450 D, &c.: the nearest Latin equivalents
appear to be ratiocinandi ars
(cf. Cic. Tusc. 1.2
, 5), dinumeratio
iii. in Smith's Lat. Dict.
means “the art of calculation” as opposed to the “theory
(q. v.). Neither, of course, can
exist without the other; but as the operations of arithmetica
were generally performed by means of geometrical
figures, which were found more suggestive as representing not numbers only,
but magnitudes generally, the customary numerical symbols and the operations
in which they were used were deemed to belong to logistica,
and are more properly treated in this article. We shall
divide the subject accordingly into two parts, dealing first with the
representation of numbers, and secondly with calculations.
I. NUMERAL SIGNS.
From the general use among Aryan peoples of
a denary or vigesimal notation, it may be inferred, with as much certainty
as can ever be obtained about pre-historic culture, that these nations at a
very early time used the fingers and toes as symbols of number (cf. A. F.
&c., Halle, 1847, and
Sprachverschiedenheit etc. an den Zahlwörtern,
Halle, 1867; Tylor, Primit. Culture,
i. ch. 7). A relic of a
yet earlier notation, the quinary, survives in the words πεμπάζειν, πεμπάζεσθαι, πεμπαστής
(Hom. Od. 4.412
; Aesch. Pers. 981
, &c.), which imply that 5 was at one
time the limit of the units in ordinary counting. At this time, and indeed
for long after the denary notation was adopted, the Greeks clearly used both
hands to count no higher than 10 (cf. Hdt. 6.63
; Arist. Problem.
no doubt this simple practice was never lost. But the references to
fingerreckoning in literature are very scanty until a late date (Plut.
174 b; D. C.
; Anth. Pal.
11.72, &c.), when a far
more complicated system, common to Greece, Italy, and the East, is found in
use. (See Roediger in Jahresb. der Deutsch. Morgenl.
1845, pp. 111-129.) This is fully described by Nicolaus
Smyrnaeus (called also Rhabda or Artabasda) in a work entitled ἔκθρασις τοῦ δακτυλικοῦ μέτρου,
probably in the 13th or 14th century, and printed by N. Caussinus in his
book De Eloquentia Sacra et Humana
ch. viii. pp. 565-568, Paris, 1636; also in Schneider's Eclog.
p. 447). In this system, units and tens were represented
on the left hand, hundreds and thousands on the right. The thumb and
forefinger of the left hand were devoted to tens, those of the right to
hundreds; the remaining fingers of the left hand belonged to the units,
those of the right to thousands. The fingers might be straight (ἐκτεινόμενοι
), bent (συστελλόμενοι
), or closed (κλινόμενοι
). In the left hand, bending
the fourth finger marked 1; bending the third and fourth, 2; the middle,
third, and fourth, 3; the middle and third only, 4; the middle only, 5; the
third only, 6. Closing
the fourth finger gave 7; the fourth
and third fingers, 8; the middle, third, and fourth, 9. The same motions on
the right hand indicated thousands, from 1000 to 9000. The motions of the
forefinger and thumb in representing tens and hundreds, on the left and
right hands respectively, are more difficult to describe. The reader is
referred to Roediger's article, above cited; to Friedlein‘s
Zahlzeichen und Elem. Rechnen der Gr. u. Römer,
p. 6; and to Prof. Palmer's art. in Journal of Philology,
vol. ii. p. 247 sqq.,
where a plate is given.
Martianus Capella (De Nuptiis Philol.
&c., bk. vii.
p. 244 of Grotius' ed. 1599) says, “Nonnulli Graeci etiam μυρία
adjecisse videntur,” and adds,
apparently in reference to this usage, “quaedam brachiorum contorta
saltatio fit,” of which he does not approve. The motions were
probably the same as those described by Bede in the tract De loquela per gestum digitorum
(Opera, Basileae, 1563, col.
171-173). Various positions of the left hand on the left breast and hips
indicated the ten thousands, corresponding positions of the right hand on
the right side the hundred thousands, and the hands folded together
represented a million. There is no means of ascertaining the origin or the
time of introduction of this method of finger-numeration. It is thought by
some commentators that Aristophanes alludes to it in Vesp.
656, but it is observable in that passage that Philocleon only concludes
from his “easy” calculation, that 150 talents are less than a
tenth of 2,000, so that he probably used his fingers in the ordinary way to
divide the latter number by 10. The more complicated system was obviously of
no use in calculation, save as a memoria technica
cases where the mind might be embarrassed by the consideration of several
numbers at once. It was probably, at first, only a means of communication
between buyers and sellers who were ignorant of each other's language. The
same or a similar system is still used for secret transactions in Persia
(cf. De Sacy in Journal Asiat.
vol. ii., and Tylor,
i. p. 246, n.).
