A History of Mathematical Notations/Volume 1/Romans

by Florian Cajori

ROMANS

40. We possess little definite information on the origin of the Roman notation of numbers. The Romans never used the successive letters of their alphabet for numeral purposes in the manner practiced by the Syrians, Hebrews, and Greeks, although (as we shall see) an alphabet system was at one time proposed by a late Roman writer. Before the ascendancy of Rome the Etruscans, who inhabited the country nearly corresponding to modern Tuscany and who ruled in Rome until about 500 B.C., used numeral signs which resembled letters of their alphabet and also resembled the numeral signs used by the Romans. Moritz Cantor^[1] gives the Etrurian and the old Roman signs, as follows: For 5, the Etrurian 𐌡 or 𝖵, the old Roman V; for 10 the Etrurian 𐌢 or +, the old Roman X; for 50 the Etrurian 𐌣 or ↆ, the old Roman Ψ or ↆ or ⫝ or Ʇ or L; for 100 the Etrurian ⊕, the old Roman ⊝; for 1,000 the Etrurian 𐌚, the old Roman ↀ. The resemblance of the Etrurian numerals to Etrurian letters of the alphabet is seen from the following letters: 𐌖, 𐌗, 𐌣, 𐌏, 𐌚. These resemblances cannot be pronounced accidental. “Accidental, on the other hand,” says Cantor, “appears the relationship with the later Roman signs, I V, X, L, C, M, which from their resemblance to letters transformed themselves by popular etymology into these very letters.” The origins of the Roman symbols for 100 and 1,000 are uncertain; those for 50 and 500 are generally admitted to be the result of a bisection of the two former. “There was close at hand,” says G. Friedlein,^[2] “the abbreviation of the word centum and mille which at an early age brought about for 100 the sign C, and for 1,000 the sign ʍ and after Augustus^[3] M.” A view held by some Latinists^[4] is that “the signs for 50, 100, 1,000 were originally the three Greek aspirate letters which the Romans did not require, viz., Ψ, ⊙, ↀ, i.e., χ, θ, ϕ. The Ψ was written Ʇ and abbreviated into L; ⊙ from a false notion of its origin made like the initial of centum; and ↀ assimilated to ordinary letters CIↃ. The half of ↀ, viz., D, was taken to be ½ 1,000, i.e., 500; X probably from the ancient form of ϴ, viz., ⊗, being adopted for 10, the half of it V was taken for 5.”^[5]

47. Our lack of positive information on the origin and early history of the Roman numerals is not due to a failure to advance working hypotheses. In fact, the imagination of historians has been unusually active in this field.^[6] The dominating feature in the Roman notation is the principle of addition, as seen in II, XII, CC, MDC, etc.

48. Conspicuous also is the frequent use of the principle of subtraction. If a letter is placed before another of greater value, its value is to be subtracted from that of the greater. One sees this in IV, IX, XL. Occasionally one encounters this principle in the Babylonian notations. Remarks on the use of it are made by Adriano Cappelli in the following passage:

“The well-known rule that a smaller number, placed to the left of a larger, shall be subtracted from the latter, as ↀIↃↃ=4,000, etc., was seldom applied by the old Romans and during the entire Middle Ages one finds only a few instances of it. The cases that I have found belong to the middle of the fifteenth century and are all cases of IX, never of IV, and occurring more especially in French and Piedmontese documents. Walther, in his Lexicon diplomaticum, Gettingen, 1745–47, finds the notation LXL=90 in use in the eighth century. On the other hand one finds, conversely, the numbers IIIX, VIX with the meaning of 13 and 10, in order to conserve, as Lupi remarks, the Latin terms tertio decimo and sexto decimo.”^[7] L. C. Karpinski points out that the subtractive principle is found on some early tombstones and on a signboard of 130 B.C., where at the crowded end of a line 83 is written XXCIII, instead of LXXXIII.

