The Encyclopaedia contains a short treatise of Arithmetic, constructed in the ordinary way. But a philosophical exposition of the principles of numerical calculation has long been a desideratum in our systems of elementary instruction. The operations of Arithmetic, from being so common, are apt to be reckoned vulgar, and hence abandoned to the blind and mechanical application of certain rules. Yet the study of numbers, if rightly directed, is admirably fitted for opening the mind, and training it to habits of accurate thought, which are of such vast importance to the pupil, whether he means afterwards to engage in the details of business, or aspires to the ulterior pursuits of science. But the theory of numeration acquires a higher interest, from its being closely interwoven with the texture of human society. The diligent inquirer may trace, in the obscure vestiges of ancient language and practice relative to numbers, the slow progress of the mind, from the first dawning of reason to the full illumination of day. What a vast interval, in the range of intellectual achievement, from the knots or shells which the ferocious savage employs in reckoning his prisoners, to that magic play of figures by which the astronomer, instructed in all the refinements of calculation, and aided by the sublime invention of logarithms, penetrates through time and space, and determines with unerring certainty the changes of the heavens in future ages!

We purpose, therefore, in this article, to give a philosophical view of the Elements of Arithmetic. But our object is, only to sketch the great outlines of the science, without descending to all the subordinate or practical details. The exposition of those principles in a simple form may proceed with ease, and will derive additional interest from the variety and importance of the collateral discussions which it involves. We shall endeavour to elucidate the origin of numbers, to trace their early application, and to connect the progress of arithmetical computation with the history of civil society. But to succeed in the performance of this task, it would require much patient and laborious research in quarters which have hitherto remained almost unexplored. Where so many difficulties must be surmounted, the first attempt will need some portion of indulgence. We shall endeavour to maintain perspicuity, though at the risk of appearing tedious. But while intently pursuing the chief object, we may sometimes fail in observing the due proportions in the distribution of the subordinate parts of the discourse.

Arithmetic may be viewed under two very distinct forms, that would require separate appellations.

1. Palpable Arithmetic, in which numbers are exhibited by counters, or abbreviated representatives of the objects themselves; and, 2. Figurate Arithmetic, in which numbers are denoted by help of certain symbols, or artificial characters, disposed after a particular order. The progress of Arithmetic is analogous to that of writing, but it has followed the advances and transitions of this sublime art at a great distance. The numeration by counters, balls, or strokes, evidently resembles hieroglyphics or picture-writing; while the invention of the alphabet, so happily contrived for the rapid transmission of thought, probably led the way to the subsequent discovery of the science of Figurate Arithmetic, founded nearly on similar principles. These capital divisions of Arithmetic we shall consider in succession.

PALPABLE ARITHMETIC.

The idea of number, though not the most easily acquired, remounts to the earliest epochs of society, and must be nearly coëval with the formation of language. The very savage, who draws from the exercise of fishing or hunting a precarious support for himself and family, is eager, on his return home, to count over the produce of his toilsome exertions. But the leader of a troop is obliged to carry farther his skill in numeration. The systematic practice of war and murder has ever distinguished our species from other animals of prey. The chieftain who prepares to attack a rival tribe, marshals his followers; and, after the bloody conflict has terminated, he reckons up the slain, and marks his unhappy and devoted captives. If those numbers were small, they could easily be represented by very portable emblems, by round pebbles, by dwarf-shells, by fine nuts, by hard grains, by small beans, or by knots tied on a string. But to express the larger numbers, it became necessary, for the sake of distinctness, to place those little objects or counters in regular rows, which the eye could comprehend at a single glance; as, in the telling of money, it would soon have become customary to dispose the rude counters, in two, three, four, or more ranks, as circumstances might suggest. The attention would then be less distracted, resting chiefly on the number of marks presented by each separate row.

Language insensibly moulds itself to our wants. But it was impossible to furnish a name for each particular number: No invention could supply such a multitude of words as would be necessary, and no memory could ever retain them. The only practical mode of proceeding was to have recourse, as on other occasions, to the powers of classification. By conceiving the individuals of a mass to be distributed into successive ranks and divisions, a few component terms might be made sufficient to express the whole. We may discern around us traces, accordingly, of the progress of numeration, through all its gradations.

The earliest and simplest mode of reckoning was by pairs, arising naturally from the circumstance of both hands being employed for the sake of expedition. It is now familiar among sportsmen, who use the names of brace and couple, words that signify pairing or yoking.—To count by threes was another step, though not practised to the same extent. It has been preserved, how- ever, by the same class of men, under the term leash, meaning the strings by which three dogs and no more can be held at once in the hand.—The numbering by fours has had a more extensive application: It was evidently suggested by the custom, in rapid tale, of taking a pair in each hand. Our fishermen, who generally reckon in this way, call every double pair of herrings, for instance, a throw or cast; and the term warp, which, from its German origin, has exactly the same import, is employed to denote four, in various articles of trade.

Those simple arrangements would, at their first application, carry the reckoning but a very little way. To express larger numbers, it was necessary to repeat the process of classification. The ordinary steps, by which language ascends from particular to general objects, might point out the right path. A collection of individuals forms a species; a cluster of species makes a genus; a bundle of genera composes an order; and a group of orders perhaps constitutes a class. Such is the method indispensably required in framing the successive arrangement of the almost unbounded subjects of Natural History. A similar mode is pursued in the subdivision and distribution of the members of a vast army.

In following out the classification of numbers, it seemed easy and natural, after the first step had been made, to repeat the same procedure. If a heap of pebbles were disposed in certain rows, it would evidently facilitate their enumeration, to break each of those rows into similar parcels, and thus carry forward the successive subdivision till it stopped. The heap, so analysed by a series of partition, might then be expressed with a very few low numbers easily formed, and capable of being distinctly retained. The particular system adopted, would soon become clothed with terms borrowed from the vernacular idiom.

Let us endeavour to trace the steps by which a child or a savage, prompted by native curiosity, would proceed in classing, for instance, thirty-nine similar objects. He might be conceived first to arrange them by successive pairs. Selecting thirty-nine of the smallest shells or grains he could find, he would dispose these in two rows, each containing nineteen counters, with one over. Having thus reduced the number to nineteen, he might subdivide this again, by representing only one of the rows with shells twice as large as before. He would consequently obtain two rows of nine each, with an excess of one. Instead of these shells, were he to employ shells of a double size, it would be sufficient to denote one of the rows, or to dispose it into two rows. These rows contain only four counters with one over. Again, by adopting counters of a double size, the last row might be represented by one pair, each containing only two marks exactly. These again could be denoted by a single pair of counters, having twice the former dimensions. As a final analysis, one counter, of double dimensions, will express the last row.—Hence the number thirty-nine, decomposed by repeated pairing, would be denoted by one counter of the sixth order, one of the third, one of the second, and another of the first.

Suppose a person should attempt to represent the same number, by triple rows of shells or counters. He would first have thirteen of the smallest shells in each row, and no more. Then, expressing one of the rows by shells of three times the size or value, it would be again resolved into three rows, each containing four counters, with an excess of one. Taking the last of these rows, therefore, and employing counters of triple size, it would be represented by a column of single counters, with one over. Finally, this column would be marked by a single tripled counter.—The number thirty-nine is thus expressed, on the system of triplication, by a counter of the fourth order, one of the third, and another of the second.

Lastly, conceive the number thirty-nine to be reckoned by double pairs or quadruple rows. Each row would then contain only nine counters, with an excess of three. But a single row would express the same as all these four, if each counter in it were changed into another of quadruple size or value; and, consequently, this row might be again distributed into four ranks, each consisting of only two marks, with one left. Retain one of these ranks, and substitute counters of quadruple effect, and two such will express the whole amount.—Hence thirty-nine, analysed by the system of double pairs or warps, would be represented by two counters of the third order, one of the second, and three of the first.

In the ruder periods of society, a gradation of counters, accommodated to such a process of numerical analysis, was supplied by pebbles, grains, or shells of different sizes. This series, however, is very limited, and would soon confine the range of decomposition. To reach a greater extent, it was necessary to proceed by a swifter analysis; to distribute the counters, for instance, successively into ten or twenty rows, and to make pebbles, shells, or other marks, having their size only doubled perhaps or tripled, to represent values increased ten or twenty Beyond this stage in the progress of numeration, none of the various tribes dispersed over the vast American Continent seem ever to have passed. In the Old World, it is probable that a long pause of improvement had ensued among the nations which were advanced to the same point in the arts of life. But the necessity, in such arithmetical notation, of employing the natural objects to signify a great deal more than their relative size imports, would lead at last to a most important step in the ascent. Instead of distinguishing the different orders of counters by their magnitude, they might be made to derive an artificial value from their rank alone. It would be sufficient, for that purpose, to employ marks all of the same kind, but disposed on a graduating series of vertical bars or columns. The augmented value which these marks acquire in rising through the successive bars, would evidently be quite arbitrary, depending, in every case, on a key to be fixed by convention. This point in the chain of discovery was attained by the Greeks at a very early period, and communicated to the Romans, who continued, during their whole career of empire, to practise a sort of tangible arithmetic, which they transmitted to their successors in modern Europe. The Chinese also have, from the remotest antiquity, been accustomed to employ a similar mode of calculation, which they are said to manage with singular skill and address.

Resuming, therefore, the number thirty-nine; if it were distributed by successive pairing, it would be thus denoted on a series of six vertical bars.

The same number, decomposed by a repeated trisection, will assume a simpler appearance, being comprised in four bars.

But thirty-nine, when analysed by a succession of double pairs, would require for its expression six counters arranged on three bars.

Suppose the objects to be reckoned were so numerous, that one hundred and sixty-five counters might be required to represent them. Placed in a single row, these counters would only give the very confused idea of multitude. But, if counted by pairs, or divided into two rows of eighty-two each, with an odd one, they would become a little more distinct. Were every counter now, in each row to denote a pair, a single row of them would have the power of both. Let this row be reckoned again by pairs, and it will change into two higher rows, each consisting of forty-one counters, without any excess. But one of these third rows would be sufficient alone, if each counter in it were esteemed equal to a pair in the second rows, or equal to a duplicate pair, or four in the first row. Again, tell one of the third rows over by pairs, and the forty-one counters will be converted into two fourth rows, containing twenty each, and one over. In like manner, each counter of a fourth row, being conceived equal to a pair in the third row, or a triplicate pair, that is, eight on the first row, a Palpable single row, including twenty of those higher counters, would have the same effect. Now, this twenty would be reduced to a pair of rows of ten counters each. Let each counter in the fifth row have the power of two in the fourth row, or of a quadruplicate pair, or sixteen in the first row; and ten such counters would be sufficient. But ten would give five pairs of sixth rows, one of which might denote the whole, if each counter in it were held equal to two in the preceding row, or to a quintuplicate pair, or thirty-two on the first row. Again, the last five counters would be divided into two pairs and one counter, and these two pairs into a single pair of a higher order.

This analysis appears tedious when so detailed, but it would proceed with great ease and rapidity in practice. The number one hundred and sixty-five would, therefore, on the system of successive pairings, be expressed by one septuplicate pair, one quintuplicate pair, one duplicate pair, and one. The language seems very uncouth, merely from its novelty and inaptness to our idiom; but its elements are extremely clear and simple. If a few cognate words had been devised to express the several combinations of pairs, or the ascending scale of the powers of two; it would have removed every objection.

This arrangement, whereby a number is analysed into certain elements by the operation of distributing it and its sections into successive pairs or duads, may be called the Binary Scale, of which two is the root or index. This scale, resting on so narrow a basis, expands slowly, and is therefore not very fit for expressing large numbers by words. But it is well adapted for the simplicity of emblematic exhibition. Suppose marks or counters were placed in perpendicular rows or parallel bars, proceeding from the right hand to the left, such that a counter on any bar should be equivalent to two laid on the bar immediately before it. Instead of putting the hundred and sixty-five counters on the first bar, it would be the same thing to leave one on that bar, and place eighty-two on the next. But instead of eighty-two counters on the second bar, the effect would be the same, to pass over it, and put forty-one on the third bar. Counting these forty-one also by pairs, we should leave one on the third bar, and carry twenty to the next. But twenty divided into successive pairs, would leave the fourth and fifth bars vacant, and throw five on the next. The five again would, after bisection, leave one on the sixth bar, and transfer two to the seventh; or, passing over this, they would carry one to the eighth bar. The decomposition thus effected would appear as below; where only four counters, and as many blanks, are sufficient to exhibit the number one hundred and sixty-five. By this elementary arrangement, a very distinct idea is conveyed: The eye can easily catch the picture, and the memory preserve it.

**BINARY SCALE**

A similar effect would be produced, though much less clearly, by the combination of strokes, or of Runic sculpture, as thus represented. The advantages of the binary scale of notation were prodigiously extolled by the celebrated Leibnitz, who condescended even to write a discourse on its use and mode of adaptation. This very learned, original, and indefatigable philosopher, had the satisfaction to discover, from the researches of ingenious Missionaries, and particularly from the letters of Father Bouvet, some feeble traces of the Binary Notation in the early history of China. Fou-hi, the first Emperor and founder of that vast monarchy, is venerated in the East as a promoter of Geometry, and the inventor of a science, the knowledge of which has been since lost. The emblem of this occult science consists of eight separate clusters of three parallel lines or trigrams, drawn one above another, after the Chinese manner of writing, and represented either entire or broken in the middle. Those varied trigrams were called Koua or suspended symbols, from the circumstance of their being exposed in the public places. In the composition of such varied clusters, it was not difficult for the sagacity of Leibnitz to perceive the application of the Binary Scale carried only to three ranks, or as far as the number eight. The entire lines signify one, two, or four, according to their order, while the broken lines are void, and serve merely to indicate the rank of the others. Before this invention, the only mode of reckoning used in China was by knots tied on a single cord. In Plate XXVII. the Emperor is figured, pointing with a stylus at the two-fold progression; being copied exactly from the third volume of the Mémoires sur les Chinois, omitting only the grotesque portrait of the Sage. Those emblems were besides intended to convey a store of mystical or mythological knowledge. In the trigrams of Fou-hi, the pious and orthodox Missionaries could not fail to discover certain edifying traces of the mystery of the Trinity, at an epoch perhaps as remote as that of the Deluge.

The next Emperor, Chen-noung, devoted his attention chiefly to the improvement of agriculture. But he likewise extended the Kona or the dualic system of his predecessor, employing hexagrams instead of trigrams. These hexagrams were formed with six parallel lines, broken or entire, and placed one above the other. In short, they were the first six bars of the binary scale, of which the last term would represent thirty-two, and all the terms together would reach to sixty-four. Chen-noung contrived that the hexagrams should likewise exhibit sixty-four varieties of combination, by placing them in eight rays issuing from the same central space which was occupied by one of the eight trigrams. See Mémoires sur les Chinois, Tome II. p. 189, Planche viii.

The Ternary Scale of numeration, which reckons by successive threes or triads, advances with more speed. Thus, suppose, as before, that one hundred and sixty-five were to be exhibited on it. Counted by threes, or, in the sportsman's phrase, by leashes, no counter would be left on the first bar, but fifty-five thrown to the next bar or that of the simple triads. These fifty-five counters, being again told by threes, would leave one on the second bar, and carry eighteen to the third bar, or that of duplicate triads. This eighteen, counted twice in succession, would pass over the fourth and fifth bars, and throw two marks to sixth bar, or that of quintuple triads. The original number so decomposed, might therefore be denominated two quintuple triads, and one single triad. It is denoted by three counters, as in the form here annexed.

The number might likewise be readily expressed, though less perfectly, by combined strokes or points.

It is apparent that the Ternary Scale, though more powerful than the Binary, requires two sets of marks or counters. In the example now taken, each counter on the ascending bars represents three, nine, twenty-seven, or eighty-one; and the number itself is consequently split into two eighty-ones and three.

Let still the same number be arranged on the Quaternary Scale which proceeds by Fours or Tetrad. One hundred and sixty-five, told over by double pairs or warps, would leave one counter on the first bar, and carry forty-one to the next. This forty-one again, reckoned by warps or throws, would drop one counter on the second bar, and transfer ten counters to the third bar. The ten being now counted, would leave two counters on the next bar, and carry two to the fourth bar. The original number would therefore be described as containing two triplicate tetrads, two duplicate tetrads, one tetrad and one; and it would be designated in this manner:

Or, if the less satisfactory mode of strokes were employed, one hundred and sixty-five would be thus exhibited.

This number is analysed into twice sixty-four, twice sixteen, four, and one. Three sets of counters would evidently be required to fit this scale for its application.

The Quaternary Scale may be considered as a reduplication of the Binary, each bar of the former comprising two bars of the latter. This effect will appear more conspicuous by comparing the same number as it was exhibited on both scales. It is alleged that the Guaranis and Lulos, two of the very lowest races of savages which inhabit the boundless forests of South America, count only by fours; at least that they express five by four and one, six by four and two, and so forth. We may gather from Aristotle, that a certain tribe of Thracians were accustomed to use the quaternary scale of numeration; for he says that they proceeded no farther than four.*

*Mosse ἀν δεκαπέντε τῶν Ὀρχαίων γίνονται τῆς ΤΕΤΤΑΠΑ. Arist. Problemata. xv. 3. which they would doubtless continue to repeat. If such was the historical fact, those simple people must have never advanced beyond the early practice of reckoning successively by casts or warps. It seems probable that Pythagoras was acquainted with the quaternary system, which he brought from Egypt and India. Hence perhaps the mystical veneration which the followers of that philosopher professed to entertain for the tetractys or quaternion, the root of the scale, which contains besides, within itself, the number denoting the elementary musical proportions. Near the end of the seventeenth century, Weigelius seriously proposed, in Germany, the adoption of the Tetractys or Quaternary numeration, which he explained, with copious detail, in a learned work entitled Aretologistica.*

To advance a step farther, let the same number be represented on the Quinary Scale, which reckons by the series of fives or pentads. Classed in this way, it would pass over the first bar, and throw thirty-three counters to the second. Told over again, it would leave three counters on the second bar, and carry six to the third; and the six, reckoned by fives, would drop one counter on the third bar, and advance one to the next. One hundred and sixty-five would therefore be denominated one triplicate pentad, one duplicate pentad, and three single pentads; and it would be thus denoted:

This number might also be exhibited on the same scale by the combination of strokes with dots:

By this classification, the number one hundred and sixty-five is divided into one hundred and twenty-five, twenty-five, and three fives. The root or index of the scale being five, it would require four sets of counters to adapt it for practice.

The first bar of the Quinary Scale is actually used in this country among traders. In reckoning articles delivered, the person who takes charge of the tale, having traced a long horizontal line, continues to draw, alternately above and below it, a warp or four vertical strokes, each set of which he crosses by an oblique score, and generally, we believe, calls out tally! as often as the number five is completed.

The Quinary system has its foundation in nature, being evidently derived from the practice of counting over the fingers of one hand. It appears accordingly, at a certain stage of society, to have been adopted among different nations. Thus, the Omaguas and the Zanucas of South America reckon generally by fives, which they call hands. The Toupinambos, a most ferocious and warlike race that inhabit the wilds of Brazil, would seem, according to the relation of Lery, to use the same kind of numeration. To denominate six, seven, and eight, those tribes only join to the word hand, the names for one, two, and three. The same mode, as we learn from Mungo Park, is practised by some African nations; particularly the Yolofs and Foulahs, who designate ten by two hands, fifteen by three hands, and so progressively. The Quinary Numeration seems likewise, at a former period, to have obtained in Persia, for the word pentcha, which denotes five, is obviously derived from the radical term pendji, signifying a hand.

