Chapter 11
Historic Systems of Notation

85. Egyptian and Phœnician. This written symbolism did not assume the complicated character it might have had, had counting with written strokes and not with the fingers been the primitive method. Perhaps the written strokes were employed in connection with counting numbers higher than 10 on the fingers to indicate how often all the fingers had been used; or if each stroke corresponded to an individual in the group counted, they were arranged as they were drawn in groups of 10, so that the number was represented by the number of these complete groups and the strokes in a remaining group of less than 10.

At all events, the decimal idea very early found expression in special symbols for 10, 100, and if need be, of higher powers of 10. Such signs are already at hand in the earliest known writings of the Egyptians and Phoenicians in which numbers are represented by unit strokes and the signs for 10, 100, 1000, 10,000, and even 100,000, each repeated up to 9 times.

86. Greek. In two of the best known notations of antiquity, the old Greek notation—called sometimes the Herodianic, sometimes the Attic—and the Roman, a primitive system of counting on the fingers of a single hand has left its impress in special symbols for 5.

In the Herodianic notation the only symbols—apart from certain abbreviations for products of 5 by the powers of 10—are $I$, $\Gamma$ ($\pi \acute{𝜖}\nu \tau 𝜖$, 5), $\Delta$ ($\delta \acute{𝜖}\kappa \alpha$, 10), $H$ ($\grave{𝜖}\kappa \alpha \tau ó\nu$, 100), $\chi$ ($\chi \acute{\iota }\lambda \iota o\iota$, 1000), $M$ ($\mu \nu \rho \acute{\iota }o\iota$, 10,000); all of them, except $I$, it will be noticed, initial letters of numeral words. This is the only notation, it may be added, found in any Attic inscription of a date before Christ. The later and, for the purposes of arithmetic, much inferior notation, in which the 24 letters of the Greek alphabet with three inserted strange letters represent in order the numbers 1, 2, …10, 20, …100, 200, …900, was apparently first employed in Alexandria early in the 3d century B. C., and probably originated in that city.

87. Roman. The Roman notation is probably of Etruscan origin. It has one very distinctive peculiarity: the subtractive meaning of a symbol of lesser value when it precedes one of greater value, as in $IV$ = 4 and in early inscriptions $IIX=8$. In nearly every other known system of notation the principle is recognized that the symbol of lesser value shall follow that of greater value and be added to it.

In this connection it is worth noticing that two of the four fundamental operations of arithmetic—addition and multiplication—are involved in the very use of special symbols for 10 and 100, for the one is but a symbol for the sum of 10 units, the other a symbol for 10 sums of 10 units each, or for the product 10 $\times$ 10. Indeed, addition is primarily only abbreviated counting; multiplication, abbreviated addition. The representation of a number in terms of tens and units, moreover, involves the expression of the result of a division (by 10) in the number of its tens and the result of a subtraction in the number of its units. It does not follow, of course, that the inventors of the notation had any such notion of its meaning or that these inverse operations are, like addition and multiplication, as old as the symbolism itself. Yet the Etrusco-Roman notation testifies to the very respectable antiquity of one of them, subtraction.

88. Indo-Arabic. Associated thus intimately with the four fundamental operations of arithmetic, the character of the numeral notation determines the simplicity or complexity of all reckonings with numbers. An unusual interest, therefore, attaches to the origin of the beautifully clear and simple notation which we are fortunate enough to possess. What a boon that notation is will be appreciated by one who attempts an exercise in division with the Roman or, worst of all, with the later Greek numerals.

The system of notation in current use to-day may be characterized as the positional decimal system. A number is resolved into the sum:

The positional idea is very old. The ancient Babylonians commonly employed a decimal notation similar to that of the Egyptians; but their astronomers had besides this a very remarkable notation, a sexagesimal positional system. In 1854 a brick tablet was found near Senkereh on the Euphrates, certainly older than 1600 b. c., on one face of which is impressed a table of the squares, on the other, a table of the cubes of the numbers from 1 to 60. The squares of $1$, $2$, $\dots$ $7$ are written in the ordinary decimal notation, but $8^2$, or $64$, the first number in the table greater than $60$, is written $1$, $4$ ($1\times 60+4$); similarly $9^2$, and so on to $59^2$, which is written $58$, $1$ ($58\times 60+1$); while $60^2$ is written $1$. The same notation is followed in the table of cubes, and on other tablets which have since been found. This is a positional system, and it only lacks a symbol for $0$ of being a perfect positional system.

