The power to form a direct and immediate conception of numbers is very limited; but the relative magnitudes of numbers, large as well as small, can be expressed with the utmost accuracy, and so as to .be clearly understood. The system of notation in common use, whereby we express not merely numbers, but parts of numbers, supplies us with means of comparing arithmetically together either the greatest or the minutest magnitudes, to which there is absolutely no limit. The proportion, for instance, that the circumference of a circle bears to the diameter, though it cannot be stated with arithmetical exactness, has been calculated to upwards of 200 decimal places,—a nicety for which the vast dimensions that science discloses in the physical universe furnish no means of comparison whatever. For let it be supposed that a circle were described, with a point on the earth’s surface as centre, so as to extend beyond the most distant star that can be discerned by the most powerful existing telescope, that the radius of the circle were known, and that the circumference were computed from it, it does not appear that an error in the thirtieth decimal place would be of such magnitude that the keenest vision, aided by the most powerful microscope, could detect it.

In all systems of number, with the exception, perhaps, of the very rudest, numerical conceptions are aided by the introduction, usually at a very early stage, of methods of grouping. In nearly every case the methods adopted connect themselves with the number of the fingers, either of one or, more usually, of the two hands. Having reached 5 or 10, the reckoner proceeds by adding to these the prior numbers; and when a second 5 or 10 is reached, a new word or sign is employed, the significance or derivation of which is generally well marked in the name or form it bears. Similarly when (say) five fives or ten tens are reached, a fresh start is made. These processes of grouping are of great importance, conveying clearer conceptions than could otherwise be obtained of the relative magnitudes of numbers. An additional evidence of the value of the principle is to be found in certain numerical combinations which are not additive, as such combinations mostly are, but subtractive. Thus, if a conception be formed of 10, and again of 20, 19 and even 18 will connect themselves more readily with the latter than with the former, and so we have such forms as duodeviginti and undeviginti; and there is little doubt that similar considerations, in combination with a regard for brevity of expression, have led to the use of the subtractive forms IX., XL., &c.

2. There are two ways in which numbers are represented, —either in words or by particular characters or symbols. It is with the latter that arithmetic has more especially to do, the numerous important and interesting questions that relate to word-numerals falling rather within the domain of language. It is only in so far as they arise out of the systems of grouping already referred to, suggesting in their formation such processes as addition or multiplication, that these call for any notice here. Dr Tylor {Primitive Culture, chap, vii.) gives a variety of remarkable collocations of this sort, as well as of descriptive numerals, that are in use among different races and tribes.

There is also a common use of characters to represent numbers which does not properly belong to this subject. The letters A, B, C, &c., or α, β, γ, or the like, the order of whose succession is fixed, are often employed to indicate numerical order, either singly or in combination with numerals proper. This is not, however, an arithmetical use of the characters; they are merely ordinals, and cannot furnish a basis for calculation.

The origin of the various characters which have been employed to indicate numbers proper, and by means of which computations can be effected, is in most cases alphabetical. In the Hebrew and Greek notations, for instance, the letters were taken in the main in their alphabetical order, being divided into three groups, of which the first represented units, the second tens, and the third hundreds; and very similar Systeme have been in use among other nations. As the Hebrew alphabet contained no more than twenty-two letters, the numbers from 500 to 900 were represented by five final forms, and sometimes by additive combinations with ∏, 400. The following is the usual Hebrew notation, numbers exceeding ten being made up by placing the larger numeral characters first :— [table]

The ordinary Greek notation has not merely a general resemblance of structure to the Hebrew, but corresponds to it closely, character for character, up to 80. The Greek alphabet consisting of twenty-four letters, three additional characters were introduced. The first, for 6, occupying the place of 1, is ς- (named στaυ), which was afterwards used as a contraction for στ; the other3, for 90 and 900, were named κόππα and σαμπι respectively, and wτitten? or and ft). The notation is thus as follows :— [table]

To distinguish the numeral letters an accent was written after the last,—thus, 38 was λη, while thousands were indicated by writing the accent below,—thus z δ was 4000. The letter M (for μυpioi) increased the numeral 10,000 times. Fractions were separated by a space from the integers they were affixed to, and the denominator was written like an index or power in our notation,—thus, μγ θ ιa was 43 1 ¾ r .

