On the Systems of Numerical Signs used by different Nations, and on the Origin of the Expression of Value by Position in the Indian Numbers.

By Alexander von Humboldt.

[Read in a Class-Session of the Royal Academy of Sciences, in Berlin, the 2d of March, 1829.] Translated from the German, and communicated by J. W.

In our researches upon numerical figures (the only hieroglyphics which, among the nations of the old continent, have been preserved, besides the alphabetical figures used to express the different sounds of spoken language,) our attention has, hitherto, rather been directed to the characteristic physiognomy of the figures and their peculiar formation, than to the spirit of the methods by which human sagacity has succeeded in expressing quantities with a greater or less degree of simplicity.

These researches have been entered into with views as narrow and as contracted as those made on languages.

The latter have, for a length of time, been compared rather according to the frequency of certain sounds and terminations, or to the form of their roots, than to the organic formation of their grammars.

For many years I have been occupied constantly, and with particular predilection, in endeavouring to bring under a general view the different systems of numerical figures used by the different nations of ancient and modern times, and in this way I have succeeded in throwing some light on the origin of what is called the Arabic numerical system. Many circumstances concurred to enable me to effect it. I myself have acquired, on my travels, a knowledge of the numerical systems of the Aztekes (Mexicans), and of the Muyscas (the inhabitants of the elevated plain of Cundinamarca); Thomas Young made the discovery of the Egyptian numerical figures, which (as is now known) do not, all of them, express the multipla of the groups by juxtaposition; the Arabic Gobar, or Dust-figures, were discovered by Silvestre de Sacy, in a Parisian manuscript; the peculiar nature of this system induced me to compare it with the numerical figures of the Mexicans and Chinese; by the publication of a great number of Indian grammars we have lately obtained the certainty, that in India within the Ganges, as well as without that river, not only different figures and alphabetical signs are used in expressing numbers, but also that the systems themselves of computation differ from one another,--in some of them value being expressed by position, in others not; lastly, a peculiar Indian method, till now quite unknown, has been preserved and discovered in a scholion of the Greek monk Neophytus.

In an Essay composed by me, and read in the Session of the Academie des Inscriptions et Belles Lettres, in 1819, I have already tried to shew, that some nations shorten the unartificial method of juxtaposition by writing exponents, or indicators over the figures, (as it is used by the Mexicans, in the ligatures of four times 13, or 52 years, by the Chinese, the Japanese, and the Hindoos, speaking the Tamul language,) and also the manner in which, by means of these indicators, and the suppression of the group-signs written horizontally or vertically, the excellent Indian system, expressing value by position, might have been arrived at.

The spreading of this system must considerably have been favoured by the very ancient use of a particular contrivance, the strings of reminiscence and calculation.

These are either loose cords, as the quippos of the Tartars, Chinese, Egyptians, Peruvians , and Mexicans, which have been transformed into Christian beads, or ritual calculation-machines; or they are extended and fixed in a frame.

In the last form they are found in the Suanpan, used in all the internal countries of Asia, in the abacus of the Romans and ancient Tuscans , and in the instruments of the palpable arithmetic used by the Slavonian tribes .

On the opinion that the numerical figures of the Muyscas (which at the same time are the hieroglyphics of the moon-days of the increasing age of the moon) have some connexion with the face of the moon, increasing by degrees according to its different phases, see Humboldt, Vues des Cord.

et Monumens des Peuples indigenes de l'Amerique, t. ii., p. 237--243.

Pl. XLIV.

On the quippos used for counting the sins in the confessional, see Acosta, Hist.

Natural de las Indias, lib. vi., cap. 8. El Inca Garcilaso, lib. vi., cap. 9. Freret, Mem. de l'Acad. t. vi., p. 109.

Klaproth's Asiat.

Magaz., t. ii., p. 78.

Otfried Müller's Etrusker, t. ii., p. 318.

In the Russian language chotki signifies the beads, and choty the calculationboard, with its fixed cords.

The files of cords or wires in the very simple Asiatic Suanpan represent the higher or lower groups of a numeral system, whether tenths, hundredths, and thousandths, or, in the sexagesimal division, degrees, minutes, and seconds.

The manner of calculating is always the same.

The pearls or beads again are the indicators of the groups, and an empty file indicates a cypher, consequently the empty sunya (sansu), sifr, or properly sifron, sihron (Arabic, according to Meninski prorsus vacuum).

I am not able to prove in a strict historical manner, that the Indian system of nine figures, indicating different values by the places they occupy, has been invented in the way I have indicated, but I believe I have discovered the way in which this invention may have been gradually brought about.

That, perhaps, is as much as can be done: for the history of the first steps in the development of the mental faculties, and of the scale of civilization, is involved in darkness; and commonly imparts to us nothing more than the knowledge of possibilities.

But it is for that very reason that this part of history is so interesting.

Only a short extract of the Essay read by me in the Academie des Inscriptions has been published, and that in a place where it hardly will be sought for . The manuscript itself is in the hands of M. Champollion, who intended to publish it, with some new and important facts respecting the different methods of the Egyptian figures discovered by him in Turin.

Since that time I have continued, from time to time, to render my first work more complete; but as I have no hope left to find the necessary leisure for publishing it in its whole extent, I intend to put together in this essay the most important results.

Perhaps this is also a favourable time for doing it; for, now that the researches respecting languages and monuments have taken a different and more useful direction, and that our intercourse with the nations inhabiting the southern and eastern countries of Asia is on the increase, it may be of some utility to bring into discussion problems, which are in close connexion with the intellectual march of the human mind, and (by the latest comparisons of numerical hieroglyphics, and the simple graphic method,) with our glorious progress in the mathematical sciences.

One of the greatest mathematicians of our times and of all times, the author of the "Mecanique Celeste ," says, "The idea of expressing all quantities by nine figures, whereby is imparted to them both an absolute value, and one by position, is so simple, that this very simplicity is the reason for our not being sufficiently aware, how much admiration it deserves.

But it is this simplicity, and the facility which calculations acquire by it, that raises the arithmetical system of the Indians to the rank of the most useful inventions.

How difficult it was to discover such a method may be inferred from the circumstance, that it has escaped the talents of Archimedes and of Apollonius of Perga, two of the most profound geniuses of antiquity."

The following observations, I hope, will shew, that the Indian method of calculation may very possibly have arisen by degrees out of others, which were used earlier, and even now continue to be used, in the eastern countries of Asia.

Gay Lussac et Arago, Annales de Chimie et de Physique, t. xii., p. 93, among the Monthly Notices of the Transactions of the Institute.

Humboldt, Essais Pol. sur la Nouvelle Espagne (2d. edit.), t. iii., p. 122--124.

Laplace, Expose du systeme du Monde (5th edit.), p. 325. The assertion of Delambre, (Histoire de l'Astron. Ancienne, t. i., p. 543,) in his contest respecting the merit of the ancient Indian arithmetic, as it is explained in Bhascara Acharya's Lilawati, is in very strange opposition to the opinion of Laplace.

But language alone can hardly prompt me to suppress the figures of the groups.

As language, in general, affects the manner of writing, and writing again reacts on language, but only under certain conditions which have been inquired into by Silvestre de Sacy, and my brother, so also the different modes of computation used by different nations, and their numerical hieroglyphics, reciprocally act upon one another.

Yet no very great consequence is to be attached to this alternate action.

Numerical figures do not always follow the same groups of unities like languages; and moreover in languages we do not always discover the same resting-points (the same quinary intermediate stops) as in numerical signs.

But if we bring together under one view, the language (names of numbers) and the numerical figures used in the remotest parts of the earth, as the common product of human intelligence applied to quantitative relations, we discover in the written numbers of one race, the isolated seeming peculiarities of language of another race; we may even add, that a certain awkwardness in the numerical command of language and writing is a very false standard of what is called the state of cultivation of mankind.

