are too numerous, both in the numerals and in other fragments of elementary names, to be regarded as merely accidental. And considering the remote period alluded to, being above four thousand years, the mind is naturally led to the conclusion that the fragments of these primeval names are derived by each language from one of its cognates, or by all from one common source.

Little is known respecting the origin and the early history of arithmetic of the ancient Hebrews or Syrians. It has been conjectured that they were indebted to the Phoenicians, their neighbours, for what they knew of the art of numbering. The most ancient books-the writings of Moses-afford no evidence of the use of any numerical system of notation. In the text of the writings of Moses all numbers are expressed in words at length, and the counting is made by tens, hundreds, &c.

It is clear from the second chapter of the Second Book of the Chronicles that the Hebrews had commercial intercourse with the Phoenicians above a thousand years before the times of the Messiah. And the ancient tradition of the Greeks also tends to favour the opinion that Cadmus [7], a man from the East, was the first who introduced the use of letters into Greece from the Phoenicians.

And

it may be added that Proclus, in his Commentary on the First Book of Euclid's Elements, states that the Phoenicians, by reason of their traffic and commerce, were accounted the first inventors of arithmetic.

The ancient Hebrew and Samaritan alphabets consisted of twentytwo letters, and were employed to denote the nine digits, the nine tens, and the first four of the nine hundreds. The remaining five hundreds were represented by combining the symbols of the first four hundreds. In later times the final caph, mem, nun, pe, tsadi were added to make up the nine simple characters for the hundreds. All other numbers were expressed by placing together the simple characters denoting the component numbers required to make up their amount, with some few exceptions. The number 15 is denoted by, or 9 and 6, and not by n, 10 and 5; because, Jah, being one of the names of God, it was imagined that such a use of the name would infringe the third commandment. For the same reason, perhaps, the number 1030 was not expressed by the characters, which form another of the names of the Deity.

The following are the characters of the Hebrew and Greek Alphabets, as they are applied to denote numbers:

:

,Hebrew. 0, Greek.

3, Hebrew.

Greek.

บ

Hebrew.

2, Greek.

1 In the third volume of the new series of the Journal of Sacred Literature, Dr. W. Wright, the Professor of Arabic at Cambridge, has explained in his notice (pp. 128-130) of the Anecdota Syriaca of Dr. Land, a system of arithmetical notation employed in many of the oldest Syrian manuscripts not later than the ninth century. There are simple characters to denote 1, 2, 5, 6, 10, 20, 100; and these appear to have been combined to express other numbers, in some respects like the Roman notation.

A comparison of the Greek with the Hebrew numerical letters will suggest at once their common origin, and that the alphabets of both languages are derived from the same source, or that one is derived from the other. The fact of the correspondence of the order, the powers, and the names of the letters of both alphabets, is an argument in favour of this opinion. But the difference in form of the characters presents a difficulty, and it is uncertain whether the Greeks derived their numeral system from the same source as their alphabet. The system of the Greeks possesses an interest which does not attach either to that of the Romans or of the Hebrews, both from the improvements and extensions it received. A knowledge of these is essential for every one who may wish to read and understand the mathematical and astronomical writings of the Greeks.

The founders of the Roman name in Italy appear to have derived their descent from a colony of Pelasgi, who transported their language thither in its earliest and rudest form. For many ages, among a people incessantly occupied with war and conquest, their dialect continued almost unchanged till after the Punic wars, and the structure of the Latin language carries us back to a period anterior to any distinct vestige of the Greek language. The names of numbers must necessarily have been formed before any regular system of abbreviated numerical notation could have existed. There are no ancient writings extant which afford any satisfactory account of the origin of the Roman numerical symbols. From the large existing remains of Roman literature, besides monumental and other inscriptions, there is ample evidence of their universal use wherever the Roman arms and Roman language prevailed. Niebuhr informs us (vol. i. p. 134) that "what we call the Roman numerals are Etruscan, and they frequently occur on their monuments. They are remnants of a hieroglyphical mode of writing, which was in use before the age of the alphabetical, and like the numerals of the Aztecans, they represent certain objects: that were associated with particular numbers. They are indigenous, and belong to the time when the west was subsisting with all its original peculiarities, before it received any influence from Asia." Notwithstanding this opinion of Niebuhr, that the Etruscan signs of numbers existed prior to the age of alphabetical writing, it must be considered doubtful, as little beyond conjecture can be attempted in the absence of evidence with respect to their origin and primary meaning. It is scarcely possible to discover what alterations in the signs took place after their adoption, or what additions or substitutions were made, except so far as they appear in the inscriptions on ancient monuments. The Romans employed seven elementary characters, as the primary signs of number, whose values are successively increased fivefold and twofold, beginning with unity. This was. probably suggested by the two hands and the five fingers on each hand; or, as it is expressed by Ovid, Fast. iii., 126,

"Seu quia tot digiti per quos numerare solemus."

