ed. Hiller, pp. 27-31; Cantor, op. cit. p. 135); some square, τετράγωνοι, ἴσακις ἴσοι (Eucl. vii. def. 18); some ἑτερομήκεις, falling under the formula n (n + 1). All these, according to Theon (loc. cit.), are founded on summations of arithmetical series: viz. the sum of the first n numbers is triangular; that of the first n odd numbers is square; that of the first n even numbers is ἑτερομήκης (Cantor, p. 135). In Plato, a number which is the product of two unequal factors is προμήκης (Theaet. 148 A) or “oblong.” The word δύναμις, which occurs first with mathematical application in the extracts from Hippocrates of Chios preserved by Eudemus (Fragm. ed. Spengel, p. 128), seems generally to mean a square number, not geometrically conceived--“a number raised to the power of 2,” so to say; but in the Theaetetus (loc. cit.) it is applied to a line which is the geometrical representative of a surd. Any odd number was also called “gnomonic” (2 n + 1), because this added to n2 produces the next highest square (cf. Ar. Phys. 3.4, 203 a, which is explained to mean that square numbers are found by adding gnomons to unity, sc. in the series 1 + 3 + 5 + &c., according to the statement of Theon mentioned above. For the gnomon, see Eucl. ii. def. 2). A few terms, not derived from geometry, may be added. “Perfect” numbers (τέλειοι, Eucl. vii. def. 22 and prop. 36) are those which are equal to the sum of their aliquot parts, as 6, 28, 496 (e. g. 6 = 1 + 2+3: 28 = 1 + 2 + 4 + 7 + 14). “Excessive” (ὑπερτέλειοι) and “defective” (ἐλλιπεῖς) are those of which the aliquot parts are respectively greater or less than the numbers. Aristotle, however (Metaph. 1.5), calls 10 a perfect number. It is unknown what definition he attached to τέλειος. “Friendly” numbers (φίλιοι) are pairs, of which one number is the sum of the aliquot parts of the other, as 220 and 284 (Iamblich. in Nicom. ed. Tennulius, pp. 47, 48; Cantor, p. 141). By the time of Hypsicles, probably circa 180 B.C., “polygonal” numbers were also distinguished, on which Diophantus afterwards wrote a short treatise (Nesselmann, pp. 463-469). These are the sums of arithmetical series, of which the common difference is 3 or more. The same nomenclature is used in the second book (on ἀριθμοὶ σχηματογραφηθέντες) of Nicomachus (Introd. Arithm. ed. Hoche, 1866), but with different applications (e. g. a “polygonal” number is one which, when represented by dots, can be arranged in the form of a polygon, and a “solid” number is the sum of several polygonal); but the distinctions of later ἀριθμητική have no literary interest, and the reader who desires more information must be referred to Nicomachus himself, and the commentary on him written by Iamblichus (ed. Tennulius, Arnheim, 1668; cf. Cantor, pp. 362-370).

Similar ratios ( τῶν λόγων δμοιότης) pro duce [p. 1.189]a proportion (ἀναλογία, Eucl. v. def. 8). The terms of a proportion are in general called δροι, but the middle terms are specially called μεσότητες. The latter name was originally applied to proportions in general, ἀναλογία meaning specially geometrical proportion (Nesselmann, p. 210, n. 49; and Cantor, p. 206). By the time of Nicomachus, ten kinds of proportion were distinguished, three of which are ascribed to Eudoxus by the Eudemian fragment preserved in Proclus (ed. Friedlein, p. 67; Bretschneider, Geom. vor Eukl. p. 30), and four are of later origin (Iambl. in Nicom. pp. 141 sqq., 159, 163, 168). The first four, according to Iamblichus (loc. cit.), were invented or introduced from Babylon, by Pythagoras. These are the arithmetical (when a--b = c--d), the geometrical (when a : b :: c: d), the harmonical or ὑπεναντία (when a--b: b--c: : a: c), and the musical or τελειοτάτη,) which is created between two numbers and their arithmetical and harmonical means (a a+b/2::2ab/a+b:b, as 6: 9:: 8: 12). Harmonical proportion is said to have been so called, because a string, if stopped at 2/3 of its length, gives the fifth, and, if stopped in the middle, the octave of the note which is produced by the whole string, and 1-2/3:2/3-1/2::1:1/2. (Hankel, Zur Gesch. der Math. p. 105. Dr. Allman in Hermathena, vii. p. 204, says that these proportions were never applied to number; but cf. Hankel, p. 114, and Arist. An. Post. 1.5, 74.)