When the index of the scale is larger, it often becomes inconvenient to place so many counters as are wanted on the same bar. But this notation may be abridged in the case where the index is even, by adopting a counter of greater dimensions, to signify the half of it. Suppose one hundred and sixty-five to be disposed in the Senary Scale, which proceeds by successive sixes or hextads. Parted into six rows, that number would leave three counters on the first bar, and cast twenty-seven to the next; this twenty-seven, being reckoned by sixes, would drop three counters on the second bar, and transfer four to the third bar. The original number would hence be described as four duplicate hextads, three single hextads, and three. But instead of placing three counters on the first and second bars, and four on the third bar, one large counter may supply the place of three; as in the form here annexed.

The Senary arrangement has few advantages to recommend it; yet it seems at one period to have been adopted in China, at the mandate of the Emperor Che-hoang-ti. This capricious tyrant, who murdered the literati and burnt their books, having conceived an astrological fancy for the number six, commanded this to be used in all concerns of business or learning throughout his vast Empire. He directed a sort of arithmetic to be composed, with six for its basis; and he enjoined, that all weights and measures should be arranged on the same scale. He divided China into six times six, or thirty-six, provinces; and was so much enamoured of this favourite number, as to order his chariot to be just six feet long, and to be drawn by six horses, with only six attendants.

* This writer even goes so far as to invent names for the several orders of his Tetractys System. They will appear to have a sufficiently German air, though not harsher than the terms we now use.

Second Order, or 4, - Erff. Third Order, or 16, - Zwerrf. Fourth Order, or 64, - Secht. Fifth Order, or 256, - Schock.

Sixth Order, or 1024, - Erff Schock. Seventh Order, or 4096, - Secht Schock. Eighth Order, or 16384, - Schock mahl schock.

Weigelii, Aretologistica vel Logistica Virtutum Genetrix. Norimbergae, 1687. Let this Octary Scale, which proceeds by successive eights or octads, be treated in a similar way. The original number, being reckoned by eights, would leave five counters on the first bar; and throw twenty to the next; and this twenty being told over again by eights, would leave four counters on the second bar, and carry two to the third. If, therefore, the large counter signify half the index, or four, one hundred and sixty-five will be thus denoted:

The number is denominated two duplicate octads, four single octads, and five; and it has been decomposed into two sixty-fours, four eights, and five.

Suppose now that the Denary Scale of Notation were employed. The same number, reckoned by tens or decades, would leave five counters on the first bar, and cast sixteen to the next bar; which, being told again, would leave six counters on the second bar, and carry one to the third. This arrangement furnishes the denomination of one duplicate decad, six single decades, and five; which is simpler than any of the former appellations, and yet it sounds uncouth, owing merely to our want of familiarity with the terms. It would be marked in this way:

We are thus conducted by successive advances to that system of numeration which has prevailed among all civilized nations, and become incorporated with the very structure of language. This almost universal consent clearly bespeaks the influence of some common principle. Nor is it difficult to perceive that the arrangement of numbers by tens would naturally flow from the practice so familiar in the earlier periods of society—that of counting by the fingers on both hands. Aristotle in his Queries points at this origin, but with a less decided tone than might have been expected from him. The philosopher even hints at other concurring causes, some of which appear to be very fanciful. Such, for instance, is his conjecture, that the root of the denary scale might be derived from the summation of the numbers one, two, three, and four, included in the Pythagorean tetractys. If the original import and composition of the Greek terms ἑκατόν, ἑκατόν, ἑκατόν, and μύριος for ten, a hundred, a thousand, and ten thousand, could be safely traced, we might discern the influence of the denary system in the formation of those words. The Roman terms for numerals proceed not farther than mille, a thousand; but they are evidently of the same family, with some of the slighter modifications.

But the origin of the names imposed on the radical numbers, appears most conspicuously displayed in the nakedness of the savage dialects. The Muyseca Indians, who formerly occupied the high plain of Bogota in the province of Grenada, were accustomed to reckon first as far as ten, which they called quihicha or a foot, meaning no doubt the number of toes on both feet, with which they commonly went bare and exposed; and, beyond this number, they used terms equivalent to foot one, foot two, &c., corresponding to twelve, thirteen, &c. Another tribe, who likewise inhabits South America, the Sabiconos, call ten, the root of the scale, tunca, and repeat only the same word, to signify an hundred and a thousand, the former being termed tunca-tunca; and the latter tunca-tunca-tunca.

Etymology, guided by the spirit of philosophy, furnishes a sure instrument for disclosing the monuments of early conception, preserved though disguised, in the structure of language. Our own dialect as immediately derived from its Gothic stem, betrays a composition not less rude or expressive than the simple articulation of the Sabiconos. According to the authorities collected by a celebrated German philologist, the late very laborious and accurate Aedelung, the word eleven was most anciently written ciilif or ciilicin, being compounded of ei or one and the verb liban, to leave, and therefore signified merely one, leave; that is, retain one, and set aside no doubt ten, the root of the scale. Twelve has the same etymon. The names twenty, thirty, forty, &c., have the terminating syllable ty, which corresponds to zig in German, and zig or zuch among the oldest writers of that parent tongue. This termination is derived from the verb ziehen, to draw, and hence twenty means simply two drawings, thirty, three drawings, &c., intimating evidently, that so many tens are separated from the heap. The term hundred, which also runs unvaried through all the filiations of the Gothic, is formed of hund, which anciently signified only ten, and red or ret, a participle from the verb recitan, to reckon or place in rows: The compound would therefore intimate as much as ten times told, that is the reduplication of ten, or ten added ten times.

In the Gothic Translation of the Gospels made by Ulphilas, in the fourth century, one hundred is expressed by tackund tackund, or the word for ten merely doubled; exactly like the tunca tunca of the Sabiconos of South America. But in the Anglo-Saxon version, which was made about three centuries later, one hundred is denoted by hund teontig, meaning ten of ten drawings. In the same curious monument of our early language, hund seafontig, or ten of

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*Διὰ τί πάντες ἀδιάφοροι ὑπὸ Βαβυλῶνος ἐκ ἀληθείας ἐξ ἡμῶν καταδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται, ἢ ἐκ τῆς ἀληθείας ἐξ ἡμῶν ἀποδιώκονται.* And, after throwing out some loose conjectures, he subjoins: "*ἡ ὁτὶ πάντες ὑπήρχαν ἀθροίσαντες, ἐκοντές δέκα δακτύλους; ὅτι ἐν ἑπτάς ἐκοντές τῷ εἰκαί ἀριθμοῦ, τῷ τῷ πλάτει καὶ ταλάλα ἀριθμοῦ." Arist. Problemata xv. 3. It seems probable, that *hund* and *ten* or *teon* were only variations of the same word. The term *thousand* is merely an abbreviation of *divis-hund*, its earliest form. The prefix *divis* is the same as the word *twice*, and *hund* was probably contracted for *one hundred*. The combined expression would, therefore, signify a redoubled hundred, or one hundred repeated ten times.

It is remarkable that the Peruvian language was actually richer in the names for numerals than the polished dialects of ancient Rome or Greece. The Romans, we have seen, went not farther than *mille*, a thousand; and the Greeks made no distinctive word beyond *myriar*, or ten thousand. But, the inhabitants of Peru, under the Incas, following the Denary System, had the term *hue*, to denote one; *chacan*, ten; *pachae*, one hundred; *huarancay*, a thousand; and *hunu*, a million. These words are either original, or have been formed, like our numerical terms, by the abbreviation of certain compound expressions.

Philosophers, from Aristotle down to Locke, have too hastily given credit to the vague reports of travelers, concerning the very scanty knowledge which savages possess of the art of numeration. Even the cautious and discriminating historian Dr Robertson, thinks himself warranted to conclude, from authorities which he quotes, that certain tribes of native Americans could reckon no further than ten; or perhaps twenty, while others advanced not beyond three. But, it is utterly repugnant to congruity, to suppose that any people should be found so destitute of mental faculties, as not to rise above those low numbers. It seems far more probable, that such numbers were merely the roots of the systems of classification adopted by the several tribes, and mistaken by unphilosophical travellers for the whole extent of their numeration. Every savage nation must surely be supposed capable of reckoning up the bands of warriors which they can bring into the field, amounting generally to hundreds, and often to thousands.

Roger Williams, one of the earliest settlers of New England, a man of sense and observation, though deeply tinged with the enthusiasm of the age, published in 1643, a small Key to the language and manners of the Indian nations who then surrounded that infant colony, in which he gives an ample list of the numerals employed by the native tribes. It hence appears, that those people used the Denary Scale of arrangement, and had a set of distinct words to express the numbers as far as a thousand, and could even advance as high as one hundred thousand, by help of combined terms. Thus, one they named *niqui*; ten, *piuck*; an hundred, *piwuck*; and a thousand, *mittonug*. But these words are apparently compound, and would doubtless be found to throw much light on the subject of numeration, if we had any means of analysing them. The same author assures us, that the Indians, employing grains of corn for symbols, were very expert in their computations.

According to the testimony of Leems, a respectable Moravian missionary, the Laplanders, in their computations, join very significantly the cardinal to the ordinal numbers. Thus, to express eleven and twelve, they say *auft nubbe lokkai*, and *gouft nubbe lokkai*; that is, one to the second ten, and two to the second ten. In like manner, to signify twenty-one and twenty-two, those rude people use the expression *auft gouft nubbe lokkai*; and *gouft gouft nubbe lokkai*, meaning one added to the third ten, and two added to the third ten. This procedure affords a curious and very happy illustration of the principle of numeral arrangement.

The Duodenary System of arrangement was introduced at a more advanced stage of society. It plainly drew its origin from the observation of the celestial phenomena, there being twelve months or lunations commonly reckoned in a solar year. The Romans adopted that index, to mark their subdivision of the unit of measure or of weight. They distinguished the foot into three hand-breaths or palms, and each palm into four lengths of the thumb-joint or digit; and, in like manner, they first bisected the pound, next bisected this again, and then divided the quarter into three final portions. The twelfth part of a foot and that of a pound were alike termed uncia, which has branched into the modern words inch and ounce, applied more discriminately with us to the subdivisions of measure and of weight.

The mode of reckoning by twelves or dozens has been very generally adopted in the wholesale trade. Nor is its application confined to the first term of the progression; but extends to the second or even the third. Twelve dozen, or an hundred and forty-four, makes the long or great hundred of the Northern Nations, or the Gross of traders. Twelve times this again, or seventeen hundred and twenty-eight, forms the Double Gross.

Let the same number, one hundred and sixty-five, as before, be reckoned on the Duodenary Scale. It contains thirteen dozen and nine, or one gross one dozen and nine, and consequently, will be represented as here annexed.

Next to the Denary Scale itself, the system of counting by progressive scores or twenties, derived from the same source, appears to have been the most prevalent. The savage who had reckoned the fingers on both hands, and then the toes on both feet, making twenty in all, might seem to have reached the utmost limit of natural calculation. The Guaranis, a very simple and inoffensive tribe, who live on the shores of the Maranon, are accordingly said to proceed no farther in their direct numeration. When these people want to signify an hundred, they only place in a row five heaps of maize, each composed of twenty grains. The Mexicans, however, being more advanced in society, were accustomed to employ the higher terms of the same progression, thus combining the Denary with the Binary scales. In the ancient hieroglyphic paintings of that unfortunate race, units, as far as a score, are exhibited by small balls; and twenty is denoted by a figure, which some authors, and particularly Clavigero, have mistaken for a club, but which was really a small standard or flag. In the same curious monuments, twenty scores, or four hun- Palpable dred, is signified by a spreading open feather; probably, because the grains of gold, lodged in the hollow of a quill, represented, in some places, money or the medium of exchange. This symbol has, from the rudeness of the drawing, been taken at times for a pine-apple, an ear of maize, or even the head of a spear; but its application to intimate the duplicate scores is certain and invariable. A sack or bag, was also painted by those ingenious people, to represent twenty times twenty scores, or eight thousand. It was of the same form as a purse called xiquipilli, and supposed to hold eight thousand grains of cacao. To avoid the multiplication of the balls, and other symbols, the Mexicans sometimes divided the flag denoting a score by two cross lines, and coloured the one half of it to signify ten, or covered three quarters of it with colour to mark fifteen. This mode of abbreviating the signs was evidently capable of farther extension. Plate XXVII. exhibits the series of Mexican hieroglyphic numerals, copied from Humboldt's splendid work entitled Vues des Cordilleres; and to illustrate their application, we have represented the current year 1816, by help of them, having its circle prefixed as indicating a complete revolution of the four seasons.

Traces of numeration by scores or twenties still exist in the old continent. The expression three score and ten, in our own language, is more venerable than seventy; and the compound quatre-vingts et dix, is the ordinary mode in French for signifying ninety. The Arabians, according to Sylvestre de Sacy, counted no farther than four hundred or twenty scores, before the sixth century of the Hejira, which corresponds to about the middle of the eleventh century of the Christian era. To denote nine hundred, they had recourse to the clumsy expedient of redoubling the alphabetic character of four hundred, and joining to it the character of one hundred. The inhabitants of the province of Biscay, and of Armorica, people descended from the ancient Celts, are said to reckon like the Mexicans, by the powers of twenty, or the terms of progressive scores.

Another progression, still swifter in its operation than that by scores, and long familiar to Astronomers, was introduced into the Alexandrian school by the famous Ptolemy, who had the merit of digesting the results of celestial observations into a body of regular science. We allude to the Sexagenary Scale, which proceeds, as its name implies, by the successive powers of Sixty. This arrangement of numbers appears quite artificial, and was no doubt suggested by the division of the circle founded on astronomical phenomena. Since the year is composed of twelve months, containing each about thirty days, the round number, three hundred and sixty, was chosen as the most convenient for subdividing the ecliptic into degrees. But the radius of the circle, was naturally employed as a standard for the measuring of lines and arcs in general; and this being nearly the sixth part of the circumference, and comprising, therefore, about sixty degrees, the arcs themselves, or their multiples, when expressed by degrees, came to be reckoned by Sixties. The notation was effected by annexing a single dash for Sixty, two dashes for its duplicate, and three dashes for its triplicate.

Since, in the notation by numeral scales, the import of a counter depends on the position of the bar on which it stands, any alteration of the place of units must produce a proportional change on the value of the whole amount of an expression. Thus, in the binary classification, the shifting of the bar of units one place lower, would, in effect, double all the preceding terms, and a second shift would double these again. In like manner, to carry the beginning of the scale a bar lower would, in the denary system, convert the units into tens, the tens into hundreds, and the hundreds into thousands; thus augmenting the whole tenfold. On the contrary, if the units were moved to a bar higher, the amount of any expression, would, in the binary scale, be reduced to one-half, and, in the denary, to the tenth part of its former value. Hence, to multiply or divide by the index of any scale or its powers, we have only to change the names of the bars, or to shift the place of the units.

The systems of progressive numeration are as well adapted to represent a descending as an ascending series; a property which greatly facilitates and simplifies the exhibition of fractions. Suppose, for example, it were sought to express eleven-sixteenths on the Binary Scale. Since a counter depressed by four bars will signify only the sixteenth of its original value, so eleven, reckoning upwards from the low bar, will express the value required.

But the analysis of the fraction might be performed otherwise. It is evident, that eleven-sixteenths on the first bar, or the bar of units, are only equivalent to twenty-two such parts, that is, one counter and sixteen-sixteenths, placed on the next descending bar. This excess again corresponds to twelve parts on the second bar, or twenty-four carried to the third bar, making one counter and eight-sixteenths. But these eight-sixteenths are equivalent to a whole counter on the fourth bar, where the exhibition stops. Hence the fraction proposed appears to consist of one-half, one-eighth, and one-sixteenth.

On the Quaternary Scale, this fraction would require only two bars, since sixteen is only the second power of the index four.

But the second mode of decomposition is, on the whole, simpler. The eleven-sixteenths of a counter on the bar of units, are equivalent to forty-four, or two counters, and twelve such parts on the next bar; and these twelve-sixteenths correspond to forty-eight, which compose three entire counters on the third bar. The original fraction is thus analysed into two-fourths and three-sixteenths.

To express the fraction on the Senary Scale, more bars are wanted. It is now equivalent, on the bar immediately after that of units, to sixty-six-sixteenths, or four counters and two parts. This small excess corresponds to twelve on the third bar, or seventy-two on the fourth bar, making four counters and eight The fraction is thus resolved into four-sixths, four-two-hundred and sixteenths, and three-twelve-hundred and ninety-sixths.

Let the same fraction be represented on the Denary Scale. It might be found, that sixteen, multiplied by six hundred and twenty-five, produces ten thousand, or the fourth power of the index, ten. Wherefore eleven times six hundred and twenty-five, or six thousand, eight hundred and seventy-five, reckoned upwards from the fourth bar; below the rank of units, will denote the fraction proposed.

But the expression of this fraction on the descending scale is more easily and directly obtained, by the process of successive decomposition. Ten times eleven sixteenths, or one hundred and ten parts on the bar immediately below the units, make six counters and fourteen parts. These fourteen parts are equivalent to one hundred and forty parts, or eight counters and twelve parts on the third bar. This excess again gives one hundred and twenty parts, or seven counters and eight parts, for the fourth bar. And lastly, these remaining eight parts are represented by five counters placed on the fifth bar. The fraction eleven sixteenths is thus analysed into sixteenths, eight-hundredths, seven-thousandths, and five-ten-thousandths.

There is not the same facility, however, in decomposing all fractions, and reducing them to the terms of a descending scale. It often happens that the expressions for these will run through many bars, or even maintain a perpetual circulation, without ever drawing to a termination. Suppose, for example, that four sevenths were to be represented on the Binary Scale: It would correspond to eight-sevenths, or a counter and one part of the second bar; which excess would be equivalent to two parts on the third bar; four on the fourth, eight, or a counter and one part on the fifth bar. This bar would therefore leave out the same portion as the second bar; and consequently the notation would be continually repeated at the interval of three bars.

It is curious to remark with regard to the present example, that if the expression on the Denary Scale were multiplied by any number whatever, the same counters, and after the same order, would always appear to occupy the inferior bars: Thus, when doubled, it will assume this form, commencing like the fourth bar of the single expression. If the expression for four-sevenths be tripled, it will show the same aspect, only beginning at the third bar:

A little reflection will discover the case to be an unexpected result. Since the perpetual succession of six bars, or only one less than the denominator, was necessary to represent the value of the given fraction; it follows, that every possible remainder arising from the repeated subdivision or decomposition by seven, must have some corresponding bar from which the same recurrence will take place. Thus, in the analysis of four-sevenths, the remainders on the successive bars were five, one, three, two, four and six; and consequently every fraction having seven for its denominator would belong to one of these.

To reduce a fraction to any descending scale, may therefore prove a tedious, and often impossible task. But the converted expression approximates rapidly to its true value, and a very few terms will be sufficient for every practical use. The numerical scales are thus equally fitted by their constitution for ascending or descending,—whether for exhibiting huge multitudes or the minutest subdivision of parts. But men were, in general, very slow to perceive or to avail themselves of this most valuable though distinguishing property of such progressions. The Chinese are the only people who have for ages been accustomed to employ the descending terms of the Denary Scale, or to reckon by Decimal parts in all their commercial transactions. The same uniform system directs the whole subdivision of their weights and measures; an advantage of the highest importance, since it gives to the calculations of those ingenious traders the utmost degree of simplicity and readiness. The natives of India, who have so long been acquainted with the use of the Denary Scale of numeration, are yet ignorant of its application to denote fractions. Below the place of unity they change the rate of progression, and descend merely by a continued bisection, adopting successively the half, the fourth, the eighth, and the sixteenth; beyond which partition they seldom advance.