These seven characters are I, V or A, X, L, C, D or IO, M: denoting respectively 1, 5, 10, 50, 100, 500, 1000. Other numbers are expressed in different ways by means of these seven symbols. A symbol repeated two or three, &c., times, denotes the double or treble of its value, as III stands for 3, and XX for 20. The symbols I, X, C aro in general found repeated not more than four times; but in

inscriptions I is found repeated six times, as IшIVIR, for sevir or sextumvir. V and L are not found repeated. A symbol of less value postfixed to one of greater value increases the greater by that value, as VI means 5 increased by 1; and LX, 50 increased by 10; but if prefixed, it diminishes the greater by that number, as IV stands for 5 diminished by 1; and XL, 50 diminished by 10. This mode of notation by deficit was peculiar to the Romans, and is in accordance with the forms of their numerical words. Instead of octodecim for 18, their writers use duodeviginti, and undeviginti instead of novendecim. The letter M, the initial of mille, denoting 1000; 2000, 3000, 4000, &c., were denoted by IIM, IIIM, IVM, &c.; and by placing a line over the symbols their value was increased a thousand times, thus: I, L, U, M, &c., denote respectively 1000, 50,000, 100,000, 1,000,000, &c.

The latest improvement in the Roman notation was devised at a late period for the expression of large numbers. The method of proceeding was perfectly analogical. Taking the symbol C for 100, and IO for 500, by postfixing O once, twice, thrice, &c., to IO; the symbols I, I, I, &c., were assumed to denote 5000, 50,000, 500,000, &c, respectively; and by prefixing C once, twice, thrice, &c., to 10, 100, 1000, &c., respectively, their values become doubled, and the symbols CIO, CCOO, COCIDO, COCCOD, &c., denoted 1000, 10,000, 100,000, 1,000,000, &c.

The Roman numerals are incapable of any material improvement. They could serve to register numbers, but could not afford the slightest aid in performing numerical calculations. In fact, they never were employed for that purpose. In the calculations which their accountants (calculatores, rationarii) had occasion to make, they were obliged to have recourse to a mechanical process, employing pebbles or counters. A box (loculus) of pebbles (calculi), and a board (tabula) on which the pebbles were placed in rows, formed their instruments of calculation. (Hor. Lib. I. Sat. vi. 74.) The terms calculate, calculation, are closely related to calculus, and in their primary meaning had reference to counting by means of pebbles. The board on which arithmetical operations were performed was also called abacus, and was divided from the right to the left by lines or grooves, on which the pebbles were placed to denote units, tens, &c. The operations of the abacus were rendered more commodious by substituting small beads strung on parallel threads, and sometimes by pegs stuck along grooves. With such an instrument, it is not difficult to perceive how addition and subtraction might be performed with ease and expedition, but to perform multiplication and division must have been a work of tedious labour.

The Roman notation was employed throughout the extent of the Empire for recording all accounts, whether fiscal or mercantile, and continued in use in Europe after its dissolution. The Roman notation was sanctioned by the almost universal employment of the Latin language in all subjects of literature and science, which for ages continued to be the language of the learned. It was employed in England from the time the Romans held possession of the island, and during the rule both of the Saxon and the Norman. The accounts in the Domesday Book are registered in the Roman characters. The dotation of the colleges of Oxford and Cambridge, and the annual accounts, were all recorded in the Roman characters long after the Indian notation had been introduced. It still continues to hold a

position of considerable importance in the recording of dates, indexes, and monumental inscriptions in Western Europe.

In the early period of the history of the Hellenic race, before their wants and necessities called for the use of large numbers, the initial letters of the names of numbers were employed to express the numbers themselves. Thus, the letters, I, II, ▲ H, X, м, being the initial letters of the words "Ios (for εις), Πέντε, Δέκα, Ἡεκατὸν, Χίλιοι, Μύριοι, were employed to express 1, 5, 10, 100, 1000, 10,000. The other numbers were expressed by repeating or combining these six characters. Abbreviated combinations were also employed; as when any of these numeral letters were written within the capital letter II, they denoted a number five times as great, as A written within II denoted 50. This method was probably the first step towards a system of numerical notation, and, except for inscriptions, was superseded for the more perfect system formed with the letters of the alphabet. At a subsequent period the Greeks employed the letters of their alphabet with three supplementary symbols to express the first order of digits 1 to 9; the second, 10 to 90; and the third, 100 to 900. The fourth order of thousands was formed by subscribing an to each of the units of the first order. The fifth, sixth, &c., orders were formed by affixing M. or Mu. for Múpio, 10,000, to each character of the first, second, &c., orders.