Here the same complicated and contrasted relations to each other take place, as with nations, of which some possess an alphabet, or mere ideographical signs, some have the most luxuriant abundance of grammatical forms and flexions, rising out of the root in systematical progression, whilst others use languages destitute of forms and flexions, as if benumbed in their very birth; and all this in the most different gradations of intellectual culture and political institutions.

So the one race of mankind finds itself driven in the most opposite directions by the alternate action of the internal and external world, (an alternate action whose first decisive efforts are wrapped up in the mythological darkness of the remotest antiquity); keeping most stedfastly to its old nature, even when great revolutions of the world bring geographically near to each other races the most heterogeneous in language:

whilst the similarity of the sounds which re-echo from the remotest zones, in grammatical forms of language, and graphic attempts to express large numbers, proves the unity of the old stock--the ascendancy of that which springs out of the internal intelligence, out of the common organization of mankind.

Travellers observing that some nations formed heaps of five or twenty pebbles or grains of seed, when they were about to make a computation, asserted that these nations were not capable to count farther than till five or twenty .

Pauw, Recherches Philos. sur les Americains, t. ii., p. 162. (Humboldt, Monumens Americains, t. ii., p. 232--237.)

As well may it be asserted that the Europeans are not capable of counting farther than ten, because seventeen is compounded of seven and ten.

In the language of the most cultivated nations of the west, for instance in those of the Greeks and Romans, some expressions are yet preserved, which refer evidently to such heaps or groups, psephizein, ponere calculum, calculum detrahere.

Groups of unities procure resting-places in counting; and as in the different nations the members of the body are similarly formed (the four extremities are divided fivefoldly) they stop either after having counted the fingers of one hand, or those of both, or they add also the two feet to the hands.

This difference in proceeding produces different resting-places, and thus are formed groups of 5, 10, or 20. It is, however, worth observing, that among the nations of the new continent, as among the Mandingas in Africa, the Biscayans, and the Gaelic tribes of the old continent, groups of twenty are in prevailing use .

Instances of such numeral groups of twenty unities are found in America among the Muyscas, the Otomites, the Aztekes, the Cora Indians, &c.

In the Chibcha language of the Muyscas, (who, like the inhabitants of Japan and of Thibet had an ecclesiastical and a laical chief; and whose method of intercalating the 37th month like the inhabitants of North India, has been published and explained by me ), 11, 12, 13, are called foot one (quihieha ata), foot two (quihieha bosa), foot three (quihieha mica), from quihieha or quhieha (foot), and the first three unities, ata, bozha or bosa, and mica.

The arithmetical signification of foot is ten, because the foot begins to be taken into account, when both hands are passed through.

To express twenty, the Muyscas use in their arithmetical language the expression foot ten, or the small house (gueta), perhaps because they used, in counting, grains of maize, and such a heap of maize reminded them of the barn, where maize was laid up.

By means of the expression small house (or barn), and twenty (both feet and hands) they formed the expressions for 30, 40, 80, by joining them together, as, twenty plus ten; twice twenty; four times twenty.

Quite similar are the Celtic expressions which have passed into the languages of Roman origin, as, quatre vingt, and quinze vingt, or those more rarely met with, as six vingt, sept vingt, huit vingt.

Deux vingt, and trois vingt are not used in French; but in the Gaelic or Celtic dialect of West Britany, through which I passed a few years ago, twenty is called ugent, forty daou-ugent, or two twenty; sixty tri-ugent, or three twenty.

It is even said deh ha nao ugent, or ten over nine twenty = 190 .

Monum. Americains, t. ii., p. 250--253.

The Muyscas had some stones covered with arithmetical hieroglyphs, which, by being placed in a certain order, facilitated to the priests (xeques) the intercalation of the ritual year.

The figure of a stone which served for that purpose may be seen in my work, Tab. XLIV.

Davies's Celtic Researches, 1804. p. 321. Legodinec Grammaire Celto-bretonne, 1807, p. 55.

In the Celtic or Kymrich dialect of Wales, 5 is called pump; 10 deg; 20 ugain; 30 deg ar ugain; 40 deugain; 60 trigain.

(William Owen's Dictionary of the Welsh Language, vol. i., p. 137.)

The same system of twenty unities is used in the language of the Biscayans:

bi is 2; lau, 4; amar, 10; oguai, 20; birroguai, 40; lauroguai, 80; berrgouai-tamar, 50, namely forty and ten (amar).

Larramendi, urte della Lengua Bascongada, 1729, p. 38. (The numeral figures of the Biscayan and Gaelic languages are not mixed together in my Monum. vol. ii., p. 237, but they are placed near one another to facilitate the comparison; by an error of print, however, les premiers is said instead of les deux or les uns et les autres.)

It is by no means a difficult task for me to give a still greater number of remarkable instances of the analogy existing between language and numerical hieroglyphics in juxtaposition, in the subtraction of unities by prefixing them, in writing, before the group, and in the intermediate landing-places of 5 and 15 in those nations who have adopted groups of 10 or 20.

In the languages of some very rude American tribes, for instance in those of the Guarinis and Lulos, six, seven, and eight are expressed by four with two, four with three, five with three.

The more civilized Muyscas say twenty (or house) with ten, instead of thirty.

The Cymri, in Wales, use in such a case deg (ten), or ugain (with twenty); as the French use soixante et dix, for seventy.

We find addition effected by juxtaposition, chiefly among the old Tuscans, Romans, Mexicans, and Egyptians; subtractive or diminishing expressions among the Indians in the Sanscrit , where 19 is called unavinsati, 99 unasata; among the Romans in undeviginti (unus de viginti), for 19; undeoctoginta, for 79; duodequadraginta, for 38: among the Greeks in eikosi deonta enos, 19; and pentekonta duoin deontoin, 48; that is, wanting two to make up fifty.

Such subtractive expressions have passed into the writing of numbers, and in such a case the figures expressing unities to be subtracted are prefixed to the signs of the groups of five and ten, and even to their multiplicates, for instance, to 50 or 100 (IV, and XI, and XL, and XT, for four-andforty among the Romans and ancient Tuscans; though in the last nation the numerical figures probably were entirely derived from the alphabet, according to the researches lately made by Otfried Müller).

In some rare Roman inscriptions collected by Marini , even four unities are found before ten, for instance, IIIIX=6.

We shall soon see, that among some tribes inhabiting the East Indies, methods are found in which the joining of figures, which among the Romans and ancient Tuscans indicated only addition or subtraction, expresses addition or multiplication, according to the place or the direction of the signs.

In these Indian systems, (to use Roman figures,) IIX is twenty, and XII is twelve.

M. Bopp quotes even 95 (or hundred --5) in the words pantschonam satam --contracted from pantscha (five) and una (less).

Otfried Müller, Etrusker, ii., p. 317--320.

Iscrizioni della Villa di Albano, p. 193. Hervas Aritmetica delle Nazioni, 1786, p. 11, 16.

In many languages the groups of 5, 10, and 20, are called a hand, two hands, and hand and foot (in the language of the Guaranis mbombiade).

When the fingers of both extremities, viz., fingers and toes, are gone through in counting, then it is considered that the number comprehends the whole body, thus: the word man becomes the symbol for twenty.

Therefore, in the language of the Yaruros (of which tribe I found populous villages, erected by the missionaries on the river Apure, which falls into the Orinoco) forty are called two men, noeni pume, from no eni two, and pume man.

It is known, that in the Persian language the word pencha signifies fist, and pendj five, derived from the Sanscrit word pancha.

"The last is (according to the ingenious observation of M. Bopp) to be considered as the original of the Roman quinque, as the Indian chatur of quatuor.