--Under this head
may be included all the representative signs used with the reckoning-board,
). These were generally
small stones or balls, or dots marked in sand, and the signs varied in value
according to the row of the abacus in which they were placed. (Hdt. 2.36
; [p. 2.72]D. L. 1.59
: cf. Becker-Goll,
2.67 ff.; and see below under Roman
“pebblesigns,” p. 74.) [ABACUS
) Written Characters.
says (in Nicom. Arithm.,
ed. Tennulius, p. 80), without
citing any authority, that among the earliest Greeks numbers were
represented in writing by repeated strokes. In one inscription (Franz,
p. 347; Boeckh, C. I. G.
2919, vol. ii. p. 584) from Tralles, ἔτεος
||||||| is found, but Boeckh suspects this to be a forgery of imperial
times. With some limitations, however, the statement of Iamblichus may be
true. It is possible that with the Greeks, as with the Phoenicians and
Egyptians, the signs of the units, tens, &c. were at an early date
repeated nine times without any intermediate compendia. (Cf. Pihan,
Exposé des Signes de Numération,
&c. Paris, 1860.)
But the earliest known system of written numerical symbolism in Greek is that
which used to be called after Herodianus, a Byzantine grammarian of the 3rd
century, who alleged that these “Herodianic” signs occurred in
laws of the Solonian period and other ancient documents, coins or
inscriptions, seen by him. (See App. Gloss. to Steph. Thesaurus,
vol. xii.; Valpy's ed. p. 690.) His statement has
since been most abundantly corroborated, especially in Athenian
inscriptions, and the system of numeration is now generally called Attic.
For our present purpose, however, the old name is more convenient. Upon this
system strokes served for units less than 5, and the chief higher numbers
are represented by their initial letters, [drachm5] for πέντε,
[drachm10] for δέκα,
[drachm100] for ἑκατόν,
[drachm1000] for λίλιοι,
[drachm1000] for μυρίοι,
with further compendia, [drachm50] for 50,
[drachm500] for 500, &c. (See C. I. A.
and ii.; or Hicks, Gr. Inscr.
passim; or Boeckh, Att.
p. 547 sqq.
curious Boeotian variations, see Franz, Epig. Graec.
ch. i. p. 348.) These signs alone are used in all the known Athenian
inscriptions of any date B.C. (in other words, in all the Inscr. of
C. I. A.
vols. i. and ii.). Outside Attica they certainly
remained in use along with the alphabetical signs, to be next described, and
are found with them on papyrus-rolls preserved in Herculaneum, which cannot
have been written before Cicero's time. The two styles are there used, as we
use Roman and Arabic numerals together, on occasions when arithmetical
division proceeds on two distinct principles, e. g. to mark the books of an
author as distinguished from the number of lines in the whole work.
(Ritschl, Die Alex. Bibliotheken,
pp. 99, 100, 123, n.) But
at some date which, as will be shown directly, cannot now be ascertained,
the letters of the alphabet with some additions came to be used in the
Semitic manner as numeral signs. It has been well pointed out (Cantor,
Vorles. über Gesch. der Math.
i. p. 108) that
the change was, for all purposes except brevity, a mistake. With the
Herodianic signs many patent analogies were exhibited which were wholly
obscured by the new symbolism. To take a very simple instance,
[drachm10] multiplied by p<*> gave [drachm50], and
[drachm100] multiplied by p<*> gave [drachm500] but on
the new system ι′
and none of
these signs contained in itself the least clue to its meaning. Hence, at
every arithmetical operation with alphabetical symbols, the mind was really
strained, first to interpret the signs, then to effect the calculation, and
lastly to express the result in signs again. We shall see later how cumbrous
the process was.
When and how the arithmetical use of the alphabet was adopted in Greece, is a
subject of the greatest difficulty. It is the custom to say that the
practice was originally Semitic (cf. Nesselmann, Algebra der
p. 72 sqq.
), but no such
practice appears on the Phoenician inscriptions at present known, and it is
not found on any Hebrew coins before 141-137 B.C. (Cf. Schröder,
quoted by Hankel, Zur
Gesch. der Mathem.
p. 34, and Dr. Euting there cited. Also
Madden, Coins of the Jews,
p. 67, temp.
Simon Maccabaeus.) On the other hand, the Hebrew cabbalistic
practice of gematria
(i. e. of treating as
interchangeable, for purposes of interpretation, words whose letters,
regarded as numerals, amount to the same total) is said to be as old as the
7th century B.C., and, if so, points to the
numerical use of the alphabet at that time (Cantor, Vorles.
i. pp. 87, 104, 105, quoting Lenormant, La Magie chez les
p. 24: cf. also Rev. 13.18, and Dr.
