Digitale Ausgabe

In our researches upon numerical figures (the only hiero-glyphics which, among the nations of the old continent, havebeen preserved, besides the alphabetical figures used to expressthe different sounds of spoken language,) our attention has,hitherto, rather been directed to the characteristic physiognomyof the figures and their peculiar formation, than to the spiritof the methods by which human sagacity has succeeded inexpressing quantities with a greater or less degree of simplicity.These researches have been entered into with views as narrowand as contracted as those made on languages. The latterhave, for a length of time, been compared rather according tothe frequency of certain sounds and terminations, or to theform of their roots, than to the organic formation of theirgrammars. For many years I have been occupied constantly,and with particular predilection, in endeavouring to bringunder a general view the different systems of numerical figuresused by the different nations of ancient and modern times, and inthis way I have succeeded in throwing some light on the originof what is called the Arabic numerical system. Many circum-stances concurred to enable me to effect it. I myself haveacquired, on my travels, a knowledge of the numerical systemsof the Aztekes (Mexicans), and of the Muyscas * (the inha-bitants of the elevated plain of Cundinamarca); ThomasYoung made the discovery of the Egyptian numerical figures,which (as is now known) do not, all of them, express themultipla of the groups by juxtaposition; the Arabic Gobar,or Dust-figures, were discovered by Silvestre de Sacy, in a

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* On the opinion that the numerical figures of the Muyscas (which at thesame time are the hieroglyphics of the moon-days of the increasing age of themoon) have some connexion with the face of the moon, increasing by degreesaccording to its different phases, see Humboldt, Vues des Cord. et Monumensdes Peuples indigènes de l’Amérique, t. ii., p. 237—243. Pl. XLIV.|301| Parisian manuscript; the peculiar nature of this system in-duced me to compare it with the numerical figures of theMexicans and Chinese; by the publication of a great numberof 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 inexpressing numbers, but also that the systems themselves ofcomputation differ from one another,—in some of them valuebeing expressed by position, in others not; lastly, a peculiarIndian method, till now quite unknown, has been preservedand discovered in a scholion of the Greek monk Neophytus.In an Essay composed by me, and read in the Session of theAcadémie des Inscriptions et Belles Lettres, in 1819, I havealready tried to shew, that some nations shorten the unartificialmethod of juxtaposition by writing exponents, or indicatorsover the figures, (as it is used by the Mexicans, in the ligaturesof four times 13, or 52 years, by the Chinese, the Japanese,and the Hindoos, speaking the Tamul language,) and also themanner in which, by means of these indicators, and the sup-pression 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 systemmust considerably have been favoured by the very ancient useof a particular contrivance, the strings of reminiscence and cal-culation. These are either loose cords, as the quippos of theTartars, Chinese, Egyptians, Peruvians *, and Mexicans,which have been transformed into Christian beads, or ritualcalculation-machines †; or they are extended and fixed in aframe. In the last form they are found in the Suanpan, usedin all the internal countries of Asia, in the abacus of theRomans and ancient Tuscans ‡, and in the instruments ofthe palpable arithmetic used by the Slavonian tribes §. The files of cords or wires in the very simple Asiatic

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* 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 calculation-board, with its fixed cords.|302| Suanpan represent the higher or lower groups of a numeralsystem, whether tenths, hundredths, and thousandths, or, in thesexagesimal division, degrees, minutes, and seconds. Themanner of calculating is always the same. The pearls orbeads again are the indicators of the groups, and an emptyfile indicates a cypher, consequently the empty sunya (sansu),sifr, or properly sifron, sihron (Arabic, according to Meninskiprorsus vacuum). I am not able to prove in a strict historical manner, that theIndian system of nine figures, indicating different values bythe places they occupy, has been invented in the way I haveindicated, but I believe I have discovered the way in whichthis invention may have been gradually brought about. That,perhaps, is as much as can be done: for the history of thefirst steps in the development of the mental faculties, and ofthe scale of civilization, is involved in darkness; and com-monly imparts to us nothing more than the knowledge ofpossibilities. But it is for that very reason that this part ofhistory is so interesting. Only a short extract of the Essay read by me in the Aca-démie des Inscriptions has been published, and that in a placewhere it hardly will be sought for *. The manuscript itself isin the hands of M. Champollion, who intended to publish it,with some new and important facts respecting the differentmethods of the Egyptian figures discovered by him in Turin.Since that time I have continued, from time to time, to rendermy first work more complete; but as I have no hope left tofind the necessary leisure for publishing it in its whole extent,I intend to put together in this essay the most important re-sults. Perhaps this is also a favourable time for doing it; for,now that the researches respecting languages and monumentshave taken a different and more useful direction, and thatour intercourse with the nations inhabiting the southern andeastern countries of Asia is on the increase, it may be of someutility to bring into discussion problems, which are in closeconnexion with the intellectual march of the human mind, and

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* 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.|303| (by the latest comparisons of numerical hieroglyphics, and thesimple graphic method,) with our glorious progress in themathematical sciences. One of the greatest mathematiciansof our times and of all times, the author of the “MécaniqueCéleste *,” says, “The idea of expressing all quantities bynine figures, whereby is imparted to them both an absolutevalue, and one by position, is so simple, that this very sim-plicity is the reason for our not being sufficiently aware, howmuch admiration it deserves. But it is this simplicity, andthe facility which calculations acquire by it, that raises thearithmetical system of the Indians to the rank of the mostuseful inventions. How difficult it was to discover such amethod may be inferred from the circumstance, that it hasescaped the talents of Archimedes and of Apollonius of Perga,two of the most profound geniuses of antiquity.” The fol-lowing observations, I hope, will shew, that the Indian methodof calculation may very possibly have arisen by degrees out ofothers, which were used earlier, and even now continue to beused, in the eastern countries of Asia. As language, in general, affects the manner of writing, andwriting again reacts on language, but only under certain con-ditions which have been inquired into by Silvestre de Sacy, andmy brother, so also the different modes of computation usedby different nations, and their numerical hieroglyphics, recipro-cally act upon one another. Yet no very great consequenceis to be attached to this alternate action. Numerical figuresdo not always follow the same groups of unities like languages;and moreover in languages we do not always discover the sameresting-points (the same quinary intermediate stops) as innumerical signs. But if we bring together under one view,the language (names of numbers) and the numerical figures usedin the remotest parts of the earth, as the common product ofhuman intelligence applied to quantitative relations, we discoverin the written numbers of one race, the isolated seeming pecu-liarities of language of another race; we may even add, that