The plural of chatur is chatvaras, and approaches very near the Doric-AEolian tettares. The Indian ch is pronounced as the English in words of Saxon origin, but in Greek it becomes a t; therefore chatvaras is changed into tatvaras, as pancha into panta (the Greek pente, AEolice pempe, whence pempazein to count by fingers, or up to five).

But in the Latin language, the Indian ch is expressed by qu and thus chatur is changed into quatuor, and pancha into quinque.

Pancha itself never signifies in India hand, but only the number five.

But panchasakha is a descriptive expression for hand, as a limb with five branches ."

On the numerals in Sanscrit, compared with those of the Greek, Latin and Gothic languages, a very interesting essay, in manuscript, was communicated to me by M. Bopp, in Paris, in 1820; and it was composed by him for the purpose of being inserted in my work, On the Numerical Signs of Nations.

As in languages, and with peculiar naivete in those of South America, the groups of five, ten, and twenty, are distinguished by peculiar expressions, so, in the same manner, these groups are easily to be recognized in numeral hieroglyphics.

The Romans and Tuscans had single figures to express 5, 50, and 500.

In the language of the Aztekes (natives of Mexico), we find not only the group-signs of a flag for 20, a quill filled with grains of gold (which in some districts of Mexico were used as money) for the quadrate of 20 or 400, and a bag (xiquipilli) with 8000 cocoa nuts (which likewise served as a medium of barter) for the cube of 20 or 8000, but even (in the instances where the flag is divided into four equal parts, and the half or three quarters of it are coloured) numerical signs for half-twenty or 10, and for [Formel] twenty or 15, two hands and a foot, as it were .

Respecting the Tuscan figure for 500, see Otfried Müller, Etrusker, sec. iv., fig. 2.

Humboldt, Monum.

Americ., tom. i., p. 309.

But the most remarkable of all proofs of the alternate effect between numerical language and numerical figures, is to be met with in India.

Here the value of the unities expressed by their position has, in Sanscrit, been even transplanted into the language.

Thus the Indians have a figurative method of expressing numbers by the names of objects, of which a certain number is known.

For instance, surya (the sun) expresses 12, because, according to the mythology of the Hindoos, there were twelve suns in the order of the twelve months.

The two Aswinas (Castor and Pollux), who also are met with in the Nakchatras, are used to express the number two.

Manu signifies fourteen, taken from the menus of their mythology.

Now it will become intelligible, how the word surymanu, in which the symbols of twelve and fourteen are combined, signifies the year 1214.

These facts I owe to the kind communication of the learned Colebrooke.

Probably, according to these principles, 1412 will be expressed by Manusurya, and 314 by Aswinimanu.

We find besides so complete a numeration in Sanscrit, that a single word, koti, is used to express 10 millions.

In the qquichau language of Peru, which does not count by groups of twenty, a single word stands for a million (hunu).

If it be true that we count by decimals, quia tot digiti, per quos numerare solemus, as Ovid says, we probably should have adopted a duodenary scale , if our extremities had been divided sixfoldly.

Such a scale has the great advantage, that its groups can be divided by 2, 3, 4 and 6 without leaving a fraction, and for that reason it has been adopted by the Chinese for their measures and weights, from the earliest times.

Debrosses, vol. ii., p. 158.

The preceding observations regard the relation between language and writing, between the numerals and their figures.

I shall proceed to consider the latter by themselves.

I remind the reader that this essay is only an extract from my larger and unfinished work, and that I shall not speak of the heterogeneous forms of the simple elements (numerical) figures, but of the spirit of the different methods of expressing numeral quantities, which are adopted by different nations.

I, therefore, shall only take notice of the figure or form of the numerical signs, whenever it affects my conclusions respecting the identity or heterogeneousness of the methods themselves.

The mode of proceeding adopted for the purpose of expressing the pure or mixed multipla of the denary fundamental groups n (for instance 4n, 4n2, or 4n + 7, 4n2 + 6n, 4n2 + 6n + 5) is very different in different nations.

Sometimes it is effected by forming a row (that is by giving different value to the figures according to their position), after the manner of some nations of the Hindoos; sometimes by mere juxtaposition, as among the ancient Tuscans, the Romans, Mexicans, and Egyptians.

Some nations use for that purpose coefficients, as that tribe of the Hindoos, inhabiting India within the Ganges, which uses the Tamul language; others use exponents or indicators, placed over the figures of the groups, as the Chinese, Japanese, and also the Greek in employing myriads; others again, use an inverted method.

These place over the nine numerical figures a number of cyphers or points, to indicate the relative value of them, and in this system, the cyphers or points are the signs of the groups placed over the unities.

The last method is found in the Arabic Gobar writings, and in an Indian numerical system, preserved and explained in a scholion of the Greek monk Neophytus.

These five different methods are in no way dependent on the forms of the numerical figures, and to prove this fact more evidently, I shall, in this treatise, employ no other signs than the arithmetical and algebraical, which are in common use.

In this way the attention is more directed to the essential point, the spirit of the method.

I already have used such a mode of pasigraphic notations in another work, treating of a very heterogeneous subject, the regular stratification and often periodical filation of the minerals (in the appendix to the Essai Geognostique sur le Gisement des Roches) , and there I have tried to shew, how by this method our abstract views on an object can be rendered more general.

Proceeding in this way, all observations respecting the form and composition of individuals, which in themselves may be very useful and true, but would turn off the attention, are suppressed, in order to place in a purer and clearer light a phenomenon, which we wish to investigate in preference to others.

This is an advantage which in some way may be justified by the chilly insipidity of such abstractions.

Ed. 1823, p. 364--375.

As regards the manner of writing numbers adopted by different nations, it is usual to distinguish the figures not depending on alphabetical letters, from the alphabetical letters which indicate numerical value, either by forming a certain row, or by short lines or points added to them; or lastly, by the initials of the numerals (in reference to language) .

The Arabic figures called Diwani, are compounded of mere monograms or abbreviations of the initial letters of the numerals, and are the most complicated instance of such initials.

Whether the C and M, used by the ancient Tuscans and Romans, are, in fact, initials taken from the Tuscan and Roman languages, is much more subject to doubt than is commonly believed.

(Leslie, Philos. of Arith., p. 7 --9, 211. Debrosses, t. i., p. 436. Hervas, p. 32--35.

Otfried Müller, Etrusker, p. 304--318.)

The cross at right angles, used by the Greek, and quite similar to the Chinese figure of ten, signifies in the most ancient inscriptions a thousand (Boeckh, Corpus Inscript.

Graec., vol. i., p. 23), and is only the oldest form of ch. (Nouveau Traite de Diplom. par deux Religieux de St. Maur, vol. i., p. 678.)

It is beyond all doubt, that the nations of Hellenic, Semitic, and Aramaic origin (among the last even the Arabs, before they received the numerical figures from the Persians, in the fifth century of the Hegira ), used the same signs to express letters and numerals, even in the period of their ripe civilization.

On the other hand we find, in the new continent, at least two nations, the Aztekes and the Muyscas, who used numerical figures without having letters.

Among the Egyptians, the numerical hieroglyphics commonly used to express units, tenths, hundreds, and thousands, seem likewise to have had no relation to the phonetic hieroglyphics.

Quite different from the alphabet are likewise the ancient Persian figures of the Pehlwi, for the first nine unities, as those of the ancient Tuscans, the Romans, and even the Greeks in the most ancient times.

Anquetil has already observed , that the alphabet of the Zend language, which, being composed of 48 elements, could have facilitated the expressing of numbers, by alphabetical letters, never uses them as numerical figures, and that in the books written in the Zend language, numbers are always expressed by the figures of the Pehlwi, and at the same time with the numerals of the Zend language.