The terms of a progression, like those of a proportion, were called ὅροι, and the progression itself was perhaps called ἔκθεσις (Cantor, p. 135, and Bienaymé in Comptes Rendus de l'Acad. des Sci. Oct. 3rd, 1870). The ratio or common difference seems to be called ἀπόστασις in Plat. Tim. 43 B. The summation of a geometrical progression is effected by implication in Eucl. 9.35. The summation of an arithmetical series is exhibited in the first three propositions of the ἀναφορικός of Hypsicles (Paris, 1867), but a quite distinct formula is used incidentally by Archimedes (περὶ ἑλίκων, Torelli's ed. pp. 226-228) in a proposition which, in effect, amounts to the summation of a series of the first n square numbers. The theory of the extraction of square roots is well enough given by Theon of Alexandria, in his commentary on the Almagest (Nesselmann, pp. 111, 112), but the practice seems to have been rough and empirical, λογιστικῶς. Thus Eutocius (Torelli's Archim. p. 208) refers to Heron and other writers for rules ὅπως δεῖ σύνεγγυς τῆν δυναμένην πλευρὰν εὑρεῖν.

In Latin literature, arithmetica appears only in technical writers. The fragment in the Arcerian MS. of gromatici, attributed to Epaphroditus and Vitruvius Rufus (Cantor, Agrimensoren, p. 214 sqq.), contains one or two propositions not found in Greek writers (e. g. the summation of the first n cubic numbers. Cantor, Vorles. p. 473). The Arithmetica of Boethius is expressly said to be founded on that of Nicomachus (Friedlein‘s ed., pp. 4 and 5).

It should be added that the great work of Diophantus is also entitled ἀριθμητικά. It treats of the solution of algebraic equations, determinate and indeterminate, simple, quadratic or cubic, with one unknown. This sort of mathematical problem appears for the first time, though without any algebraic symbolism, in Heron of Alexandria (e.g. Geometria, 101, 7-9, p. 133 in Hultsch's ed.), who was above all things practical. The arithmetical puzzles of the Greek anthology, attributed chiefly to Metrodorus of the time of Constantine, show the popularity of the study, but its first philosophical application appears in the ἐπάνθημα of Thymaridas, obscurely described by Iamblichus (in Nicom. Ar. ed. Tennulius, p. 36), and admirably expounded by Nesselmann (Alq. der Griechen, pp. 232 sqq.; also Cantor, Vorles. über Gesch. der Mathem. p. 371). Diophantus himself does not pretend to be the inventor of a new method of inquiry, but only a systematiser, a στοιχειώτης. Thus in his definitions he uses καλεῖται, ἐστίν, where an inventor would have said καλείσθω, ἔστω, and he entirely omits to mention the rule, which he frequently uses, that the product of two minus quantities is plus. His problems are of the following kind, e. g. to find three numbers such that the sums of any two of them and of the three shall be a square (3.7). It is observable that he reasons entirely with algebraic and arithmetical symbols, only one proposition (5.13) being treated geometrically. His symbolism does not occur in any other author, and does not therefore fall within the scope of this article. His works are most fully summarised and criticised by Hankel and Nesselmann (opp. citt.). The references given above indicate all the best modern authorities on every branch of the subject.

[J.G]

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