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* Laplace, Exposé du système du Monde (5th edit.), p. 325. The assertion ofDelambre, (Histoire de l’Astron. Ancienne, t. i., p. 543,) in his contest respectingthe merit of the ancient Indian arithmetic, as it is explained in Bhascara Acharya’sLilawati, is in very strange opposition to the opinion of Laplace. But languagealone can hardly prompt me to suppress the figures of the groups.|304| a certain awkwardness in the numerical command of languageand writing is a very false standard of what is called the stateof cultivation of mankind. Here the same complicated andcontrasted 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 formsand flexions, rising out of the root in systematical progression,whilst others use languages destitute of forms and flexions, asif benumbed in their very birth; and all this in the most dif-ferent gradations of intellectual culture and political institu-tions. So the one race of mankind finds itself driven in themost opposite directions by the alternate action of the internaland external world, (an alternate action whose first decisiveefforts are wrapped up in the mythological darkness of theremotest antiquity); keeping most stedfastly to its old nature,even when great revolutions of the world bring geographi-cally near to each other races the most heterogeneous inlanguage: whilst the similarity of the sounds which re-echofrom the remotest zones, in grammatical forms of language, andgraphic attempts to express large numbers, proves the unity ofthe old stock—the ascendancy of that which springs out ofthe internal intelligence, out of the common organization ofmankind. Travellers observing that some nations formed heaps of fiveor twenty pebbles or grains of seed, when they were about tomake a computation, asserted that these nations were notcapable to count farther than till five or twenty *. As well may it be asserted that the Europeans are notcapable of counting farther than ten, because seventeen is com-pounded of seven and ten. In the language of the most cul-tivated nations of the west, for instance in those of the Greeksand Romans, some expressions are yet preserved, which referevidently to such heaps or groups, ψεφίζειν, ponere calculum,calculum detrahere. Groups of unities procure resting-placesin counting; and as in the different nations the members ofthe body are similarly formed (the four extremities are dividedfivefoldly) they stop either after having counted the fingers of

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* Pauw, Recherches Philos. sur les Américains, t. ii., p. 162. (Humboldt,Monumens Américains, t. ii., p. 232—237.)|305| one hand, or those of both, or they add also the two feet to thehands. This difference in proceeding produces different rest-ing-places, and thus are formed groups of 5, 10, or 20. It is,however, worth observing, that among the nations of the newcontinent, as among the Mandingas in Africa, the Biscayans,and the Gaelic tribes of the old continent, groups of twentyare in prevailing use *. In the Chibcha language of the Muyscas, (who, like theinhabitants of Japan and of Thibet had an ecclesiastical and alaical chief; and whose method of intercalating the 37th monthlike the inhabitants of North India, has been published andexplained by me †), 11, 12, 13, are called foot one (quihiehaata), 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 significationof foot is ten, because the foot begins to be taken into account,when both hands are passed through. To express twenty, theMuyscas use in their arithmetical language the expression footten, or the small house (gueta), perhaps because they used, incounting, grains of maize, and such a heap of maize remindedthem of the barn, where maize was laid up. By means of theexpression small house (or barn), and twenty (both feet andhands) they formed the expressions for 30, 40, 80, by joiningthem together, as, twenty plus ten; twice twenty; four timestwenty. Quite similar are the Celtic expressions which havepassed 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 notused in French; but in the Gaelic or Celtic dialect of WestBritany, through which I passed a few years ago, twenty iscalled ugent, forty daou-ugent, or two twenty; sixty tri-ugent,or three twenty. It is even said deh ha nao ugent, or ten overnine twenty = 190 ‡.

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* Instances of such numeral groups of twenty unities are found in Americaamong the Muyscas, the Otomites, the Aztekes, the Cora Indians, &c.† Monum. Américains, t. ii., p. 250—253. The Muyscas had some stonescovered with arithmetical hieroglyphs, which, by being placed in a certain order,facilitated to the priests (xeques) the intercalation of the ritual year. The figureof 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,|306| It is by no means a difficult task for me to give a still greaternumber of remarkable instances of the analogy existing betweenlanguage and numerical hieroglyphics in juxtaposition, in thesubtraction of unities by prefixing them, in writing, beforethe group, and in the intermediate landing-places of 5 and 15in those nations who have adopted groups of 10 or 20. Inthe languages of some very rude American tribes, for instancein those of the Guarinis and Lulos, six, seven, and eight areexpressed 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 casedeg (ten), or ugain (with twenty); as the French use soixanteet dix, for seventy. We find addition effected by juxta-position, chiefly among the old Tuscans, Romans, Mexicans,and Egyptians; subtractive or diminishing expressions amongthe Indians in the Sanscrit *, where 19 is called unavinsati,99 unasata; among the Romans in undeviginti (unus deviginti), for 19; undeoctoginta, for 79; duodequadraginta,for 38: among the Greeks in εἴκοσι δεόντα ἕνος, 19; andπεντεκόντα δυοῖν δεοντοῖν, 48; that is, wanting two to make upfifty. Such subtractive expressions have passed into thewriting of numbers, and in such a case the figures expressingunities to be subtracted are prefixed to the signs of the groupsof five and ten, and even to their multiplicates, for instance,to 50 or 100 (IV, and XI, and XL, and XT, for four-and-forty among the Romans and ancient Tuscans †; though inthe last nation the numerical figures probably were entirelyderived from the alphabet, according to the researches latelymade by Otfried Müller). In some rare Roman inscriptions

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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. (WilliamOwen’s Dictionary of the Welsh Language, vol. i., p. 137.) The same systemof 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, namelyforty and ten (amar). Larramendi, urte della Lengua Bascongada, 1729, p. 38.(The numeral figures of the Biscayan and Gaelic languages are not mixed toge-ther in my Monum. vol. ii., p. 237, but they are placed near one another to facili-tate the comparison; by an error of print, however, les premiers is said instead ofles deux or les uns et les autres.)* 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.|307| collected by Marini *, even four unities are found before ten,for instance, IIIIX=6. We shall soon see, that among sometribes inhabiting the East Indies, methods are found in whichthe joining of figures, which among the Romans and ancientTuscans indicated only addition or subtraction, expressesaddition or multiplication, according to the place or the direc-tion of the signs. In these Indian systems, (to use Romanfigures,) IIX is twenty, and XII is twelve. In many languages the groups of 5, 10, and 20, are calleda hand, two hands, and hand and foot (in the language ofthe Guaranis mbombiade). When the fingers of both ex-tremities, viz., fingers and toes, are gone through in counting,then it is considered that the number comprehends the wholebody, thus: the word man becomes the symbol for twenty.Therefore, in the language of the Yaruros (of which tribe Ifound populous villages, erected by the missionaries on the riverApure, 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. “Thelast is (according to the ingenious observation of M. Bopp)to be considered as the original of the Roman quinque, as theIndian chatur of quatuor. The plural of chatur is chatvaras,and approaches very near the Doric-Æolian τέτταρες. TheIndian ch is pronounced as the English in words of Saxonorigin, but in Greek it becomes a t; therefore chatvaras ischanged into tatvaras, as pancha into panta (the Greek πέντε,Æolice πέμπε, whence πεμπάζειν to count by fingers, or upto five). But in the Latin language, the Indian ch is ex-pressed by qu and thus chatur is changed into quatuor, andpancha into quinque. Pancha itself never signifies in Indiahand, but only the number five. But panchasakha is adescriptive expression for hand, as a limb with five branches †.” As in languages, and with peculiar naïveté in those of