Ginsburg's art. Kabbalah
in Encycl. Brit.,
edit. vol. xiii.). And there is a peculiarity in the Hebrew and Greek
alphabetical numerals which suggests some connexion between them. In both
cases the proper alphabet is deficient, and is supplemented up to the same
limit. The Hebrew alphabet of 22 letters gives numbers only up to 400. The
deficiency is supplied, up to 900, by using the final
forms of letters, the medial
which (cf. Greek ς
) had already been used to represent 20, 40, 50, 80, and 90.
The Ionic alphabet of 24 letters, which was formally adopted at Athens in
403 B.C., could give numbers only as far as 600.
Three letters are wanting to complete the hundreds, and for this purpose the
three ἐπίσημα, ς,
, two of which had certainly been used
in older alphabets, but are omitted in the Ionic, are introduced. But these
unlike the Hebrew finals, do
not occur together, but stand for 6, 90, and 900 respectively, at widely
distinct places in the series. Now ς
doubt represents the old Vau
), and both this and κόππα
(ϟ) occur at the proper places of those letters in the alphabet,
yet the last sign #5
whether it represent
the Phoenician shin
) or tzade,
occurs, in either case, out of its place and is clearly resumed into the
alphabet for arithmetical purposes. But if we consider the difficulty of
reviving a long-forgotten letter at all, and remember that ς
and ϟ occur in their proper order, we
should conclude that the Greek numerical alphabet, if it was settled by
custom only, was settled at a very early time indeed, possibly before the
Hebrew. It is even conceivable that the non-Phoenician letters, ϝ, θ, χ, ψ, ω,
were originally invented for
purely arithmetical purposes, and were afterwards adopted as alphabetical
But against these suppositions there is a most formidable array of facts. In
the first place, the inscriptions at present known do not disclose the
existence, for literary purposes, of so full an alphabet as that used in
numeration. There is none in which both ϝ
and ϟ occur side by side [p. 2.73]
with both i and
(See the charts appended to
Kirchhoff, Zur Gesch. des Griech. Alph.
3rd edit., 1877, and
pp. 157-160 of the text. The transcript in Hicks, Gr. Inscr.
No. 63, p. 117 sqq.,
is misleading. The original in
1871, p. 39 sqq.,
contains neither μ
) Secondly, the common alphabetic
numerals do not appear on inscriptions proper (exclusive, that is, of coins
and MSS. to be mentioned presently) before the 2nd century B.C., and, among these, only on the Asiatic. The
oldest specimen is probably one of uncertain place (printed in C. I.
vol. iv. pt. xxxix. No. 6819), which is assigned by E.
Curtius to about 180 B.C. (Franz, Epigr. Gr.
p. 349, cites,
as oldest, one of Halicarnassus, C. I. G.,
No. 2655, which
Boeckh thinks to be little earlier than the Christian era.) A (not yet
published) Rhodian inscription in the British Museum, assigned to about the
same time, still uses the Herodianic signs. It should be added, also, that
the earliest Asiatic inscriptions, which contain alphabetic numerals,
arrange them generally with the lowest digit first, reversing the usual
order (e. g. μκ, ζκ,
&c. in No. 6819
above cited). It has already been mentioned that no Attic inscriptions
before imperial times contain alphabetic numerals at all. (It is, no doubt,
purely accidental that #5
does not occur in
any inscription: Franz, Epigr. Gr.
p. 352.) It may be
admitted that public inscriptions would be the last place into which a new
system of numerals would force its way, but it is hardly likely that the
Herodianic signs would have survived in public documents several centuries
after the alphabetic had come into general use among merchants, &c.
Thirdly, the earliest numerical or quasi-numerical use of the Greek
alphabet, of which we can be quite sure, is not the same as that now in
question. The tickets of the 10 panels of Athenian heliastae
were marked with letters from α
(Schol. to Ar. Plut.
No. 119, p. 202; Franz, Epigr.
p. 349.) The books of Homer, as divided by Zenodotus, are headed
with the 24 letters of the Ionic alphabet, omitting ς
and ϟ. The books of the Ethics,
of Aristotle are numbered in the
same way; and that this division is ancient is evident from Alex.
Aphrodisiensis, who (in Metaph.
9, 81 b, 25) quotes from
ζ̓ τῶν Νικομ.
a series of definitions
which are now found in the 6th book. It should be mentioned finally, to
complete the perplexity of the subject, which, considering its importance,
has been strangely neglected, that there is no evidence (it would, of
course, be hard to find) of a time when a short alphabet was used as far as
it would go, and the remaining hundreds were represented by double letters
or Herodianic signs; nor any evidence of fluctuation in the value of the
letters. ϟ for instance, might be expected to have sometimes its
Semitic value 100, instead of 90, or Σ
might occasionally represent 100, instead of Ρ.