In addition to this another entirely different notation is found in Greek inscriptions. It more nearly resembles the Roman system, the numeral characters (except the first) being the initials of the numeral names, and being repeated till the number to be signified was expressed. I represents, 1, ∏ (πAτ∈) 5, A (δtκα) 10, Η (cκaτo'v, the aspirate being written as an H) 100, X (χίλιοι) 1000, M (∕ιυpioι) 10,000. Characters enclosed within three lines (forming a ∏) are thereby multiplied by 5. To give an example, in this notation ∣ II ∣ HH ]~∆1 ΔΠIII represents 768.

The Roman system, with which we are still familiar nearly in its completeness, employs the letters of the alphabet, but-is not based upon alphabetical order. Many attempts have been made to account for the symbols, and the complete solution of the problem of their origination is perhaps unattainable. Sir John Leslie, following some writers of the 16th and 17th centuries (see in particular the Cursus Mathematicus of Dechâles, vol. i. 1674, 2d. ed. 1690), advocated the opinion that, one line or stroke being taken to represent the unit, when ten of these were set down, a stroke would be drawn across them in a slanting direction to cancel them, and the unit stroke and cancelling stroke would thus give the form X for 10; that a repetition of this proceeding would give a third stroke when 100 was reached, and the three might take the form C or C; and that, similarly, the combination of four strokes would give M for 1000. This explanation is perhaps too ingenious. It has the merit of accounting for the X (the crux of the method), which may have been introduced in some such way. But one does not readily see Iιow the CZ could be formed from the X, or the M from the Q i and it appears far more likely that the signs for 100 and 1000 are merely the initial letters of Centum and Mille, — all the more that the very ancient notation noticed above as found in Greek inscriptions has evidently an origin of this kind. In any case V, L» and D appear to be respectively the halves of X, the angular C (C), and the rounded M (co). The ancient forms of D and M, viz., IO and CIO, have ceased to be familiar. By an extension of this style of characters, IOO denoted 5000; CCIOO, 100,000; CCCIOOO, 1,000,000, &c. To represent two, three, <fcc., millions, the CCCIOOO was repeated the required number of times.

The Roman notation employs fewer characters than the Greek, and makes greater use of combinations. One is repeated up to four; to the new character for five, ones are repeated up to nine; ones are added to the ten character; at fifteen the five enters, and so at twenty-five, &c.; for the tens up to forty the ten is repeated, and so on,—the symbols I, V, X, L, C, D expressing all numbers by regular combinations up to M, a thousand. The subtractive collocations, IV, IX, XL, XC, now the ordinary, were originally alternative forms, as were also the rarer combinations IIX for 8, XIIX for 18, XXC for 80, <tc. To the extent of these subtractive forms, the values of the characters depended on their position, a smaller number being added to a greater when it followed, and subtracted from it when it preceded it. This element of position is, however, an irregular and exceptional one; and, instead of being of such advantage as the local value of the Arabic notation, is rather a hindrance in calculation.

Far superior to all the ancient systems, and indeed to every other system that exists, is the arithmetical notation that is in common use. The Arabic numerals, aa they are called, are ten in number, nine of them representing the first nine numbers, and the tenth, the cipher or zero, indicating the absence or negation of numerical value. The significance of these figures or digits depends on their relative position, and the great merit of the system is due to this element of local value. Standing singly, the figures denote simply one, two, three, &c.; but in combinations of them every removal towards the left increases the value of the figure ten times. In 5673, for instance, the 3 denotes three, the 7 seven tens (70), the 6 six times ten tens (600), the 5 five times ten hundreds (5000). Should any of the series of tens be wanting, as in nine thousand and forty-eight, where hundreds do not occur, the place is supplied by a cipher, which throws back the digit that expresses thousands into its proper place, thus 9048. It is by this use of the cipher in supplying blanks, and so regulating the places of the significant digits, that the principle of local value is carried out,. See further the sections on notation and numeration below, p. 527.