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* Iscrizioni della Villa di Albano, p. 193. Hervas Aritmetica delle Nazioni,1786, p. 11, 16.† On the numerals in Sanscrit, compared with those of the Greek, Latin andGothic languages, a very interesting essay, in manuscript, was communicated tome by M. Bopp, in Paris, in 1820; and it was composed by him for the purposeof being inserted in my work, On the Numerical Signs of Nations.|308| South America, the groups of five, ten, and twenty, aredistinguished by peculiar expressions, so, in the same manner,these groups are easily to be recognized in numeral hiero-glyphics. The Romans and Tuscans * had single figures toexpress 5, 50, and 500. In the language of the Aztekes(natives of Mexico), we find not only the group-signs of aflag for 20, a quill filled with grains of gold (which in somedistricts of Mexico were used as money) for the quadrate of20 or 400, and a bag (xiquipilli) with 8000 cocoa nuts (whichlikewise served as a medium of barter) for the cube of 20 or8000, but even (in the instances where the flag is divided intofour equal parts, and the half or three quarters of it arecoloured) numerical signs for half-twenty or 10, and for\( \frac{3}{4} \) twenty or 15, two hands and a foot, as it were †. But the most remarkable of all proofs of the alternate effectbetween numerical language and numerical figures, is to bemet with in India. Here the value of the unities expressedby their position has, in Sanscrit, been even transplanted intothe language. Thus the Indians have a figurative methodof expressing numbers by the names of objects, of which acertain number is known. For instance, surya (the sun) ex-presses 12, because, according to the mythology of the Hindoos,there were twelve suns in the order of the twelve months. Thetwo Aswinas (Castor and Pollux), who also are met with inthe Nakchatras, are used to express the number two. Manusignifies fourteen, taken from the menus of their mythology.Now it will become intelligible, how the word surymanu, in whichthe symbols of twelve and fourteen are combined, signifies theyear 1214. These facts I owe to the kind communication ofthe learned Colebrooke. Probably, according to these prin-ciples, 1412 will be expressed by Manusurya, and 314 byAswinimanu. We find besides so complete a numeration inSanscrit, that a single word, koti, is used to express 10 millions.In the qquichau language of Peru, which does not count bygroups of twenty, a single word stands for a million (hunu). If it be true that we count by decimals, quia tot digiti, per

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* Respecting the Tuscan figure for 500, see Otfried Müller, Etrusker, sec. iv.,fig. 2.† Humboldt, Monum. Améric., tom. i., p. 309.|309| quos numerare solemus, as Ovid says, we probably should haveadopted a duodenary scale *, if our extremities had beendivided sixfoldly. Such a scale has the great advantage, thatits groups can be divided by 2, 3, 4 and 6 without leavinga fraction, and for that reason it has been adopted by theChinese for their measures and weights, from the earliesttimes. The preceding observations regard the relation betweenlanguage and writing, between the numerals and their figures.I shall proceed to consider the latter by themselves. I remindthe reader that this essay is only an extract from my largerand unfinished work, and that I shall not speak of the hete-rogeneous forms of the simple elements (numerical) figures,but of the spirit of the different methods of expressingnumeral quantities, which are adopted by different nations.I, therefore, shall only take notice of the figure or form of thenumerical signs, whenever it affects my conclusions respectingthe identity or heterogeneousness of the methods themselves.The mode of proceeding adopted for the purpose of expressingthe pure or mixed multipla of the denary fundamental groupsn (for instance 4n, 4n2, or 4n + 7, 4n2 + 6n, 4n2 + 6n + 5) isvery different in different nations. Sometimes it is effected byforming a row (that is by giving different value to the figuresaccording to their position), after the manner of some nationsof the Hindoos; sometimes by mere juxtaposition, as amongthe ancient Tuscans, the Romans, Mexicans, and Egyptians.Some nations use for that purpose coefficients, as that tribeof the Hindoos, inhabiting India within the Ganges, which usesthe 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, usean inverted method. These place over the nine numericalfigures a number of cyphers or points, to indicate the relativevalue of them, and in this system, the cyphers or points arethe signs of the groups placed over the unities. The lastmethod is found in the Arabic Gobar writings, and in anIndian numerical system, preserved and explained in a scho-

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* Debrosses, vol. ii., p. 158.|310| lion of the Greek monk Neophytus. These five differentmethods are in no way dependent on the forms of the nu-merical figures, and to prove this fact more evidently, I shall,in this treatise, employ no other signs than the arithmeticaland algebraical, which are in common use. In this way theattention is more directed to the essential point, the spiritof the method. I already have used such a mode of pasi-graphic notations in another work, treating of a very hete-rogeneous subject, the regular stratification and often peri-odical filation of the minerals (in the appendix to the EssaiGeognostique sur le Gisement des Roches) *, and there Ihave tried to shew, how by this method our abstract viewson an object can be rendered more general. Proceeding inthis way, all observations respecting the form and compositionof individuals, which in themselves may be very useful andtrue, but would turn off the attention, are suppressed, inorder to place in a purer and clearer light a phenomenon,which we wish to investigate in preference to others. Thisis an advantage which in some way may be justified by thechilly insipidity of such abstractions. As regards the manner of writing numbers adopted by dif-ferent nations, it is usual to distinguish the figures not de-pending on alphabetical letters, from the alphabetical letterswhich indicate numerical value, either by forming a certainrow, or by short lines or points added to them; or lastly, bythe initials of the numerals (in reference to language) †. It is beyond all doubt, that the nations of Hellenic, Semitic,and Aramaic origin (among the last even the Arabs, beforethey received the numerical figures from the Persians, in thefifth century of the Hegira ‡), used the same signs to express

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* Ed. 1823, p. 364—375.† The Arabic figures called Diwani, are compounded of mere monograms orabbreviations of the initial letters of the numerals, and are the most complicatedinstance of such initials. Whether the C and M, used by the ancient Tuscans andRomans, are, in fact, initials taken from the Tuscan and Roman languages, is muchmore 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 tothe 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 Traité de Diplom. par deux Religieux de St. Maur, vol. i., p. 678.)‡ Silvestre de Sacy, Gramm. Arabe, 1810, t. i., p. 74, Note 6.|311| letters and numerals, even in the period of their ripe civiliza-tion. On the other hand we find, in the new continent, atleast two nations, the Aztekes and the Muyscas, who usednumerical figures without having letters. Among the Egyp-tians, the numerical hieroglyphics commonly used to expressunits, tenths, hundreds, and thousands, seem likewise to havehad no relation to the phonetic hieroglyphics. Quite differentfrom the alphabet are likewise the ancient Persian figures ofthe Pehlwi, for the first nine unities, as those of the ancientTuscans, the Romans, and even the Greeks in the most ancienttimes. Anquetil has already observed *, that the alphabet ofthe Zend language, which, being composed of 48 elements,could have facilitated the expressing of numbers, by alphabet-ical letters, never uses them as numerical figures, and that inthe books written in the Zend language, numbers are alwaysexpressed by the figures of the Pehlwi, and at the same timewith the numerals of the Zend language. Should, by furtherresearches, this want of numerical figures in the Zend languagebe confirmed, it must induce us to suppose that the Zendnation, which, in its language, discovers a very intimate affinityto 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 often, hundred, and thousand are derived from letters in thePehlwi language. Dal is 10; re and za, connected together,100; re and ghain, connected together, 1000. When wetake together, in one view, all that is known to us of thenumerical figures used by the different nations, little as itis, it seems sufficient to oblige us to confess, that thedistinction of alphabetical numerals and of numerical figures,independent of the alphabet, is as uncertain and uselessas that of the languages in such as are composed of mono-syllables, and in those using polysyllabic words, a distinctionlong ago abandoned by truly philosophical investigators oflanguages. The numerical figures used by the inhabitants ofsome southern districts of India within the Ganges, who speakthe Tamul language, do not express value by position, and