49. Alexander von Humboldt^[8] makes the following observations:

“Summations by juxtaposition one finds everywhere among the Etruscans, Romans, Mexicans and Egyptians; subtraction or lessening forms of speech in Sanskrit among the Indians: in 19 or unavinsati; 99 unusata; among the Romans in undeveginti for 19 (unus de viginti), undeoctoginta for 79; duo de quadraginta for 38; among the Greeks eikosi deonta henos 19, and pentekonta düoin deontoin 48, i.e., 2 missing in 50. This lessening form of speech has passed over in the graphics of numbers when the group signs for 5, 10 and even their multiples, for example, 50 or 100, are placed to the left of the characters they modify (IV and IΛ, XL and XT for 4 and 40) among the Romans and Etruscans (Otfried Müller, Etrusker, II, 317–20), although among the latter, according to Otfried Müller’s new researches, the numerals descended probably entirely from the alphabet. In rare Roman inscriptions which Marini has collected (Iscrizioni della Villa di Albano, p. 193; Hervas, Aritmetica delle nazioni [1786], p. 11, 16), one finds even 4 units placed before 10, for example, IIIIX for 6.”

50. There are also sporadic occurrences in the Roman notations of the principle of multiplication, according to which VM does not stand for 1,000−5, but for 5,000. Thus, in Pliny’s Historia naturalis (about 77 A.D.), VII, 26; XXXIII, 3; IV praef., one finds^[9] LXXXIII.M, XCII.M, CX.M for 83,000, 92,000, 110,000, respectively.

51. The thousand-fold value of a number was indicated in some instances by a horizontal line placed above it. Thus, Aelius Lampridius (fourth century A.D.) says in one place, “CXX, equitum Persarum fudimus: et mox X in bello interemimus,” where the numbers designate 120,000 and 10,000. Strokes placed on top and also on the sides indicated hundred thousands; e.g., |X|CLXXXDC stood for 1,180,600. In more recent practice the strokes sometimes occur only on the sides, as in |X|·DC.XC., the date on the title-page of Sigüenza’s Libra astronomica, published in the city of Mexico in 1090. In antiquity, to prevent fraudulent alterations, XXXM was written for 30,000, and later still CIↃ took the place of M.^[10] According to Cappelli^[11] “one finds, often in French documents of the Middle Ages, the multiplication of 20 expressed by two small x’s which are placed as exponents to the numerals III, VI, VIII, etc., as in IIIIˣˣ=80, VIˣˣXI=131.”

52. A Spanish writer^[12] quotes from a manuscript for the year 1392 the following:

“M
IIII, C
IIII, LXXIII florins” for 4,473 florins.

“III C IIII III florins" for 3,183 (?) florins.

In a Dutch arithmetic, printed in 1771, one finds^[13]

c
𝔦 𝔵𝔵𝔦𝔦𝔧 for 123, c
𝔦 m
𝔵𝔵𝔦𝔦𝔧 c
𝔦𝔦𝔦𝔧 𝔩𝔳𝔧 for 123,456.

53. For 1,000 the Romans had not only the symbol M, but also I, ∞ and CIↃ. According to Priscian, the celebrated Latin grammarian of about 500 A.D., the ∞ was the ancient Greek sign Χ for 1,000, but modified by connecting the sides by curved lines so as to distinguish it, from the Roman X for 10. As late as 1593 the ∞ is used by C. Dasypodius^[14] the designer of the famous clock in the cathedral at Strasbourg. The CIↃ was a I inclosed in parentheses (or apostrophos). When only the right-hand parenthesis is written, IↃ, the value represented is only half, i.e., 500. According to Priscian,^[15] “quinque milia per I et duas in dextera parte apostrophos, IↃↃ. decem milia per supra dictam formam additis in sinistra parte contrariis duabus notis quam sunt apostrophi, CCIↃↃ.” Accordingly, IↃↃ stood for 5,000, CCIↃↃ for 10,000; also IↃↃↃ represented 50,000; and CCCIↃↃↃ, 100,000; (∞), 1,000,000. If we may trust Priscian, the symbols that look like the letters C, or those letters facing in the opposite direction, were not really letters C, but were apostrophes or what we have called parentheses. Through Priscian it is established that this notation is at least as old as 500 A.D.; probably it was much older, but it was not widely used before the Middle Ages.