The ordinary numerals are called Arabic, and it appears to have been through the Arabians that they were introduced into Europe; but they are now generally acknowledged to be of Indian origin. As may be imagined, they have passed through a great variety of forms, one of the earliest types of them being the Devaηagari, a species of Sanskrit numerals. In the early Indian treatise of Bhascara and Brahmegupta, translated by Mr Colebrooke (see article Algebra, p. 517 of vol. i,), as well as in the still earlier writings of Arya-Bhatta, there are both indications and illustrations of the use of the nine digits and the cipher; and the system is thus traced back to the 7th, and in all probability to the 5th century of the Christian era. Even then it was evidently no novelty, but is alluded to as holding an established position; and the Hindu writers nowhere lay claim to the invention of it, but constantly assign to it a supernatural origin. The method was known to the Arabians in the 9th century; and in the course of the 10th it seems to have come into general use among them, especially in their astronomical writings and tables. It was probably in the following century that the Arabs introduced the notation into Spain; but in regard to this we have no explicit information, and different accounts are given of the earliest instances of the use of the system in Europe. On the one hand, it is alleged that the figures first occur in a translation of Ptolemy, of the date 1136, while others maintain that they were introduced (about 1252) by means of the celebrated astronomical tables published by and named from Alphonso the Wise. That their use was known in Italy at the commencement of the 13th century, appears to be satisfactorily established, for there is no good reason to doubt the genuineness of the MS. writings of Leonardo of Pisa, copies of which have been found bearing the dates 1202 and 1220. Numerous other instances are given of the early use of the nine figures and the cipher, especially by astronomers, and in calendars. The great superiority of this to earlier modes of numerical expression became gradually apparent, and in course of time it came into almost universal use among civilised nations.

For a time there was, not unnaturally, considerable inexactness or confusion in the employment of the notation. In early writings such combinations are found, for example, as X2 for 12, 301 for 31, &c. In the latter case the law of local value is lost sight of, and the characters 30 are used as equivalent to thirty, irrespective of their position.

3. Calculation or computation by means of numerical characters is what is ordinarily regarded as the distinctive province of arithmetic, and the worth of a system of notation is to be estimated by the facilities it affords for the operations of reckoning. The methods in common use will be detailed, and the principles on which they depend briefly expounded, in subsequent sections of this article.

Computation of a comparatively rude kind was often carried on in ancient times, and is practised still in some countries, by what are called palpable methods,—as, for instance, by means of counters, or by balls strung on rods or running in grooves. Of this the abacus of the Romans (used too, but not so generally, among the Greeks), the swan-pan, still in constant use among the Chinese, &c., aro examples. (See Abacus, vol. i. p. 4.)

The methods that preceded the adoption of the Arabic numerals were all comparatively unwieldy, and very simple processes involved great labour. The notation of the Bomans, in particular, could adapt itself so ill to arithmetical operations, that nearly all their calculations had to be made by the abacus. One of the best and most manageable of the ancient systems is the Greek, though that too is very clumsy, as may be seen from an elementary example.

Let it be required to muliply 862 by 523, i.e., <Mβ , by φκγ'. The product of ω and φ is 400,000, i.e., ∕χ κ ; of ξ and φ, 30,000, or γ κ ; of β ω ⅛ ß and φ, 1000, or y α∙ Similarly the multipli- Φ κ y cation of ω by κ gives 16,000, or a M/ r; of V*y*μ ξ by κ, pσ' l of β by κ, μ; of ω by γ, βυ', a *,S^ of ξ by γ, ρπ; and of β by γ, These aσ μ may be arranged in some such order as in β v the margin, and summed. The result is pπ 5^

μ lt ε lt ωκ√, or 450,856.