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* Mém. de l’Acad. des Belles Lettres, t. xxxi., p. 357.|312| are quite different from those used in Sanscrit manuscripts, ifthe figure of 2 is excepted. Who can say whether thesefigures are not to be derived from the Tamul letters? It istrue the figure for the group of a hundred is not to be foundamong these letters, but the sign of the group of ten is to berecognized in the letter ya, and the two in the letter u. Thenumerical figures in the Teloogoa language, likewise spoken inthe southern districts of India within the Ganges*, which indi-cate value by position, are in a strange manner different fromall other Indian figures, as far as they are known, in the signsfor 1, 8, and 9, whilst they agree with them in the figures of2, 3, 4, and 6. The necessity of expressing quantities bywriting has probably been felt sooner than that of writingwords, and therefore we may consider numerical figures as themost ancient written characters. The instruments of palpable arithmetic, as they are calledin an ingenious work, the Philosophy of Arithmetic, by Mr.Leslie, (1817,) in opposition to the figurative or graphic, areboth hands, heaps of pebbles (calculi, ψηφοι), grains of seed,loose strings with knots (arithmetical strings, the quippos ofthe Tartars and Peruvians), the suanpan put in a frame, thetable of the abacus, and the calculating machine of the Slavo-nian nations with balls or grains of seed in files. All these in-struments furnish to the eye the first graphic notations ofgroups of a different degree. One hand, or a string withknots or moveable balls, indicates the unities up to 5, or 10,or 20. How often, by shutting of the single fingers, one handis gone through (πεμπάζεσθαι), is indicated by the other hand,of which every finger, that is, every unit, expresses a group offive. Two loose strings with knots stand in the same relationto 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 Egyptiansat the time of the Pythagorean league,) was brought to the

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* 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 nativesit is named Trilinga, or Telenga. With the table of the numerical figures inCampbell’s grammar, other varieties of Indian numerical characters to be foundin Wahl’s General History of the Oriental Languages, 1784, Tab. I. may be com-pared.|313| nations of the west as abax or tabula logistica, are used in thesame 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 bythe Chinese, and even the magic drawings (ralm) of centralAsia and Mexico, which exhibit knotty parallels, often brokenoff, almost in the manner of musical notes, seem only to begraphic projections of these calculation and reflection-strings *. In the Asiatic suanpan, and in the abacus, which was muchmore used by the Romans, on account of the inconvenientfigures adopted by them, than by the Greeks, who had beenmuch more successful in their mode of writing numbers †, thequinary rows are preserved together with the denary ones,forming geometrical progressions upwards and downwards.Outside of every calculation-string, indicating a group ororder, (n, n2, n3,) a shorter string was placed, on which everyball expressed the amount of five balls of the longer string.By this contrivance the number of the unities was diminishedto such a degree, that the principal string needed only to con-tain four balls, and the accessary only one ‡. It seems that among the Chinese, from the most ancienttimes, the custom had prevailed to consider arbitrarily anyone of the parallel strings as containing the units, and thatthus they obtained, upwards and downwards, decimal frac-tions, entire numbers, and powers of ten §. How late (in the

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* In the East, ralm is called the negromantic art of the sand. Uninterruptedlines, 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.) InDresden a remarkable manuscript is preserved. It was brought from Mexico, andexhibits nothing but figures like musical notes. I have published it in myMonumens Améric. 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 trulyAmerican kouas and linear drawings, I discovered after that time in someAztetic hieroglyphic manuscripts, and on the sculptures of Palenque, in the repub-lic of Guatimala. In the ancient Chinese numerical figures, the sign of thegroup of ten is a pearl on a string, and evidently an imitative drawing of thequippu.† Nicomachus, in Ast. Theologumena Arith. 1817, p. 96. In the financialsystem of the middle age, the account table (abax) became the exchequer.‡ So the Roman abacus was contrived. In China five balls were placed on thefirst, and two balls on the last. The balls, which were not employed in a calcula-tion, were pushed aside.§ On the first attempts to establish the decimal system, made by MichaelStifelius of Eslingen, Stevenus of Brugge, and Bombelli of Bologna, see Leslie,Philos. of Arithmetic, p. 134.|314| beginning of the sixteenth century?) has the knowledge of thedecimal fractions been introduced into the countries of theWest, which the nations of the East had been taught longago by their palpable arithmetic! The descending scale fromthe unit downwards, was known to the Greeks only in thesexagesimal system for degrees, minutes, and seconds; but asthey had not n—1, that is, 59 numerical figures, the value byposition could only be expressed by layers of two figures. If we direct our attention to the origin of numbers, we findthat they could be written and read with great exactness, whenthey were indicated by heaps of pebbles, or by the balls on thestrings of the calculation-machines. The impression whichthese proceedings left behind on the mind, has everywhereaffected the manner of writing numbers. In the historical,ritual, and negromantic hieroglyphics of the Mexicans, pub-lished by me, the units up to nineteen (the first simple figurefor a group is twenty) are exhibited as great round colouredgrains, and, what deserves to be mentioned, they are countedfrom 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 ancientGreek monuments, and in the Tuscan sepulchral inscriptions,the units are expressed by vertical lines; the same customprevailed among the Romans and Egyptians (which, respect-ing the last, has been proved by Thomas Young, Jomard,and Champollion). The Chinese use horizontal lines up tothe number four; and such lines are likewise found in somePhœnician coins described by Eckhel (t. iii., p. 410). TheRomans sometimes omitted the quinary figure in inscriptions,and, therefore, we find even eight lines as unities placedtogether. Many instances of this kind are collected by Marini,in his Monumenti dei Fratelli Arvali *, a work which deservesattention. The heads of the nails, which anciently were em-ployed by the Romans to indicate the years (annales antea inclavis fuerunt, quos ex lege vetusta figebat prætor maximus,says the elder Pliny, vii. 40.) could have led them to theunity-points of the Mexicans, and, in fact, we find such points

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* T. i., p. 31.; t. ix., 675—for instance in Octumvir.|315| used in the subdivision of ounces and feet, together with thehorizontal lines (used by the Chinese and Phœnicians) *.These points and lines, nine or nineteen in number, in thedenary or vicesimal scale of the old and new continent, arethe most simple of all notations in the system of juxtaposition.Here the unities are properly more counted than read. Theseparate existence, the individuality, if I may say so, of thenumerical figures as signs of numbers, is first to be recognizedin the numerical letters of the Semitic and Hellenic tribes, andamong the inhabitants of Thibet, and the Indian tribes, whoexpress 1, 2, 3, 4, by ideographic, distinct figures. In theancient Persian Pehlwi a remarkable transition is to be ob-served from the inartificial method of expressing numbers bythe repetition of the figure of the unity, to that of using com-pound ideographic hieroglyphics, in numbers greater than theunity. There the first nine figures are evidently formed by asmany notches or teeth as they contain unities; five and nineare even merely the numbers 2, 3, and 4 twisted together,without the repetition of the figure of one. In the system ofthe Devanagari, which is truly of Indian origin, in the Per-sian and Arabic-European figures, we are only able to discovera contraction of 2 and 3 units, in the figures of 2 and 3 †,certainly not in the higher figures, which in India within theGanges are written very differently from one another. As I mention here the Indian numerical figures, and shallbe obliged to do it frequently in this essay, I feel myselfbound to make some observations on this expression. At thesame time I shall take the opportunity of declaring myselfagainst the old prejudices, that in India only one set of nu-merical figures are employed in expressing numbers, and neveralphabetical letters in their place, and that in every district ofthat extensive country, a knowledge of a system, a differentvalue to the different position of the figures is met with,likewise that there never are peculiar figures used to indicatethe groups. As, according to what has been repeatedly saidby my brother William von Humboldt, the Sanscrit is not