54. While the Hindu-Arabic numerals became generally known in Europe about 1275, the Roman numerals continued to hold a commanding place. For example, the fourteenth-century banking-house of Peruzzi in Florence—Compagnia Peruzzi—did not use Arabic numerals in their account-books. Roman numerals were used, but the larger amounts, the thousands of lira, were written out in words; one finds, for instance, “lb. quindicimilia CXV (symbol characters) V (symbol characters) VI in fiorini” for 15,115 lira 5 soldi 6 denari; the specification being made that the lira are lira a fiorino d’oro at 20 soldi and 12 denari. There appears also a symbol much like (symbol characters), for thousand.^[16]

Nagl states also: “Specially characteristic is . . . . during all the Middle Ages, the regular prolongation of the last I in the units, as (symbol characters), which had no other purpose than to prevent the subsequent addition of a further unit.”

55. In a book by H. Giraua Tarragones^[17] at Milan the Roman numerals appear in the running text and are usually underlined; in the title-page, the date has the horizontal line above the numerals. The Roman four is IIII. In the tables, columns of degrees and minutes are headed “G.M.”; of hour and minutes, “H.M.” In the tables, the Hindu-Arabic numerals appear; the five is printed (symbol characters), without the usual upper stroke. The vitality of the Roman notation is illustrated further by a German writer, Sebastian Frank, of the sixteenth century, who uses Roman numerals in numbering the folios of his book and in his statistics: “Zimmet kumpt von Zailon .CC.VÑ LX. teütscher meil von Calicut weyter gelegen. . . . . Die Nägelin kummen von Meluza / für Calicut hinaussgelegen vij·c. vnd XL. deutscher meyl.”^[18] The two numbers given are 260 and 740 German miles. Peculiar is the insertion of vnd (“and”). Observe also the use of the principle of multiplication in vij·c. (=700). In Jakob Köbel’s Rechenbiechlin (Augsburg, 1514), fractions appear in Roman numerals; thus, II^C/IIII^C.LX stands for 200/460.

56. In certain sixteenth-century Portuguese manuscripts on navigation one finds the small letter b used for 5, and the capital letter R for 40. Thus, xbiij stands for 18, Ri ij for 43.^[19]

Fig. 15.—Degenerate forms of Roman numerals in English archives (Common Pleas, Plea Rolls, 637, 701, and 817; also Recovery Roll 1). (Reduced.)

A curious development found in the archives of one or two English courts of the fifteenth and sixteenth centuries^[20] was a special Roman numeration for the membranes of their Rolls, the numerals assuming a degraded form which in its later stages is practically unreadable. In Figure 15 the first three forms show the number 147 as it was written in the years 1421, 1436, and 1466; the fourth form shows the number 47 as it was written in 1583.

57. At the present time the Roman notation is still widely used in marking the faces of watches and clocks, in marking the dates of books on title-pages, in numbering chapters of books, and on other occasions calling for a double numeration in which confusion might arise from the use of the same set of numerals for both. Often the Roman numerals are employed for aesthetic reasons.

58. A striking feature in Roman arithmetic is the partiality for duodecimal fractions. Why duodecimals and not decimals? We can only guess at the answer. In everyday affairs the division of units into two, three, four, and six equal parts is the commonest, and duodecimal fractions give easier expressions for these parts. Nothing definite is known regarding the time and place or the manner of the origin of these fractions. Unlike the Greeks, the Romans dealt with concrete fractions. The Roman as, originally a copper coin weighing one pound, was divided into 12 unciae. The abstract fraction was called deuna (= de uncia, i.e., as [1] less uncia [1/1 2]). Each duodecimal subdivision had its own name and symbol. This is shown in the following table, taken from Friedlein, in which S stands for semis or "half" of an as.