Should, by further researches, this want of numerical figures in the Zend language be confirmed, it must induce us to suppose that the Zend nation, which, in its language, discovers a very intimate affinity to the Sanscrit, must have been separated from the Hindoos, before the last had arrived at expressing value by position.

But, the nine unities excepted, the figures of the groups of ten, hundred, and thousand are derived from letters in the Pehlwi language.

Dal is 10; re and za, connected together, 100; re and ghain, connected together, 1000.

When we take together, in one view, all that is known to us of the numerical figures used by the different nations, little as it is, it seems sufficient to oblige us to confess, that the distinction of alphabetical numerals and of numerical figures, independent of the alphabet, is as uncertain and useless as that of the languages in such as are composed of monosyllables, and in those using polysyllabic words, a distinction long ago abandoned by truly philosophical investigators of languages.

The numerical figures used by the inhabitants of some southern districts of India within the Ganges, who speak the Tamul language, do not express value by position, and are quite different from those used in Sanscrit manuscripts, if the figure of 2 is excepted.

Who can say whether these figures are not to be derived from the Tamul letters?

It is true the figure for the group of a hundred is not to be found among these letters, but the sign of the group of ten is to be recognized in the letter ya, and the two in the letter u. The numerical figures in the Teloogoa language, likewise spoken in the southern districts of India within the Ganges, which indicate value by position, are in a strange manner different from all other Indian figures, as far as they are known, in the signs for 1, 8, and 9, whilst they agree with them in the figures of 2, 3, 4, and 6. The necessity of expressing quantities by writing has probably been felt sooner than that of writing words, and therefore we may consider numerical figures as the most ancient written characters.

Silvestre de Sacy, Gramm.

Arabe, 1810, t. i., p. 74, Note 6.

Mem. de l'Acad. des Belles Lettres, t. xxxi., p. 357.

Campbell's Grammar of the Teloogoa language, (Madras,) 1816. p. 4. 208. This is what formerly, but falsely, was called the Gentoo language.

By the natives it is named Trilinga, or Telenga.

With the table of the numerical figures in Campbell's grammar, other varieties of Indian numerical characters to be found in Wahl's General History of the Oriental Languages, 1784, Tab. I. may be compared.

The instruments of palpable arithmetic, as they are called in an ingenious work, the Philosophy of Arithmetic, by Mr. Leslie, (1817,) in opposition to the figurative or graphic, are both hands, heaps of pebbles (calculi, psephoi), grains of seed, loose strings with knots (arithmetical strings, the quippos of the Tartars and Peruvians), the suanpan put in a frame, the table of the abacus, and the calculating machine of the Slavonian nations with balls or grains of seed in files.

All these instruments furnish to the eye the first graphic notations of groups of a different degree.

One hand, or a string with knots or moveable balls, indicates the unities up to 5, or 10, or 20. How often, by shutting of the single fingers, one hand is gone through (pempazesthai), is indicated by the other hand, of which every finger, that is, every unit, expresses a group of five.

Two loose strings with knots stand in the same relation to one another.

The calculation-strings, with moveable balls, extended and fixed in a frame, or the ancient Asiatic suanpan, which, in very ancient times, (perhaps by the Egyptians at the time of the Pythagorean league,) was brought to the nations of the west as abax or tabula logistica, are used in the same manner, only that the strings indicate groups of second, third, and fourth order upwards and downwards.

The kouas, which are more ancient than the characters used at present by the Chinese, and even the magic drawings (ralm) of central Asia and Mexico, which exhibit knotty parallels, often broken off, almost in the manner of musical notes, seem only to be graphic projections of these calculation and reflection-strings .

In the East, ralm is called the negromantic art of the sand.

Uninterrupted lines, and others broken off, present the elements of it, and direct the negromant.

(Richardson and Wilkins, Diction.

Persian and Arabic.

1806. t. i., p. 482.)

In Dresden a remarkable manuscript is preserved.

It was brought from Mexico, and exhibits nothing but figures like musical notes.

I have published it in my Monumens Americ.

Pl. xliv. When at Paris, I was visited by a learned Persian, who, at the first view, recognized in it an oriental ralm.

Quite similar, and truly American kouas and linear drawings, I discovered after that time in some Aztetic hieroglyphic manuscripts, and on the sculptures of Palenque, in the republic of Guatimala.

In the ancient Chinese numerical figures, the sign of the group of ten is a pearl on a string, and evidently an imitative drawing of the quippu.

In the Asiatic suanpan, and in the abacus, which was much more used by the Romans, on account of the inconvenient figures adopted by them, than by the Greeks, who had been much more successful in their mode of writing numbers , the quinary rows are preserved together with the denary ones, forming geometrical progressions upwards and downwards.

Outside of every calculation-string, indicating a group or order, (n, n2, n3,) a shorter string was placed, on which every ball expressed the amount of five balls of the longer string. By this contrivance the number of the unities was diminished to such a degree, that the principal string needed only to contain four balls, and the accessary only one .

Nicomachus, in Ast.

Theologumena Arith. 1817, p. 96.

In the financial system of the middle age, the account table (abax) became the exchequer.

So the Roman abacus was contrived.

In China five balls were placed on the first, and two balls on the last.

The balls, which were not employed in a calculation, were pushed aside.

It seems that among the Chinese, from the most ancient times, the custom had prevailed to consider arbitrarily any one of the parallel strings as containing the units, and that thus they obtained, upwards and downwards, decimal fractions, entire numbers, and powers of ten . How late (in the beginning of the sixteenth century?) has the knowledge of the decimal fractions been introduced into the countries of the West, which the nations of the East had been taught long ago by their palpable arithmetic!

The descending scale from the unit downwards, was known to the Greeks only in the sexagesimal system for degrees, minutes, and seconds; but as they had not n--1, that is, 59 numerical figures, the value by position could only be expressed by layers of two figures.

On the first attempts to establish the decimal system, made by Michael Stifelius of Eslingen, Stevenus of Brugge, and Bombelli of Bologna, see Leslie, Philos. of Arithmetic, p. 134.

If we direct our attention to the origin of numbers, we find that they could be written and read with great exactness, when they were indicated by heaps of pebbles, or by the balls on the strings of the calculation-machines.

The impression which these proceedings left behind on the mind, has everywhere affected the manner of writing numbers.

In the historical, ritual, and negromantic hieroglyphics of the Mexicans, published by me, the units up to nineteen (the first simple figure for a group is twenty) are exhibited as great round coloured grains, and, what deserves to be mentioned, they are counted from the right to the left hand, like the Semitic writings.

This is evident in 12, 15, 17, where the first row contains ten, and the second is not quite filled up.

In the most ancient Greek monuments, and in the Tuscan sepulchral inscriptions, the units are expressed by vertical lines; the same custom prevailed among the Romans and Egyptians (which, respecting the last, has been proved by Thomas Young, Jomard, and Champollion).

The Chinese use horizontal lines up to the number four; and such lines are likewise found in some Phoenician coins described by Eckhel (t. iii., p. 410).

The Romans sometimes omitted the quinary figure in inscriptions, and, therefore, we find even eight lines as unities placed together.

Many instances of this kind are collected by Marini, in his Monumenti dei Fratelli Arvali , a work which deserves attention.

The heads of the nails, which anciently were employed by the Romans to indicate the years (annales antea in clavis fuerunt, quos ex lege vetusta figebat praetor maximus, says the elder Pliny, vii. 40.) could have led them to the unity-points of the Mexicans, and, in fact, we find such points used in the subdivision of ounces and feet, together with the horizontal lines (used by the Chinese and Phoenicians) . These points and lines, nine or nineteen in number, in the denary or vicesimal scale of the old and new continent, are the most simple of all notations in the system of juxtaposition.