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* Marini, t. i., p. 228.† Abel Remusat Langues Tartares, p. 30. On the strange numerical figuresused in Java, see Crawfurd, vol. ii., p. 263.|316| well distinguished by the name of “Indian and ancient-Indianlanguage,” because in that country many more very ancient lan-guages are found which do not derive their origin from the San-scrit; so likewise the expression, “Indian and ancient-Indianfigures,” is too indefinite, not only as far as it regards the formof the figures, but also respecting the spirit of the methods.For in India the principal groups of n, n2, n3, and their mul-tipla, 2n, 3 n ..... are sometimes expressed by juxtaposition,sometimes by coefficients, and sometimes merely by the placeof the figures. Even the existence of a distinct figure for thecypher, is, in the Indian system, no necessary condition of themethod of expressing value by position, as it is proved by thescholion of the monk Neophytus. In India within the Gangesthe most extended languages are the Tamul and the Teloogou.The tribes who speak the first use figures different from theiralphabet, of which only two, the two and the eight, exhibit aslight similarity with the Indian (Devanagari) figures of twoand five *. Much more different from the Indian figuresare those of the Cingalese †. In both the Tamul and theCingalese languages the different value of the figures is notindicated by position, they have also no distinct figure for thecypher, but distinct hieroglyphics for the groups n, n2, n3 ....The Cingalese make use of juxtaposition, the Tamuls of co-efficients. In India without the Ganges, in the empire of theBurmese, we find the value expressed by position, and a dis-tinct figure for the cypher, but the figures used by them donot resemble the Arabic, Persian, and Devanagari Indianfigures ‡. The Persian figures, used also by the Arabs, are allof them quite different from the Devanagari figures §; 7 islike a Roman 5; 8 like a Tuscan 5. Among the figureswhich we call Arabic, only 1, 2, 3, resemble the figures of theDevanagari of the same value; the 4 of the Devanagari is our8; our 9 is the 7 of the Devanagari. Our 7 is the Persian 6.

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* 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 Bur-mese 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.|317| In Bengali the 5 is expressed by the figure of a crescent, and2, 5, 6, 8, and 9 are quite different from the Devanagarifigures *. The numerical figures of Guzerath are only dis-torted Indian Devanagari figures †. I shall make no observation on the influence of the earliestnumerical figures on the form of the alphabetical letters, norof the distortion of the letters purposely introduced in orderto distingush them from the numerical signs, nor even on thedifference of the place which a figure used in both respectsoccupies sometimes (as in the aboudjet of the Semitic tribesin Asia and Africa) ‡. Such observations do not belong to thesubject of this essay, and have been the origin of manygroundless hypotheses in comparing the alphabetical letterswith the numerical hieroglyphics. I myself was once of theopinion that the Indian figures, notwithstanding the form oftwo and three, were the letters of an obsolete alphabet, ofwhich yet some traces are found among the Phœnician, Sama-ritan, Palmyrian, and Egyptian characters, (the last on themummies). Even the old Persian monuments of NakshiRustan seem to exhibit them §. How many characters inthese inscriptions resemble in a striking manner the nume-rical figures known under the name of Indian! Many otherscholars have likewise asserted, that the numerical figurescalled the Indian, are derived from the Phœnician alphabet ||,and the sagacious Eckhel has already observed that the simi-larity between the letters of the Phœnicians and the numericalfigures was so great, that the word Abdera is expressed by19990, and by 15550 ¶. But the origin of the numericalfigures, as well as that of the alphabetical letters, is envelopedin an obscurity, to dissipate which, by a philological investi-gation, founded on historical facts, is rendered impossible by

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* Graves Chamney Houghton, Rud. of Bengali Grammar, 1821, p. 133.† Robert Drummond, Illustrations of the Grammatical Parts of the Guzerathand Mahratt Language, 1818, p. 25.‡ Silvestre de Sacy, t. i., p. 10.§ Silvestre de Sacy, Antiquités de la Perse, Pl. I., n. 1. Compare the nume-rical 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 Mor-genl. Litteratur, p. 601, 630. Fundgruben des Orients, b. iii., p. 87.¶ Doctrina Numerorum veterum, 1794, t. iii. p. 396—404, 421, 494.|318| the scarcity of the materials, if we do not wish to content our-selves with a few negative results. We have seen that some nations, in expressing numbers bywriting, mix together alphabetical letters and ideographicfigures arbitrarily chosen. Likewise we find, that respectingthe mode of expressing the multipla of the fundamental groupsthe most heterogeneous methods are used. We discover eventhat one system completely developes what, in another, is onlyslightly indicated. The same incongruity obtains in lan-guages. In one language, some grammatical forms do appearonly in a few instances, and are slightly expressed, whilstanother has developed them with a peculiar predilection, andwith every effort of mental power. Should I, therefore, ex-plain the numerical systems singly, as they are used by dif-ferent nations, the similarity of their methods would be ren-dered obscure, and we should lose the track on which thehuman mind proceeding, at last arrived to discover the master-piece of the Indian arithmetic, in which every figure has adouble value, an absolute and a relative, of which the last isincreasing, in a geometrical progression, from the right to theleft. In my following observations I shall therefore abandonthe ethnographical order, and only consider the different meansemployed by nations to express in writing the groups of theunits. First method. Juxtaposition is effected by simple addi-tion in numerical figures as well as in alphabetical signs. Itwas in use among the ancient Tuscans, the Romans, among theGreeks only up to a myriad, among the Semitic tribes, theMexicans, and also in the greatest part of the Pehlwi calculations.This method renders the computation extremely difficult, whenthe multipla of the groups (2n, 3n, 2n2 ....) are not expressedby distinct signs. The Tuscans and the Romans repeated thefigure of ten as far as fifty; the Mexicans, whose first figureof a group was 20 (a flag), repeated this hieroglyphic upto 400. The Greeks, however, have in the rows of the tenthsand hundreds, which begin with iota and rho, distinct figuresfor 20, 30, 400 and 600. The three episemes (letters of anobsolete alphabet), bau, koppa, and sanpi, serve to express 6,90, and 900. The two last terminate the rows of the tenths|319| and the hundreds, and in this manner the numerical value ofthe Greek alphabetical figures approaches a little nearer to thesemitic aboudjed *. M. Böckh, in his learned observations onthe digamma, has shewn that bau is the wau of the Semites,(the F of the Latins,) koppa, the Semitic koph (9), and sanpithe Semitic shin †. The row of the unities beginning withalpha, and ending with theta, forms in the Greek system theoeot-numbers (ϖυθμένες), and Apollonius had invented a con-trivance ‡, by the help of which they were reduced, in the lastresults, to the corresponding members of the second and thirdrow (the analogues). Second Method. Multiplication or diminution of thevalue by signs placed over or under the figures. In the fourthrow of the Greek notations, the pythmenes return by analogy,but increased a thousand times by a line placed under thefigures. In this way the Greeks arrived, in their numericalsystem, 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 β withone, two or three lines would have expressed 20, 200, and 2000,and thus the Greek system would have approached, as we shallsee afterwards, the system of the Arabic Gobar figures, whichis very little known, and at the same time the system expressingvalue by position. But, unhappily, the Greeks did not adoptthis notation for the tenths and hundreds, applied it only forthe thousands, and did not try to employ it in higher groups. As a line added under the figures increases their value athousand times, thus a vertical line, in the Greek system, addedover the figure, indicates a fraction, whose numerator is theunit, and whose denominator is expressed by the figure itself.Thus in Diophantus, γ´ = \( frac{1}{3} \), δ = μ; but if the numerator isgreater than the unity, it is expressed by the principal figure,and the denominator is written like an exponent γᵟ = ¾ §. In

