shall be taken to form a class of a still higher order, and so on; advancing onwards from class to class as far as occasion may require. The number of units at first fixed upon to form a class is quite arbitrary: it might be five, it might be ten, it might be twelve, or any other number: but this being once fixed, the same number of each of the classes must be taken to form a class of the next higher denomination. § 11. To explain the Decimal or Denary Scale of Notation. In the decimal scale the number ten is arbitrarily fixed upon as the basis of computation; and the cipher, which is the name given to nothing, and the first nine natural numbers are represented by the following symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. When the number ten is reached, this is considered as a new unit of a superior order; and the succeeding numbers are formed by successive combinations of the first nine natural numbers with ten, with two tens, with three tens, &c., until ten classes of ten each are gone through, when the last number in the last of these classes is called one hundred. This now becomes a unit of the next superior order, and another series of numbers is formed by combining one hundred with the numbers just enumerated; and when ten classes of hundreds are gone through, the last number is called not ten hundred, but one thousand. Neither tens of thousands nor hundreds of thousands have a separate name assigned to them': but the same process of ascending by classes, ten of which form one of the next order, is continued until one thousand thousand is reached, and this is called a million. Proceeding onwards in the same way, a million million is called a billion; a million billion is called a trillion; a million trillion is called a quadrillion, and so on. § 12. To explain the principle of Local Value. In order to represent numbers higher than nine, i.e. numbers which contain tens, hundreds, &c., the following device has been invented. No symbols besides those above enumerated are used, but it is agreed that each figure besides its individual shall have a local value, namely, a value depending 1 Would not the assignment of distinct names to "tens of thousands" and "hundreds of thousands" facilitate the apprehension of high numbers, and render more obvious the law that a new unit is formed when ten of any class is reached? upon the place it occupies; and that, while any figure standing simply by itself retains the individual value assigned to it, any figure standing to the left of another shall thereby be increased ten-fold. Thus when it is arranged that a figure standing in any particular place1 shall represent so many units, the figure to the left of this will represent tens, or units of the second class: the figure to the left of this ten teus, or hundreds; the figure to the left of this, ten hundreds, or thousands, and so on; the local value of each figure continually increasing in a ten-fold degree as we advance one place further to the left. If in writing a number, any class, as units, tens, hundreds, &c. be wanting, the cipher" is used; which although without signification when standing by itself, serves when combined with other figures to fill up the vacant places, and so to give the significant figures their required local value. [Obs. The peculiarity of decimal notation must not be confused with the local value assigned to the figures; the two things are perfectly distinct, and do not in any way depend upon each other. Indeed while many nations have used a decimal notation, very few traces of local value can be found in any system except the Hindoo-Arabic which we use. The decimal scale probably originated in the practice of counting on the fingers, whence the name digits for the symbols representing the first nine numbers. Had any nation counted only on one hand, such a system would have been the quinary, and six would have been the unit of the next superior order. Had they counted on fingers and toes the 1 In whole numbers the figure on the extreme right is said to be in the units place; the next figure to the left, in the tens place; the next figure to the left, in the hundreds place; the next, in the thousands place; and so on. But in writing decimal fractions (which, as will be shewn afterwards, afford the means of extending the decimal sca'e below unity) it is not the right-hand figure, but the figure to the left of the decimal point which stands in the place of units. 2 The word cipher; Tippa, cifra, is from the Arabic terin Tsaphara, "quod vacuum aut inane est," blank, or void. At the end or in the middle of any number the cipher is of use to keep the significant digits in their proper rank, when the units or the hun dreds or any other denomination may be wanting, eg. 60 means 6 tens followed by no units: 606 means 6 hundreds, with no tens, but 6 units. At the beginning of a number ciphers would be useless: if so placed they could only indicate the absence of any higher class; e.g. 096 means only 9 tens and 6 units; the cipher showing that there are no hundreds, which is equally intelligible if the cipher be omitted. The use of the word cipher led to the digits being all called ciphers, and so introduced the use of the verb to cipher. system would have been vicenary. Some traces of both these systems are to be found, for instance, in our reckoning by scores1. The quinary, the denary, and the vicenary are the only natural systems; and it will be found that no other than these have ever prevailed in common use. The duodecimal scale, with 12 for the base, would present some peculiar advantages, as 12 is exactly divisible by 2, 3, 4, and 6; while 10 is so divisible by only 2 and 5: but in the infancy of any nation the method of reckoning by one of the natural systems seems to have been always first established, and not to have been afterwards disturbed by any more artificial arrangement. When the practical method