as.. deunx. dextans (decunx) dodrans... bes.... septunx. semis... quincunx. triens.... quadrans... sextans .... sescuncia 1.. uncia.. 1 11 ... 2000/204/0 11 1 S S == or S:: S Sor Sor S: 11 1 TABLE 1 - or S ::. S or S. S or S1 or S. 1 . . or = == or = or 1 or :. = or Z or : 12=1-LELEL or or on bronze abacus In place of straight lines occur also curved ones. = or ::. Sor :: ¹ Op. cit., Plate 2, No. 13; see also p. 35. (de uncia 1-1) ((de sextans 1-1) (decem unciae) (de quadrans 1-4) (duae assis sc. partes) (septem unciae) (quinque unciae) Page:A History Of Mathematical Notations Vol I (1928).djvu/57

 

1. Vorlesungen über Geschichte der Mathematik, Vol. I (3d ed.), p. 523, and the table at the end of the volume.
2. Die Zahlzeichen und das elementare Rechnen der Griechen und Römer (Erlangen, 1869), p. 28.
3. Theodor Mommsen, Die unteritalischen Dialekte (Leipzig, 1840), p. 30.
4. Ritschl, Rhein. Mus., Vol. XXIV (1869), p. 12.
5. H. J. Roby, A Grammar of the Latin Language from Plautus to Suetonius (4th ed.; London, 1881), Vol. I, p, 441.
6. Consult, for example, Friedlein. op. cit., p. 26–31; Nesselmann, op. cit., p. 86–92; Cantor, Mathematische Beiträge zum Kulturleben der Völker, p. 155–67; J. C. Heilbronner, Historia Matheseos universae (Lipsiae, 1742), p. 732–35; Grotefend, Lateinische Grammatik (3d ed.; Frankfurt, 1820), Vol. II, p. 163, is quoted in the article “Zahlzeichen” in G. S. Klügel’s Mathematisches Wörterbuch, continued by C. B. Mollweide and J. A. Grunert (Leipzig, 1831); Mommsen, Hermes, Vol. XXII (1887), p. 596; Vol. XXIII (1888), p. 152. A recent discussion of the history of the Roman numerals is found in an article by Ettore Bortolotti in Bolletino della Mathesis (Pavia, 1918), p. 60–66, which is rich in bibliographical references, as is also an article by David Eugene Smith in Scientia (July–August, 1926).
7. Lexicon Abbreviaturarum (Leipzig, 1901), p. xlix.
8. “Über die bei verschiedenen Völkern üblichen Systeme Von Zahlzeichen, etc.,” Crelle’s Journal für die reine und angewandte Mathematik (Berlin, 1829), Vol. IV, p. 210, 211.
9. Nesselmann. op. cit., p. 90.
10. Confer, on this point, Theodor Mommsen and J. Marquardt, Manuel des antiquités romaines (trans. G. Humbert), Vol. X by J. Marquardt (trans. A. Vigié; Paris, 1888), p. 47, 49.
11. Op. cit., p. xlix.
12. Liciniano Saez, Demostración Histórica del verdadero valor de Todas Las Monedas que corrían en Castilla durante el reynado del Señor Don Enrique III (Madrid, 1796).
13. De Vernieuwde Cyfferinge van M^r. Willem Bartjens. Herstelt, . . . . door M^r Jan van Dam, . . . . en van alle voorgaande Fauten gezuyyert door . . . . Klaas Bosch (Amsterdam, 1771), p. 8.
14. Cunradi Dasypodii Institutionum Mathematicarum voluminis primi Erotemata (1593), p. 23.
15. “De figuris numerorum,” Henrici Keilii Grammatici Latini (Lipsiae, 1859), Vol. III, 2, p. 407.
16. Alfred Nagl, Zeitschrift für Mathematik und Physik, Vol. XXXIV (1889), Historisch-literarische Abtheilung, p. 164.
17. Dos Libros de Cosmographie, compuestos nueuamente por Hieronymo Giraua Tarragones (Milan, M.D.LVI).
18. Weltbůch / spiegel vnd bildtnis des gantzen Erdtbodens . . . . von Sebastiano Franco Wördernsi . . . . (M.D. XXXIIII), fol. ccxx.
19. J. I. de Brito Rebello, Livro de Marinharia (Lisboa, 1903), p. 37, 85–91, 193, 194.
20. Antiquaries Journal (London, 1926), Vol. VI, p. 273, 274.