Ptolemy, and succeeding Astronomers, having adopted the Sexagenary Scale in their measures and calculations of the lines about a circle, likewise employed its descending terms to express the fractional parts. This sexagesimal subdivision, called sometimes logistic arithmetic, is still retained in expressing arcs by degrees, minutes, and seconds; though it has at length been superseded by the decimal system, in the exhibition of chords, or of the sines and tangents which came to displace these in the modern Trigonometry. Nor was the latter improvement embraced, or even contemplated at once. The Arabs, though well acquainted with the advantages of the Denary Scale, appeared satisfied with the sexagenary numeration derived from their Greek instructors. The great restorer of mathematical science in Europe, George Purbach of Vienna, a man of original and extensive genius, who died at an early age in 1462, in the tables for sines which he computed to every ten minutes of the quadrant, distinguished the radius into six hundred thousand parts; thus blending the sexagesimal with the decimal notation. His disciple and successor, John Müller, commonly styled Regiomontanus from Königsberg, the name of his place of birth, after some hesitation laid aside this subdivision of the radius in 1464, and enlarged it to a million of parts, having recalculated the sines, and likewise joined to them tables of tangents. But his work lay many years after his decease in manuscript, and was not printed until 1533. A very long period still elapsed before mathematicians were trained to the use of decimals. Simon Stevinus, a celebrated Flemish geometer and engineer, was the first who composed, in 1582, a distinct treatise on the theory of those fractions.

We might suppose that the properties of the Denary Scale, whether in ascending or descending, would soon be discerned and reduced to actual practice. But the use of fractions is associated with a more advanced state of society. Men very seldom take large and connected views of things; they generally grope their way step by step, as actual wants and circumstances chance to direct them. In the subdivision of the unit, they have often proceeded by bisection, sometimes by trisection, and frequently by joining together these modes, or combining them with decimation. This irregular and unsteady procedure is apparent in our own dissection of weights and measures. The simplest division is that of halving, which seems very naturally to spring from the competition between seller and buyer.—Our dry measure is, accordingly, broken down by a series of bisections. Thus, the chaldron is divided by a redoubled bisection into four quarters, each of these by a triple bisection into eight bushels, each of these again by a redoubled bisection into four pecks, each peck bisected into two gallons, each gallon by a redoubled bisection divided into four quarts, and each of these bisected into two pints. The avoirdupois weight, which is the sort generally used in business, appears both to ascend and descend, from a common point. The pound is divided by a quadruple bisection into sixteen ounces, and each of these again into sixteen drams. But were it desired that a hundred pounds should constitute the quintal, this would be reduced, by a successive quartering, to six and a fraction. Wherefore the nearest round number, seven, was taken, which being doubled gives fourteen pounds for a stone of horseman's weight, and this doubled again makes twenty-eight for the quarter, which consequently furnishes one hundred and twelve pounds for the nominal hundred weight. Twenty times this composes the ton. In the subdivision of money, the progression is still more irregular. The pound is Palpable Arithmetic is first divided into twenty shillings, next each of those into twelve pence, and then each penny into four farthings.

All the calculations in theoretical mathematics are now conducted invariably on the decimal scale. The same kind of subdivision has, to a certain extent, been introduced, for the sake of its great convenience, into the practice of gauging and land-surveying. But we may despair of ever seeing the decimal progression adopted in the general intercourse of society. The French have, indeed, set an example to the rest of Europe, in this case, at least, perfectly harmless, and highly worthy of imitation,—by framing a consistent and universal system of measures, weights, and coins, drawn from nature itself, and disposed with admirable simplicity and elegance. Yet, even in France, that seductive plan has, amidst the collision of opposite views and interests, experienced only a partial success. National prejudice, inflamed by ignorance, is too often opposed to every species of improvement; and the influence of a country, now shorn of her glory, is not likely to extend beyond her own frontiers.

When the index of a Numerical Scale is large, the notation may be conveniently abridged, by marking only what counters are wanted to complete any bar, or render its expression equivalent to that of an additional counter placed on the bar immediately before it. Thus, instead of eight counters on a particular bar, it would be sufficient to join one to the preceding bar, and put two defective or open counters in the Denary Scale, or four such counters on the Duodenary Scale. The same mode of contraction is alike applicable to the expression of integers, or of fractions. For the sake of illustration, let some of the former examples be resumed. The number one hundred and sixty-five was thus represented on the Octary Scale; there being one counter, and the substitute for four on the first bar, the same substitute on the second bar, and two counters on the first bar. Now add a counter to the second bar, and lay three open counters on the third bar, and this will be changed into the form annexed.

This expression again is, by a similar process, converted into another, consisting of three open counters on the first and on the second bar, with three full counters on the third bar. Hence the number proposed, must consist of three duplicate octads, abating three octads and three, or is equal to one hundred and ninety-two, diminished by twenty-seven.

On the Denary Scale, the same number was denoted as under I. and will be successively changed into two other forms.

The last expression, therefore, signifies two hundred, abating thirty-five.

Again, the number one hundred and sixty-five, which, on the Duodenary Scale, stood as annexed, will be reduced to the simpler expression; by changing the three counters, and six on the first bar for three open counters, and an additional counter on the second bar, intimating one gross and two dozens, abating three.

This method of employing open or deficient counters is applicable likewise to the notation of fractions. Thus, eleven-sixteenths expressed on the Quaternary Scale, may have these three different forms:

On the Senary Scale this fraction will admit of more varieties:

VOL. I. PART II. On the Denary Scale, this fraction may have its first expression changed, by successive modifications, into a simpler form.

The fraction four-sevenths, which, on the Denary Scale, formed a perpetual recurrence, may be abbreviated in the same way. The first expression is converted into another more commodious one, by changing the counters that exceed five on any bar into deficient counters.

We shall now investigate some general properties of those Numerical Scales. Suppose, in the Binary Progression, there was standing but a single counter on a high bar. It is obvious, that this might be removed, and two placed on the inferior bar. But one of these might likewise be removed, and two counters substituted for it on the next bar lower. The same process could be pursued through any number of bars, the removed counters being always marked by a dot, and the one which is finally rejected placed on the outside of the last bar with two dots over it. It hence appears, that one placed on any bar of the Binary Scale, is equal in value to one joined to the sum of a series of units running through all the inferior bars. Thus, in the example now produced, one counter occupying the sixth bar, and therefore indicating the number thirty-two, is equal in effect to one annexed to the sum of sixteen, eight, four, two, and one.

Let similar modifications be introduced into the Ternary Scale. Suppose a single counter to stand by itself, all the rest of the bars being empty. It may be taken away, and three counters substituted for it on the next inferior bar. But one of these may be withdrawn, and three others placed on the following bar: Of this triplet, the undermost might again be removed, and three substituted for it on the next bar. The same process, it is evident, could be repeated, till the change reached the lowest bar, leaving out an excess of one, marked by two dots to signify its being transferred. Hence the single counter on the Ternary Scale is, by successive mutations, converted into two rows of counters extending through all the inferior counters, and leaving an excess of one. Thus the number two hundred and forty-three, the value of a counter placed on the fifth bar of the Ternary Scale, is equal to one added to double the sum of eighty-one, twenty-seven, nine, three, and one, the values of counters occupying all the inferior bars.

In like manner, if a solitary counter in the Quaternary Scale be withdrawn, four counters may be substituted on the next bar. Remove the undermost of these, and set four more on the succeeding bar. Take away one of these again, and put other four counters on the adjacent bar. Proceed in the same way, till the quaternion reaches the last bar, and is reduced to a triplet, by the exclusion of one counter. By this analysis, therefore, the simple counter is resolved with an excess of one, into three rows of counters which run through the whole of the lower bars. In the present instance, the number two hundred and fifty-six, the import of a counter on the fifth bar of the Quaternary Scale, is equivalent to one joined to triple the sum of sixty-four, sixteen, four, and one, the values of all the succeeding bars.

It may seem scarcely necessary to pursue this investigation farther; but we shall extend it likewise to the Quinary Scale. A single counter, it is obvious, may now be removed, and five substituted for it on the next bar. The undermost of these again may be withdrawn, and five placed instead of it on the following bar. One of these may then be taken away, and five substituted for it on the adjacent bar. The same procedure is repeated to the last bar, leaving four rows, with an excess of one counter. In the present instance, a single counter on the fourth bar, and corresponding, therefore, on this scale the number to six hundred and twenty-five, is equivalent to one added to four times the sum of one hundred and twenty-five, twenty-five, five, and one, the values of all the inferior bars. We may hence conclude in general, that if one be taken from any power of the index of any numerical scale, the remainder will be equal to all the inferior powers repeated as often as the index, when diminished by one, contains unit. Wherefore if the counters, reckoned as mere units, be separated from any compound expression, the whole will be converted into as many trains of counters, occupying all the inferior bars as correspond to the index of the scale diminished by one. Suppose, for example, the expression here noted, which is disposed on five bars, of the Quaternary Scale, and is equivalent to seven hundred and eleven.

By decomposing separately each successive bar, and placing the excluded counters close beside the place of units, it will be changed into this regular but complex form, which consists of three rows of double counters, leaving out two; three rows of triple counters, leaving out three; a single counter, leaving out one; and three counters on the last bar left out. If we omit the nine excluded counters, and take only each single row of the rest, the result will be greatly simplified. This reduced expression corresponds to two hundred and thirty-four. But seven hundred and eleven, diminished by nine, the number of counters at first employed, leaves seven hundred and two; and this again, being divided by three, gives two hundred and thirty-four.

Applying this principle to the Denary Scale, it must follow, that if the sum of the integers be subtracted from any number, the remainder will be divisible by nine, and the quotient will be equal to the sum of the separate numbers denoted by each of those integers repeated through all the inferior places.

Another similar property, belonging to numerical Palpable scales, may be deduced from the combination of the deficient or open counters. Let us begin with the Binary System. Suppose a solitary counter to occupy the sixth bar. It may be removed, if two counters be placed in its stead on the next bar. But, without changing its value, we may to this pair evidently join another, composed of a full and an open counter, which perfectly balance each other. Withdraw the open counter that stands undermost, and substitute for it two open counters on the fourth bar.

To these again, add a balanced pair, consisting of an open and a full counter. Take away the undermost counter, set two similar counters on the third bar, and to this pair annex a full and an open counter. By continuing this process, the single counter will be decomposed into three rows of counters, alternately full and open; with an excluded counter, which is open when the number of the bars, as in the present case, is even, but a full counter if that number be odd. Thus thirty-two, the value of the counter on the sixth bar, is equal to three times all the inferior alternating bars, or the excess of sixteen above eight, joined to the excess of four above two, and together with one, that is, eleven; abating, however, the excluded counter which is here open; but thrice eleven, omitting one, is thirty-two.

Let a similar analysis be applied to the Ternary Scale. Change the solitary counter for three counters laid on the next bar, and to these join a balanced pair consisting of a full and an open counter. Remove this open counter, and substitute three open counters for it on the succeeding bar, and to the triplet annex an open and a full counter. Take away the full counter, and place three such counters on the following bar. Repeat the procedure, till the first bar comes to be occupied; and there will evidently emerge four rows of counters extending through all the inferior bars, and alternately full and open, with an excluded counter of an opposite character to those which terminate the decomposition. In the present instance, where the single counter stood on the fifth bar, a counter of the ordinary kind is left out. Wherefore the number eighty-one is equal to four times the amount of the excess of twenty-seven above nine, and of that of three above one, or twenty, together with the unit excluded.

It may be deemed sufficient for grounding a general conclusion, to repeat the same process on the Palpable Arithmetic.

Quaternary Scale. Instead of the solitary counter, place four counters on the next bar, and conjoin with these a balanced pair of counters, composed of a full and an open one. Remove the undermost of these, and substitute four open counters on the following bar. To these again, add an open and a full counter, which will not affect the value of the column. Pursue the operation till all the bars are occupied by counters, and one excluded. It is evident therefore that, on the Quaternary Scale, the decomposition of a single counter produces five rows of alternating counters through all the inferior bars, with an excluded counter of an opposite nature to that of the row which completes the analysis. In the example now given, the number two hundred and fifty-six, the value of the counter on the fifth bar, is equal to five times the amount of the excess of sixty-four above sixteen, and of four above one, or fifty-one, together with the unit left out.

The conclusion may, therefore, be extended to any progressive scale: If the value of unit on a separate bar be increased or diminished by one, according as the rank is even or odd, the remainder will be divisible by one greater than the index of the scale, and the quotient will be equal to the amount of the values of units alternating in excess and defect through all the inferior bars.

The same property, it is evident, could be transferred to any compound expression: Nothing is wanted but to change the counters on each successive bar into alternating rows. Thus, in the Binary Scale, the expression which signifies one hundred and seven, will be converted into another composed of triple and alternating rows, with an exclusion of two counters in excess, and three in defect, or the balance of an open counter.

If this arrangement be now divided by three, the result, after restoring the deficient unit, will be

By collecting and condensing the counters on each bar, the whole will be reduced to another very simple form; which indicates thirty-six, the third part of one hundred and eight.

Let another example be selected from the notation of the Quaternary Scale. The counters on these six bars will represent the number two thousand three hundred and fifty-four. The excluded counters will stand thus; and, consequently, there is on the whole an excess of four counters which must be rejected. The original number is therefore reduced to two thousand three hundred and fifty, of which the fifth part will be denoted by repeating the counters alternately full and open on the lower bars.

This last expression being condensed and abridged will stand thus, In all these decompositions, we have stopped at the bar of units; but if we pursue the analysis through the descending bars, we shall discover trains of equivalent fractions which never terminate. Thus, to begin with the Binary Scale: A counter on the bar of units may be taken away, and two counters placed instead of it on the following bar. Of this pair again, one may be removed, and another pair substituted for it on the next lower bar. One of these again may be withdrawn, and two placed on the following bar. The same operation of exchange, it is obvious, may be repeated for ever. Wherefore, the value of a single counter is here the same as that of a single row of counters, extending indefinitely over the lower bars. But the counter on the bar immediately below the place of units, indicates one-half, that on the next one-fourth, that on the following bar one-eighth, and so forth continually. Wherefore the sum of the fractions one-half, one-fourth, one-eighth, one-sixteenth, extended without limit, must always approach to one.

Let a similar transformation be carried through the Ternary Scale. Suppose a half counter to stand on the bar of units: It may be removed, and three half counters, or a whole counter and half of one substituted on the next bar. Take away this half counter, and set three such, or a counter and a half on the succeeding bar. Repeat the same process continually, and the half counter on the bar of units, will be converted into a single row of entire counters, extending without limitation through all the inferior bars. But these successive counters signify one-third, one-ninth, one-twenty-seventh, &c. Whence the fraction one-half is equal to the sum of one-third, one-ninth, one-twenty-seventh, &c. continued ad infinitum.

In the Quaternary Scale, let the third of a counter occupy alone the bar of units. It may be withdrawn, and four such parts, or a whole counter, and the third of one placed in its stead on the next bar. This third again may be removed, and a counter, with another third, substituted for it on the following bar. The same procedure being repeated, the third part of a counter in the place of units will be changed into a row of entire counters running through all the inferior bars. It therefore follows, that the fraction one-third, is equal to the sum of the infinite series one-fourth, one-sixteenth, one-sixty-fourth, &c.

Again, let similar modifications be carried through the Quinary Scale. The fourth of a counter on the bar of units may be exchanged for five such parts, or one counter and a quarter on the following bar; and this quarter may now be removed, and five quarters, or one counter and a quarter set on the next bar. The process of decomposition may thus be continued perpetually, leaving, instead of the fourth of a counter, an unlimited range of counters stretching over the interior bars. Consequently the fraction one-fourth is equal to the aggregate terms of the progressive one-fifth, one-twenty-fifth, one-hundred and twenty-fifth, one-six hundred and twenty-fifth, continued without termination.

From these very simple analyses, we may therefore conclude in general, that the fraction of unit, which has for its denominator one less than the index of any numerical scale, is equal to the sum of all the descending powers, or the value of a single row of counters, extending indefinitely through the inferior bars. Thus one-ninth is equal to a tenth, a hundredth, a thousandth, &c. or one-eleventh is equal to a twelfth part, a hundred and forty-fourth, a thousand seven hundred and twenty-eighth, &c.

But the summation of a descending series, whose terms alternate, may with equal facility be discovered, by introducing the admixture of deficient counters. Thus, in the Binary Scale, if the third part of a counter stood on the bar of units, it could be removed, and two-thirds of a counter, that is, a whole counter abating the third part, substituted for it on the next bar. But this open part of a counter might be taken away, and a whole open counter diminished by the third part of a full counter placed on the following bar. This third again could be withdrawn, and a full counter with a deficient third part of one set on the next bar. This alternate exchange might easily be repeated over the suc- cessive bars. The third part of a counter is thus converted into an unlimited row of full and open counters. Hence the fraction one-third is equal to the sum of one-half abating one-fourth, of one-eighth abating one-sixteenth, of one-thirty-second abating one sixty-fourth, or equal to the excess of the continued series one-half, one eighth, one thirty-second, &c. above the corresponding series one-fourth, one-sixteenth, one sixty-fourth, &c.

The substitution may, in like manner, be performed on the Ternary Scale. Suppose the fourth part of a counter to occupy the bar of units. Remove this fragment and substitute three-quarters, or a whole counter abating one-quarter, on the next bar. Instead of this deficient quarter again, place three such, or one open counter, conjoined with the quarter of a full one, on the succeeding bar. Pursue the same procedure, and the quarter of a counter will be transformed into a single row of counters, alternately full and open, extending without limitation over the lower bars. Therefore the fraction one-fourth is equal to the sum of one-third abating one-ninth, of one twenty-seventh abating one eighty-first of one two hundred and forty-third, abating one seven-hundred and twenty-ninth, &c.; or one-fourth is equal to the excess of the perpetual series one-third, one twenty-seventh, &c. above the similar series, one-ninth, one eighty-first, &c.

Another instance will be sufficient to point out the general principle. Let one-fifth part of a counter stand on the bar of units in the Quaternary Scale. It may be removed, if four-fifths, or one counter wanting a fifth, were placed on the next bar. Instead of this open fifth, let four open fifths, or one open counter, joined to a fifth of excess, be put on the following bar. Change this fifth again for a full counter, with a deficient fifth on the following bar. A series of alternating counters is thus successively formed. Whence the fraction of one-fifth is equal to the continued series one-fourth, abating one-sixteenth, one sixty-fourth, abating one two hundred fifty-sixth, &c.

We may hence infer generally, that the fraction of unit, divided by one greater than the index of a numerical scale, is equal to the amount of all the descending powers taken alternatively as additive and subtractive. Thus, one-eleventh part is equal to the excess of a tenth above a one-hundredth, that of a thousandth above a ten-thousandth, that of a hundred thousandth above a millionth, and so forth continually. In like manner, on the Duodenary Scale, the thirteenth part is equal to a twelfth, abating the hundred and forty-fourth, the seventeen-hundred and twenty-eighth, abating the twenty thousand seven hundred and thirty-sixth, &c.

In all these transformations of fractions, arising from the index of the numerical scale, increased or diminished by one, the operation is repeated or alternated at each successive bar. But similar changes may be made on fractions derived from the same modifications of the powers of the index, which will regularly circulate along the bars at a corresponding interval. Thus, on the Binary Scale, the fraction one-third, or the second power of the index two diminished by one, will form by decomposition an intermitting row, or a perpetual circulation, passing over the successive alternate bars. For one-third of a counter on the bar of units is equivalent to two-thirds on the following bar, which again are equal to four-thirds, or an entire counter and a third, on the next bar. Pursuing the same analysis, a row of counters emerges on the alternate bars. In reality, if the intermediate bars, which here serve only for the transit of the pair of thirds, were left out altogether, the notation would pass into that of the Quaternary Scale, and obey the general rule.

Again, on the same Binary Scale, the fraction one-seventh, or the reciprocal of the third power of two, diminished by one, will be found to circulate at every third bar. Thus, one-seventh of a counter on the bar of units gives two such parts for the second bar, four for the third and eighth, or a whole counter and an excess of one-seventh for the fourth counter; and if this kind of decomposition be carried forward, another counter will appear on the seventh bar, a third on the tenth bar; and so forth in perpetual succession.

