Page:A History Of Mathematical Notations Vol I (1928).djvu/52

32 49. Alexander von Humboldt 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 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, “$\overline{CXX}$, equitum Persarum fudimus: et mox $\overline{X}$ 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., |$\overline{X}$|$\overline{CLXXX}$DC 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. According to