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* Hervas, Aritmetica delle Nazioni, p. 78. On the ancient order of the lettersin 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.§ Delambre, t. ii. p. 11. The line added over the alphabetical letters to indicatethat they are used as numerical figures, ought not to be confounded with the signof fractions. The first is also never vertical in the oldest manuscripts of mathe-|320| the Roman inscriptions, a horizontal line added over them,increases their value a thousand times, and here it may be con-sidered only as a means of abbreviating and of saving space. The method of Eutocius for expressing myriads is more im-portant. In it we find among the Greeks the first trace of thesystem of exponents, or rather indicators, which rose to sucha degree of importance in the East. Mα, Mβ, Mγ, indicate10,000, 20,000, 30,000. Here we find these indicators usedonly with the myriads. But the Chinese and the Japonese,which last received their civilization from the first about 200years before our æra, both use them for the multipla of allgroups. Three horizontal lines under the figure of ten, signifythirteen, but if they are placed over it they express thirty.According to this method, 3456 is written, (I use the Romanfigures 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 denotesa thousand, are used to express 2000 or 4000 *. Among theAztekes, or Mexicans, I found for 312 years the sign of thevinculum, with six unities as exponent (6×52=312), and Ihave published it in my work of the American Monuments.Among the Chinese, Aztekes, and Egyptians, the signs ofgroups are always under the exponent, as if \( \overset{@}{\text{X}} \) were writteninstead of 50; but in the Arabic Gobar figures the signs of thegroups are placed over the indicators. For in the Gobar sys-tem the signs of the groups are points, consequently cyphers;for in India, in Thibet, and in Persia, cyphers and points areidentical. These Gobar signs, which, since 1818, have attracted

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matical works, but horizontal, and thus the mistaking of it for the sign of fractionis obviated. Bast. de Usu Litterarum ad Numeros indicandos, in GregoriiCorinthii Liber de Dialectis Linguæ Græcæ. 1811. p. 850.* Kosegarten, de hierogl. Aegypt. p. 54. The assertion of Gatterer adoptedby him from Bianchini (Dec. 1, cap. iii., p. 3), from Goquet (vol. i. p. 226), andfrom Debrosses (vol. i. p. 432), that among the Egyptians the figures receivedvalue by their position in a perpendicular row, has not been confirmed by modernresearches. Gatterer, Weltgeschichte bis Cyrus, p. 555, 586.|321| my peculiar attention, were discovered in a manuscript in thelibrary of the old Abbey of St. Germain du Près, by my friendand instructor, M. Silvestre de Sacy. This great orientalistsays, “Le Gobar a un rapport avec le chiffre indien, mais il n’apas de zero *.” To me, however, it seems, that the figure of the cypher thereis found; it is, however, not placed aside the figures, but overthem, as in the scholion of Neophytus. It is, indeed, the signsof the cypher or the points, which have caused to be given tothese characters, the strange name of gobar or dust characters.He who sees them for the first time is doubtful whether theyrepresent a transition from figures to letters, or not. It is onlywith 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 usedby the Greeks †, but not frequently met with, and be-ginning only with the myriad. Here, α ‥ is used for 10,000,and β :: for 200 millions. One point, which, however, is neveremployed, serves to express 100 in this system of geometricalprogressions. In Diophantus and Pappus, a point is found be-tween the alphabetical figures, instead of the initial Μυ (My-riad). In this method, therefore, a point multiplies to figuresto the left 10,000 times. It would seem that some obscureideas of notation, by points and cypher, had been brought fromthe East into Europe by the Alexandrines. The figure of thecypher is, indeed, used by Ptolomæus, and even as an indi-cation of something that is wanting. He employs it in thedescending sexagesimal scale to indicate the wanting degrees,minutes, and seconds. Delambre even pretends to have foundthe figure of the cypher in the manuscripts of Theon, in hiscommentary on the Syntaxis of Ptolomæus‡. The cypher,

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* See Gramm. Arabe, p. 76, and the observations added to pl. viii.† Ducange, Paleogra, p. xii.‡ Histoire de l’Astronomie Ancienne, t. i., p. 547; t. ii., p. 10. The passage ofTheon is not to be found in his printed works. Delambre is inclined to attributethe origin of the Greek figure of the cypher sometimes to an abbreviation of ούδέν|322| consequently, was known in the west long before the invasionof the Arabs *. We have seen, that the Chinese, by placing indicators per-pendicularly over or under the groups, indicate the differencebetween \( \underset{2}{\text{X}} = 12 \), and \( \overset{2}{\text{X}} = 20 \). The same effect is producedamong the Greeks, Armenians, and those tribes of the Hin-doos, who speak the Tamul language, by adding figures inhorizontal direction. Diophantus and Pappus wrote β Μ υfor twice ten thousands, whilst α Μ υβ (where β is placed to theright of the initials of the myriad) signifies ten thousands, plustwo or 10,002. In the same manner, the Tamul figures areused, as, for instance: 4X = 40, and X4 = 14. In the ancientPersian Pehlwi, according to Anquetil, and among the Arme-nians, according to Cerbied †, multiplicators employed to ex-press the multipla of the hundred, are found placed to theleft. To these examples may yet be added, the above-men-tioned point of Diophantus, which is used instead of Μυ, andmultiplies the figures to the left a thousand times. For, as faras regards the method, it belongs to this class ‡. Fourth Method. — Multiplication and diminution, inascending and descending direction, brought about by dividingthe figures in layers, of which the value decreases in geome-trical proportion. Both Archimedes in the Octades, and Apollonius in theTetrades, make use of this notation, but only in numbers above(10,000)2, and in a hundred of millions, or a myriad of my-riads @§. Here, as well as in the descending sexagesimal seale

