#### LOGI´STICA

LOGI´STICA [ sc. Plat. Gorg. 450 D, &c.: the nearest Latin equivalents appear to be (cf. Cic. Tusc. 1.2, 5), dinumeratio (Id. Rep. 3.2, 3), (cf. iii. in Smith's Lat. Dict.), or computatio (post. Aug.）] means “the art of calculation” as opposed to the “theory of numbers,” arithmetica (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.

a.) Greek.--(1.) Finger-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. Pott, Zählmethode, &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 (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, 65; Arist. Problem. xv.), and no doubt this simple practice was never lost. But the references to fingerreckoning in literature are very scanty until a late date (Plut. Apophth. 174 b; D. C. 71.32.1; 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. Gesellsch. 1845, pp. 111-129.) This is fully described by Nicolaus Smyrnaeus (called also Rhabda or Artabasda) in a work entitled written probably in the 13th or 14th century, and printed by N. Caussinus in his book (lib. ix. ch. viii. pp. 565-568, Paris, 1636; also in Schneider's Eclog. Physic. 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 (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 in 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, Primit. Culture, i. p. 246, n.).

b) Pebble-signs--Under this head may be included all the representative signs used with the reckoning-board, abacus, ἄβαξ or ἀβάκιον (q.v.). 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, Charikles, 2.67 ff.; and see below under Roman “pebblesigns,” p. 74.) [ABACUS]

c.) Written Characters.--Iamblichus 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, Epig. Graeca, 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. 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 [drachm10] for δέκα, [drachm100] for [drachm1000] for [drachm1000] for with further compendia, [drachm50] for 50, [drachm500] for 500, &c. (See C. I. A. vols. i. and ii.; or Hicks, Gr. Inscr. passim; or Boeckh, Att. Seewesen, p. 547 sqq. &c. For curious Boeotian variations, see Franz, Epig. Graec. App. II. 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 [multi] ε gave ϝ′, and ρ [multi] ε gave θ́, 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 Griechen, 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, Phönihische Spr., 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 Chaldéens, p. 24: cf. also Rev. 13.18, and Dr. Ginsburg's art. Kabbalah in Encycl. Brit., 9th 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 forms of which (cf. Greek ς and 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, ϟ, and #5, 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 no doubt represents the old Vau (ϝ), 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 (Gr. σάν, Hdt. 1.139) 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 signs.

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 ψ. (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 Rhein. Mus. 1871, p. 39 sqq., contains neither μ nor ω.) 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. G. 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 to κ, omitting ς. (Schol. to Ar. Plut. 277; Hicks, Inscr. No. 119, p. 202; Franz, Epigr. Gr. 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, Politics, and Topics 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 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 Matheseos (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.

 ZZZ

or

 ZZZ

is 1,

 ZZZ

or

 ZZZ

is 10,

 ZZZ

or

 ZZZ

is 100,

 ZZZ

or

 ZZZ

is 1000,

 ZZZ

is 1111 in the vertical form, and similarly for the other numbers: e. g.

 ZZZ

is 7744,

 ZZZ

is 7766,

 ZZZ

is 9999. The work of Noviomagus has been at last identified by Friedlein (Zahlz. p. 12) as 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 ( 5.285, ed. Luard). Secondly, Greek arithmetic has no cipher. The 0 which Delambre (Astron. Anc. i. p. 547, ii. pp. 14, 15) found in the Almagest is a contraction of and 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′, including ς for 6, represent the units, ι--ϟ′ the tens, ρ--#5′ 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 [υπριμε]α-[υπριμε]θ represent 1000--9000. For 10,000 Μυ or M, the initial letter of 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. for 10,000, [υπριμε]κ for 20,000 in Geminus, or , β̔́, &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. A., 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) and 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) in 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.

 κα ιζ

or

 κα′ ιζ

or ιζκακα″. 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 ,

 ZZZ

or

 ZZZ

C′ and S, and for 2/3 , ω″, are found. Brugsch (Numerorum Demot. Doctr., Berlin, 1849, p. 31) gives, on the authority of Greek papyri, the signs

 ZZZ

 ZZZ

for subtraction, and

 ZZZ

for a total. (See the plate appended to Friedlein, op. cit., and references there given.)