∕x m ∈ m ωκy

The notation employed here extends to hundreds of millions. A scheme proposed by Archimedes, the most distinguished mathematician of antiquity, in his work entitled ⅛aμμhrη<s, Ar enamus, goes far beyond this. Taking the limit of the ordinary system, viz., a myriad of myriads, or 100,000,000, as the basis of a new scale, he treats of numbers formed by the continued product of that amount repeated eight times, extending to sixty-four places in our notation. These he divided into what we should call periods, of eight figures each, to which he gave the name of octades. He thus appears to have to some extent anticipated the modern method of grouping, though, from the want of knowledge of the principle of local value, the practical advance he made was unimportant. Apollonius of Perga, who flourished a little after the time of Archimedes, took the myriad as the basis of his system, and repeated the sign Mυ for each product of a myriad; thus λδMυ.Mυ.Mυ indicates what we would write as 34,000,000,000,000. But his most important contribution to arithmetical science was his grouping the tens in multiplication, so as to connect large numbers, as far as possible, with those we represent by the nine digits. He thus endeavoured, and with some measure of success, to remedy the defect of the general system arising from the want of apparent connection of such characters as λ and τ, for example, or π and ω, with what he called their bases, γ and η. It is chiefly in the commentaries of Eutocius on the works of Archimedes and Apollonius that the examples of the ancient Greek arithmetic which we possess are preserved.

The operations of the Greek arithmetic involving fractions were necessarily very complicated. A simpler system was introduced (by Ptolemy, according to his commentator Theon, but it appears to have existed before his time) in what are known as sexagesimals, which are precisely analogous to the duodecimals by which we sometimes calculate areas. The division of the circle into 360 degrees arose, no doubt, from 360 approximating the number of days in the year, and containing a large number of divisors. The radius of the circle, or side of the inscribed hexagon, subtended at the centre 60 of these degrees, and from this the degree was again divided into 60 parts called minutes, the minute into 60 seconds, <fcc., just as the foot is divided into 12 inches, the inch into 12 seconds, &c., in duodecimals. The sexagesimal system, though applied in the first instance to circular, was also employed in linear, measurements. As an example, the square of λζ δ v∈, i.e., of 37° 4' 55', will be found to be z ατoε δ ιδ ι κε, i.e., 1375° 4' 14" 10" ’ 25"". The numerals beyond 60 or £ being dispensed with in this system, the next letter, o, was used to denote zero; and it has been conjectured that this may have been the origin of the form of our cipher.

The most important step in the progress of modern arithmetic was the introduction of decimal fractions, and the extension of the Arabic notation to the expression of them. The first writer who advocated and exemplified the use of decimals was Simon Stevin de Bruges, better known as Stevinus, in a paper (Zα Disme) published about 1585; but he employed an awkward notation, and it was only after a considerable period that they assumed the form— entirely consistent with the notation of integers—with which we are familiar. From a form which Lord Napier employs in his Rabdologia (1617), the introduction of the decimal point (as a comma) has been ascribed to him, but apparently without sufficient evidence.

The following writers on arithmetic may be named, in addition to those already mentioned:—Diophantus, who flourished in or about the 4th century; Maximus Planudes, who died about 1350; Lucas Pacioli (de Burgo, or di Borgo), whose Summa de Arithmetica (1494) was the first work on algebra printed, and one of the earliest on arithmetic; Bishop Tonstall, whose De Arte Supputandi (1522) was the first work on the subject printed in English; Robert Recorde (died 1558), whose Grounds of Arles and Whetstone of Witte were arithmetical works of great value; Nicolo Tartaglia (died 1559); Michael Stiefel or Stifelius (1486-1567), said to have been the inventor of the signs + and Peter Ramus (1515-1572); Albert Girard (died 1634); William Oughtred (died 1660); Edward Cocker, whose Arithmetic (1st ed. 1677), a posthumous work, and probably a forgery, is of slight merit, though it passed through many editions; Kästner (Geschichte der Mathematik, 4 vols. 1796-1800); Montucla (Histoire des Mathématiques, voL...