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sometimes to a peculiar relation in which the letter omicron stands to the sexage-simal 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, placedaside 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 iswritten under 12, 13, 22, 132? In Greek inscriptions the signs of cyphers indi-cate oboles. (Bockh, Staatshaushaltung der Athener, b. ii., p. 379.)* Planudes, Treatise on the Arithmoi Indikoi, third method, Multiplication of theValue by Coefficients.† Grammaire Arménienne, 1825, p. 25.‡ This mode of separating numbers by points, though it is otherwise used in avery inconsistent manner, expresses properly the value by position. It is alsoused in three passages of the elder Pliny (vi., 24, 33; xxx., 3), which have givenrise to many controversies.§ Delambre, Hist. de l’Astron. Ancienne, t. i., p 105; t. ii., p. 9.|323| of the astronomer of Alexandria, where degrees, minutes, andseconds are indicated, the value is evidently expressed by theposition of the figures. They follow one another in differentlayers, and thus, they express an absolute and a relative value.But, as in the last mentioned scale, every layer is composed oftwo figures (for want of n — 1 or 59 figures); the value expressedby position does here not procure the advantages accruing fromthe Indian figures. When the three hundred and sixty partsof a circle are considered as so many entires, the minutes aresixtieths of them, the seconds are sixtieths of the minutes, &c.Considering them as fractions, Ptolomæus distinguished themby the sign of fraction, a line placed above, and, in order to ex-press their descending progression, by which every layer of twofigures has sixty times less value than the preceding, the frac-tion-signs were increased in number from layer to layer.According to these principles, the minutes are indicated by oneline, the common designation of fractions (the numerator ofwhich is the unit), the seconds by two such lines, the terces bythree; but, the degrees themselves, as being entires, were notdistinguished by a line, but, perhaps, by nought (οὐδέν), or acypher * — I say, perhaps, for in the writings of Ptolomæusand Theon, the figure of the cypher is not yet used to indicatedegrees. The simple enumeration of the methods used by thosenations who did not know the Indian system of position, in ex-pressing the multipla of the fundamental groups, shews, in myopinion at least, the way in which the Indian system probablyhas been invented by degrees. When the number 3568 iswritten either in a perpendicular or in a horizontal direction, bymeans of indicators: \( \overset{3}{\text{M}}\overset{5}{\text{C}}\overset{6}{\text{X}}\overset{8}{\text{I}} \), it is evident that the figures ofthe groups M C, &c., may be omitted. For our Indian figuresare only the multiplicators of the different groups. This mode

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* On the use of the cypher, see Leslie, p. 12, 135; Germanen und GriechenHist., v. ii., p. 2—33; Ducange, Glossar. Mediæ Græcitatis, t. ii., p. 572. Mannertde Nummorum quos Arabicos vocant Origine Pythagor. p 17. In the Greekarithmetic Mο signifies the unity; μόνας, as a delta (Δ), when a cypher (properlyomicron) is placed over it, signifies tetartos. Bast, Gregor., Corinth. p. 851. Thus,we find in Diophantus, Mοκα = 21. The Indian grammatical sign, anuswara, hasindeed the figure of the Indian cypher. It indicates, however, nothing but a modi-fication in the accentuation of the vowel placed nearest to it, and is in no wayconnected with the sunya.|324| of expressing every number only by unities (multiplicators) ismoreover suggested by the suanpan, of which the strings indi-cated the thousands, hundreds, tenths and unities in fixedorder. To express the above-mentioned number the stringscontained 3, 5, 6 and 8 balls. No sign of groups is here to beobserved. 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 in-vented in either way, by the figurative as well as by the pal-pable arithmetic. If a string was empty, or in writing a layerwas not occupied by a figure, if consequently a group (a mem-ber of the progression) was wanting, the empty place in writingwas filled up by the hieroglyphic of emptiness, a circle open inthe middle, sunya, sifron, cyphra *. That the notation of numerical quantities has been improvedand brought to perfection in India only by degrees, is provedby the numerical figures of the Tamul language, in whichevery numerical quantity is expressed by the nine figures ofthe unities, by distinct figures for the groups of 10, 100, and1000, and by multiplicators added to the left hand. The sameis also proved by the remarkable arithmoi indikoi, in thescholion of the monk, Neophytus, which is preserved in thelibrary of Paris (Cod. Reg. fol. 15), and was communicatedto me by the kindness of Professor Brandes. The nine figuresof Neophytus are, the four excepted, all of them like those ofthe Persians. The figures of 1, 2, 3, and 9 are even found insome Egyptian inscriptions containing numbers †. The unitiesare multiplicated, a ten, a hundred, and a thousand times,by writing over them one, two, or three cyphers, as \( \overset{\circ}{2}=20 \),\( \overset{\circ}{24}=240 \), \( \overset{\circ\circ}{5}=500 \), \( \overset{\overset{\circ}{\circ\circ}}{6}=6000 \). If, instead of cyphers, points areused, the Arabic Gobar figures are obtained. I shall givehere a literal translation in Latin of the scholion itself, andobserve only, that the monk erroneously calls the expressiontzüphron an Indian word.

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* In English, the expression cypher for nullity has been preserved; whilst, inthe other languages of the west, this word is used to indicate the numerical figuresin general; in them the cypher is called zéro (sifron, siron). According to Wilsonany numerical quantity is called in Sanscrit sambhara.† Kosegarten, p. 54.|325| Tzyphra est et vocatur id, quod cuivis litteræ inde a decadeet insequentibus numeris quasi ο̄μικρὸν inscribitur. Significatautem hac Indica voce tale analogiam numerorum. Ubi igiturscriptum est simile primæ litteræ ἄλφα, pro unitate scriptæ,atque superimpositum habet vel punctum, vel quasi ο̄μικρὸν,addita altera figura litteræ Indicæ, differentiam et augmentumnumerorum declarat. E. g. pro primo Græco numero, ᾱ scripto,apud Indos | sive linea recta perpendicularis, quando nonhabet superimpositum punctum vel ο̄μικρὸν, ipsum hoc denotatunitatem, ubi vero superimpositum sit punctum atque alteralittera adscripta sit, figura quidem similis priori, significat XI,propter additamentum similis litteræ atque superimpositumunum punctum. Similiter etiam in reliquis litteris, quemad-modum adspectus docet. Si vero plura habet puncta, pluradenotat. Quod intelligas, lector, et supputes unumquidque. In this system we do not find that value is expressed by positionany more than in the system of the Gobar figures. The number3006 was written \( \overset{\overset{\circ}{\circ\circ}}{36} \). But in using it, it must soon have beenobserved, that the same figures often expressed different value,and that (when all the groups were filled up) in \( \overset{\overset{\circ}{\circ\circ}}{3} \) \( \overset{\circ\circ}{4} \) \( \overset{\circ}{6} \) 7, thepoints or cyphers, by decreasing regularly in number, becamesuperfluous. The cyphers, as it were, served only to facilitatethe pronouncing of the number. If we now suppose that thecustom of writing the cyphers aside the figures instead ofplacing them above, became prevalent, the Indian notation asused at present, was introduced for the unmixed groups, as\( \overset{\overset{\circ}{\circ\circ}}{3}=3000 \). If further, to \( \overset{\overset{\circ}{\circ\circ}}{3}=3000 \) were to be added \( \overset{\circ}{4}=40 \), thatplace of the cypher was filled up which was assigned to 40 bythe exponent indicating the group. Thus 3040 was obtained,and two of the three cyphers, which were required to expressthe thousands, and which had previously been placed on oneline with the unities, remained there to indicate the emptyplaces. According to the scholion of Neophytus, therefore,the figures of the cypher are (like the points over the Gobarfigures) indicators for the notation of the ascending groups.From the observations made on this system it is easy to per-|326| ceive, how the cyphers have been placed in the row of thefigures, and have preserved that place, when the value byposition was adopted. In reviewing once more the different methods used by thedifferent 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 indicatinggroups, and those almost only for n2, n3, n4 .... not for 2n,3n, and 2n2, 2n3 .... as among the Romans and ancient Tus-cans *, 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 greatnumber of figures of groups, not only to express n, n2, (iotaand rho among the Greek alphabetic figures) but also to ex-press 3n or 4n2 (in λ and υ), by whose application a greatheterogeneity of the elements is produced in expressing 2+2n+ 2n2 (for instance, σκβ for 222); we find lastly, that themultipla of the fundamental groups and their powers (2n,3n, 4n2, 5n2,) are, by others, expressed either by the additionof indicators over or under the figures of the groups (by theChinese \( \overset{2}{\text{X}} \), \( \overset{3}{\text{X}} \), \( \overset{4}{\text{C}} \), \( \overset{5}{\text{O}} \), by the Hindoos speaking the Tamullanguage 2X, 3X, 4C, 5C,) or by placing over the figures ofthe first nine unities a progressive number of points, thatis α̇ = 10, β̇ = 20, α̈ = 100, α@ = 1000, δ@ = 40,000, as in the Gobar-figures, in the scholion of Neophytus, and in the descendingsexagesimal scale of the astronomers of Alexandria, for\( \frac{1}{60} \), \( \frac{1}{60^2} \), \( \frac{1}{60^@} \), in 1° 37′ 37″ 37‴ .... We have seen in whatmanner the indicators (multiplicators) used by the nations ofEastern Asia, and by the inhabitants of the southern districtsof India within the Ganges, or, where originally figures ofgroups did not exist, in what manner the placing of pointsover the pythmenes in the Gobar-system and in the scholionof Neophytus; and lastly, in what manner even the strings ofthe suanpan, in which different value is expressed by the