(b.) --(1.) Finger-signs. 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, Mil. Glor. 2.3, “dextera digitis rationem computat,” but the meaning of this is not very clear. Pliny indeed (H. N. 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. 1.9).

(2.) Pebble-signs.--The Romans used at least two forms of abacus, one in which buttons (claviculi) moved in grooves (alveoli), 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 accompanying is not mentioned by any writer of classical times. The MSS. of the 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. Friedlein, Zahlz. pp. 22-27).

(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 already mentioned (lib. i. cap. 10). It begins:--

(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. cit., and Cantor, Vorles. Math.) The more common theories of recent times are that L, C, and M or [Psi] are corruptions of Ψ (the Chalcidian form of χ, written ⊥), Θ and Φ, 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. 1869, xxiv. p. 12; and Mommsen, Unter It. Dial. pp. 19-34; Roby, Lat. Gram. App. D, ii.; Friedlein, op. cit. 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, 1.39-41.)

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, Zahlz. pp. 28-31, where the forms attributed to Pliny are specially discussed; and Marquardt, Röm. Alt. 3.2, 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

 ZZZ

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 and It should be remembered that the as was, for all purposes, the type of unity. Thus Balbus (ad Celsum 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, 5) 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. 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 is also very often called scrupulus and and that the book of the 3rd century gives, besides the unusual fractions drachma ( 1/8 ), tremissis (1/16), and the name hemisescla for semisextula or (Friedlein, p. 41). Other fractions were, of course, expressible (e.g. sc. &c.), and after the time of Constantine new terms appear as translations of Greek or adaptations of older Roman names (e.g. superdimidius, supertertius, &c., for &c.: Friedlein, pp. 41-43, 97, 98), but the divisions of the as and uncia given in the Appendices are the only fractions for which special signs are found. The signs from uncia to quincunx are merely arrangements of horizontal strokes or dots, as., :,

 ZZZ

,

 ZZZ

,

 ZZZ

. Semis 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 uncia are usually L or Ε or L, sicilicus [CC], sextula \, ~ or 2, [CI] or χ, scriptulum [CF] or [CI][CI] (Cf. Bede, 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. (Vide, beside the references already given, Hankel, Zur Gesch. der Math. pp. 56-63; Cantor, Vorles. p. 445.)

II. CALCULATION.

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).

(a.) Greek.--Addition (σύνθεσις) 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 however, notably the Salaminian table (see Cantor, Vorles. 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, 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 (Circ. Dimens., 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 side.

 k δ′ 3013 1/2 1/4 k δ′ 3013 1/2 1/4

 #5 γ 9000000, 39000, 1500, 750. γ 30000, 130, 5, 2 1/2 . [uprime]q lq ak κδ′ 9000, 39, 1 1/2 , 1/2 , 1/4 . k δ′ η′ 1500, 6 1/2 , 1/4 , 1/8 , ψν γδ′ η′ ι#2′ 750, 3 1/4 , 1/8 , 1/16. #5η Μ [υπριμε]βχπθ ι#2′. 9082689 1/16.

(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 (ed. 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. (

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 (), 10,000-9999,9999 to the second group (), and so on, so that a certain simplicity of description was gained; e. g. 1,0001,0001 would be described as α of the 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--

and

Eutocius, however (ad Arch Circ. Dim. loc. cit.), speaks of the ὠκυτόκιον of Apollonius (MSS. 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 (q. v.), was probably rough and empirical. The theory of finding a G. C. M. or a L. C. M. is exhibited in Euclid, 7.2, 3, 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′ 33″.

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

(b.) --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, iii. p. 115; Friedlein, Zahlz. pp. 88-90, and plate 21 shows the use of the abacus above described for various purposes of elementary calculation). The Calculus of 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. Friedlein, 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 for ), Africa 1/5 + [frac160], of the whole earth. If T be the earth, E Europe, As Asia, and Af Africa, then T = E + As + Af. As = 2/3 E. Af = 6/13 E: therefore T = (1 + 2/3 + 6/13) E = (5/3 + 6/13) E = 8 3/39 E. Therefore 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 7. bis 13. Jahrhundert, 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.

[J.G]

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