The Greek inscriptions already collected are so numerous that the statements
here made are not likely ever to want correction in any important detail.
The fact, at present indisputable, to which they point, is that alphabetic
numerals do not appear at all until long after ς
and ϟ had disappeared from the literary alphabet, and
that these letters are nevertheless used, and used in their right places,
for numeration. The revival of these letters and of #5
implies, under the circumstances, a degree of antiquarian
learning such as cannot be attributed to the public at large. It looks like
the work of some scholar, backed by the influence of paramount political
authority. It will be conceded that Alexandria is the most likely place, in
the first three centuries B.C., to find kings and
scholars in co-operation, and to find some mutual influence of Greek and
Semitic literary usages. It remains only to add, what has been reserved for
this place, that by far the most ancient and certain evidence of alphabetic
numeration comes from Egypt under the first Ptolemies. The oldest
Graeco-Egyptian papyrus (at Leyden, No. 379: v. Robiou, quoting Lepsius, in
Acad. des Inscr. Suj. div.,
1878, vol. 9), which is dated
257 B.C., contains the numerals κθ̓
(=29). Still earlier evidence is furnished by
coins, especially a great number of Tyrian coins of Ptol. II. Philadelphus,
assigned to 266 B.C. (The κ
on some coins of
Ptol. I. Soter, and the double signs AA, BB, &c., on those of
Arsinoë Philadelphi, are of doubtful signification.) From this time
onwards the evidence of Ptolemaic coins and papyri is abundant. It is not
unreasonable to suppose that the ordinary Greek alphabetic numeration was
first used at Alexandria on coins, for which its brevity, its sole
advantage, would make it especially useful. Jewish usage may have suggested
it or been suggested by it; but, however that may be, Alexandrian commerce
and the fame of Alexandrian learning would be sufficiently potent agencies
to disseminate the new system throughout the Hellenic East.
Before proceeding to exhibit the Greek use of alphabetic numerals, it will be
well here to mention briefly two facts, of some interest in themselves,
which need not further concern us. Heilbronner, in his Historia
(pp. 735-737), cites from Hostus, who refers to
Noviomagus, a system of numeral signs in which arms, as it were, are
attached to a central line according to a fixed plan, which may best be
exhibited by an example.
is 1111 in the vertical form, and similarly for the other numbers:
is 9999. The work of Noviomagus has been at last identified by
p. 12) as De
libri ii. (Cologne, 1539) book i. ch. 15. This style is
said by Noviomagus to be used by “Chaldaei et astrologi.” It
was known to John of Basingstoke, who learnt it in Athens about 1240, and is
described by Matthew Paris (Chronica,
ed. Luard). Secondly, Greek arithmetic has no cipher. The 0 which Delambre
i. p. 547, ii. pp. 14, 15) found in the
Almagest is a contraction of οὐδέν,
occurs only in the measurements of angles which contain no degrees or no
minutes. It stands, therefore, always alone, and is not used as a digit of a
high number. The stroke which Otfried Muller found on an Athenian
inscription, and which Boeckh thought to be a cipher, is clearly explained
by Cantor as the iota,
the customary sign of
10. (See Cantor, Math. Beitr.
p. 121 sqq.,
and pl. 28; Nesselmann, p. 138, n. 25; Hultsch,
Scriptores Metrol. Graeci,
Vorrede, [p. 2.74]
pp. v., vi.; Friedlein, p. 82.) The numerical values attributed
to each letter in the Greek alphabet are stated in every Greek grammar.
Suffice it here to say that the letters α′--θ′,
for 6, represent the units, ι′
--ϟ′ the tens,
the hundreds. For the thousands the alphabet
recommences, but the stroke or acute accent which marks the numerical use of
a letter is now placed in front of the letter, and rather below it, so that
1000--9000. For 10,000 Μυ
or M, the initial
letter of μύριοι,
was generally used on the
Herodianic principle; and with multiples of 10,000 the coefficient might be
placed before, after, or over this M. If the co-efficient were placed first,
the M was sometimes omitted and a dot substituted. Other devices appear in
MSS., e. g. [υπριμε]ι,
for 20,000 in Geminus, or
&c. (See Hultsch,
Metrol. Script. Rellig.,
vol. i. pp. 172, 173; and
Ritschl, op. cit.
p. 120; Nicomachus, ed. Hoche,
Introd. p. x.) In the case of high numbers, accents were usually omitted and
a stroke was drawn over all the component letters (cf. C. I.
vol. iii. Nos. 60 and 77, for the two styles); and as these were
arranged in the modern order, with the highest on the left and the lowest on
the right, the distinguishing mark of the thousands was also often omitted
and the value of the letter was indicated by its place, e. g. βτε
is 2305. The improved nomenclatures invented
by Archimedes (in the ψαμμίτης
Apollonius (exhibited by Pappus, Math. Coll.,
bk. ii.) may
have been originally accompanied by improved symbolisms, but no trace of
them now remains.