But the same conclusion might also be drawn from the general principle, if we consider that the Binary Progression, by omitting always two consecutive bars, is converted into the Octary Scale.

It is not difficult to perceive, that every fraction is capable of being either exactly represented on any given scale, or of being denoted by an expression which circulates after an interval of fewer bars than the denominator of the fraction contains units. In fact, the moment the same set of fractional counters comes to appear a second time, the whole expression must evidently recur in the same order. But all the possible variations or series of remainders must ever lie within the number itself, which constitutes the di- visor. Thus, it was found that the expression for any fraction having the denominator seven, circulates on the Binary Scale; at the interval of three bars. The same fraction represented on the Quaternary Scale has a like recurrence; but, on the Octary Scale, the expression is renewed at every two bars, while it does not circulate till after passing over six bars, in the Ternary, Quaternary, Quinary, and Denary Scales. Employing a similar decomposition, it will appear that a fraction, with eleven for its denominator, will, in the Quaternary and Quinary Scales, circulate on five bars, but will embrace no fewer than ten bars, by its circulation in the Ternary, Quaternary, Senary, Octary, and Denary Scales.

We may hence reverse the operation, and determine the absolute value of any circulating expression, or discover the fraction of which it is only the continued expansion. Suppose the group of counters here presented to circulate perpetually, at an interval of six bars on the Quinary Scale. If these counters were all transferred to the seventh, or last bar, they would signify four thousand four hundred and sixty-four.

But another scale might be substituted, such that each single bar shall correspond in effect to this condensed seventh bar. Consequently the index of this new scale, or the sixth power of five, being diminished by one, will denote the denominator of the absolute fraction; which is therefore four thousand four hundred and sixty-four divided by fifteen thousand six hundred and twenty-four, or by reduction two-sevenths.

In like manner, on the Denary Scale, let a set of counters always circulate, spreading over six successive bars. Conceive them transferred to the last bar, and they will express seventy-six thousand nine hundred and twenty-three. But if another scale were adopted, each bar of which had a power equal to six of these bars, its index would be a million. One less than this is consequently the denominator of the fraction, which is therefore equal to one thirteenth.

Having considered at some length the properties of numerical scales, and their various transformations, we have now to explain the ordinary operations performed on numbers themselves. These operations are all reducible to two very simple changes,—the conjoining and the separating of numbers. When two or more numerical expressions are conjoined, that is, condensed into a single expression, or collected into one sum, the process is called Addition. But palpable when one numerical expression is separated or drawn out from another, leaving only a difference or remainder, the process is called Subtraction. If the addition should be employed merely in repeating the same number, it admits of abbreviation, and is then termed Multiplication. On the contrary, if the subtraction be limited to the continued withdrawing of the same number from another, the process becomes capable of abridgment, and is termed Division.

**Addition.**

The whole operation consists in collecting and condensing the separate expressions. Beginning with the lowest bar, the counters are gathered together, and if they exceed the index of the scale, this excess only is retained, and one counter annexed or carried to the next bar. But if the counters on any bar should contain the index more than once, the number of repetitions is transferred a place higher, while the remainder of the reckoning is left as it stood. A very few examples will render the mode of proceeding quite clear. Let it be required to collect the expressions here disposed on the Quaternary Scale.

On the bar of units, two counters occur, which are brought together at the bottom. Two counters are likewise found on the next bar, and each of the three following bars has each three counters. But four counters are found on the next bar, which are equal to one transferred to the highest bar, leaving a blank in their own place. In this case very little change in the notation is wanted; and except in a single transfer to a higher bar, the counters are merely brought closer together, and form the expression at the bottom. This again might be somewhat abridged by introducing open counters, as represented here, below the other. The sum of the three compound numbers thus represented on the Quaternary Scale, is therefore by reduction nine thousand two hundred and ten.

Suppose the same numbers were converted into the Denary Scale. They would be respectively three thousand five hundred and twenty-eight, one thousand and seventy-three, and four thou- Palpable sand six hundred and nine; which are thus exhibited successively on four bars. Now, collecting all the single counters on the first bar to the right, they amount to ten, which joined to the two fives make twenty; they consequently, in their summation, leave that bar void, and throw two counters to the next. These two counters now joined to the four which occur on the second bar, make six, and again with the five, eleven, leaving one counter on the second bar, and furnishing one to the next. This counter, with another already on the third bar, leaves two; while the two fives on it deliver one on the fourth bar. The result, as before, is nine thousand three hundred and ten. The final expression may be somewhat abridged, by employing deficient counters; it gives ten thousand, abating one thousand, and two hundred and ten, the same result as before.

But perhaps it is more natural and more like the spontaneous practice of uninstructed men, to proceed by successive steps, and only conjoin two numbers at the same time. Nor is it requisite, in this mode of proceeding, that the numbers to be severally added should be actually expressed; it may be sufficient, at each step, to retain them mentally. Thus, when the second number is to be added to the first, the three counters which should occupy the first bar being joined, the three single ones already noted make six, or leave one, and change the other five into ten, which furnishes one counter to the second bar; and one counter on the fourth bar increases its stock to four. Again, when the third number is added, it would give nine counters to the first Palpable bar, and hence change it into ten, leaving the Arithmetic bar void, and throwing a counter to the next bar. But the six counters on the third bar, convert the five counters into ten, and thus furnish one to the next bar, while they leave two counters behind them.

The working of this example would be simplified by employing the notation of deficient counters. Thus the several numbers are changed into other expressions as here expressed. On the first bar, the full and the open counters, being three of each kind, exactly balance each other; but, on the second bar, the full counters leave an excess of one. The full counters on the third bar, being eight in all, exceed the open ones by two; while, on the fourth bar, they leave a defect of one.

Let another example be taken of addition from numbers arranged on the Quinary Scale. The counters on the first bar amount to six, which leaves one counter, and furnishes one to the next bar. This bar now holds eight counters, leaving three consequently, and throwing one to the third bar. The third bar has then ten counters, or is left void, and gives two counters to the next bar. This bar, holding eleven counters, leaves one, and supplies two to the highest bar. The total amount of the numbers is therefore three quadruplicate pentads, one triplicate pentad, three pentads and one, or by reduction, two thousand and sixteen. By introducing deficient counters, the process might be simplified and abridged. The first bar continues the same as before; but in the next bar, after one had been carried to the second bar, and joined to the full counters, there yet is an excess of three full counters. In the third bar, the opposite counters, being two each, are quite balanced, and leave a void; but the fourth bar yields an excess of one full counter, while the fifth collects three full counters. The result is every way the same as before.

Let these numbers be transformed on the Denary Scale. They will stand as here expressed, occupying three bars. The first number furnishes to the successive bar two, three, and six counters; the next number gives seven, eight, and two; the third number delivers four to the third bar, passes over the second, and deposits seven on the first; and the last number, supplies five, nine, and two to the successive bars. In the first bar, the single counters collected together make six, and the two fives contribute one to the next bar. The second bar, holding four fives with a surplus counter, leaves one, and throws two to the next bar. This increase raises the third bar to four fives, and therefore, leaving it void, advances two counters to the fourth bar. The amount of the addition is therefore, as before, being two thousand and sixteen.

If the operation were performed in its utmost simplicity, by annexing one of the numbers always after the other, the progress of summation would be thus exhibited. The first number being actually represented by counters, the second, which is only retained in the memory, changes by its absorption the several bars, and produces one thousand and seventeen. To this again is joined mentally the third number, which gives one thousand four hundred and twenty-four, as another step in the progress. And, lastly, the fourth number, being annexed to this, completes the transformation, and produces the final result, two thousand and sixteen.

But, in most cases, the work of summing different numbers may be sensibly shortened, by a judicious application of open counters; which, by their intermixture, effecting in a long operation a sort of balance, will, for the most part, save the trouble of carrying to the higher bars. Whenever the full counters on a bar exceed half the index of the scale, it is preferable to substitute for them open counters. Thus, in the last example, the four numbers to be added may be denoted as here expressed. In the first bar, there is, on the whole, an excess of four open counters; in the second, an excess of two full counters; but, in the third bar, the opposite counters are exactly balanced, and leave a void. The result is consequently two thousand and twenty, abating four. It is evident that all these operations are conducted in the same way, whether the counters occupy the bars of an ascending or of a descending series. The position of the bar of units determines the values of all the rest. Decimal fractions are, therefore, no more entitled to a separate discussion than those derived from any other scale.

**Subtraction**

is the process by which a number is separated or extracted from another; the difference sought being the remainder left after this operation. If the counters representing the larger number should exceed on each bar those denoting the subtrahend, we have only to mark the several excesses. But, if the upper number has fewer counters on any bar than the subtrahend, it will be necessary to take one from the higher bar, and augment the expression of the other, by joining as many counters as the index of the scale contains units. Suppose it were required on the Senary Scale to subtract two thousand seven hundred and forty-five, from six thousand three hundred and eighteen. The numbers will appear thus arranged. The upper bar of units being vacant, we put six on it, and withdraw or borrow from the next bar an equivalent counter marked with a dot. The three counters on the corresponding bar of the subtrahend, being then set aside, leave other three counters for the difference. On the next bar, a counter, answering to the one below, being removed from the two remaining counters, gives a difference of one. On the third bar, the four counters of the subtrahend cannot be taken away from the single counter above, till it has been augmented to seven, at the expense of one counter withdrawn from the next bar, and therefore marked with a dot. The fourth bar leaves all its four counters, and the fifth bar drops its excess of two.

The operation is varied and shortened, by adopting open counters. To take away the three open counters on the bar of units from the vacancy above it, substitute for that void the counterpoise of three full and three open counters, and then separating the last, there are left three full counters. On the second bar, one counter only remains. The single counter of the third bar may have annexed to it two full and two open counters, and the latter being removed, will consequently leave three full counters.

On the fourth bar, conceive an open and a full Palpable counter to be joined, and then, omitting the latter, there will remain two open counters. The single open counter of the fifth bar may likewise have a pair of open and of full counters combined with it without changing its import, and consequently three open counters will be left. On the sixth bar the single counter is left standing. The same expression is easily denoted by full counters, and gives, on reduction, the number three thousand five hundred and seventy-three, as the result of the subtraction.

Suppose the same numbers were expressed on the Denary Scale. Beginning with the bar of units, the five counters corresponding to those of the subtrahend being set aside, leave three. In the next bar, the single counter is first increased by ten, which are equivalent to a counter withdrawn from the third bar, and then four being separated from these, gives seven for a remainder. In the third bar, a single counter borrowed from the next, furnishes two fives, one of which, with the remaining two counters, being taken away, leave five. The counters on the last bar, now reduced to five, leave only three, after the separation of the two in the subtraction. The remainder is consequently three thousand four hundred and seventy-three.

In every instance where a counter is borrowed from a higher bar, the effect would evidently be unaltered, if a counter were added on the same bar to the number below. This modification of the process is what has been generally termed carrying. It is Multiplication is nothing but a process of repeated Addition. When the terms, however, to be multiplied are complex, and the index of the Numerical Scale is large, the operation will admit of being very considerably abridged. It has been already shown, that a number is virtually multiplied by the index of the scale, by advancing its expression one bar; that it is multiplied by the second power of that index by advancing it two bars; and so forth continually, according to the progressive powers. Again, if any term of the multiplier be great, it is preferable, instead of repeating the counters of the multiplicand, to collect them mentally, and only mark the result. The ready performance of multiplication depends entirely on the right application of these two principles. A few examples will elucidate the process.

Suppose it were required to multiply the number forty-five by twenty-three, that is, to add the units contained in forty-five twenty-three times. First, let those numbers be disposed on the Binary Scale. The counter on the unit bar of the multiplier shows that the whole of the multiplicand is to be set down once as it stands.

The next counter indicates, that the whole of the upper range of counters must be advanced one bar; the counter on the third bar intimates, that the same row should be carried another bar higher. The last counter, occupying the fifth bar, shows that the counters of the multiplicand should be advanced four bars, or that the first of them should be placed on the same bar below it. These counters, again, collected into a single row, give by reduction, one thousand and thirty-five.

Next, let the same numbers be arranged on the Ternary Scale. Following the terms of the multiplier, therefore, the multiplicand must be repeated twice as it stands, then once on a bar higher, and again twice on the bar above this. But twice two make four, leaving one on the third bar, and throwing one to the fourth bar, which produces three, or leaves that bar empty, and delivers one to the next bar. Again, the multiplicand is repeated on a bar higher, and doubled on a bar two steps higher. Collecting then the counters on each successive bar, amounting to one on the seventh bar, one on the sixth, two on the fourth, and one on the second, the result will, after being reduced, appear the same as before.

It is obvious, that the units contained in forty-five must give the very same amount when repeated twenty-three times, as those in twenty-three after a repetition of forty-five times. But the interchange of multiplier and multiplicand, would, in this case, rather simplify the process. The upper counters are here advanced at once two bars and redoubled, and then repeated once on the next set of bars. Being collected, there is still the same amount. Again, suppose the same numbers to be referred to the Senary Scale. The three counters on the unit bar of the multiplicand being repeated five times, make fifteen, or leave three on that bar, and give two to the next. But five times the simple counter on the next bar, together with the two carried, make seven or leave one, and furnish one to the third bar; and the single counter on this repeated five times, joined to the counter transferred, make six, which, passing over the third, furnish one to the fourth bar. Now, three times the three counters give nine, or leave an excess of three for the second bar, and send one to the third. Then the single counter on the second bar is repeated three times and one added to it, and the next counter likewise repeated three times, but without addition. Collecting now the counters on the several bars, there appear four on the fourth, the third, and the second bars, and only three on the first.

This process would be materially simplified, by adopting deficient counters. In this case, the first open counter of the multiplier indicates that the whole of the multiplicand is to be taken away, or changed into opposite counters. The next two deficient counters also imply, that the multiplicand is to be taken twice away, or converted into open counters on the bars one place higher. The collective amount is twelve hundred and ninety-six, diminished by the sum of two hundred and sixteen, thirty-six, six and three, or two hundred and sixty-one; that is, one thousand and thirty-five.

Lastly, let the multiplication of those numbers be performed on the Denary Scale. The five of the multiplicand being repeated three times, as indicated by the counters of the multiplier, leaves five, and furnishes one counter to next bar, and the four counters of the second bar being tripled, and acquiring the one carried, leave three, with another counter for the third bar. The five being next doubled by the two counters of the second bar of the multiplier, give ten for the place of the second bar, or one for the third; and the four counters being doubled and augmented by this transferred one, give nine for the third bar. Collecting these several counters, they represent, as before, one thousand, and thirty-five.

Let us take another example where the numbers to be multiplied are rather larger, being three hundred and eight, and by one hundred and thirty-two.

Quaternary Scale. Beginning with the single counter which occupies the second bar of the multiplier, it throws all In the Denary Scale, more counters will be required, though a large one be adopted, for the sake of conciseness, to represent five, the half of the index. Beginning as usual at the right hand, the terms of the multiplicand are first doubled mentally, as intimated by the two counters of the multiplier; and as twice eight makes sixteen, the six is placed on the bar of units, and one carried to the next bar; then the double of three gives six counters for the third bar. Again, those terms are tripled; as signified by the counters on the second bar of the multiplier; but thrice eight are twenty-four, leaving four on the second bar, and throwing two to the third, while the triple of three gives nine, for the fourth bar. And, lastly, the whole of the multiplicand is set down two bars in advance. The several rows of counters being collected, give still the same result, or forty thousand, six hundred and fifty-six.

The operation may be somewhat abridged by employing, as circumstances will permit, open counters. Thus, instead of eight in the multiplicand, we may assume ten wanting two, or one full counter on the second bar, and two open counters. In this instance, the operation would be greatly abridged, by using open counters. The multiplicand is two gross, and two dozen, abating four; and the multiplier only one gross, abating one dozen. The open counter on the second bar of the multiplier changes in effect the whole character of the multiplicand into the opposite counters, and the full counter on the third bar repeats the same counters, and advances them a bar higher. The result is, consequently, two triple gross and four dozen, abating five gross.

The application of open or deficient counters will be found useful, in varying and simplifying the process of multiplication, even where smaller numbers are concerned. Thus, to confine our views to the common Denary Scale, let it be required to multiply eight by seven. The former, it is obvious, may be denoted by one counter on the second bar, and two open counters on the first; and the latter by one counter on the second, and three open counters on the first bar. These open counters show that the terms of the upper number should be sub- This philosophical trick cannot fail to appear striking to young practitioners, and may prove really useful, by helping to fix thoroughly and accurately in their memory the ordinary multiplication table. But the same principle can be extended farther. Suppose it were required to multiply ninety-seven by ninety-six. These numbers are merely one hundred, abating three and four. The product is, therefore, of hundreds, one hundred fold, abating four, and then three; that is, nine thousand, three hundred; together with three times four units, or twelve.

But instead of seven open counters on the third bar, the defect from ten, or three full counters may be placed, and an open counter set on the fourth bar. This result is consequently ten thousand, three hundred and twelve, abating one thousand.

When fractions are expressed on the same numerical scale, their multiplication proceeds with equal facility as that of integers; it being only requisite to begin with the bar of units, and to descend with the lower bars. Thus, if it were sought to multiply thirteen and a quarter, by nine and a half. Let these quantities be first arranged on the Binary Scale, and they will stand thus:

Begin therefore with the counter on the bar of units, and mark the upper row in the same ranks in which it stands; or, what must evidently give the same result, take the extreme counter which occupies the next lower bar, and depress all the counters in the row by the interval of a whole bar. Proceed in like manner with the other counters. The collective amount is one hundred and twenty-five, and seven-eighths. Suppose the same fractional quantities were transferred to the Quaternary Scale. Instead of commencing with the bar of units, begin with the lowest counters, and therefore double the whole of the terms of the multiplicand, placing them a bar lower, and observing that twice three makes six, which leaves two counters, and carries one to a higher bar. The result is the same as before, since the fraction fourteen-sixteenths is equal to seven-eighths; and that one triplicate tetrad, three duplicate tetrads, three single tetrads and one, amount to one hundred and twenty-five.

Let those fractional numbers be now represented on the Denary Scale. They will be thus expressed.

Beginning with the last counters of the multiplier, or the five on the bar next that of units, its product into the upper five, making twenty-five, should be placed on a bar lower. This leaves five, and throws two to the higher bar; and five times two making ten, the two counters carried are left, and another counter is thrown to the multiplication of the three, making sixteen, which drops six, and likewise changes into six the counters placed on the next bar. Again, the five of the multiplicand repeated nine times, leaves five, and delivers four to the next bar, increasing the counters on it from eighteen to twenty-two, or leaving two and transferring two. The multiplication into the three leaves nine, and gives over two, converting the counters on the next bar to eleven, or one counter left and one advanced. Collecting the counters on the different bars, we get one hundred and twenty-five, and eight hundred and seventy-five thousandth parts.

But the operation will be materially simplified, by introducing an open counter into the second term of the multiplier, as here expressed. The first step of the multiplication by five is the same as before; but the open counter on the next bar indicates, that the whole of the multiplicand is to be set down with opposite counters; and the full counter on the highest bar shows, that it must be repeated in advance without alteration. Collecting the counters together, and observing their mutual influence, the general result is one hundred and twenty-five, with nine hundred and five thousandth parts, abating thirty; that is, as before, one hundred and twenty-five, with eight hundred and seventy-five thousandth parts.

Division is the opposite process to Multiplication, and consists in finding how often the same number can be separated or drawn out from another. In the rudest way, therefore, this operation would be performed, by telling over a certain number of counters repeatedly from the same heap. But instead of a slow process of repeated subtraction, the number to be severed, or the divisor, may be first multiplied to approach the mass to be shared, or the dividend. The remainder can again be treated in the same manner, and the operation renewed, till nothing is left of the dividend, or a difference less than the divisor itself. Those multipliers, collected together, will express the quotient, or the number of subtractions required to exhaust the mass. A few examples shall be taken for illustration.