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* For brevity’s sake I here take no notice of the figures of the groups of thequinary system (V, L, D...) which form intermediate sections.|327| relative position of the strings, could lead men to invent thatsystem, in which value is expressed by position. Whether the simple Indian system, expressing value byposition, was brought into the west by the learned astronomer,Rihan Muhammed eben Ahmet Albiruni *, who remained along time in India, or by Moorish custom-house officers inthe ports of North Africa, and their intercourse with theItalian merchants, I do not presume to decide. Further,though mental culture was doubtless very early disseminatedin India, it remains doubtful whether the numerical systemexpressing value by position, which has so powerfully affectedthe progress of the mathematical sciences, had already beeninvented and adopted by that nation, when the Macedonianconqueror invaded their country. In how different a con-dition, in how much more perfect a state would the mathe-matical sciences have been transmitted to the learned epochof the Hashimides by Archimedes, Apollonius of Perga andDiophantus, if the western countries of the old continent hadreceived the Indian numerical system twelve or thirteen cen-turies sooner, at the time of Alexander’s expedition. Butthat part of upper India, which was then overrun by theGreeks, the Penjab, as far as Palibothra, was, according to thelearned researches of M. Lassen, inhabited by nations verylittle advanced in civilization. Those who lived farther tothe east, called them even barbarians. Seleucus Nicator wasthe 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 †. The old Indian numerical figures of the Tamul language,which express the quantities 2n, 3n2 .... by the addition ofmultiplicators, and consequently besides the figures for the firstnine unities, have distinct ones for n, n2, n3, .... prove evidentlythat, in India, besides that system which exclusively has ob-tained the name of Indian (or Arabic) figures, and in whichvalue is expressed by position, there yet, at the same time,

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* According to an observation of the orientalist Sedillot, not less acquaintedwith the Greek, than with the Arabian astronomy.† Lassen, Comment. Geog. de Pentap., p. 58.|328| existed others, which did not express value by the same method.Now it may be the case, that Alexander and his successors inBactria, in their temporary incursions, did not have intercoursewith any tribes, among whom the knowledge of the system ex-pressing 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 in-creasing zeal, by philologists, who have opportunity of examin-ing either Greek, Persian, or Arabian manuscripts *. Themanner in which old manuscripts of the Sanscrit literatureare paged, can sometimes bring us to important observa-tions and discoveries. To give an instance, hardly anyperson would have expected to find in India, besides thedecimal system with position, a sedecimal system withoutposition. It seems, however, that some Indian tribes hadadopted in their calculations groups of sixteen, as the nativesof America, and the Gauls and Biscayans, those of twenty.For such a remarkable numeration has been discovered, morethan ten years ago, by M. Bopp, in a manuscript of the oldIndian poem, Mahabharata (Cod. Reg., Paris, p. 178). Hehad the kindness to communicate it to me for publication, whenI laid before the Académie des Inscriptions et Belles Lettresmy first essay on the numerical signs of the different nations.The first sixty-five pages are paged with Indian alphabetic let-ters, but only the consonants of the Sanscrit alphabet are used(k for 1, kh for 2 ....). This refutes the opinion till now ge-nerally prevailing, that the Hindoos always used ideographicfigures to express numbers, and never alphabetical letters, asthe Semitic tribes and the Greeks †. On the sixtieth page begins the extraordinary sedecimal

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* Among the Arabian manuscripts, those especially are to be recommended topeculiar attention, which treat of custom-house or financial affairs, or of arithmeticin general; as, for instance, Abu Jose Alchindus de Arithmetica Indica; Abdel-hamid Ben Vasee Abulphadl de Numerorum Proprietatibus; Amad Ben OmarAlkarabisi Liber de Indica Numerondi Ratione; the Indian Algebra by Katka;Mohammed Ben Lara de Numerorum Disciplina (Casari Biblioth. Arabico-His-pana, t. i. p. 353, 405, 410, 426, 433.)† Si l’arithmétique de position n’est pas originaire de l’Inde, elle doit au moins yavoir existé de temps immémorial; car on ne trouve chez les Indiens aucune traced’une notation alphabétique telle que la notation des Hebreux, des Grecs et desArabes.—(Delambre, Histoire de l’Astronomie Ancienne, t. i., p. 543.)|329| notation. Of the pythmenes, fifteen in number, hardly twofigures are found among those we know. (The aspirated tof the Sanscrit alphabetical letters is used for 3 and d for12.) They are likewise quite different from the Indian (Ara-bic) 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, signifieseight. They form, as it were, resting-places, intermediatelandings, of the sedecimal system for \( \frac{1}{4} \) and \( \frac{1}{2} \)n. But, \( \frac{3}{4} \) of n(12), is not indicated by a cypher, but by a peculiar hieroglyph,similar to the Arabic four. To express the principal funda-mental group itself (16), and the multipla of it (2 n, 3 n ....),the known Bengali figures are used; so that the Bengali 1, pre-ceded by a curved line, signifies 16; the Bengali 2 is used for32; the 3 for 48. The multipla of n are consequently indicatedonly as groups of first, second, third, .... order. The numbers2 n + 4, or 3 n + 6, (that is 36 and 54 in the sedecimal sys-tem,) are expressed by the Bengali figure of 2, and the addedMahabharatan figure of 4, or by the Bengali figure of 3, andthe added Mahabharatan figure of 6 *. It is, indeed, a very regular, but at the same time a very in-convenient and complicated mode of counting, the origin ofwhich is the more difficult to guess, as it presupposes theknowledge of the Bengali figures.