The representation of fractions (λεπτά
MSS. is also various, but the most common methods are either to write the
denominator over the numerator, or to write the numerator once with one
accent and the denominator twice with two acdents, e. g.
or ιζ′ κα″
Fractions of which the numerator is unity (
“sub-multiples,” as they are sometimes called) are the most
common. With these the numerator is omitted and the denominator is written
above the line, or is written once with two accents. (See for special
details Nesselmann, pp. 112-115; Hultsch, op. cit.,
vol. i. pp. 172-175; Friedlein, Zahlzeichen,
pp. 13, 14.)
Special signs for 1/2 ,
C′ and S, and for 2/3 , ω″,
are found. Brugsch (Numerorum Demot.
Berlin, 1849, p. 31) gives, on the authority of Greek
papyri, the signs
for subtraction, and
for a total. (See the plate appended to Friedlein, op. cit.,
and references there given.)
The later mode of representing numbers on the fingers seems to have been the
same among the Romans as the Greeks. The best known reference is Juvenal,
10.248 (where see Prof. Mayor's note). The oldest is possibly Plautus,
2.3, “dextera digitis rationem
computat,” but the meaning of this is not very clear. Pliny indeed
34.16) says that Numa set up a statue of Janus
with the fingers so arranged as to represent 355, the number of days in a
year (cf. Macrob. Conviv. Sat.
--The Romans used at least two forms of
abacus, one in which buttons (claviculi
) moved in
), another in which the stones were
loose. A drawing of a very elaborate abacus of the first kind is given by
Friedlein in Zeitschr. f. Math. und Phys.,
1864, vol. iv. pl.
v. (cf. Zahlz.
p. 22). It is capable of representing whole
numbers up to 999,999, all fractions with 12 for denominator, and some
others. It employs 45 buttons in 19 grooves. Seven vertical grooves at the
bottom of the instrument contain four buttons each, those in the left-hand
groove representing a million, the values descending towards the right down
to the units. Opposite these grooves, at the top of the board, are seven
smaller grooves, containing 1 button each, representing 5,000,000, 500,000,
&c., down to 5. The eighth lower groove contains 5 buttons, each
representing 1/12; the eighth upper groove contains 1 button, representing
6/12. Three grooves at the side contain a button for 1/24 at the top,
another for 1/48 in the middle, and two for 2/72 at the bottom of the board
respectively. It is possible also that some abaci
had balls moving on wires or strings, similar to those
still used in schools. In these, of course, the lines would be held
horizontally, and not vertically. The so-called Pythagorean abacus, with its
is not mentioned by any
writer of classical times. The MSS. of the Geometria,
attributed to Boethius, in which it is first
described, cannot be considered earlier than the 11th century, and no trace
of any such abacus appears elsewhere before the 9th century. It need not
therefore be discussed in this article (v.
(3.) Written characters.
--There are some signs that the Romans
occasionally used their alphabet for numerical purposes; but the practice
was neither general nor reduced to any fixed rule, and the dates of our
authorities for it, where known, are all late. Some verses on the subject
appear, with slight variations in several MSS. One version of them is given
by Noviomagus in the work De Numeris,
mentioned (lib. i. cap. 10). It begins:--
Possidet A numeros
Atque trecentos B per se retinere
Non plus quam centum C litera fertur
habere, Litera D velut A quingentos
(See Friedlein, Zahlz.
pp. 20, 21.) But it is unlikely that an
alphabet so short and so capable of disturbance as the Roman certainly was,
could ever have been used, in the Greek manner, for numerical purposes.
The ordinary Roman numerals are too well known, and are still in too common
use, to require detailed exhibition. The well-known theory that 10 was
represented by two strokes (X), 100 by three (C), and 1000 by four (M), and
that V, L and [D ] or D are the halves of these signs (Nesselmann, pp.
89, 90; Key's Latin Grammar,
§ 251), has the
advantage of symmetry, but does not account for the more ancient forms of
these symbols. (See the plates appended to Friedlein, op.
and Cantor, Vorles. Math.