Suppose sixteen thousand and six were to be divided by fifty-three.

Let these numbers be arranged first on the Ternary Scale, the dividend being lowermost, and space left for putting the quotient immediately under the divisor. Beginning at the left hand, the divisor is evidently contained once in the first four bars; place a counter then for the quotient, on the last of these bars; set the divisor immediately under the dividend, and note the difference, which is a counter on both the seventh and eighth bars. Of this difference, with the remaining counters brought down, four bars are less than the divisor, but it may be contained twice in five bars. Passing over one bar, therefore, two counters, joined to the quotient, are placed on the fourth bar of the range. The multiplicand is then doubled, and set down. But four bars of the remainder being less than the divisor, it may be found once in five bars, and consequently, omitting the third bar, one counter is placed on the second bar; and the remainder of the multiplicand is so great, that two counters may be set on the last bar; which duplication of the divisor finally exhausts the dividend. The quotient is hence three hundred and two.

Let the same operation be performed on the Quinary Scale. The divisor, which here occupies three bars, is evidently not contained in the three first bars of the dividend, but it is contained twice in four of those bars.

Two counters, representing this quotient, are therefore set down on the last of those bars, and the counters of the divisor, being doubled, are placed under the higher terms of the dividend. The remainder on three bars, being four and two counters, again contain the divisor twice on a lower bar. The double of the divisor being subtracted, the residue, or four counters, is not divisible, without including likewise the two lowest bars, when it gives two for the rest of the quotient. The divisor being now doubled, equals the remainder exactly.

The operation will be shorter on the Denary Scale. The divisor, consisting of five counters on the second bar and three on the first, which includes a counter on the fifth and fourth, and six on the first, is contained thrice in the three first bars of the dividend. On the third bar, therefore, place three counters. The divisor being then tripled, leaves a remainder of one counter, which, being extended, the last bar becomes again divisible, yielding Suppose the division were to be performed on the Duodenary Scale. Transferred to that scale, the divisor would, in the mercantile phrase, be called four dozen and five, and the dividend nine double gross, three gross, one dozen and ten.

In the two first bars of this number, the divisor is evidently contained twice; therefore, two counters are set down on the third bar, and the divisor doubled and subtracted, leaving a remainder of five gross, one dozen and ten. Two bars of this again contains the divisor once; and one counter, to denote it, being placed on the second bar, the subtraction is made, giving a remainder of eight dozen and ten, exactly double of the divisor, and consequently marked by two counters. The quotient is thus two gross one dozen and two, or three hundred and two.

This process, then, though it requires larger counters, is yet not less simple than the former. The operation of the larger scale is visibly quicker than the other.

In this instance, the divisor is complete; but it will often happen that a remainder is left, and consequently, that the process may be continued on the descending bars, expressing an excess of a fractional quotient, which either terminates, or constantly recurs again in a perpetual circle. Suppose, for example, it were proposed to divide three hundred and thirty by twenty-five. These numbers would be thus arranged on the Binary Scale:

Under the fourth bar, the divisor is contained once, leaving a counter on the eighth and one on the second bar. Under the third bar, it is again found once, with the remainder of four consecutive counters. Passing over the second bar, the divisor is contained once under the first; but not again till after an interval of two bars, when there is left, as under the second bar, four consecutive counters. At this point, therefore, a circulation must take place, since the third bar below that of units, thus corresponds to the next above it. The same sequence will be continually supported: First an empty bar, then a counter, followed by two empty bars.—To indicate this circle of renovation, the mark ρ for Aries, the first Sign of the Ecliptic, is adopted, as intimating the birth of the revolving year; and, therefore, by extension, the recommencement of a cycle. Let the division be now performed on the Ternary Scale. Under the third bar, the divisor is evidently contained once; and once again successively under the second and the first bar. But the remainder is now so small, that the divisor must be depressed to the second bar below that of units, to be found in it. On the next bar lower, it is contained twice, with the remainder of a counter, a blank, and again a counter; the same as what occurred under the second bar. Here the circulation commences, and forms this perpetual series.

The symbol $\varphi$ is placed to denote the alternate counters which intimate the commencement of the sequence.

Suppose the numbers twenty-five and three hundred and thirty, were now arranged on the Quaternary Scale. The divisor is contained thrice in the dividend, under the second bar, and once again below the place of units in the

ternary scale.

After this division, there is left only two consecutive counters, noted by the sign $\varphi$, in which the divisor, at the interval of two bars, is contained three times, with the same remainder. It is evident, therefore, that the fractional part of the quotient will be afterwards continually repeated, being exhibited by a column of three counters alternating with a vacant bar.

On the Quinary Scale, this example of division becomes almost intuitive. The divisor being here expressed by a single counter on the third bar, the quotient must evidently be the same as the dividend, only thrown two bars lower. The result of the division is consequently thirteen and a fifth.

Lastly, let the operation be performed on the Denary Scale. The divisor twenty-five is obviously contained once under the bar of tens; in the remainder, it is contained three times, under the next Another example shall conclude this part of the subject. Let it be required to find the quotient of three hundred and sixty-five by thirty-two. And to avoid prolixity, we may confine it to the Denary Scale. Under the bar of tens, the divisor is evidently contained once, and in the remainder once again below the units. Under the next bar, it occurs four times; after which, the remainder, being only two, does not admit of division, till after passing over a whole bar. Under the third descending bar, the divisor is found six times, with an excess of eight. On the next bar, it occurs twice; and the remainder finally contains it five times on the last bar. The quotient is therefore eleven, with fourteen-tenths, six thousandths, two ten-thousandths and five hundred-thousands, or the fraction forty-thousand six hundred and twenty-five hundred thousandths.

With a very little attention, it is easy to follow these various transformations; and it therefore appears superfluous to give any more commentary. Such is the natural process of analysing numbers, and of variously combining and separating them; and such are by consequence the simpler modes of abridging the labour of computation. Palpable arithmetic has, in the course of improvement, long preceded the invention and use of numeral characters. It was retained in Europe for a very considerable time after the adoption of this more convenient and powerful instrument, and might even at present be employed in practice to a certain extent with obvious advantage. The exhibition of numbers by counters appears happily fitted for unfolding the principles of calculation. In the schools of ancient Greece, the boys acquired the elements of knowledge by working on a smooth board with a narrow rim,—the Abacus; so named, evidently from the combination of A, B, Γ, the first letters of their alphabet, resembling, except perhaps in size, the tablet likewise called A, B, C, on which the children with us used to begin to learn the art of reading. The pupils, in those distant ages, were instructed to compute, by forming progressive rows of counters, which, according to the wealth or fancy of the individual, consisted of small pebbles, of round bits of bone or ivory, or even of silver coins. From ἀποκεῖται, the Greek word for a pebble, comes the verb, ἀποκεῖσθαι, to compute. But the same board served also for teaching the rudiments of writing and the principles of Geometry. The Abacus being strewed with green sand, the pulvis eruditus of classic authors, it was easy, with a radius or small rod, to trace letters, draw lines, construct triangles, or describe circles.—Besides, the original word ἀβάξ, the Greeks had the diminutive ἀβαξίας; and it seems very probable, that this smaller board was commonly used for calculations, while the larger one was reserved among them for the purpose of tracing geometrical diagrams.

To their calculating board, the ancients make frequent allusions. It appears, from the relation of Diogenes Laërtius, that the practice of bestowing on pebbles an artificial value according to the rank or place which they occupied, remounts higher than the age of Solon, the great reformer and legislator of the Athenian commonwealth. This sagacious observer and disinterested statesman, who was however no admirer of regal government, used to compare the passive ministers of kings or tyrants, to the counters or pebbles of arithmeticians, which are sometimes most important, and at other times quite insignificant. Ἀρχιμήνη, in his oration for the Crown, speaking of balanced accounts, says, that the pebbles were cleared away, and none left. His rival, Demosthenes, repeating his expression, employs farther the verb ἀποκεῖσθαι, which means to take up as many counters as were laid down. It is evident, therefore, that the ancients, in keeping their accounts, did not separately draw together the credits and the debts, but set down pebbles for the former, and took up pebbles for the latter. As soon as the board became cleared, the opposite claims were exactly balanced. We may observe, that the phrase to clear one's scores or accounts, meaning to settle or adjust them, is still preserved in the popular language of Europe, being suggested by the same practice of reckoning with counters, which prevailed indeed until a comparatively late period.

The Romans borrowed their Abacus from the Greeks, and never aspired higher in the pursuit of science. To each pebble or counter required for that board, they gave the name of calculus, a diminutive formed from calx, a stone; and applied the verb calculare, to signify the operation of combining or separating such pebbles or counters. Hence innumerable allusions by the Latin authors. Ponere calculus—subducere calculum, to put down a counter, or to take it up; that is, to add or subtract; vocare aliquod ad calculum, ut par sit ratio acceptorum et datorum—to submit anything to calculation, so that the balance of debtor and creditor may be struck. The Emperor, Helvius Pertinax, who had been taught, while a boy, the arts of writing and casting accounts, is said, by Julius Capitolinus, to be litteris elementariis et calculo imbutus. St Augustine, whose juvenile years were devoted to pleasure and dissipation, acquaints us, in his extraordinary Confessions, that to him no song ever sounded more odious than the repetition or cantio, that one and one make two, and two and two make four. The use of the Abacus, called sometimes likewise the Mensa Pythagorica, formed an essential part of the education of every noble Roman youth:

Nec qui abaco numeros, et secto in palceae metas Seit risisse vafer. ——— Petr. Sat. i. 152.

From Martianus Capella, we learn that, as refinement advanced, a coloured sand, generally of a greenish hue, was employed to strew the surface of the abacus.

Sie abacum perstare jubet, sic termino glauce Pandere pulvereum formarum ductibus sequor. Lib. vii. De Arithmetica.

A small box or coffer, called a Loculus, having compartments for holding the calculi or counters, was a necessary appendage of the abacus. Instead of carrying a slate and satchel, as in modern times, the Roman boy was accustomed to trudge to school, loaded with his arithmetical board, and his box of counters:

Quo pueri magnis e centurionibus orti, Lexvo suspensi loculos tabulamque lacerto. Horat. Sat. i. 8.

In the progress of luxury, tali or dies made of ivory, were used instead of pebbles, and small silver coins came to supply the place of counters. Under the Emperors, every patrician living in a spacious mansion, and indulging in all the pomp and splendour of eastern princes, generally entertained, for various functions, a numerous train of foreign slaves or freedmen in his palace. Of these,

* Ἐλέγξει δὲ τὰς στάσεις τῶν παραπλησίων δυναμίων ταυταπλησίων ἐν ταῖς ὑφέσεις ΤΑΙΣ ΕΠΙ ΤΟΝ ΛΟΤΤΕΜΟΝ. Ἡ γενετικὴ ἐκδίκησις τοῦ μετὰ ΠΛΑΤΩΝ ἐπικείμενος, ἢ ἡ ἘΠΙΤΤΙ. Diog. Laërt. in vitâ Solonis.

† Καὶ καταβάσει ἐκεῖ ἀπὸ ὑψών καὶ μηδὲ σειρά. Demosthenes pro Corona. Palmable the librarior or miniculator, was employed in teaching the children their letters; but the notarius registered expenses, the rationarius adjusted and settled accounts, and the tabularius or calculator, working with his counters and board, performed what computations might be required. Sometimes these laborious combiners of numbers were termed reproachfully cancules or calculones. In the fervour of operation, their gestures must often have appeared constrained and risible.

Compotat, ac cevet. Ponatur calculus adiut Cum tabulae pueri.— Jas. Sat. ix. 40.

The nicety acquired in calculation by the Roman youth, was not quite agreeable to the careless and easy temper of Horace.

Romani pueri longis ratiomina assent, Discunt in partes centum dividere: dicat Filias Albini, si de quindecim remota est Uncia, quid superat? potens dixisse Triensium Rem potest servare tuam, redit uncia, quid fit? Semis.— Epist. ad Pisoneum.

It was a practice among the ancients to keep a diary, by marking their fortunate days by a lapillus, or small white pebble, and their days of misfortune by a black one. Hence the frequent allusions which occur in the Classics:

O! dilem lactum, notandumque mihi condidissimo calcule. Plin. Epist. vi. 11.

diesque nobis Signa salva multiformibus lapillis Mart. ix. 53.

Hunc, Macrini, dies numerar meliore tegillo, Qui tibi labecles apposit candidus annos. Pers. Sat. iii. 1, 2.

To facilitate the working by counters, the construction of the abacus was afterwards improved. Instead of the perpendicular lines or bars, the board had its surface divided by sets of parallel grooves, by stretched wires, or even by successive rows of holes. It was easy to move small counters in the grooves, to slide perforated beads along the wires, or to stick large knobs or round-headed nails in the different holes. To diminish the number of marks required, every column was surmounted by a shorter one, wherein each counter had the same value as five of the ordinary kind, being half the index of the Denary Scale. The abacus, instead of wood, was often, for the sake of convenience and durability, made of metal, frequently brass, and sometimes silver. In Plate XXVII. we have copied, from the third volume of the Supplement added by Polenius to the immense Thesaurus of Gravins, two varieties of this instrument, as used by the Romans. They both rest on good authorities, having been delineated from antique monuments,—the first kind by Ursinus, and the second by Marcus Velesrus. In the one, the numbers are represented by flatish perforated beads, ranged on parallel wires; and, in the other, they are signified by small round counters moving in parallel grooves. These instruments contain each seven capital bars, expressing in order units, tens, hundreds, thousands, ten thousands, hun-

dred thousands, and millions; and above them are Palmable shorter bars following the same progression, but Arithmetic having five times the relative value. With four beads on each of the long wires, and one bead on every corresponding short wire, it is evident that any number could be expressed, as far as ten millions.

In all these, the Denary Scale is followed uniformly; but there is, besides, a small appendage to the arrangement founded on the Duodenary System. Immediately below the place of units, is added a bar, with its corresponding branch, both marked O, being designed to signify ounces, or the twelfth parts of a pound. Five beads on the long wire, and one bead on the short wire, equivalent now to six, would therefore denote eleven ounces. To express the simpler fractions of an ounce, three very short bars are annexed behind the rest; a bead on the one marked S or 5, the contraction for Semissis, denoting half-an-ounce; a bead on the other, which is marked by the inverted O, the contraction for Sicilicum, signifying the quarter of an ounce; and a bead on the last very short bar, marked Q, a contraction for the symbol Q or Bince Sextula, intimating a duella or two-sixths, that is, the third part of an ounce. The second form of the abacus differs in no essential respect from the first, the grooves only supplying the place of parallel wires.

We should observe that the Romans applied the same word abacus, to signify an article of luxurious furniture, resembling in shape the arithmetical board, but often highly ornamental, and destined for a very different purpose,—the relaxation and the amusement of the opulent. It was used in a game apparently similar to that of chess, which displayed a lively image of the struggles and vicissitudes of war. The infamous and abandoned Nero took particular delight in this sort of play, and drove along the surface of the abacus with a beautiful quadriga, or chariot of ivory.

The civil arts of Rome were communicated to other nations by the tide of victory, and maintained through the vigour and firmness of her imperial sway. But the simpler and more useful improvements survived the wreck of empire, among the various people again restored by fortune to their barbarous independence. In all transactions wherein money was concerned, it was found convenient to follow the procedure of the Abacus, in representing numbers by counters placed in parallel rows. During the middle-ages, it became the usual practice over Europe for merchants, auditors of accounts, or judges appointed to decide in matters of revenue, to appear on a covered bank or bench, so called, from an old Saxon or Franconian word, signifying a seat. Hence those terms were afterwards appropriated to offices for receiving pledges, chambers for the accommodation of money-dealers, or courts for the trying of questions respecting property or the claims of the Crown. Hence also the word bankrupt, which occurs in all the dialects of Europe. The term scaccarium, from which was derived the French, and thence the English, name for the Exchequer, anciently signified merely a chess-board, being formed from scaccum, denoting one of the moveable pieces in that intricate game. The reason of this application of the term is sufficiently obvious. The table for accounts was, to facilitate the calculations, always covered with a cloth, resembling the surface of the *scaccarium* or *abacus*, and distinguished by perpendicular and chequered lines. The learned Skene was therefore mistaken in supposing that the Exchequer derived its name from the play of chess, because its suitors appear to fight a keen and dubious battle.

The Court of Exchequer, which takes cognizance of all questions of revenue, was introduced into England by the Norman conquest. Richard Fitz-Nigel, in a treatise or dialogue on the subject, written about the middle of the twelfth century, says that the *scaccarium* was a quadrangular table about ten feet long and five feet broad, with a ledge or border about four inches high, to prevent anything from rolling over, and was surrounded on all sides by seats for the judges, the tellers, and other officers. It was covered every year, after the term of Easter, with fresh black cloth, divided by perpendicular white lines, or distinctures, at intervals of about a foot or a palm, and again parted by similar transverse lines. In reckoning, they proceeded, he says, according to the rules of arithmetic,† using small coins for counters. The lowest bar exhibited *pence*, the one above it *shillings*, the next *pounds*; and the higher bars denoted successively *ten*, *twenties*, *hundreds*, *thousands*, and *ten thousands* of pounds; though, in those early times of penury and severe economy, it very seldom happened that so large a sum as the last ever came to be reckoned. The first bar, therefore, advanced by *dozens*, the second and third by *scores*, and the rest of the stock of bars by the multiples of *ten*. The teller sat about the middle of the table; on his right hand, *eleven* pennies were heaped on the first bar, and a pile of *nineteen* shillings on the second; while a quantity of pounds was collected opposite to him, on the third bar. For the sake of expedition, he might employ a different mark to represent half the value of any bar, a silver penny for ten shillings, and a gold penny for ten pounds.

In early times, a chequered board, the emblem of calculation, was hung out, to indicate an office for changing money. It was afterwards adopted as the sign of an inn or *hostelry*, where victuals were sold, or strangers lodged and entertained. We may perceive traces of that ancient practice existing even at present. It is customary in London, and in some provincial towns, to have a chequer, diced with red and white, painted against the sides of the door of a chop-house.

The use of the smaller *abacus* in assisting numerical computation was not unknown during the middle ages. In England, however, it appears to have scarcely entered into actual practice, being mostly confined to those "slender clerks" who, in such a benighted period, passed for men of science and learning. The calculator was styled, in correct Latinity, *algebrae*; but, in the Italian dialect, *abbacista*, or *abbachiere*. A different name came afterwards to be imposed. The Arabians, who, under the appellation of Saracens or Moors, conquered Spain, and enriched that insulated country by commendable industry and skill, had likewise introduced their mathematical science. Having adopted a most refined species of numeration, to which they gave the barbarous name of *algarismus*, *algorismus*, or *algorithmus*, from the definite article *al*, and the Greek word *ἀριθμός*, or number, this compound term was adopted by the Christians of the West, in their admiration of superior skill, to signify calculation in general, long before the peculiar mode had become known and practised among them. The term *algarism* was corrupted in English into *angrim* or *angrym*, as printed by Wynkyn de Worde, at the end of the fifteenth century; and applied even to the pebbles or counters used in ordinary calculation. In confirmation of this remark, we shall not scruple to quote a passage from our ancient poet Chaucer, who flourished about a century before, and whose verses, however rude, are sometimes highly graphic.

"This clerk was cleped hende Nicholas; Of derne love he coude and of solas; And thereto he was sile mad ful prive, And like a maiden neke for to se. A chambre bad he in that hostelrye, Alone withouten any compaignie, Full fetisly ydight with herbes sote, And he himself was swete as is the rote Of licoris, or any setewale. His almageate and bokes grete and smale, His astrolabe, longing for his art, His angrym-stones, laues faire apart. On shelves couched at his beddes heed, His presse ycovered with a falding red. And all about there lay a gay santrie, On which he made on nightes melodie So sweetyly, that all the chambre rong. And Angelus ad verginem he song, And after that he song the kings note; Ful often blessed was his merry throte."

