South America." 1 This system was used also by many of the North Siberian and African tribes. Traces of it are found in the languages of peoples who now use the decimal scale; for example, in Homeric Greek. The Roman notation reveals traces of it; viz., I, II, ... V, VI, ... X, XI, ... XV, etc. It is curious that the quinary should so frequently merge into the vigesimal scale; that savages should have passed from the number of fingers on one hand as an upper unit or a stopping-place, to the total number of fingers and toes as an upper unit or resting-point. The vigesimal system is less common than the quinary, but, like it, is never found entirely pure. In this the first four units are 20, 400, 8000, 160,000, and special words for these numbers are actually found among the Mayas of Yucatan. The transition from quinary to vigesimal is shown in the Aztec system, which may be represented thus, 1, 2, 3, 4, 5, 5+1, ... 10, 101, ... 10+5, 10+5+1,... 20, 20+1,... 20 +10, 20+ 10+1, 40, etc.2 Special words occur here for the numbers 1, 2, 3, 4, 5, 10, 20, 40, etc. The vigesimal system flourished in America, but was rare in the Old World. Celtic remnants of one occur in the French words quatre-vingts (4 × 20 or 80), six-vingts (6 × 20 or 120), quinzevingts (15 × 20 or 300). Note also the English word score in such expressions as three-score years and ten. ... Of the three systems based on human anatomy, the decimal system is the most prevalent, so prevalent, in fact, that according to ancient tradition it was used by all the races of the world. It is only within the last few centuries that the other 1 CONANT, op. cit., p. 592. For further information see also POTT, Die quinäre und vigesimale Zählmethode bei Völkern aller Welttheile, Halle, 1847; POTT, Die Sprachverschiedenheit in Europa an den Zahlwörtern nachgewiesen, sowie die quinäre und vigesimale Zählmethode, Halle, 1868. 2 TYLOR, op. cit., Vol. I., p. 262. ་ two systems have been found in use among previously unknown tribes. The decimal scale was used in North America by the greater number of Indian tribes, but in South America it was rare. In the construction of the decimal system, 10 was suggested by the number of fingers as the first stopping-place in counting, and as the first higher unit. Any number between 10 and 100 was pronounced according to the plan b(10) + a(1), a and b being integers less than 10. But the number 110 might be expressed in two ways, (1) as 10 x 10 +10, (2) as 11 x 10. The latter method would not seem unnatural. Why not imitate eighty, ninety, and say eleventy, instead of hundred and ten? But upon this choice between 10 × 10 + 10 and 11 × 10 hinges the systematic construction of the number system.2 Good luck led all nations which developed the decimal system to the choice of the former; 3 the unit 10 being here treated in a manner similar to the treatment of the lower unit 1 in expressing numbers below 100. Any number between 100 and 1000 was designated c(10)2 + b(10)+a, a, b, c representing integers less than 10. Similarly for numbers below 10,000, d(10)3 + c(10)2 + b(10)1 + a(10)°; and similarly for still higher numbers. Proceeding to describe the notations of numbers, we begin with the Babylonian. Cuneiform writing, as also the accompanying notation of numbers, was probably invented 1 CONANT, op. cit., p. 588. 2 HERMANN HANKEL, Zur Geschichte der Mathematik in Alterthum und Mittelalter, Leipzig, 1874, p. 11. Hereafter we shall cite this brilliant work as HANKEL. 