of numeration had been fixed, the numerical language to express it would be afterwards formed; and this would be succeeded by the invention of written symbols. The Hebrews, Phoenicians, and Syrians used the letters of their alphabets for numerical symbols; and the Greeks, who derived their alphabet from the Phoenicians, borrowed from the same source their system of numerical notation. From what source the Roman numeral symbols originated is a point which has given rise to much conjecture; one explanation, namely that the system was made up from signs used in reckoning by single units, will be noticed below. For the symbols which we now use no other origin has been suggested than that of arbitrary invention: the shape of several of the figures has been considerably modified in course of time; but the use of nine figures with zero, and the principle of local value2 were introduced among the nations of Europe from the Arabs, first into Spain in the 12th century, and especially into Italy in the beginning of the 13th century. The Arabs derived them from India, where the Hindoos had used them from a period anterior to all written records, and attributed the invention of them to the Deity, "the invention of nine figures with the device of places to make them suffice for all numbers, being ascribed to the beneficent Creator of the universe 3." The use of this method among the Hindoos can be traced up to the 5th century after 1 The word score itself, the long notch on the tally, shews the method of counting which was most common among our forefathers. 2 The Chinese possess a system of decimal Arithmetic not only of very great antiquity, but one in which a very close approximation is made to local value; they use however symbols for the superior units (hundreds, thousands, &c.) which in our system are expressed by position alone. 3 Note 2 to page 4 of Colebroke's Translation of Bháscara's Lilavati: where it is stated that 'the place, where no figure belongs to it, is shewn by a blank; which to obviate mistake, is denoted by a dot or small circle.' Christ; among the Arabs to the 9th century. It appears to have been communicated about 1136 by the Moors in Spain to the Spaniards, but at first to have been little used except in Astronomical works and calendars; its more general adoption was introduced into Italy by the writings of Leonardo Pisano in 1202; but Roman numerals still continued to be most commonly used throughout Europe for a long period subsequent to this; and indeed merchants' accounts were so kept until the middle of the 16th century.] § 13. It will be useful here to explain the methods of notation used by the Greeks and the Romans. The system of the Greeks will serve to illustrate the manner of representing numbers by the letters of an alphabet; while the peculiarities of the Roman numerals, still commonly adopted among ourselves, as in inscriptions, &c., ought to be well understood. The Greeks then, in order to denote numbers, used the 24 letters of their alphabet, with three additional signs, which signs, as ordinary letters, had become obsolete at an early period: these were the Baû or Digamma, originally the 6th letter of the alphabet, which under the form 5 (called rò èñíσημον Βαϋ) denoted the number 6 ; the guttural Κόππα, which originally followed wî in the alphabet, written or 4, called TÒ ÉπíσημOV KÓTπа, and as a numerical sign denoting 90; and the arbitrary symbol Ɛauri (compounded from the old letter Σáv from the Hebrew Zain, and πî) written, and denoting 900. Their numbers therefore were represented as follows: a', B', y', d', é', 5', §', n', 0', 1, 2, 3, 4, 5, 6, 7, 8, 9, ί, ια, ιβ'...κ, κα, κβ' ..Χ, μ, ν, ξ', ό, π', 100, 200, 300, 400, 500, 600, 700, 800, 900, or r 4, 0, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, The word "air" (a = 1, '= 10, p = 100) will help the memory to retain the first letters of the lines of units, tens, and hundreds. Besides this notation there was an older method of expressing numbers (a method found on ancient inscriptions, &c.) by means of the initial letters of Ios for eἷς, Πέντε, Δέκα, = = 50; [] = πεντάκις Ηεκατόν, Χίλιοι, and Μύριοι. In this system I = 1, II=2, III = 3, IIII = 4, II = 5, III = 6, пIIII 9, A 10, AI - 11, ΔΔ = 20, ΔΑΔ 30, H = 100, HH 200, X = 1000, XX 2000, M = 10000. Also abbreviated combinations of II with other letters were used; thus H = πεντάκις δέκα ἑκατόν = 500; Η = πεντάκις χίλιοι = 5000. Also by writing M beneath any letter its value was increased ten thousand fold: Thus y was 30000; k was 220000. In writing fractions, eithery, alone meant,; or else the denominator was written above the numerator, like an index in algebra, as M § 14. Various conjectures have been made concerning the origin of the Roman numerals, and among others the following hypothesis has been put forward: Suppose that a person who counted on his fingers wrote a stroke for each successive unit up to ten; and when he had advanced as far as ten strokes, that he drew a cross line through them to denote that he had come to the end of his handful: his marks would be I, II, III, ... IHHHHHIL If now he shortened his mark for ten into a single unit with a cross line drawn through it, he would have X for ten: for one hundred he might adopt the unit with two cross lines, as ; for one thousand he would require a unit with three cross lines, or four strokes, which might be written M, or m, or even o; next, if he halved these symbols he would have half X or V for five; half or L for fifty; half o or D or I for five hundred. Whether this hypothesis be correct or no, at any rate the Romans represented numbers by combinations of these symbols; they had a certain principle of local value as far as this, namely that a smaller symbol, standing before a larger one, in numbers less than one hundred, was to be subtracted, but standing after it was to be added. Their notation therefore was as follows: |