*Miller's Tale*, v. 135-136.

The *abacus*, with its store of counters, wanted the valuable property of being portable, and was at all times evidently a clumsy and most inconvenient implement of calculation. In many cases, it became quite indispensable to adopt some sure and ready method of expressing at least the lower numbers. The Digital Ns ancients employed the variously combined inflexions meration of the fingers on both hands to signify the numerical series, and on this narrow basis they framed a system of considerable extent. In allusion to the very ancient practice of numbering by the arbitrary play of the fingers, Orontes, the son-in-law of Artaxerxes, having incurred the weighty displeasure of that monarch, is reported by Plutarch to have exclaimed in

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† He calls it *Arismetica*: In the *Myrrour of the Worlde*, printed by Caxton, in 1481, it is strangely named *Ars Metrike*, a proof of the total ignorance of Greek at that period in England. terms exactly of the same import as those before ascribed to Solon, that "the favourites of kings resemble the fingers of the arithmetician, being sometimes at the top and sometimes at the bottom of the scale, and are equivalent at one time to ten thousand, and at another to mere units."

Among the Romans likewise, the allusions to the mode of expressing numbers by the varied inflexion of the fingers, are very frequent. Hence the classical expressions, computare digitis, and numerare per digitos; and hence the line of Ausonius,

"Quod ter lectatus cum police computat index."

In this play of the fingers great dexterity was acquired; and hence the phrase which so frequently occurs in the Classics—micare digitis. It was customary to begin with the left hand, and thence proceed to the right hand, on which the different combined inflexions indicated exactly one hundred times more. Hence the peculiar force of this passage from Juvenal:

"Rex Pylius, magno si quiequam credis Homero, Exemplum vitae fit a cornice secunda; Felix minium, qui tot per seculos mortem Distulit, alque suos jam dextrâ computat annos."

Sat. x. 246—250.

Many such allusions to the mode of indicating numbers by the varied position of the fingers or the hands, occur in the writings of Cicero and Quintilian. The ancients, indeed, for want of better instruments, were tempted to push that curious art to a very great extent. By a single inflexion of the fingers of the left hand, they proceeded as far as ten; and by combining another inflexion with it, they could advance to an hundred. The same signs on the right hand, being augmented, as we have seen, an hundred fold, carried them as far as ten thousand; and by a farther combination, those signs, being referred successively to different parts of the body, were again multiplied an hundred times, and therefore extended to a million. This kind of pantomime outlived the subversion of the Roman empire, and was particularly fitted for the slothful religious orders who fattened on its ruins, and, relinquishing every manly pursuit, recommended silence as a virtue, or enjoined it as an obligation. Our venerable Bede has explained the practice of manual numeration at some length; and, in Plate XXVII., we have given a small specimen of such inflexions and digital signs.

These signs were merely fugitive, and it became necessary to adopt other marks, of a permanent nature, for the purpose of recording numbers. But of all the contrivances adopted with this view, the rudest undoubtedly is the method of registering by tallies, introduced into England along with the Court of Exchequer, as another badge of the Norman conquest. These consist of straight well-seasoned sticks, of hazel or willow, so called from the French verb tailler, to cut, because they are squared at each end. The sum of money was marked on the side with notches, by the cutter of tallies, and likewise inscribed on both sides in Roman characters, by the writer of the tallies. Arithmetic.

The smallest notch signified a penny, a larger one a shilling, and one still larger a pound; but other notches, increasing successively in breadth, were made to denote ten, a hundred, and a thousand. The stick was then cleft through the middle by the deputy-chamberlains, with a knife and a mallet; the one portion being called the tally, or sometimes the scachia, stipes, or kancia; and the other portion named the counter-tally, or folium.

After the union with Scotland had been concluded in 1707, a store of hazel-rods for tallies was sent down to Edinburgh, being intended, no doubt, as a mighty refinement on the Scottish mode of keeping accounts. Their advantages, however, were not perceived or acknowledged, and they have since been suffered, we believe, to lie as so much useless lumber. But the case is very different in England, where a blind and slavish attachment to ancient forms, however ridiculous they may through time have become, is almost constantly opposed to the general progress of society. Were a sensible traveller from India or China to visit our metropolis, and report, on his return home, that a nation highly polished, enlightened, and opulent, yet keep their accounts of the public revenue, surpassing annually many millions of pounds, by means of notches cut on willow rods,—he would certainly not be credited, but supposed to use the licence of substituting a description of the practice of the most savage tribes of the American Continent.

The Chinese have, from the remotest ages, used in all their calculations, an instrument called the Swan-Pan, or Computing Table, similar in its shape and construction to the abacus of the Romans, but more complete and uniform. It consists of a small oblong board surrounded by a high ledge, and parted lengthwise near the top by another ledge. It is then divided vertically, by ten smooth and slender rods of bamboo, on which are strung two small balls of ivory or bone in the upper compartment, and five such balls in the lower and larger compartment; each of the latter on the several bars denoting unit, and each of the former, for the sake of abbreviation, expressing five. See Plate XXVII., where the balls are actually set to signify the numbers annexed.

The system of measures, weights, and coins, which prevails throughout the Chinese empire, being entirely founded on the decimal subdivision, the swan-pan was admirably suited for representing it. The calculator could begin at any particular bar, and reckon with the same facility either upwards or downwards. This advantage of treating fractions exactly like integers was, in practice, of the utmost consequence. Accordingly, those arithmetical machines, but of very different sizes, are constantly used in all the shops and booths of Canton and other cities, and are said to be handled by the native traders with such rapidity and address as quite astonish the European factors.

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* Καθάπερ ἐν τῶν ἀριθμητικῶν ΔΑΚΤΤΑΙΟΙ ῥῦν μὲν ΜΠΡΙΑΔΑΣ, ῥῦν δὲ ΜΟΝΑΔΑ πιθήκαι δυαῖαι, ῥῦν αὐλός καὶ τῶν βασιλέων φύλαξ, ῥῦν μετὰ τῶν ὑποστατῶν, ῥῦν δὲ τῆλε ἀγγείων. Plut. Apothegm. But the Chinese have also contrived a very neat and simple kind of digital signs for denoting numbers, greatly superior, both in precision and extent, to the method practised by the Romans. Since every finger has three joints, let the thumb-nail of the other hand touch those joints in succession, passing up the one side of the finger, down the middle, and again up the other side, and it will give nine different marks, applicable to the Denary Scale of arrangement. On the little finger, those marks signify units, on the next finger tens, on the mid-finger hundreds, on the index thousands, and on the thumb hundred thousands. With the combined positions of the joints of the one hand, therefore, it was easy to advance by signs as far as a million. To illustrate more fully this ingenious practice, we have, immediately below the kosa of the Emperor Fou-hi, copied, in plate XXVII., from a Chinese elementary treatise of education, the figure of a hand, noted at the several joints of each finger, by characters along the inside, corresponding to one, two, and three, down the middle by those answering to four, five, and six; and again up the outside by characters expressing seven, eight, and nine. It is said that the merchants in China are accustomed to conclude bargains with each other by help of those signs; and that often, from selfish or fraudulent views, they conceal the pantomime from the knowledge of bystanders, by only seeming to seize the hand with a hearty grasp.

Having thus treated at sufficient length the principles and application of Palpable Arithmetic; we shall now proceed to consider the second branch of the subject, which may be denominated Figurate Arithmetic; since, to denote the range of numbers, it employs conventional symbols, or certain artificial characters disposed in a particular order, and performs, by help of these, all the ordinary operations implied in the combining and separating of numbers.

**FIGURATE ARITHMETIC**

Includes, first the mode of Notation, and next the various operations themselves which are required in compounding and resolving numbers. We shall consider these subdivisions in their order.

**NOTATION.** The various attempts of men in every state of society at representing numbers, all spring from the same feeling so strongly implanted in our breasts, which unceasingly prompts them to seek the approbation and applause of their fellows. Other passions are spent in the mere support and continuance of the race; it is the love of distinction alone that elevates the human character, infuses life and action through the mass, and vigorously impels it to undertake and achieve mighty projects. The daring and restless spirit of improvement often produces much transient misery by its failures; yet, amidst all the vicissitudes which chequer and darken the tide of human affairs, it opens cheering prospects or the permanent amelioration of our species. The inspiring leader of a successful band, or the petty legislator of a rising community, is anxious to preserve the memory of the exploits he performed, or the benefits he conferred. But he is not content with obtaining the applause of his contemporaries; this fleeting existence is insufficient to fill his imagination; he looks anxiously beyond the grave, and sighs for the admiration of generations yet unborn. Hence the anxiety among all people to erect monuments of high achievements or illustrious characters. In the early periods of society, a vast mound of earth, or a huge block of stone, was the only memorial of any great event. But, after the simpler arts came to be known, efforts were made to transmit to posterity the representations of the objects themselves. Sculptures of the humblest kind occur on monumental stones in all parts of the world, sufficient to convey tolerably distinct images of the usual occupation and employments of the personages so commemorated. The next step in the progress of society was to reduce and abridge those rude sculptures, and thence form combinations of figures approaching to the hieroglyphical characters. At this epoch of improvement, the first attempts to represent numerals would be made. Instead of repeating the same objects, it was an obvious contrivance to annex to the mere individual the simpler marks of such repetition. Those marks would of necessity be suited to the nature of the materials on which they were inscribed, and the quality of the instruments employed to trace them. In the historical representations, for instance, which the Mexicans and certain Tartar herdes painted on skins, a small coloured circle, as exhibiting the original counter, shell, or pebble, was repeated to denote numbers. But, on the Egyptian Obelisks, the lower numerals, at least, are expressed by combined strokes. None but straight lines, indeed, are fitted for carving on pillars of stone, and especially for cutting on wooden posts. Curve lines were not admitted in the earliest rudiments of writing. Even after the use of hieroglyphics had been laid aside, and the artificial system of alphabetic characters adopted, the rectilineal forms were still preferred, as evidently appears in the Greek and Roman capitals, which, being originally of the lapidary sort, are much older than the small or current letters. One of the most ancient set of characters, the Runic, in which the northern languages were engraved, combines almost exclusively simple strokes at different angles.

The primary numeral traces may, therefore, be regarded as the commencement of a philosophical and universal character, drawn from nature itself; and alike intelligible to all ages and nations. They are still preserved, with very little change, in the Roman notation. Those forms, prior to the adoption of the alphabet itself, were no doubt imported by the Grecian colonies that settled in Italy, and gave rise to the Latin name and commonwealth. Assuming a perpendicular stroke I to signify one, another such II would express two, the junction of a third III three, and so repeatedly till the reckoner had reached ten. See Plate XXVII. The first class was now completed, and to intimate this, the carver threw a dash across the stroke or common unit; that is, he employed two decussating strokes X to Figurate denote ten. He next repeated this mark to express twenty, thirty, and so forth, till he finished the second class of numbers. Arrived at an hundred, he would signify it, by joining another dash to the mark for ten, or by merely connecting three strokes thus C. Again, the same spirit of invention might lead him to repeat this character, in denoting two hundred, three hundred, and so forth, till the third class was completed. A thousand, which begins the fourth class on the Denary Scale, was therefore expressed by four combined strokes M; and this was the utmost length to which the Romans first proceeded by direct notation.

But the division of these marks afterwards furnished characters for the intermediate numbers, and hence greatly shortened the repetition of the lower ones. Thus, having parted in the middle the two decussating strokes X denoting ten, either the under half A, or the upper half V, was employed to signify five.

Next, the mark C for an hundred, consisting of a triple stroke, was largely divided into Γ and L, either of which represented fifty. Again, the four combined strokes M, which originally formed the character for a thousand, came afterwards, in the progress of the arts, to assume a rounded shape O, frequently expressed thus CIO, by two disparded semicircles divided by a diameter: This last form, by abbreviation on either side, gave two portions CI and IO to represent five hundred.

It was an easy process, therefore, to devise an universal character for expressing numbers. But the task was very different, to reduce the exhibition of language in general to such concise philosophical principles. This attempt seems accordingly to have been early abandoned by all nations, except the Chinese. The inestimable advantage of uniting again the whole human race, in spite of the diversity of tongues, by the same permanent system of communication, was sacrificed for the easier attainment, of representing, by artificial signs, those elementary and fugitive sounds, into which the words of each particular dialect could be resolved. Hence the Alphabet was invented, which, notwithstanding its obvious defects, must ever be regarded as the finest and happiest effort of genius. More letters were afterwards added in succession, as the analysis of the primary sounds became extended; but the alphabet had very nearly attained its present form, at the period when the Roman commonwealth was extending its usurpation over Italy. About that epoch, a sort of reaction seems to have arisen between the artificial and the natural systems, and the numeral strokes were finally displaced by such alphabetic characters as the most resembled them. See Plate XXVII. The ancient Romans employed the letter I to represent the single stroke or mark for one; they selected the letter V, since it resembles the upper half of the two decussating strokes, or symbol for five; the letter X exactly depicted the double mark for ten; again, the letter L was adopted as resembling the divided symbol for fifty; while the entire symbol, or the tripled stroke, denoting an hundred, was exhibited by the hollow square C, the original form of the letter C before it became rounded over. The quadrupled stroke for a thousand was distinctly represented by the letter M, and its variety by the compound character CLC, consisting of the letter L inclosed on both sides by C and by the same letter reversed; a portion of this again, or LC, being condensed into the letter D, expressed five hundred. The letters C and M, beginning the words Centum and Mille, might have a farther claim to represent an hundred and a thousand. But the coincidence was merely accidental, since these terms migrated probably from the corresponding Greek words ἑκατόν and χίλιον.

This was the limit of numeration among the early Romans; but, in the progress of refinement, they repeated the symbols of a thousand to denote the higher terms of the Denary Scale. Thus, CCCL was employed to represent ten thousand, and CCCCL to signify an hundred thousand; the letter L, inclosed between the CO, being, for the sake of greater distinctness, elongated. Again, each of these being divided, gives LCO for five thousand, and LOO for fifty thousand. These characters, however, were often modified and abbreviated in monumental inscriptions. By drawing a horizontal line over the letters, their value was augmented one thousand times. In the plate, so often referred to, we have endeavoured, from the best authorities, to exhibit, under the title of Lapidary Numerals, a complete specimen of the various contractions used by stone-cutters among the Romans. It was customary, with them, for the sake of abbreviation, to reckon, as rude tribes are apt to do, partly backwards. Thus, instead of octodecem, and novemdecem, the words for eighteen and nineteen, they frequently used duodeviginti and undecviginti, as more elegant and expressive. This practice led to the application of deficient numbers; an improvement scarcely to be expected from a people so little noted for invention. Instead of writing nine, thus VIII, by joining four to five, they counted one back from ten, or placed I before X. In the same way, they represented forty, and four hundred, ninety, and nine hundred, by XL, and CD, XC, and CM.

Such, we have no doubt, is the real account of the rise and progress of the Roman numerals. It perfectly agrees with the few hints left us by Aulus Gellius, who expressly says, that I and X were anciently represented by one and two strokes: though Philologers, misled by partial glimpses, have indeed given a very different statement. Priscian, the Grammarian, who flourished in the reign of the Emperor Justinian, asserts, that the mark I was only borrowed from the Athenians, being adopted by them as the principal letter of the word MIA, or one, the M of which is considered as mute; that V or U was employed by the Romans to denote five, because it is the fifth vowel in the common order; that X was assumed to represent ten, as being the tenth consonant, and likewise following the V; that L was taken to signify fifty, being sometimes interchanged with N, which, as a small letter, expressed that number among the Greeks; that C was adopted to mark Figurata an hundred, because it is the first letter of the word Centum; that D, being the next letter of the alphabet, was employed to signify five hundred; and that M was borrowed from the Greek letter X for XIAIA, or a thousand; only that it was rounded at the ends, to distinguish it from the symbol for ten.—But it would be idle to pursue these fanciful remarks, or to engage in serious refutation of random conjectures, which betray such a total want of general views.

After the system of Roman numerals, however, had acquired its full extent, the solicitude of superstition long preserved some traces of the rudest and most primitive mode of chronicling events. At the close of each revolving year, generally on the Ides of September, the Praetor Maximus was accustomed, with great ceremony, to drive a nail in the door on the right side of the temple of Jupiter, next that of Minerva, the patron of learning and the inventor of numbers. In times of public distress, when pestilence raged, and famine spread desolation abroad, the warlike, yet superstitious Romans, anxious to stay those calamities, sought to anticipate the return of the season. On such occasions, they elected a Dictator, for the sole purpose of driving the sacred nail, and beginning a more propitious year. Hence the expression of Cicero—Ex hoc die, clavum anni movebis.

As the Chinese had constructed the Swan-pan on the principles of the Roman Abacus, so they had likewise, at the remotest epoch of the Empire, framed a system of numerals, in many respects similar to those which the Romans probably derived from their Pelasgic ancestors. This will appear from the inspection of the characters engraved on Plate XXVII. It is only to be observed, that the Chinese mode of writing is the reverse of ours; and that, beginning at the top of the leaf, they descend in parallel columns to the bottom, proceeding, however, from right to left, as practised by most of the Oriental nations.

Instead of the vertical lines used by the Romans, we therefore meet with horizontal ones, in the Chinese notation. Thus, one is represented by a horizontal stroke, with a sort of barbed termination; two by a pair of such strokes; and three by as many parallel strokes; the mark for four has four strokes, with a sort of flourish; three horizontal strokes, with two vertical ones, form the mark for five; and the other symbols exhibit the successive strokes abbreviated, as far as nine. Ten is figured by a horizontal stroke, crossed with a vertical score, to show that the first rank of the Denary Scale was completed; an hundred is signified by two vertical scores, connected by three short horizontal lines; a thousand is represented by a sort of double cross; and the other ranks, ascending to an hundred millions, have the same marks successively compounded. Since the figures in Plate XXVII. were engraved from two elegant Chinese manuscripts, we have met with impressions of a more complete set of numerals, printed with metallic types in 1814, at Serampore, in the Elements of Chinese Grammar, by the Reverend Dr Marshman, one of the Baptist Missionaries, whose zeal, talents, enterprise, and indefatigable assiduity, in exploring the recondite dialects of the East, have reflected un-

The numbers eleven, twelve, &c. are represented by putting the several marks for one, two, &c. the excesses above ten, immediately below its symbol. But, to denote twenty, thirty, &c. the marks of the multiples two, three, &c. are placed above the symbol for ten. This distinction is pursued through all the other cases. Thus, the marks for two, three, &c. placed over the symbols of an hundred or of a thousand, signify so many hundreds or thousands.—The character for ten thousand, called wàn, appears to have been the highest known at an early period of the Chinese history; since, in the popular language at present, it is equivalent to all. But the Greeks themselves had not advanced farther. In China, wàn wàn signifies ten thousand times ten thousand, or an hundred millions; though there is also a distinct character for this high number. In the Eastern strain of hyperbole, the phrase wàn wàn, far out-doing a thousand years, the measure of Spanish loyalty, is the usual shout of Long Live the Emperor! The Chinese character cháo for a million, though not of the greatest antiquity, is yet as old as the time of Confucius. The characters for ten, and for an hundred, millions, are not found in their oldest books, but occur in the Imperial Dictionary.

Such is the very complete but intricate system of Chinese numerals. It has been constantly used, from the remotest times, in all the historical, moral, and philosophical compositions of that singular people. The ordinary symbols for words, or rather things, are, in their writings, generally blended with skill among those characters. But the Chinese merchants and traders have transformed this system of notation into another, which is more concise, and better adapted for the details... Figurate of business. The changes made on the elementary characters, it will be seen, are not very material. The one, two, and three, are represented by perpendicular strokes; the symbols for four and five are altered; six is denoted by a short score above an horizontal stroke, as if to signify that five, the half of the index of the scale, had been counted over; seven and eight are expressed by the addition of one and two horizontal lines; and the mark for nine is composed of that for six, or perhaps at first a variety of five, joined to that of four.