The inventors of the $0$-symbol and the modern complete decimal positional system of notation were the Indians, a race of the finest arithmetical gifts.

The earlier Indian notation is decimal but not positional. It has characters for 10, 100, etc., as well as for $1$, $2$, $\dots$ $9$, and, on the other hand, no $0$.

Most of the Indian characters have been traced back to an old alphabet³⁴ in use in Northern India 200 b. c. The original of each numeral symbol 4, 5, 6, 7, 8 (?), 9, is the initial letter in this alphabet of the corresponding numeral word (see table on page 89,³⁵ column 1). The characters first occur as numeral signs in certain inscriptions which are assigned to the 1st and 2d centuries a. d.  (column 2 of table). Later they took the forms given in column 3 of the table.

When 0 was invented and the positional notation replaced the old notation cannot be exactly determined. It was certainly later than 400 a. d., and there is no evidence that it was earlier than 500 a. d. The earliest known instance of a date written in the new notation is 738 a. d. By the time that 0 came in, the other characters had developed into the so-called Devanagari numerals (table, column 4), the classical numerals of the Indians.

The perfected Indian system probably passed over to the Arabians in 773 a. d., along with certain astronomical writings. However that may be, it was expounded in the early part of the 9th century by Alkhwarizmî, and from that time on spread gradually throughout the Arabian world, the numerals taking different forms in the East and in the West.

Europe in turn derived the system from the Arabians in the 12th century, the “Gobar” numerals (table, column 5) of the Arabians of Spain being the pattern forms of the European numerals (table, column 7). The arithmetic founded on the new system was at first called algorithm (after Alkhwarizmî), to distinguish it from the arithmetic of the abacus which it came to replace.

A word must be said with reference to this arithmetic on the abacus. In the primitive abacus, or reckoning table, unit counters were used, and a number represented by the appropriate number of these counters in the appropriate columns of the instrument; e. g. 321 by 3 counters in the column of 100’s, 2 in the column of 10’s, and 1 in the column of units. The Romans employed such an abacus in all but the most elementary reckonings, it was in use in Greece, and is in use to-day in China.

Before the introduction of algorithm, however, reckoning on the abacus had been improved by the use in its columns of separate characters (called apices) for each of the numbers 1, 2, …, 9, instead of the primitive unit counters. This improved abacus reckoning was probably invented by Gerbert (Pope Sylvester II.), and certainly used by him at Rheims about 970–980, and became generally known in the following century.

Now these apices are not Roman numerals, but symbols which do not differ greatly from the Gobar numerals and are clearly, like them, of Indian origin. In the absence of positive evidence a great controversy has sprung up among historians of mathematics over the immediate origin of the apices. The only earlier mention of them occurs in a passage of the geometry of Boetius, which, if genuine, was written about 500 a. d. Basing his argument on this passage, the historian Cantor urges that the earlier Indian numerals found their way to Alexandria before her intercourse with the East was broken off, that is, before the end of the 4th century, and were transformed by Boetius into the apices. On the other hand, the passage in Boetius is quite generally believed to be spurious, and it is maintained that Gerbert got his apices directly or indirectly from the Arabians of Spain, not taking the 0, either because he did not learn of it, or because, being an abacist, he did not appreciate its value.

At all events, it is certain that the Indo-Arabic numerals, 1, 2, …9 (not 0), appeared in Christian Europe more than a century before the complete positional system and algorithm.

The Indians are the inventors not only of the positional decimal system itself, but of most of the processes involved in elementary reckoning with the system. Addition and subtraction they performed quite as they are performed nowadays; multiplication they effected in many ways, ours among them, but division cumbrously.