3 In this connection read also MORITZ CANTOR, Vorlesungen über Geschichte der Mathematik, Vol. I. (Second Edition), Leipzig, 1894, pp. 6 and 7. This history, by the prince of mathematical historians of this century, will be in three volumes, when completed, and will be cited hereafter as CANTOR. by the early Sumerians. A vertical wedge stood for one, while and signified 10 and 100, respectively. In case of numbers below 100, the values of the separate symbols were added. Thus, for 23, <<< for 30. The signs of higher value are written on the left of those of lower value. But in writing the hundreds a smaller symbol was placed before that for 100 and was multiplied into 100. Thus, <signified 10 x 100 or 1000. Taking this for a new unit, << > was interpreted, not as 20 x 100, but as 10 x 1000. In this notation no numbers have been found as large as a million. The principles applied in this notation are the additive and the multiplicative. Besides this the Babylonians had another, the sexagesimal notation, to be noticed later. An insight into Egyptian methods of notation was obtained through the deciphering of the hieroglyphics by Champollion, Young, and others. The numerals are | (1), ₪ (10), C (100), I (1000),(10,000), (100,000), (1,000,000), e (10,000,000). The sign for one represents a vertical staff; that for 10,000, a pointing finger; that for 100,000, a burbot; that for 1,000,000, a man in astonishment. No certainty has been reached regarding the significance of the other symbols. These numerals like the other hieroglyphic signs were plainly pictures of animals or objects familiar to the Egyptians, which in some way suggested the idea to be conveyed. They are excellent examples of picture-writing. The principle involved in the Egyptian notation was the additive throughout. would be 111. Thus, eni спо 1 For fuller treatment see MORITZ CANTOR, Mathematische Beiträge zum Kulturleben der Völker, Halle, 1863, pp. 22-38. Hieroglyphics are found on monuments, obelisks, and walls of temples. Besides these the Egyptians had hieratic and demotic writings, both supposed to be degenerated forms of hieroglyphics, such as would be likely to evolve through prolonged use and attempts at rapid writing. The following are hieratic signs:1 Since there are more hieratic symbols than hieroglyphic, numbers could be written more concisely in the former. The additive principle rules in both, and the symbols for larger values always precede those for smaller values. About the time of Solon, the Greeks used the initial letters of the numeral adjectives to represent numbers. These signs are often called Herodianic signs (after Herodianus, a Byzantine grammarian of about 200 A.D., who describes them). They are also called Attic, because they occur frequently in Athenian inscriptions. The Phoenicians, Syrians, and Hebrews possessed at this time alphabets and the two latter used letters of the alphabet to designate numbers. The Greeks began to adopt the same course about 500 B.C. The letters of the Greek alphabet, together with three antique letters, 5, 9, 1, and the 1 CANTOR, Vol. I., pp. 44 and 45. The hieratic numerals are taken from Cantor's table at the end of the volume. ! symbol M, were used for numbers. For the numbers 1-9 they wrote a, B, y, 8, e, s, , n, 0; for the tens 10-90, 4, K, λ, μ, v, & ?, for the hundreds 100-900, p, σ, T, v, 4, X, Y, w, D; fol the thousands they wrote a, B, В Y, 8, e, etc.; for 10,000, M for 20,000, M; for 30,000, M, etc. = In the Roman notation we have, besides the additive, the principle of subtraction. If a letter is placed before another of greater value, the former is to be subtracted from the latter. Thus, IV 4, while VI 6. Though this principle has not been found in any other notation, it sometimes occurs in numeration. Thus in Latin duodeviginti = 2 from 20, or 18.2 The Roman numerals are supposed to be of Etruscan origin. = = Thus, in the Babylonian, Egyptian, Greek, Roman, and other decimal notations of antiquity, numbers are expressed by means of a few signs, these symbols being combined by addition alone, or by addition together with multiplication or subtraction. But in none of these decimal systems do we find the all-important principle of position or principle of 1 DR. G. FRIEDLEIN, Die Zahlzeichen und das Elementare Rechnen der Griechen und Römer, Erlangen, 1869, p. 13. The work will be cited after this as FRIEDLEIN. See also DR. SIEGMUND GÜNTHER in MÜLLER'S Handbuch der Klassischen Altertumswissenschaft, Fünfter Band, 1. Abteilung, 1888, p. 9. 2 CANTOR, Vol. I., pp. 11 and 489. |