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 (
ἡ 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::2
ab/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]