) The more common
theories of recent times are that L, C, and M or [Psi] are corruptions
(the Chalcidian form of χ,
written ⊥), Θ
while X is referred
either to [otimes], the old form of Θ,
or to the Greek X, so that all these signs would be adopted from the letters
of the Greek alphabet, for which the Romans had no use. [p. 2.75]
(See Ritschl in Rhein. Mus.
xxiv. p. 12; and Mommsen, Unter It. Dial.
pp. 19-34; Roby,
App. D, ii.; Friedlein, op.
p. 27.) The objections to this theory are, of course, that the
proposed letters are not used in their Greek order, and that the Romans and
Etruscans used, in conjunction with these very signs, a wholly peculiar mode
of representing intervening numbers. Such forms as IX, XL, XC, are so
original, as to suggest the originality also of the signs of which they are
compounded. (Still stranger forms, as XIIX for 18, are also found:
Friedlein, p. 32; Corssen, Etrusker,
A few of the more uncommon Roman numerals should be here mentioned. The sign
for 1000 being [Psi] (not M till post-Augustan times: Mommsen, op. cit.
p. 30), that for 10,000 was ([Psi]),
and that for 100,000 (([Psi])); but the ordinary sign for a million was
[drachm1000000], and any higher multiple of 100,000 was similarly
enclosed with side and top lines. But the repetition of [Psi] and the
other signs above given being found cumbrous, it was usual, with intervening
multiples of 1000, to write the coefficient with a stroke over it, or with
or. merely M appended, e. g. XIIDC,
or XII milia
DC or XIIMDC. (Cf. Friedlein,
pp. 28-31, where the forms attributed to Pliny are
specially discussed; and Marquardt, Röm. Alt.
p. 32, and 5.1, p. 98, notes 161 and 522.) Other forms are, found, but it is
to be remembered that MSS. are not safe guides to the usages of classical
times. The form
for 6, for instance, is not uncommon in MSS., but is not attested
by any coins or inscriptions older than the 6th century (Friedlein, p. 33).
The fractions generally used by the Romans were the divisions of the as
be remembered that the as
was, for all purposes, the
type of unity. Thus Balbus (ad
Celsum de Asse,
1) says, “Quidquid unum est, assem
ratiocinatores vocant” (cf. Marquardt, 3.2, pp.: 42-44), and the
fractions of the as
are applied to divisions of any
kind of magnitude. Livy (5.24
) has “terna jugera et septunces” and (6.16, 6)
“bina jugera et semisses agri.” Columella and the gromatici
(ed. Lachmann, &c., Berlin, 1848)
use the same terms for divisions of time or length. (Cf. Varro, de R.
1.10; Friedlein, pp. 34, 35; Roby, Lat. Gram.
i. App. D, vi.-xiii.) The names of the divisions of the as
from deunx to uncia,
i.e. from 11/12 to
1/12, are set out below in the Appendix, Table XIII. Those of the uncia
are given in Table XIV. It may be mentioned,
however, in this place that scrupulum
very often called scrupulus
and that the book De
of the 3rd century gives, besides duella,
the unusual fractions drachma
( 1/8 ), tremissis
(1/16), and the name hemisescla
or dimidia sextula
(Friedlein, p. 41). Other
fractions were, of course, expressible (e.g. quattuor
&c.), and after the time of Constantine new terms appear as
translations of Greek or adaptations of older Roman names (e.g. superdimidius, supertertius,
Friedlein, pp. 41-43, 97, 98), but the divisions of the as
given in the Appendices
are the only fractions for which special signs are found. The signs from
are merely arrangements of horizontal strokes or dots, as., :,
is represented by Σ,
and from this to deunx
the signs are Σ,
with those for
&c., added to the right of
it. Then as
is an upright stroke 1. The signs below
are usually semuncia,
L or Ε
\, ~ or 2, dimidia
[CI] or χ,
[CF] or [CI][CI] (Cf.
Bede, De Ratione Assis,
Opera, Basileae, i.
col. 182.) Much fuller tables are given in Friedlein, plates 13-15, and the
forms applicable to divisions of the denarius
are set out in Roby, Lat. Gram.
i., App. D, viii. It is
possible in this place only to mention the most common and interesting facts
and to refer to the authorities who treat the subject in detail. The reader
cannot expect here an adequate commentary on Frontinus or Victorius.
beside the references already given,
Hankel, Zur Gesch. der Math.
pp. 56-63; Cantor,
It has been already remarked that fingersigns are of no practical assistance
to calculation save as a mode of representing a sum, difference, product,
&c., and so relieving the memory to some extent in the processes of
mental arithmetic. The actual work of calculation was done with the abacus
or with written signs. Addition and
subtraction were always done with the former. So also were multiplications
and divisions, where the multiplier or divisor was a low number, but as a
general rule multiplication was done with written signs, and division by
both methods together. The schemes of addition and subtraction set out by
Nesselmann (p. 119) are without authority, and it is to be remarked that it
was in multiplication only that the ancients approached at all nearly to the
modern facility of using written signs (cf. Friedlein, pp. 26 and 74).
) and subtraction (ἀθαίρεσις
) seem to have involved generally some mental
arithmetic, for apparently on the ordinary abaci
only one number of several digits could be represented at a
time. The practice probably was to set out one of two numbers to be added,
to add the other mentally and set out the sum (κεφάλαιον
), removing or adding to the ψῆφοι
previously arranged as the calculation progressed.