Here the unities are properly more counted than read.

The separate existence, the individuality, if I may say so, of the numerical figures as signs of numbers, is first to be recognized in the numerical letters of the Semitic and Hellenic tribes, and among the inhabitants of Thibet, and the Indian tribes, who express 1, 2, 3, 4, by ideographic, distinct figures.

In the ancient Persian Pehlwi a remarkable transition is to be observed from the inartificial method of expressing numbers by the repetition of the figure of the unity, to that of using compound ideographic hieroglyphics, in numbers greater than the unity.

There the first nine figures are evidently formed by as many notches or teeth as they contain unities; five and nine are even merely the numbers 2, 3, and 4 twisted together, without the repetition of the figure of one.

In the system of the Devanagari, which is truly of Indian origin, in the Persian and Arabic-European figures, we are only able to discover a contraction of 2 and 3 units, in the figures of 2 and 3 , certainly not in the higher figures, which in India within the Ganges are written very differently from one another.

T. i., p. 31.; t. ix., 675--for instance in Octumvir.

Marini, t. i., p. 228.

Abel Remusat Langues Tartares, p. 30.

On the strange numerical figures used in Java, see Crawfurd, vol. ii., p. 263.

As I mention here the Indian numerical figures, and shall be obliged to do it frequently in this essay, I feel myself bound to make some observations on this expression.

At the same time I shall take the opportunity of declaring myself against the old prejudices, that in India only one set of numerical figures are employed in expressing numbers, and never alphabetical letters in their place, and that in every district of that extensive country, a knowledge of a system, a different value to the different position of the figures is met with, likewise that there never are peculiar figures used to indicate the groups.

As, according to what has been repeatedly said by my brother William von Humboldt, the Sanscrit is not well distinguished by the name of "Indian and ancient-Indian language," because in that country many more very ancient languages are found which do not derive their origin from the Sanscrit; so likewise the expression, "Indian and ancient-Indian figures," is too indefinite, not only as far as it regards the form of the figures, but also respecting the spirit of the methods.

For in India the principal groups of n, n2, n3, and their multipla, 2n, 3 n ..... are sometimes expressed by juxtaposition, sometimes by coefficients, and sometimes merely by the place of the figures.

Even the existence of a distinct figure for the cypher, is, in the Indian system, no necessary condition of the method of expressing value by position, as it is proved by the scholion of the monk Neophytus.

In India within the Ganges the most extended languages are the Tamul and the Teloogou.

The tribes who speak the first use figures different from their alphabet, of which only two, the two and the eight, exhibit a slight similarity with the Indian (Devanagari) figures of two and five . Much more different from the Indian figures are those of the Cingalese . In both the Tamul and the Cingalese languages the different value of the figures is not indicated by position, they have also no distinct figure for the cypher, but distinct hieroglyphics for the groups n, n2, n3 ....

The Cingalese make use of juxtaposition, the Tamuls of coefficients.

In India without the Ganges, in the empire of the Burmese, we find the value expressed by position, and a distinct figure for the cypher, but the figures used by them do not resemble the Arabic, Persian, and Devanagari Indian figures . The Persian figures, used also by the Arabs, are all of them quite different from the Devanagari figures; 7 is like a Roman 5; 8 like a Tuscan 5. Among the figures which we call Arabic, only 1, 2, 3, resemble the figures of the Devanagari of the same value; the 4 of the Devanagari is our 8; our 9 is the 7 of the Devanagari.

Our 7 is the Persian 6.

In Bengali the 5 is expressed by the figure of a crescent, and 2, 5, 6, 8, and 9 are quite different from the Devanagari figures . The numerical figures of Guzerath are only distorted Indian Devanagari figures .

Robert Anderson, Rudiments of Tamul Grammar, 1821, p. 135.

James Chater, Grammar of the Cingalese Language.

Colombo, 1815, p. 135.

Carey, Grammar of the Burman Language, 1814, p. 196. Only the Burmese figures of 3, 4, and 7, resemble in some manner those of 2, 5, and 7.

Compare John Shakespear, Grammar of the Hindustani Language, 1813, p. 95, and Pl. I. William Jones, Grammar of the Persian Language, 1809, p. 93. Silvestre de Sacy, Grammaire Arabe.

Pl. VIII.

Graves Chamney Houghton, Rud.

of Bengali Grammar, 1821, p. 133.

Robert Drummond, Illustrations of the Grammatical Parts of the Guzerath and Mahratt Language, 1818, p. 25.

I shall make no observation on the influence of the earliest numerical figures on the form of the alphabetical letters, nor of the distortion of the letters purposely introduced in order to distingush them from the numerical signs, nor even on the difference of the place which a figure used in both respects occupies sometimes (as in the aboudjet of the Semitic tribes in Asia and Africa) . Such observations do not belong to the subject of this essay, and have been the origin of many groundless hypotheses in comparing the alphabetical letters with the numerical hieroglyphics.

I myself was once of the opinion that the Indian figures, notwithstanding the form of two and three, were the letters of an obsolete alphabet, of which yet some traces are found among the Phoenician, Samaritan, Palmyrian, and Egyptian characters, (the last on the mummies).

Even the old Persian monuments of Nakshi Rustan seem to exhibit them . How many characters in these inscriptions resemble in a striking manner the numerical figures known under the name of Indian!

Many other scholars have likewise asserted, that the numerical figures called the Indian, are derived from the Phoenician alphabet , and the sagacious Eckhel has already observed that the similarity between the letters of the Phoenicians and the numerical figures was so great, that the word Abdera is expressed by 19990, and by 15550 . But the origin of the numerical figures, as well as that of the alphabetical letters, is enveloped in an obscurity, to dissipate which, by a philological investigation, founded on historical facts, is rendered impossible by the scarcity of the materials, if we do not wish to content ourselves with a few negative results.

Silvestre de Sacy, t. i., p. 10.

Silvestre de Sacy, Antiquites de la Perse, Pl. I., n. 1. Compare the numerical inscriptions on Mount Sinai, in Description de l'Egypte, vol. v. Pl. LVII.

Guyot de la Marne in Mem. de Trevoux, 1736, p. 160; 1740.

Mars, p. 269. John, Bibl. Archeol. b. i., p. 479. Büttner, vergl. Tafeln, 1769, St. i., p. 13. Eichhorn, Einleitung in der alte Testam. b. i., p. 197. Wahl, Geschichte der Morgenl.

Litteratur, p. 601, 630. Fundgruben des Orients, b. iii., p. 87.

Doctrina Numerorum veterum, 1794, t. iii. p. 396--404, 421, 494.

We have seen that some nations, in expressing numbers by writing, mix together alphabetical letters and ideographic figures arbitrarily chosen.

Likewise we find, that respecting the mode of expressing the multipla of the fundamental groups the most heterogeneous methods are used.

We discover even that one system completely developes what, in another, is only slightly indicated.

The same incongruity obtains in languages.

In one language, some grammatical forms do appear only in a few instances, and are slightly expressed, whilst another has developed them with a peculiar predilection, and with every effort of mental power.

Should I, therefore, explain the numerical systems singly, as they are used by different nations, the similarity of their methods would be rendered obscure, and we should lose the track on which the human mind proceeding, at last arrived to discover the masterpiece of the Indian arithmetic, in which every figure has a double value, an absolute and a relative, of which the last is increasing, in a geometrical progression, from the right to the left.

In my following observations I shall therefore abandon the ethnographical order, and only consider the different means employed by nations to express in writing the groups of the units.

First method.

Juxtaposition is effected by simple addition in numerical figures as well as in alphabetical signs.