To represent eleven, twelve, &c. in this mode, a single stroke is placed on the left of the cross for ten, and the several additions of one, two, &c. annexed on the right.

From twenty to an hundred, the signs of the multiples are prefixed to the mark for ten.

The same method is pursued through the hundreds, the marks of the several multiples being always placed on the left hand before the contracted symbol of pith, or an hundred. The additions are made on the right, with a small cipher or circle (o), called ling, when necessary, to separate the place of units. The distinction between two hundred and three, and five hundred and thirty, deserves to be particularly remarked.

A similar process extends to the notation of thousands; but, for ten thousand, the character wan is abbreviated. As a specimen of their combination, we select the following complex expression,

which denotes five hundred and forty-three millions, four hundred and seventy-five thousand, and three. The same number would be thus represented in the regular system of Chinese notation:

Where the first column on the right hand presents the marks for fifty and four, with the interjacent character wan, or ten thousand; the next column to the left has the several marks for a thousand, three, and an hundred; the middle column exhibits the symbols of forty and of seven; the adjacent column repeats the character wan, or ten thousand, and then presents those for five and a thousand; and the last column has the symbol ling, or the rest, which fills up the blank, with the mark for three.

The last expression seems abundantly complicated, and yet it is unquestionably simpler and clearer than the corresponding notation with Roman numerals, as represented below.

From such an intricate example, the imperfection of the Roman system will appear the more striking.

The abbreviated process of the Chinese traders was probably suggested by the communication with India, where the admirable system of denary notation has, from the earliest ages, been understood and practised. The adoption of a small cipher to fill the void spaces, was a most material improvement on the very complex character ling, used formerly for the same purpose.

About the close of the seventeenth century, the Jesuit missionaries Bouvet, Gerbillon, and others, then residing at the Court of Pekin, and able mathematicians, appear to have still further improved the numeral sym- times more. Thus, $\Delta$ denoted fifty thousand; and $\Pi$ five hundred thousand. See Plate XXVII.

3. But a mighty stride was afterwards made in numerical notation by the Greeks, when they distributed the twenty-four letters of their alphabet into three classes, corresponding to units, tens, and hundreds. To complete the symbols for the nine digits, an additional character was introduced in each class. The mark $\varepsilon$, called episonon, was inserted among the units immediately after $\iota$, the letter denoting five; and the kappa and sanpi, represented by $\Sigma$, $\Lambda$, or $\Theta$, terminated respectively the range of tens and of hundreds, or expressed ninety and nine hundred. This arrangement of the symbols, it is obvious, could extend only to the expression of nine hundred and ninety-nine; but, by subscribing an iota under any character, the value was augmented a thousand fold, or by writing the letter M, or the mark for a myriad, or ten thousand, under it, the effect was increased ten times more. This last modification was sometimes more simply accomplished by placing two dots over the character.

Such is the beautiful system of Greek numerals, so vastly superior in clearness and simplicity to the Roman combination of strokes. It was even tolerably fitted as an instrument of calculation. Hence the Greeks early laid aside the use of the abacus; while the Romans, who never showed any taste for science, were confined, by the total inaptitude of their numerical symbols, to the practice of the same laborious manipulation.

It should, however, be remarked, that the Greeks distinguished the Theory from the Practice of Arithmetic, by separate names. The term Arithmetic itself was restricted by them to the science which treats of the nature and general properties of numbers; while the appellation Logistic was appropriated to the collection of rules framed to direct and facilitate the common operations of calculation. The ancient systems of Arithmetic, accordingly, from the books of Euclid, to the treatise of Boethius and the verses or commentaries of Capella, are merely speculative, and often abound with fanciful analogies. Pythagoras had brought from the East a passion for the mystical properties of numbers, under the veil of which he probably concealed some of his secret or esoteric doctrines. He regarded Numbers as of divine origin, the fountain of existence, and the model and archetype of all things. He divided them into a variety of different classes, to each of which were assigned distinct properties. They were prime or composite, perfect or imperfect, redundant or deficient, plane or solid; they were triangular, square, cubic, or pyramidal. Even numbers were held by that visionary philosopher as feminine, and allied to earth; but the odd numbers were considered by him as endowed with masculine virtue, and partaking of the celestial nature. He esteemed the unit, or monad, as the most eminently sacred, and as the parent of all scientific numbers; he viewed two, or the duad, as the associate of the monad, and the mother of the elements; and he regarded three, or the triad, as perfect, being the first of the masculine numbers, comprehending the beginning, middle, and end, and hence fitted to regulate... by its combinations the repetition of prayers and libations. As the monad represented the Divinity, or the Creative Power, so the duad was the image of Matter; and the triad, resulting from their mutual conjunction, became the emblem of Ideal Forms.

But the tetrad, or four, was the number which Pythagoras affected to venerate the most. It is a square, and contains within itself all the musical proportions, and exhibits by summation all the digits as far as ten, the root of the universal scale of numeration; it marks the seasons, the elements, and the successive ages of man; and it likewise represents the cardinal virtues, and the opposite vices. The ancient division of mathematical science into Arithmetic, Geometry, Astronomy, and Music, was four-fold, and the course was therefore termed a tetractys, or quaternion. Hence Dr Barrow would explain the oath familiar to the disciples of Pythagoras: "I swear by him who communicated the Tetractys."

Five, or the pentad, being composed of the first male and female numbers, was styled the number of the world. Repeated any how by an odd multiple, it always reappeared; and it marked the animal senses, and the zones of the globe.

Six, or the hexad, being composed of its several factors, was reckoned perfect and analogical. It was likewise valued, as indicating the sides of the cube, and as entering into the composition of other important numbers.

Seven, or the heptad, formed from the junction of the triad with the tetrad, has been celebrated in every age. Being unproductive, it was dedicated to the virgin Minerva, though possessed of a masculine character. It marked the series of the lunar phases, the number of the planets, and seemed to modify and pervade all nature.

Eight, or the octad, being the first cube that occurred, was dedicated to Cybele, the mother of the Gods, whose image in the remotest times was only a cubical block of stone.

Nine, or the ennead, was esteemed as the square of the triad. It denotes the number of the Muses, and, being the last of the series of digits, and terminating the tones of music, it was inscribed to Mars.

Ten, or the decad, from the important office which it performs in numeration, was, however, the most celebrated for its properties. Having completed the cycle and begun a new series of numbers, it was aptly styled apocatastasis or periodic, and therefore dedicated to the double-faced Janus.

The cube of the triad, or the number twenty-seven, expressing the time of the moon's periodic revolution, was supposed to signify the power of the lunar circle. The quaternion of celestial numbers, one, three, five, and seven, joined to that of the terrestrial numbers, two, four, six, and eight, compose the number thirty-six, the square of the first perfect number six, and the symbol of the universe, distinguished by wonderful properties.

But it would be endless to recount all the visions of the Pythagorean school; nor should we descend to notice such fancies, if, by a perpetual descent, the dreams of ancient Philosophers had not, in the actual state of society, still tinctured our language, and mingled with the various institutions of civil life. Not to wander in search of illustration, we see the predilection for the number seven strongly marked in the customary term of apprenticeships, in the period acquired for obtaining academical degrees, and in the legal age of majority.

The Chinese appear, from the remotest epochs of their empire, to have entertained the same admiration of the mystical properties of numbers that Pythagoras imported from the East. Distinguishing numbers into even and odd, they considered the former as terrestrial, and partaking of the feminine principle Yang; while they regarded the latter as of celestial extraction, and ended with the masculine principle Yin. The even numbers were represented by small black circles, and the odd ones by similar white circles, variously disposed and connected by straight lines. See Plate XXVII. The sum of the five even numbers, two, four, six, eight, and ten, being thirty, was called the number of the Earth; but the sum of the five odd numbers one, three, five, seven, and nine, or twenty-five, being the square of five, was styled the number of Heaven. The nine digits were likewise grouped in two different ways, termed the Lo-chou, and the Ho-tou. The former expression signifies the Book of the river Lo, or what the Great Yu saw delineated on the back of the mysterious tortoise which rose out of that river: It may be conceived from this arrangement.

| Four | Nine | Two | |------|------|-----| | Three | Five | Seven |

| Eight | One | Six |

Nine was reckoned the head, and one the tail of the tortoise; three and seven were considered as its left and right shoulders; and four and two, eight and six, were viewed as the fore and the hind feet. Thenumber five, which represented the heart, was also the emblem of Heaven. We need scarcely observe, that this group of numbers is nothing but the common magic-square, each row of which makes up fifteen.

As the Lo-chou had the figure of a square, so the Ho-tou had that of a cross. It is what the Emperor Fou-hi observed on the body of the horse-dragon, which he saw spring out of the river Ho. The central number was ten, which, it is remarked by the commentators, terminates all the operations on numbers.

| Seven | Two | |-------|-----| | Five | One | | Six | | The Greek system of notation proceeded directly as far as ten thousand, comprising four terms of the Denary Scale; but by subscribing M, the initial letter of μεγάλος, it was carried over another similar period, to signify hundreds of millions. But the penetrating genius of Archimedes quickly discerned the powers, and unfolded the properties of such progressions. In a curious tract, entitled Λακυνίου or Arenarius, this philosopher amused himself, with showing, that it was possible, assuming the estimation of Aristarchus of Samos, and other Astronomers of that age, to represent the number of particles of sand which would be required to fill the sphere of the universe. He took the limit of the ordinary numeral system, or ten-thousand times ten-thousand, that is, an hundred millions, as the root of a new scale of progression, which therefore advanced eight times faster than the simple denary notation. Archimedes proposed to carry this comprehensive system as far as eight periods, which would therefore correspond to a number expressed in our mode by sixty-four digits. From the nature of a geometrical progression, he demonstrated, that proportional numbers would range at equal distances; and consequently, that the product of any two numbers must have its place determined by the sum of the separate ranks, a principle which involves the theory of logarithms.

The fine speculation of the Sicilian philosopher does not, however, appear to have been carried into effect; and without actually performing those calculations, he contents himself by pointing out the process, and stating the approximate results. But Apollonius, the most ingenious and inventive, next to Archimedes, of all the ancient mathematicians, resumed that scheme of numeration, simplified the construction of the scale, and reduced it to a commodious practice. For greater convenience, he preferred the simple myriad as the root of the system, which, therefore, proceeded by successive periods, corresponding to four of our digits. The periods were distinguished by breaks or blanks. That most important office which, in the modern system of notation, the cipher performs, by marking the rank of the digits, was indeed unknown to the earlier Greeks. They were hence obliged, when the lower periods failed, to repeat the letters Mu., or the contraction for μεγάλος, ten thousand. Thus, to express thirty-four trillions, they wrote λακυνίου, μεγάλος, μεγάλος. To signify units separately, it was customary with them to prefix the mark M°, or the abbreviation for μονάδα.

The procedure of the Greek arithmetician was necessarily slower and more timid than our simple, yet refined mode of calculation. Each step in the multiplication of complex numbers appeared separate and detached; without any concentration which the moderns obtain, by carrying forward the multiples of ten, and blending together the different members of the product. In ancient Greece, the operations of arithmetic, like writing, advanced from left to right; each part of the multiplier was in succession combined with every part of the multiplicand; and the several products were distinctly noted, or, for sake of compactness, grouped and conveniently dispersed, till afterwards collected into one general amount.

Pappus of Alexandria, in his valuable Mathematical Collections, has preserved a set of rules which Apollonius had formed, for facilitating arithmetical operations. These are, in the cautious spirit of the ancient Geometry, branched out into no fewer than twenty-seven propositions, though all comprised in the principle formerly stated by Archimedes: That the product of two integers of different ranks, will occupy a rank corresponding to the sum of the component orders. Suppose μ were to be multiplied into ρ, or forty into two hundred: Take the lower corresponding characters δ and β, or four and two, which were called σύμβολα or radicals, the one depressed ten times, and the other an hundred times; and multiply their product π or eight successively by the ten and the hundred, or at once by a thousand, and the result is γ or eight thousand.

We shall take an example in multiplication, affording more variety than such as occur in Euclid, which generally consist in the mere squaring of numbers. Let it be required to multiply eight hundred and sixty-two, by five hundred and twenty-three. The operation would be performed in this way:

\[ \begin{align*} & \omega \times \beta \\ & \phi \times \gamma \\ & \mu \times \alpha \\ & \alpha \times \sigma \mu \\ & \beta \times \nu \\ & \epsilon \times \pi \sigma \\ & \mu \times \omega \times \xi \end{align*} \]

In the first range, φ multiplied into ω, being the same as the product of eight and five, augmented ten thousand times, is consequently denoted by ρ or \( \frac{\rho}{M} \); φ multiplied into ξ gives the same result as five times six increased a thousand fold, and therefore expressed by γ or \( \frac{\gamma}{M} \); and φ multiplied into β, evidently makes a thousand or q. In the second range, χ multiplied into ω gives the same product as eight repeated twice, and then augmented a thousand times, or denoted by \( \frac{\alpha \delta}{M} \); χ multiplied into ξ is equivalent to six repeated twice, and afterwards increased an hundred fold, or expressed by \( \frac{\epsilon \sigma}{M} \); and χ multiplied by β gives forty, the value of μ. In the third range, γ multiplied into ω produces twenty-four hundred, which is denoted by \( \beta \nu \); γ multiplied into ξ makes an hundred and eighty, or \( \frac{\epsilon \sigma}{M} \); and... Lastly, \( y \) multiplied into \( \beta \) gives \( r \), the symbol for six.

Collecting the scattered members into one sum, the result of the multiplication of eight hundred and sixty-two by five hundred and twenty-three is \( \mu \times \omega \times 5 \) or four hundred and fifty thousand, eight hundred and fifty-six.

But the Greek notation was not adapted for the descending scale. To express fractions, two distinct methods were followed. 1. If the numerator happened to be unit, the denominator was indicated by an accent. Thus \( \delta \) signified one fourth, and \( \xi \) one twenty-fifth; but one-half being of most frequent recurrence, was signified by a particular character, varying in its form, \( C, \zeta, C', \) or \( \kappa \). 2. In other cases, it was the practice of the Greeks, to write the denominator, as we do an exponent, a little above the denominator, and towards the right hand: Thus, \( \beta \) intimated two-elevenths, and \( \pi \alpha \) eighty-one, of an hundred and twenty-one parts.

As an illustration of the management of fractions, we select an example somewhat complicated from the commentary which Eutocius of Ascalon wrote about the third century of our era, on the Tract of Archimedes concerning the quadrature of the circle. Let it be required to multiply the mixed number one thousand and thirty-eight, with nine-elevenths, by itself.

\[ \begin{align*} &\alpha \omega \lambda \eta \theta \\ &\alpha \omega \lambda \eta \theta \\ &\varepsilon \pi \chi \eta \omega \varepsilon \eta \beta \\ &\pi \xi \delta \beta \delta \varepsilon \nu \chi \nu \delta \\ &\gamma \beta \delta \sigma \mu \chi \delta \\ &\eta \tau \sigma \mu \xi \delta \varepsilon \\ &\omega \iota \eta \beta \\ &\chi \nu \delta \\ &\kappa \delta \\ &\varepsilon \\ &\pi \omega \\ &\tau \lambda \eta \alpha \sigma \nu \alpha \zeta \\ &\tau \lambda \eta \alpha \sigma \nu \beta \lambda \zeta \\ &\alpha \tau \varepsilon \theta \varepsilon \mu \eta \beta \lambda \varepsilon \\ &\varepsilon \mu \eta \iota \sigma \kappa \\ &\beta \lambda \varepsilon \sigma \kappa \\ &\gamma \kappa \varepsilon \\ &\alpha \tau \varepsilon \delta \iota \delta \iota \kappa \varepsilon \end{align*} \]

It is to be observed, that, to multiply the several integers by the fraction nine-elevenths, amounts to their multiplication by nine, and the subsequent division by eleven. The excesses being two, and six-elevenths, are denoted by \( \beta \varepsilon \) and \( \sigma \varepsilon \); while the product of the fraction itself gives eighty-one of an hundred and twenty-one parts, expressed by \( \pi \alpha \).

But the laborious operations that such complex fractions required, were afterwards superseded by the use of sexagesimals, which we have already observed, the astronomers, and especially Ptolemy, had introduced.

"The division of the circumference of the circle into three hundred and sixty equal parts or degrees, was no doubt originally founded on the supposed length of the year, which, expressed in round numbers, consists of twelve months, each composed of thirty days. The radius approaching to the sixth part of the circumference, would contain nearly sixty of those degrees; and after its ratio to the circumference was more accurately determined, the radius still continued to be distinguished into the same number of divisions, which likewise bore the same name. As calculation now aimed at greater accuracy, each of these sixty divisions of the radius was, following the uniform progression, again subdivided into sixty equal portions called minutes; and repeating the process of sexagesimal subdivision, seconds and thirds were successively formed. The operations with sexagesimal fractions were performed in the descending scale, on a principle quite similar to that which Archimedes had before laid down. Each period of the multiplier, still proceeding from the left hand, was multiplied into a period of the multiplicand; and this product was then thrown to a rank depressed as much as the descents of both its factors. Thus, minutes multiplied into seconds, produced thirds; and seconds multiplied into thirds, produced fifths." Edinburgh Review, xviii. p. 200.

As an exemplification of this process, we shall take the question proposed by Theon, to find the square of the side of a regular decagon inscribed in a circle, or the chord of thirty-six degrees, which, according to Ptolemy's computation, measured in sexagesimal parts of the radius, thirty-seven degrees, four minutes, and fifty-five seconds. The multiplication is thus effected:

\[ \begin{align*} &\lambda \zeta \delta \nu \varepsilon \\ &\lambda \zeta \delta \nu \varepsilon \\ &\alpha \tau \varepsilon \theta \varepsilon \mu \eta \beta \lambda \varepsilon \\ &\varepsilon \mu \eta \iota \sigma \kappa \\ &\beta \lambda \varepsilon \sigma \kappa \\ &\gamma \kappa \varepsilon \\ &\alpha \tau \varepsilon \delta \iota \delta \iota \kappa \varepsilon \end{align*} \] Here in the first line, \( \lambda \) multiplied into \( \lambda \) in the place of units, gives \( \alpha \beta \gamma \delta \) or thirteen hundred and sixty-nine degrees; \( \lambda \) into \( \delta \) on the next bar, gives \( \gamma \alpha \beta \gamma \delta \) or one hundred and forty-eight minutes; and \( \lambda \) into \( n \), on the lowest bar, gives \( \beta \gamma \delta \) or two thousand and thirty-five seconds. In the second line, \( \delta \) multiplied into \( \lambda \) gives the product \( \gamma \alpha \beta \gamma \delta \) as before; \( \delta \) multiplied into \( \delta \), both of them on the bar of minutes, gives \( \gamma \alpha \beta \gamma \delta \) or sixteen seconds; \( \delta \) into \( n \), gives \( \gamma \alpha \beta \gamma \delta \), or two hundred and twenty thirds. Lastly, in the third line, the \( n \) on the bar of seconds, multiplied successively into \( \lambda \) and \( \delta \), produce, as before, \( \beta \gamma \delta \) and \( \gamma \alpha \beta \gamma \delta \) on the bars of seconds and thirds; and \( n \), multiplied by itself, gives \( \gamma \alpha \beta \gamma \delta \), or three thousand and twenty-five fourths. These several products being reduced and collected together, formed the total amount of \( \gamma \alpha \beta \gamma \delta \) or thirteen hundred and seventy-five degrees, four minutes, fourteen seconds, ten thirds, and twenty-five fourths; but all the terms below seconds, were omitted in practice as insignificant.