(This perhaps is what Herodotus alludes to in 2.36.) Some abaci,
however, notably the Salaminian table (see Cantor,
i. pp. 111, 112), have two sets of columns at
opposite ends of the board. It is supposed by Cantor that these columns were
used by two different persons--a banker, for instance, and his customer; but
it may also be suggested that the two sets are intended for the
representation of two numbers in an addition or subtraction. Multiplication
was sometimes effected by repeated additions (cf. Lucian, Ἑρμότιμος,
48); but the process, even where
the multiplier is low, is very cumbersome when the multiplicand is high, and
some sort of a multiplication table must early have been compiled. The
fullest specimens of Greek arithmetic which we possess are a great number of
multiplications set out by Eutocius of Ascalon in his notes to Archimedes
Torelli's ed., pp. [p. 2.76]sqq.
). One of these, which is rendered in
modern figures by Nesselmann (p. 118), and with some improvements by
Friedlein (p. 76), may be here given. It is the more interesting because it
involves fractions. (The letter k
is used here
instead of the Greek sign for 1/2 .) The modern figures are given at the
Μ Μ[υπριμε]θ [υπριμε]αφ
||1 1/2 ,
|| 1/2 ,
|| 1/4 .
||6 1/2 ,
|| 1/4 ,
|| 1/8 ,
||3 1/4 ,
|| 1/8 ,
Μ [υπριμε]βχπθ ι#2′.
(Cf. also Delambre, Astr. Anc.
vol. ii. ch. 1.) The reader
sees that the process begins by taking the highest multiple of 10 in the
multiplier and multiplying therewith all the digits of the multiplicand,
beginning on the left. The second digit of the multiplier is then taken, and
so on. The treatment of the fractions should be observed. Two other very
interesting examples, taken from Heron's Geometica
Hultsch, pp. 81 and 110), are also given by Friedlein (p. 77). In the first
of these the process involves the multiplication of 63/64 by 2/62. The
product is given in the form 126/64.1/64, reduced to 1/64 + 62/64.1/64, and
is there left. (καὶ ξγ́ξδ́ξδ́ τωο̂ δύο ξδ́ ξδ́
ρκσ´ξδ́ξδ́ τῶν ξδ́ξδ́, γινόμενα καὶ ταῦτα ἑξηκοστοτέταρτον ά
καὶ ξβ́ξδ́ξδ́ τῶν ξδ́ξδ́.
The nearest approach to modern multiplication with Indian numerals is made by
Apollonius, according to the extracts preserved by Pappus in his 2nd Book
above mentioned. Apollonius recommends that with all multiples of 10 the
co-efficients alone (πυθμένες
) should be
multiplied first, and the tens or powers of ten multiplied afterwards. But
this method, as we have said, does not seem to have been accompanied by a
new symbolism, and is strictly confined to multiples of 10, with no added
units. It was accompanied by a new nomenclature, similar to that of
Archimedes, according to which numbers from 1-9999 belonged to the first
group (μυριάδες ἁπλαῖ
to the second group (μυριάδες διπλαῖ
so on, so that a certain simplicity of description was gained; e. g.
1,0001,0001 would be described as α
third group + α
of the second + α
of the first (cf. Nesselm. p. 127, and Papp.
2.27). But the invention seems, like that of Archimedes, to have been
sportive chiefly, and is certainly illustrated only by the multiplication of
the numbers symbolised by all the letters in the two lines-- Ἀρτέμιδος κλεῖτε κράτος ἔξοχον, ἐννέα
and Μῆνιν ἅειδε θεὰ Δημήτερος
Eutocius, however (ad
loc. cit.), speaks of the ὠκυτόκιον
of Apollonius (MSS. ῶκυτόβοον:
the emendation was originally Halley's) as a
great aid to multiplication. This was possibly a “ready
reckoner,” or table of calculated products. It is difficult to see
how, as Cantor suggests (Vorles.
pp. 298, 387), it can have
been connected with the new classification of numbers described by Pappus
(cf. Nesselmann, pp. 126-135).
No example of the division of wnole numbers occurs with the working-out in
any Greek author. It is obvious, however, from the expressions used and the
mode in which remainders are stated, that the practice was to take a
multiple of the divisor and subtract it from the dividend; then take another
multiple of the divisor and subtract it from the first remainder, and so on
until the last remainder was less than the divisor. The series of quotients
was then added together, and the fractional remainder, if any, was separated
into a series of “submultiples” or fractions with unity for
numerator. Thus Heron (Geom.
ed. Hultsch, p. 56), dividing 25
by 13, sets out the quotient as 1 + 1/2 + 1/3 + 1/13 + 1/78. No name for
“quotient” is found. The customary Greek expression for the
result of a division was that the divisorth part of the
dividend was so and so (Friedlein, p. 79). The theory of the extraction of
square roots is exhibited geometrically by Theon in his commentary to the
first book of the Almagest (ed. Halma, 1821: vide
also Cantor, Vorles.
p. 420; Nesselmann, pp.