It was in use among the ancient Tuscans, the Romans, among the Greeks only up to a myriad, among the Semitic tribes, the Mexicans, and also in the greatest part of the Pehlwi calculations.

This method renders the computation extremely difficult, when the multipla of the groups (2n, 3n, 2n2 ....) are not expressed by distinct signs.

The Tuscans and the Romans repeated the figure of ten as far as fifty; the Mexicans, whose first figure of a group was 20 (a flag), repeated this hieroglyphic up to 400. The Greeks, however, have in the rows of the tenths and hundreds, which begin with iota and rho, distinct figures for 20, 30, 400 and 600. The three episemes (letters of an obsolete alphabet), bau, koppa, and sanpi, serve to express 6, 90, and 900. The two last terminate the rows of the tenths and the hundreds, and in this manner the numerical value of the Greek alphabetical figures approaches a little nearer to the semitic aboudjed . M. Böckh, in his learned observations on the digamma, has shewn that bau is the wau of the Semites, (the F of the Latins,) koppa, the Semitic koph (9), and sanpi the Semitic shin . The row of the unities beginning with alpha, and ending with theta, forms in the Greek system the oeot-numbers (puthmenes), and Apollonius had invented a contrivance , by the help of which they were reduced, in the last results, to the corresponding members of the second and third row (the analogues).

Hervas, Aritmetica delle Nazioni, p. 78.

On the ancient order of the letters in the Semitic alphabet see Description de l'Egypte moderne, t. ii. P. ii. p. 208.

Staatshaushaltung der Athener. B. II. p. 385.

Delambre, Hist. de l'Astronomie Ancienne, t. ii. p. 10.

Second Method. Multiplication or diminution of the value by signs placed over or under the figures.

In the fourth row of the Greek notations, the pythmenes return by analogy, but increased a thousand times by a line placed under the figures.

In this way the Greeks arrived, in their numerical system, at a myriad,--they wrote every number up to 9999.

Had they adopted this notation with a line for all the groups, and suppressed all the figures after theta (9), the letter b with one, two or three lines would have expressed 20, 200, and 2000, and thus the Greek system would have approached, as we shall see afterwards, the system of the Arabic Gobar figures, which is very little known, and at the same time the system expressing value by position.

But, unhappily, the Greeks did not adopt this notation for the tenths and hundreds, applied it only for the thousands, and did not try to employ it in higher groups.

As a line added under the figures increases their value a thousand times, thus a vertical line, in the Greek system, added over the figure, indicates a fraction, whose numerator is the unit, and whose denominator is expressed by the figure itself.

Thus in Diophantus, g' = [Formel] , d = m; but if the numerator is greater than the unity, it is expressed by the principal figure, and the denominator is written like an exponent g = 3/4 . In the Roman inscriptions, a horizontal line added over them, increases their value a thousand times, and here it may be considered only as a means of abbreviating and of saving space.

Delambre, t. ii. p. 11. The line added over the alphabetical letters to indicate that they are used as numerical figures, ought not to be confounded with the sign of fractions.

The first is also never vertical in the oldest manuscripts of mathematical works, but horizontal, and thus the mistaking of it for the sign of fraction is obviated. Bast. de Usu Litterarum ad Numeros indicandos, in Gregorii Corinthii Liber de Dialectis Linguae Graecae.

1811. p. 850.

The method of Eutocius for expressing myriads is more important.

In it we find among the Greeks the first trace of the system of exponents, or rather indicators, which rose to such a degree of importance in the East.

Ma, Mb, Mg, indicate 10,000, 20,000, 30,000.

Here we find these indicators used only with the myriads.

But the Chinese and the Japonese, which last received their civilization from the first about 200 years before our aera, both use them for the multipla of all groups.

Three horizontal lines under the figure of ten, signify thirteen, but if they are placed over it they express thirty.

According to this method, 3456 is written, (I use the Roman figures as signs of groups, and the Indian as exponents)

M3

C4

X5

I6

Among the Egyptians the same kind of indicators are found.

Two or four unities placed over a curved line, which denotes a thousand, are used to express 2000 or 4000 . Among the Aztekes, or Mexicans, I found for 312 years the sign of the vinculum, with six unities as exponent (6x52=312), and I have published it in my work of the American Monuments.

Among the Chinese, Aztekes, and Egyptians, the signs of groups are always under the exponent, as if [Formel] were written instead of 50; but in the Arabic Gobar figures the signs of the groups are placed over the indicators.

For in the Gobar system the signs of the groups are points, consequently cyphers; for in India, in Thibet, and in Persia, cyphers and points are identical.

These Gobar signs, which, since 1818, have attracted my peculiar attention, were discovered in a manuscript in the library of the old Abbey of St. Germain du Pres, by my friend and instructor, M. Silvestre de Sacy.

This great orientalist says, "Le Gobar a un rapport avec le chiffre indien, mais il n'a pas de zero ."

Kosegarten, de hierogl. Aegypt. p. 54. The assertion of Gatterer adopted by him from Bianchini (Dec. 1, cap. iii., p. 3), from Goquet (vol. i. p. 226), and from Debrosses (vol. i. p. 432), that among the Egyptians the figures received value by their position in a perpendicular row, has not been confirmed by modern researches.

Gatterer, Weltgeschichte bis Cyrus, p. 555, 586.

See Gramm.

Arabe, p. 76, and the observations added to pl. viii.

To me, however, it seems, that the figure of the cypher there is found; it is, however, not placed aside the figures, but over them, as in the scholion of Neophytus. It is, indeed, the signs of the cypher or the points, which have caused to be given to these characters, the strange name of gobar or dust characters.

He who sees them for the first time is doubtful whether they represent a transition from figures to letters, or not.

It is only with pain, that the Indian 3, 4, 5 and 9 can be distinguished.

Dal and ha are, perhaps, the Indian figures of 6 and 2 distorted.

The indication by means of points is thus effected:--

3 instead of 30,

4.. instead of 400,

6 instead of 6000.

These points recall to our memory a mode of notation used by the Greeks , but not frequently met with, and beginning only with the myriad.

Here, a .. is used for 10,000, and b :: for 200 millions.

One point, which, however, is never employed, serves to express 100 in this system of geometrical progressions.

In Diophantus and Pappus, a point is found between the alphabetical figures, instead of the initial Mu (Myriad).

In this method, therefore, a point multiplies to figures to the left 10,000 times.

It would seem that some obscure ideas of notation, by points and cypher, had been brought from the East into Europe by the Alexandrines.

The figure of the cypher is, indeed, used by Ptolomaeus, and even as an indication of something that is wanting.

He employs it in the descending sexagesimal scale to indicate the wanting degrees, minutes, and seconds.

Delambre even pretends to have found the figure of the cypher in the manuscripts of Theon, in his commentary on the Syntaxis of Ptolomaeus.

The cypher, consequently, was known in the west long before the invasion of the Arabs .

Ducange, Paleogra, p. xii.

Histoire de l'Astronomie Ancienne, t. i., p. 547; t. ii., p. 10. The passage of Theon is not to be found in his printed works.

Delambre is inclined to attribute the origin of the Greek figure of the cypher sometimes to an abbreviation of ouden sometimes to a peculiar relation in which the letter omicron stands to the sexagesimal system, l. c., t. ii., p. 14, and Journal des Savans, 1817, p. 539. It is strange, that in the old Indian arithmetic of the Lilawati, the figure of the cypher, placed aside a number, indicates, that such a number is to be subtracted.

Delambre, vol. i., p. 540. What does the Ling (the true cypher) signify, which in Chinese figures is written under 12, 13, 22, 132?

In Greek inscriptions the signs of cyphers indicate oboles.

(Bockh, Staatshaushaltung der Athener, b. ii., p. 379.)