Of the three supplementary symbols employed in the Greek numerical notation, there being no letter corresponding to the Hebrew vau, the character is used for the number 6, and called étionμov Bau, indicating vau. The other two symbols were for 90, and for 900, the former called ἐπίσημον κοππὰ and the latter ἐπίσημον σανπὶ, that is, indicating Koph, and indicating Tsadi. By combining these symbols any other numbers could be expressed as oa denotes 41; va, 401; da, 4001; aσλd, 1234; λ”.Mv, 370,000. Neither the order nor the number of the characters had any effect in fixing the value of any number intended to be expressed. The value of the same combination of symbols is the same in whatever order they are placed; in general, however, they were written from right to left, according to their increasing value.

The oldest arithmetical writings of the Greeks which have descended to modern times are to be found in the seventh, eighth, ninth, and tenth books of the Elements of Euclid, who lived between B.c. 323 and 284. These four books contain complete treatises on numbers, their properties, proportions, commensurable and incommensurable, and their application to geometry. Diophantus, who has been placed by some writers as early as A.D. 280, by others as contemporary with the Emperor Julian, was the author of thirteen books on arithmetic and algebra. Only six of the thirteen books are known to be extant. The first four were translated into French by Simon Stevin, and the other two by Albert Girard. The six books are printed in the collected works of Stevin, revised and augmented by his friend Albert Girard in 1634.

The progress of astronomical and mathematical science occasioned the necessity of larger numbers than could be expressed by the Greek notation then in use. Archimedes, who lived between B.C. 287 and 212, improved and extended the Greek method of notation. In his work entitled fauμírns or Arenarius, he proposed to express a number which should exceed the number of the grains of sand that might be

contained in the sphere of the universe as conceived by Aristarchus. He assumed a scale of numeration whose radix or base is a myriad of myriads, or ten thousand times ten thousand, the number which then formed the limit of the Greek numerical language. All numbers less than this radix he called primary numbers, and the radix itself he made the unit of secondary numbers; he then proceeds to ternary, quaternary, and other numbers of higher orders, forming successive classes.

According to the Indian notation, the units of Archimedes, the primary, secondary, ternary, quaternary, quinary, &c., orders of numbers, will consist of 1; 1 with 8 ciphers; 1 with 16 ciphers; 1 with 24 ciphers; 1 with 32 ciphers, &c., respectively. And the unit of the ten thousand times ten thousandth order would consist of 1 with 799999992 ciphers.

His method for determining the number of places in any required number is the following. He supposes a series of numbers beginning with unity, in continued proportion, and shows that the product of any two terms of this series is equal to that term whose place reckoned from the first, is less by unity than the sum of the two numbers which indicate the places of the two terms; as the 7th term of the series is equal to the product of the 3rd and 5th terms. He then assumes the series 1, 10, 100, 1000, &c., in which each successive term increases tenfold, or where the common ratio is 10. The first 8 terms of this series (omitting the first term) are primary numbers; the next 8 terms, secondary numbers; the third 8, ternary numbers, and so on; and the question is to determine that term in the series which is equal to the product of any two assigned terms, or the term whose place is the sum (less by unity) of the two numbers which indicate the places of the two assigned terms. The classes themselves he calls octades, or periods of eight, from each class requiring eight symbols, or eight places of figures of common notation, to express the numbers included in each class. He then shows, without finding or assuming the number itself, that the number requiring for its expression not more than eight of these octades, or, in our notation, not exceeding 64 places of figures, will exceed the number of the grains of sand in the sphere of Aristarchus, each grain of sand being so small that 10,000 of them are less than one seed of poppy.

Apollonius, about 240 B.C., adopted the plan of Archimedes of classifying numbers, but instead of the octades of Archimedes, he adopted tetrads, reducing the radix from ten thousand times ten thousand, to ten thousand. The units after the first class he designated in order, the single myriad, the double myriad, the treble myriad, &c., which he denoted by Ma, M6, My, &c., respectively. His chief object, however, appears to have been to simplify the process of multiplication, and to make the multiplication of all higher numbers dependent on the product of any two of the first nine digits. The work of Apollonius has perished, and even the name of its title is unknown. It is highly probable that the substance of it was embodied in the first two books of the mathematical collections of Pappus. Only a fragment of the latter portion of the second part is known to be extant, in which are exhibited several examples of the method of Apollonius. This fragment was published by Dr. Wallis with a Latin translation and notes in 1688, and is reprinted in the third volume of his works, pp. 595-614.