108-110; Friedlein, p. 84). The practice, however, as has been said above
under the article ARITHMETICA
), was probably rough and
empirical. The theory of finding a G. C. M. or a L. C. M. is exhibited in
, and 36,
38. Compound divisions, in which the divisor and dividend contain degrees,
minutes and seconds, are given by Theon in his commentary to Ptolemy before
mentioned. (Nesselmann, pp. 142-144.) The following example is selected by
Friedlein (p. 83):--1515° 20′ 15″ is to be
divided by 25° 12′ 10″. The first quotient 60 is
found by trial. Then 60.25°=1500°.
1515°--1500° = 15° = 900′: 900′
+ 20′ = 920′: 60 . 12′ = 720′:
920′--720′ = 200′: 60.10″ =
10′. 200′ 15″--10′ = 190′
15″. The next quotient is 7′. Then
25°.07′ = 175′, 190′--175′ =
15′ = 900″: 900″ + 15″ =
915″: 12′.7′ = 84″.
915″--84″ = 831″: 10″.7′ =
70‴ = 1″ 10‴: 831″--1″
10‴ =829″ 50‴. The last quotient
33″ is a little too high, but is adopted by Theon as near enough
for his purpose. The mode of multiplication and subtraction need not be
further exhibited. The final quotient is 60° 7′
No method of extracting cube-roots is mentioned in any Greek writer, and such
an operation would, in any case, belong more to ἀριθμητικὴ
than to λοψιστική.
--Of the methods of calculation in
use among the Romans even less is known than the little which is
discoverable of Greek logistic.
What is certain is
that the Roman abacus
was adapted to higher needs
than the Greek, and that it was used in very complicated calculations (cf.
Columella, de Re Rust.
iii. p. 115; Friedlein,
pp. 88-90, and plate 21 shows the use of the abacus
above described for various purposes of
elementary calculation). The Calculus
Victorius, written in the 5th century of our era, is a ready-reckoner of
sums, [p. 2.77]
differences, products, quotients, and
reductions of extraordinary fulness (v.
pp. 93 sqq.,
and Appendix). The existence of such a
book, which provides answers to questions of great simplicity as well as to
the more difficult, seems to show that the Romans were not more adept at
arithmetic than the Greeks. The passages of Roman writers which refer
incidentally to calculations, deal almost entirely with fractions. We may
guess from Horace (A. P.
327-330) how long a time was spent
in schools in learning by heart the divisions of the as
and the differences between them. We may gather from Pliny (Plin. Nat. 6.38
) how inexact the treatment of
fractions was, and yet how difficult were the problems attempted. This
latter passage is very neatly explained by Friedlein (p. 90), whose note may
be here given. Europe, says Pliny, is rather less than 1 1/2 of Asia and 2
1/6 of Africa. It follows that ( “si misceantur omnes summae” )
Europe is rather more than 1/3 + 1/8 , Asia 1/4 + 1/16 (reading sexta decima
), Africa 1/5 + [frac160], of the whole earth. If
be the earth, E
Asia, and Af
Africa, then T = E
+ As + Af. As
= 2/3 E. Af
= 6/13 E:
= (1 + 2/3 + 6/13)
= (5/3 + 6/13) E
= 8 3/39 E.
= 39/83 T
= 117/83.3 = 83/83.3 +
34/83.3 = almost 1/3 + 1/8 , &c. It will be observed that the mode
of treating the fractions is exactly similar to the Greek. The treatment of
divisions of the as
and other monetary fractions is,
of course, far simpler, because here both numerators and denominators are
strictly limited, and the terms themselves suggest by their definition the
mode of calculating with them. Similarly any English boy, in dealing with
pounds, shillings, and pence, soon perceives that the admissible fractions
of a pound are limited to 1 9/20 or 23 9/240, of a shilling to 11/12. The
methods of arithmetic in use in the Roman empire from the time of Boethius
to that of Planudes are exhaustively discussed by Friedlein in the work
Zahlzeichen, etc. der Griechen und Römer und des
Christlichen Abendlandes vom
of which frequent use has been made in this
article. But these methods cannot be said to belong to classical antiquity;
and, if they did, they could not be conveniently summarised in this place.
We have attempted here no more than to give such facts with regard to Greek
and Roman arithmetic as are of importance to the interpretation of the
authors most generally read, to the criticism of inscriptions, or to a due
conception of ancient life and manners.