Planudes, Treatise on the Arithmoi Indikoi, third method, Multiplication of the Value by Coefficients.

We have seen, that the Chinese, by placing indicators perpendicularly over or under the groups, indicate the difference between [Formel] , and [Formel] . The same effect is produced among the Greeks, Armenians, and those tribes of the Hindoos, who speak the Tamul language, by adding figures in horizontal direction.

Diophantus and Pappus wrote b M u for twice ten thousands, whilst a M ub (where b is placed to the right of the initials of the myriad) signifies ten thousands, plus two or 10,002.

In the same manner, the Tamul figures are used, as, for instance:

4X = 40, and X4 = 14.

In the ancient Persian Pehlwi, according to Anquetil, and among the Armenians, according to Cerbied , multiplicators employed to express the multipla of the hundred, are found placed to the left.

To these examples may yet be added, the above-mentioned point of Diophantus, which is used instead of Mu, and multiplies the figures to the left a thousand times.

For, as far as regards the method, it belongs to this class .

Grammaire Armenienne, 1825, p. 25.

This mode of separating numbers by points, though it is otherwise used in a very inconsistent manner, expresses properly the value by position.

It is also used in three passages of the elder Pliny (vi., 24, 33; xxx., 3), which have given rise to many controversies.

Fourth Method. -- Multiplication and diminution, in ascending and descending direction, brought about by dividing the figures in layers, of which the value decreases in geometrical proportion.

Both Archimedes in the Octades, and Apollonius in the Tetrades, make use of this notation, but only in numbers above (10,000)2, and in a hundred of millions, or a myriad of myriads @. Here, as well as in the descending sexagesimal seale of the astronomer of Alexandria, where degrees, minutes, and seconds are indicated, the value is evidently expressed by the position of the figures.

They follow one another in different layers, and thus, they express an absolute and a relative value.

But, as in the last mentioned scale, every layer is composed of two figures (for want of n -- 1 or 59 figures); the value expressed by position does here not procure the advantages accruing from the Indian figures.

When the three hundred and sixty parts of a circle are considered as so many entires, the minutes are sixtieths of them, the seconds are sixtieths of the minutes, &c. Considering them as fractions, Ptolomaeus distinguished them by the sign of fraction, a line placed above, and, in order to express their descending progression, by which every layer of two figures has sixty times less value than the preceding, the fraction-signs were increased in number from layer to layer.

According to these principles, the minutes are indicated by one line, the common designation of fractions (the numerator of which is the unit), the seconds by two such lines, the terces by three; but, the degrees themselves, as being entires, were not distinguished by a line, but, perhaps, by nought (ouden), or a cypher -- I say, perhaps, for in the writings of Ptolomaeus and Theon, the figure of the cypher is not yet used to indicate degrees.

Delambre, Hist. de l'Astron. Ancienne, t. i., p 105; t. ii., p. 9.

On the use of the cypher, see Leslie, p. 12, 135; Germanen und Griechen Hist., v. ii., p. 2--33; Ducange, Glossar.

Mediae Graecitatis, t. ii., p. 572. Mannert de Nummorum quos Arabicos vocant Origine Pythagor. p 17.

In the Greek arithmetic Mo signifies the unity; monas, as a delta (D), when a cypher (properly omicron) is placed over it, signifies tetartos.

Bast, Gregor., Corinth. p. 851. Thus, we find in Diophantus, Moka = 21.

The Indian grammatical sign, anuswara, has indeed the figure of the Indian cypher.

It indicates, however, nothing but a modification in the accentuation of the vowel placed nearest to it, and is in no way connected with the sunya.

The simple enumeration of the methods used by those nations who did not know the Indian system of position, in expressing the multipla of the fundamental groups, shews, in my opinion at least, the way in which the Indian system probably has been invented by degrees.

When the number 3568 is written either in a perpendicular or in a horizontal direction, by means of indicators:

[Formel] , it is evident that the figures of the groups M C, &c., may be omitted.

For our Indian figures are only the multiplicators of the different groups.

This mode of expressing every number only by unities (multiplicators) is moreover suggested by the suanpan, of which the strings indicated the thousands, hundreds, tenths and unities in fixed order. To express the above-mentioned number the strings contained 3, 5, 6 and 8 balls.

No sign of groups is here to be observed.

The places themselves supply the signs of groups, and these places (the strings) are filled up with the unities (multiplicators).

Thus the Indian system may have been invented in either way, by the figurative as well as by the palpable arithmetic.

If a string was empty, or in writing a layer was not occupied by a figure, if consequently a group (a member of the progression) was wanting, the empty place in writing was filled up by the hieroglyphic of emptiness, a circle open in the middle, sunya, sifron, cyphra .

In English, the expression cypher for nullity has been preserved; whilst, in the other languages of the west, this word is used to indicate the numerical figures in general; in them the cypher is called zero (sifron, siron).

According to Wilson any numerical quantity is called in Sanscrit sambhara.

That the notation of numerical quantities has been improved and brought to perfection in India only by degrees, is proved by the numerical figures of the Tamul language, in which every numerical quantity is expressed by the nine figures of the unities, by distinct figures for the groups of 10, 100, and 1000, and by multiplicators added to the left hand.

The same is also proved by the remarkable arithmoi indikoi, in the scholion of the monk, Neophytus, which is preserved in the library of Paris (Cod. Reg. fol. 15), and was communicated to me by the kindness of Professor Brandes.

The nine figures of Neophytus are, the four excepted, all of them like those of the Persians.

The figures of 1, 2, 3, and 9 are even found in some Egyptian inscriptions containing numbers . The unities are multiplicated, a ten, a hundred, and a thousand times, by writing over them one, two, or three cyphers, as [Formel] , [Formel] , [Formel] , [Formel] . If, instead of cyphers, points are used, the Arabic Gobar figures are obtained.

I shall give here a literal translation in Latin of the scholion itself, and observe only, that the monk erroneously calls the expression tzüphron an Indian word.

Kosegarten, p. 54.

Tzyphra est et vocatur id, quod cuivis litterae inde a decade et insequentibus numeris quasi onmikron inscribitur.

Significat autem hac Indica voce tale analogiam numerorum.

Ubi igitur scriptum est simile primae litterae alpha, pro unitate scriptae, atque superimpositum habet vel punctum, vel quasi onmikron, addita altera figura litterae Indicae, differentiam et augmentum numerorum declarat.

E. g. pro primo Graeco numero, a scripto, apud Indos | sive linea recta perpendicularis, quando non habet superimpositum punctum vel onmikron, ipsum hoc denotat unitatem, ubi vero superimpositum sit punctum atque altera littera adscripta sit, figura quidem similis priori, significat XI, propter additamentum similis litterae atque superimpositum unum punctum.

Similiter etiam in reliquis litteris, quemadmodum adspectus docet.

Si vero plura habet puncta, plura denotat.

Quod intelligas, lector, et supputes unumquidque.

In this system we do not find that value is expressed by position any more than in the system of the Gobar figures.

The number 3006 was written [Formel] . But in using it, it must soon have been observed, that the same figures often expressed different value, and that (when all the groups were filled up) in [Formel] [Formel] [Formel] 7, the points or cyphers, by decreasing regularly in number, became superfluous.

The cyphers, as it were, served only to facilitate the pronouncing of the number.

If we now suppose that the custom of writing the cyphers aside the figures instead of placing them above, became prevalent, the Indian notation as used at present, was introduced for the unmixed groups, as [Formel] . If further, to [Formel] were to be added [Formel] , that place of the cypher was filled up which was assigned to 40 by the exponent indicating the group.

Thus 3040 was obtained, and two of the three cyphers, which were required to express the thousands, and which had previously been placed on one line with the unities, remained there to indicate the empty places.

According to the scholion of Neophytus, therefore, the figures of the cypher are (like the points over the Gobar figures) indicators for the notation of the ascending groups.

From the observations made on this system it is easy to perceive, how the cyphers have been placed in the row of the figures, and have preserved that place, when the value by position was adopted.

In reviewing once more the different methods used by the different nations of both continents in computing numbers, which till now have been in part so little known, we find, firstly, in some only a small number of figures indicating groups, and those almost only for n2, n3, n4 .... not for 2n, 3n, and 2n2, 2n3 .... as among the Romans and ancient Tuscans , X, C, M, and, therefore all the intermediate groups, for instance, 2n or 2n2, are to be expressed by juxtaposition, as in XX or CCCC; we find further, in others, a great number of figures of groups, not only to express n, n2, (iota and rho among the Greek alphabetic figures) but also to express 3n or 4n2 (in l and u), by whose application a great heterogeneity of the elements is produced in expressing 2+2n + 2n2 (for instance, skb for 222); we find lastly, that the multipla of the fundamental groups and their powers (2n, 3n, 4n2, 5n2,) are, by others, expressed either by the addition of indicators over or under the figures of the groups (by the Chinese [Formel] , [Formel] , [Formel] , [Formel] , by the Hindoos speaking the Tamul language 2X, 3X, 4C, 5C,) or by placing over the figures of the first nine unities a progressive number of points, that is a = 10, b = 20, a = 100, a@ = 1000, d@ = 40,000, as in the Gobarfigures, in the scholion of Neophytus, and in the descending sexagesimal scale of the astronomers of Alexandria, for [Formel] , [Formel] , [Formel] , in 1° 37' 37" 37''' ....

We have seen in what manner the indicators (multiplicators) used by the nations of Eastern Asia, and by the inhabitants of the southern districts of India within the Ganges, or, where originally figures of groups did not exist, in what manner the placing of points over the pythmenes in the Gobar-system and in the scholion of Neophytus; and lastly, in what manner even the strings of the suanpan, in which different value is expressed by the relative position of the strings, could lead men to invent that system, in which value is expressed by position.

For brevity's sake I here take no notice of the figures of the groups of the quinary system (V, L, D...) which form intermediate sections.

Whether the simple Indian system, expressing value by position, was brought into the west by the learned astronomer, Rihan Muhammed eben Ahmet Albiruni , who remained a long time in India, or by Moorish custom-house officers in the ports of North Africa, and their intercourse with the Italian merchants, I do not presume to decide.

Further, though mental culture was doubtless very early disseminated in India, it remains doubtful whether the numerical system expressing value by position, which has so powerfully affected the progress of the mathematical sciences, had already been invented and adopted by that nation, when the Macedonian conqueror invaded their country.

In how different a condition, in how much more perfect a state would the mathematical sciences have been transmitted to the learned epoch of the Hashimides by Archimedes, Apollonius of Perga and Diophantus, if the western countries of the old continent had received the Indian numerical system twelve or thirteen centuries sooner, at the time of Alexander's expedition.

But that part of upper India, which was then overrun by the Greeks, the Penjab, as far as Palibothra, was, according to the learned researches of M. Lassen, inhabited by nations very little advanced in civilization.

Those who lived farther to the east, called them even barbarians.

Seleucus Nicator was the first who passed the river Sarasvatis, and by doing so, the limits that separated the civilized and uncivilized tribes; and then he advanced towards the Ganges .

According to an observation of the orientalist Sedillot, not less acquainted with the Greek, than with the Arabian astronomy.

Lassen, Comment.

Geog.

de Pentap., p. 58.

The old Indian numerical figures of the Tamul language, which express the quantities 2n, 3n2 .... by the addition of multiplicators, and consequently besides the figures for the first nine unities, have distinct ones for n, n2, n3, .... prove evidently that, in India, besides that system which exclusively has obtained the name of Indian (or Arabic) figures, and in which value is expressed by position, there yet, at the same time, existed others, which did not express value by the same method. Now it may be the case, that Alexander and his successors in Bactria, in their temporary incursions, did not have intercourse with any tribes, among whom the knowledge of the system expressing value by position had then become prevalent.

I could wish that the traces of what is still to be discovered (and that is yet very much), might soon be pursued with increasing zeal, by philologists, who have opportunity of examining either Greek, Persian, or Arabian manuscripts . The manner in which old manuscripts of the Sanscrit literature are paged, can sometimes bring us to important observations and discoveries.

To give an instance, hardly any person would have expected to find in India, besides the decimal system with position, a sedecimal system without position.

It seems, however, that some Indian tribes had adopted in their calculations groups of sixteen, as the natives of America, and the Gauls and Biscayans, those of twenty.

For such a remarkable numeration has been discovered, more than ten years ago, by M. Bopp, in a manuscript of the old Indian poem, Mahabharata (Cod. Reg., Paris, p. 178).

He had the kindness to communicate it to me for publication, when I laid before the Academie des Inscriptions et Belles Lettres my first essay on the numerical signs of the different nations.

The first sixty-five pages are paged with Indian alphabetic letters, but only the consonants of the Sanscrit alphabet are used (k for 1, kh for 2 ....).

This refutes the opinion till now generally prevailing, that the Hindoos always used ideographic figures to express numbers, and never alphabetical letters, as the Semitic tribes and the Greeks .

Among the Arabian manuscripts, those especially are to be recommended to peculiar attention, which treat of custom-house or financial affairs, or of arithmetic in general; as, for instance, Abu Jose Alchindus de Arithmetica Indica; Abdelhamid Ben Vasee Abulphadl de Numerorum Proprietatibus; Amad Ben Omar Alkarabisi Liber de Indica Numerondi Ratione; the Indian Algebra by Katka; Mohammed Ben Lara de Numerorum Disciplina (Casari Biblioth. Arabico-Hispana, t. i. p. 353, 405, 410, 426, 433.)

Si l'arithmetique de position n'est pas originaire de l'Inde, elle doit au moins y avoir existe de temps immemorial; car on ne trouve chez les Indiens aucune trace d'une notation alphabetique telle que la notation des Hebreux, des Grecs et des Arabes.--(Delambre, Histoire de l'Astronomie Ancienne, t. i., p. 543.)

On the sixtieth page begins the extraordinary sedecimal notation.

Of the pythmenes, fifteen in number, hardly two figures are found among those we know. (The aspirated t of the Sanscrit alphabetical letters is used for 3 and d for 12.)

They are likewise quite different from the Indian (Arabic) figures.

It deserves to be noticed, that the figure of 1, with a cypher added to it, signifies four, as that figure doubled (two vertical lines), with a cypher added to them, signifies eight.

They form, as it were, resting-places, intermediate landings, of the sedecimal system for [Formel] and [Formel] n. But, [Formel] of n (12), is not indicated by a cypher, but by a peculiar hieroglyph, similar to the Arabic four. To express the principal fundamental group itself (16), and the multipla of it (2 n, 3 n ....), the known Bengali figures are used; so that the Bengali 1, preceded by a curved line, signifies 16; the Bengali 2 is used for 32; the 3 for 48. The multipla of n are consequently indicated only as groups of first, second, third, .... order. The numbers 2 n + 4, or 3 n + 6, (that is 36 and 54 in the sedecimal system,) are expressed by the Bengali figure of 2, and the added Mahabharatan figure of 4, or by the Bengali figure of 3, and the added Mahabharatan figure of 6 .

I use here the expression Mahabharatan figure only for the purpose of indicating, with a proper word, the numerical system discovered in the manuscript of that poem.

It is, indeed, a very regular, but at the same time a very inconvenient and complicated mode of counting, the origin of which is the more difficult to guess, as it presupposes the knowledge of the Bengali figures.