This calculation is laborious and intricate; yet, with a very few terms, it approaches to a considerable degree of accuracy. One of the most elegant theorems in elementary geometry demonstrates, that the side of a regular decagon, inscribed in a circle, is equal to the segment of the radius, divided in extreme and mean ratio. Therefore the square now computed, should be equal to the product of sixty, or the radius, into twenty-two degrees, fifty-five minutes, and five seconds, the smaller segment; that is, equal to thirteen hundred and seventy-five degrees, and five minutes, from which it differs only by the defect of less than one minute.

The Sexagesimal Arithmetic was, therefore, a most valuable improvement engravened on the notation of the Greeks. The astronomers of Alexandria and Constantinople continued to employ it in all their calculations, and were afterwards imitated by succeeding observers among the Arabians and Persians. The mode of working sexagesimals had thus become generally known, and reduced to practice; but we owe the first distinct treatise on those fractions to a very extraordinary character,—Barlaam, a Calabrian monk, the friend and Greek preceptor of the famous Petrarch, and a man of learning and vigorous intellect, who laboured by his writings and his missions to reunite the Eastern to the Western church. This adventurous personage, whose wayward conduct and dark features betrayed a lurking ferocity, met with a most singular fate. Being overtaken by a tremendous thunder-storm, while crossing the Adriatic Sea, he lashed himself to the mast of the bark, and was, in this situation, struck dead by a flash of lightning. The event happened in 1348: but Barlaam's tract on sexagesimals, neatly composed in six books, after the strict manner of the Ancients, and entitled generally \( \Delta \sigma \gamma \epsilon \rho \eta \sigma \alpha \tau \iota \nu \alpha \tau \iota \sigma \), or Computation, first appeared in a Latin version at Strasburg in 1572, though not published complete with the Greek text, until the year 1600, when it was edited at Paris by Chambers of Eton, from a manuscript procured from the Continent by the zeal of Sir Henry Savile.

To facilitate the operations with sexagesimals, it seemed indispensible to have a more extensive multiplication table, that should include the mutual products of all the numbers from one to sixty. This was actually constructed, about the middle of the sixteenth century, by Philip Lansberg, a Dutch Clergyman; and has been exhibited since, in various forms, by Dr Wallis and others. In the mean time, a material change had been effected in the subdivision of the radius of the circle, from which the sexagesimal system had taken its rise. Purbach, the great restorer of mathematical science, instead of making the radius to consist of 216,000 seconds, as Ptolemy and succeeding Astronomers had done, stopt short at sixty degrees, and distinguished each of these, by a repeated centesimal division, into ten thousand equal parts. Regiomontanus advanced a step farther, and rejecting the sexagesimal admixture, he divided the radius at once into a million of parts, thus following out an arrangement purely decimal. The subdivision into degrees, minutes, and seconds, was henceforth confined to the circumference itself; and when logarithms came afterwards to be adapted to those fractions, they received the appellation, once general, though now restricted, of Logistic. But the sexagesimal subdivision had nearly been rejected altogether. Our very meritorious countryman, Mr Briggs, in computing his large canon of logarithms, followed in another branch the example of Purbach, by distinguishing each degree of the circumference into an hundred minutes, and each of these again into an hundred seconds; and we cannot help regretting, that this easy and obvious improvement had not been generally embraced at the time it was proposed. The French Mathematicians have lately gone farther, and endeavoured to pursue, to its utmost extent, the decimal subdivision first introduced by Regiomontanus. They begin with dividing the quadrant into an hundred, instead of ninety, degrees; and then following the plan of Briggs, they successively divide each degree into an hundred minutes, and each minute into an hundred seconds. But the advantages which might arise from the adoption of this plan, are not sufficient perhaps to outweigh the manifest inconvenience that must attend it in the present advanced state of the science; and notwithstanding the sanguine dreams of some of its projectors, we cannot indulge the expectation of ever seeing it obtain a general and durable currency.

The Greek arithmetic, then, as successively moulded by the ingenuity of Archimedes, of Apollonius, and Ptolemy, had attained, on the whole, to a singular degree of perfection, and was capable, notwithstanding its cumbrous structure, of performing operations of very considerable difficulty and magnitude. The great and radical defect of the system consisted in the want of a general mark analogous to our cipher, and which, without having any value itself, should serve to ascertain the rank or power of the other characters, by filling up the vacant places in the scale of numeration. Yet were the Greeks not altogether unacquainted with the use of such a sign; for Ptolemy, in his Almagest, employs the small \( \varepsilon \), to occupy the accidental blanks which occurred in the notation of sexagesimals. This letter was perhaps chosen by him, because immediately succeeding to \( \varepsilon \), which denotes 60, it could not, in the sexagesimal arrangement, occasion any sort of ambiguity. But the advantage thence resulting was entirely confined to that particular case. The letters, being already sig- Figurate Arithmetic significant, were generally disqualified for the purpose of a mere supplementary notation; and the selection of an alphabetic character to supply the place of the cipher may be considered as an unfortunate circumstance, which appears to have arrested the progress towards a better and more complete system. Had Apollonius clasped the numerals by denary triads, instead of tetrads, he would have greatly simplified the arrangement, and avoided the confusion arising from the admixture of the punctuated letters, expressing thousands. It is by this method of proceeding with periods of three figures, or advancing at once by thousands instead of tens, that we are enabled most expeditiously to read off the largest numbers. The extent of the alphabet was favourable to the first attempts at enumeration; since, with the help of three intercalations, it furnished characters for the whole range below a thousand; but that very circumstance, in the end, proved a bar to future improvements. It would have been a most important stride, to have next exchanged those triads into monads, by discarding the letters expressive of tens and hundreds, and retaining only the first class, which, with its inserted epismom, should denote the nine digits. The ida, which signified ten, now losing its force, might have been employed as a convenient substitute for the cipher. By such progressive changes, the arithmetical notation of the Greeks would at last have reached its utmost perfection, and have exactly resembled our own. A wide interval, no doubt, did still remain; yet the genius of that acute people, had it continued unfettered, would in time, we may presume, have triumphantly passed the intervening boundaries. But the death of Ptolemy was succeeded by ages of languor and decline; and the spirit of discovery insensibly evaporated in miserable polemical disputes, till the fair establishment of Alexandria was finally overwhelmed under the irresistible arms of the Arabs, lately roused to victory and conquest by the enthusiasm of a new religion."

(Edinburgh Review, Vol. XVIII. p. 203.)

The ingenuity and varied resources of the ancient Greeks were the main causes which diverted them from discovering our simple denary system. Having attained a distinct conception of the powers of the geometrical progression, and even advanced so far as to employ their small o to fill the breaks of a period, nothing seemed wanting but to dismiss the punctuated letters and those for tens and for hundreds, and to retain merely the direct symbols for units, that is, the first third part of their alphabet. Here, however, those masters of science were stopt in their career, and the Eastern Empire presents a melancholy picture of the decline and corruption of human nature. Ingenuity had degenerated into polemical subtlety, and the manly virtues which freedom inspires were exchanged for meanness and self-abasement.

Some writers, misled by very superficial views of the subject, have yet ascribed the invention of the modern numeral characters to the Greeks, or even to the Romans. Both these people, for the sake of expedition, occasionally used contractions, especially in representing the numbers and fractions of weights or measures, which, to a credulous peruser of mutilated inscriptions, or ancient blurred manuscripts, might appear to resemble the forms of our ciphers. But this resemblance is merely casual, and very far indeed from indicating the adoption of a regular denary notation. The most contracted of the Roman writings was formed by the marks attributed to Tiro or Seneca, while that of the Greeks was mixed with the symbols called Sigla; both of which have exercised the patience and skill of Antiquaries and Diplomatists. In the latter species of characters, were kept the accounts of the revenues of the Empress Irene at Constantinople. But the modern Greeks appear likewise to have sometimes used a simpler kind of marks, at least for the low numbers. The continuator of Matthew Paris's History relates, that "in the year 1251 died John Basingstoke, Archdeacon of Leicester, who brought into England the numeral figures of the Greeks, and explained them to his friends." It is subjoined, that they consisted of a perpendicular stroke, with a short line inserted at different heights and at different angles, signifying units on the left, and tens on the right side. The figures themselves are scrawled on the margin of the text. But they are evidently so different in their form, and so distinct in their nature, from the modern ciphers, that one cannot help feeling surprise, to see an author of any discernment refer the introduction of the latter to Basingstoke.

It cannot be doubted, that we derived our knowledge of the numeral digits from the Arabians, who had themselves obtained this invaluable acquisition from their extended communication with the East. Those deserving people who, under the name of Moors or Saracens, had for many centuries cultivated Spain, were most ready to acknowledge their obligation to the natives of India, who, according to Alsephadi, a learned Arabian doctor, boasted of three very different inventions,—the composition of the Golalla Wadamna, or Pilpay's Fables,—the game of chess,—and the nine digital characters. Still much obscurity hangs over the whole subject. Two distinct inquiries naturally present themselves: 1. At what period did the Arabians first become acquainted with those characters? and, 2. What is the precise epoch when the knowledge of them was imparted to the Christian nations of Europe. We shall take a short review of both these questions.

1. Gatterer, the late ingenious and very learned Professor of History at Göttingen, in his Elements of their Indian Universal Diplomacy, maintains that our ciphers were only primordial letters, invented by Taaut or Theut, and known to the ancient Egyptians and Phoenicians, being still distinctly observed, as he asserts, in the inscriptions painted on the coverings of the oldest mummies; and that afterwards, along with other branches of science, they passed to the Oriental nations, among whom they were preserved, till the victorious arms of the Mussulmen penetrated to India, and brought back those precious monuments of genius. But we cannot believe that a contrivance so very simple, and so eminently useful as that of the nine digits, if once communicated, could ever again be lost or neglected. Pythagoras and Boethius merely contemplated the properties of numbers, and seem not, in their calculations, to have gone beyond the use of the Abacus. An early intercourse had no doubt subsisted between the people of Egypt and of India, and a striking resemblance may be traced in their customs, their buildings, and their religious rites. But the characters exhibited on the Egyptian monuments bear no indication of the Denary System, and are, like the Roman and Chinese numerals, abridged representations of objects, rather than arbitrary signs.

That the occupiers of Hindostan and the nations communicating with them, have for ages been acquainted with the use of the denary notation, cannot be disputed. But was this an original discovery, or at what distant epoch was it first introduced among them? The easy credulity of European visitors encouraged the Brahmins to set up very lofty pretensions respecting the antiquity of their science. Among other treasures, they boasted the possession, from time immemorial, of an elementary treatise on arithmetic and mensuration, composed in Sanscrit, and called Lilawati, of such estimable value as to be ascribed to the immediate inspiration of Heaven. But the researches of our ingenious countrymen in exploring that sacred language of India, have dispelled some illusions, and greatly abated the admiration of the public for such eastern learning. From what we have been able to gather, the Lilawati is a very short and meagre performance, loaded with a silly preamble and colloquy of the Gods. It begins with the numeration by nine digits, and the supplementary cipher or small o, in what are called the Devanagari characters; and it contains the common rules of arithmetic, and even the extraction of the square root, as far as two places of figures; but the examples are generally very easy, scarcely forming any part of the text, and only written on the margin with red ink. Of fractions, whether decimal or vulgar, it treats not at all.

The Hindus pretend, that this arithmetical treatise was composed about the year 1185 of the Christian era. The date of a manuscript, however, is always very uncertain. We know besides, that the oriental transcriber is accustomed to incorporate without scruple such additions in the text as he thinks fit. Nor will any of the criteria which might ascertain the age of a manuscript apply to the eastern writings, where the composition of the paper, the colour of the ink, and the form of the characters, have for ages continued unchanged.

If the exuberant fancy of the Greeks led them far beyond the denary notation, it seems probable, that the feebler genius of the Hindus might just reach that desirable point, without diverging into an excursive flight. Though now familiar with that system, they are still unacquainted with the use of its descending decimal scale; and their management of fractions, accordingly, is said by intelligent judges to be tedious and embarrassed. In Plate XXVII., on the left hand, and near the bottom, we have given the Sanscrit digits, and have placed over them the numeral elements from which they might be formed. These consist of a succession of simple strokes, variously combined as far as nine. The resemblance to the Devanagari characters appears very striking. From these again, the common Hindu, and the vulgar Bengalee digits, are evidently moulded, with only slight alterations of figure. The Birman numerals, which we have copied from Symes' Embassy to the Kingdom of Ava, are manifestly of the same origin; only they have a thin, wiry body, being generally written on the palmyra-leaf with the point of a needle.

It appears, from a careful inspection of the manuscripts preserved in the different public libraries of Europe, that the Arabians were not acquainted with the denary numerals, before the middle of the thirteenth century of the Christian era. They cultivated the mathematical sciences with ardour, but seldom aspired at original efforts, and generally contented themselves with copying their Grecian masters. The alphabet of the Arabians had been employed for expressing numbers exactly in the same way as that of the Greeks. The letters, in their succession, were sometimes applied to signify the lower of the ordinal numbers; but more generally they were distinguished into three classes, each composed of nine characters, corresponding to units, tens, and hundreds. Though, like most of the Oriental nations, the Arabians write from right to left, yet they followed implicitly the Greek mode of ranging the numerals, and performing their calculations. With the same deference, they received the other lessons of their great masters, and very seldom hazarded any improvement, unless where industry and patient observation led them incidentally to extend mensuration, and to rectify and enlarge the basis of Astronomy.

It seems highly probable, therefore, that the Arabians did not adopt the Indian numerals until a late period, and after the torrent of victory had opened an easy communication with Hindostan. They might derive their information through the medium of the Persians, who spoke a dialect of their language, had embraced the same religion, and were, like them, inflamed by the love of science and the spirit of conquest. The Arabic numerals, accordingly, resemble exceedingly the Persic, which are now current over India, and there esteemed the fashionable characters. But the Persians themselves, though no longer the sovereigns of Hindostan, yet display their superiority over the feeble Gentoos, since they generally fill the offices of the revenue, and have the reputation of being the most expert calculators in the East. It should be observed, however, that, according to Gladwin, these accountants have introduced a peculiar contracted mode of registering very large sums, partly by the numeral characters, and partly by means of symbols formed of abbreviated words. Yet Sir John Chardin relates, that the Persians have no proper terms to express numbers beyond a thousand, which they merely repeat, as our young arithmeticians often do, to signify a million or a billion.

The Indian origin of the denary numerals is further confirmed by the testimony of Maximus Planudes, a monk of Constantinople, who wrote, about the middle of the fourteenth century, a book on practical arithmetic, entitled Λογική ἰδεῶν, or Υπορρογικά κατὰ ἰδεῶν, ἢ λογικῶν μυστικῶν, that is, "the great Indian mode of Calculating." In his introduction, he explains concisely the use of the characters in notation. But Planudes appears neither to have received his information directly from India, nor through the medium of the Persians, the nearest neighbours on the eastern confines of the Greek Empire. It is most probable that he was made acquainted with those numerals by his intercourse with Europe, having twice visited, on a sort of embassy, the Republic of Venice; for, of two manuscripts preserved in the library of St Mark, the one has the characters of the Arabians, and the other has that variety which was first current in Europe, while neither of them shows the original characters used in Hindostan.

But the most important inquiry is, to ascertain the period at which the knowledge of our present numerals was first spread over Europe. As it certainly had preceded the invention of the art of printing, the difficulty of resolving the question is much increased by the necessity of searching and examining old and often doubtful manuscripts. Some authors would date the introduction of those ciphers as early as the beginning of the eleventh century, while others, with far greater appearance of reason, are disposed to place it two hundred and fifty years later.

Whilst the thickest darkness brooded over the Christian world, the Arabians, reposing after their brilliant conquests, cultivated with assiduity the learning and science of Greece. If they contributed little from their own store of genius, they yet preserved and fanned the holy fire. Nor did they affect any concealment, but would freely communicate to their pupils and visitors that precious knowledge which they had so zealously drawn from different quarters. Some of the more aspiring youth, in England and France, disgusted with the wretched trifling of the schools, resorted for information to Spain; and having the courage to subdue the rooted abhorrence entertained in that age against Infidels, took lessons in Philosophy from the accomplished Moors. Among those pilgrims of science, the most celebrated was Gerbert, a monk, born of obscure parents, at Aurillac, in Auvergne, but promoted by his talents successively to the Bishopricks of Rheims and of Ravenna, and finally raised to the Papal chair, which he filled during the last four years of the tenth century, under the name of Sylvester II. This ardent genius studied Arithmetic, Geometry, and Astronomy, among the Saracens; and on his return to France, charged with various knowledge, he was esteemed a prodigy of learning by his contemporaries. Nor did the malice of rivals fail to represent him as a magician, leagued with the infernal powers. Gerbert wrote largely on Arithmetic and Geometry, and gave rules for shortening the operations of the Abacus, which he likewise termed Algorithmus. In some manuscripts, the numbers are expressed in ciphers; but we are not thence entitled to infer, as many writers have done, that he had actually the merit of introducing those characters into Europe. The context of his discourse will not support such a conclusion. The figures were not, we have seen, still known to the Arabians themselves; and must have long afterwards been inserted in those copies, for the convenience of transcribers.

Nor can we safely refer the introduction of Arabic figures to our famous Roger Bacon, whose various attainments and unwearied research after genuine knowledge, raised him far above the level of his contemporaries, but who, to the disgrace of his age and country, suffered a sharp persecution, and a tedious imprisonment, on the ridiculous charge of practising the redoubted arts of magic. But the writings of Bacon really discover no proofs of his acquaintance with the denary notation; and the fact commonly stated as an irresistible evidence in his favour, bears a very different interpretation. An almanack, now preserved in the Bodleian Library at Oxford, and containing numerals in their earliest forms, has, by the credulity of after-times, been, with all other feats and inventions, ascribed, of course, to the great necromancer. But unluckily this production is marked with the date 1292, the very year on which Bacon, after a lingering illness, expired; and it besides professes to have been calculated for the meridian of Tolouse, and had consequently been imported, without doubt, from France.*

About the same period, John of Halifax, named, in the quaint Latinity then used, Sacro-Bosco, who had likewise travelled, wrote his Treatise De Sphera, in some copies of which the numbers are given in ciphers. But it appears from examination, that such abbreviations were introduced by the license of transcribers.

There is little doubt that the Arabic figures were first used by Astronomers, and afterwards circulated in the almanacks over Europe. The learned Gerard Vossius places this epoch about the year 1250; but the judicious and most laborious Du Cange thinks that ciphers were unknown before the fourteenth century; and Father Mabillon, whose diplomatic researches are immense, assures us, that he very rarely found them in the dates of any writings prior to the year 1400. Kircher, with some air of probability, seeks to refer the introduction of our numerals to the astronomical tables which, after vast labour and expense, were published by the famous Alphonso, King of Castile, in 1252, and again, more correctly, four years afterwards. But it is suspected that, in the original work, the numbers were expressed in Roman or Saxon characters. Two letters from that enlightened, but ill-requited prince, to our Edward I., which are preserved in the Tower of London, have the dates 1272 and 1278, still denoted by those ancient characters.

* Nothing appears to be worse founded than the attempts to represent the elder Bacon in the light of an original inventor. Notwithstanding the obscurity of his writings, it needs but a little criticism to dispel the conceits fomented by national partiality. Friar Bacon advances no claim even to the discovery of gunpowder, which has been so gratuitously ascribed to him. On the contrary, he admits that the boys in his time were acquainted with the use of this substance in fire-works; and he merely pretends, in a sort of anagram, to give a receipt for making it stronger and better than ordinary.

After the chief ingredient in the composition of gunpowder, under the mistaken names of natron or nitrum, and saltpetre or rock-salt, had been imported from the East, probably through the intervention of the Crusaders, its disposition to explode in the contact of inflammable matters, if not communicated along with it, could not remain for any time a secret. The explosive force was a very different, and a far more important property, which is perhaps rightly attributed to Schwartz, a German monk, who, in the course of his experiments, stumbled on it about the middle of the fourteenth century. In the tenth volume of the *Archæologia*, the Rev. Mr North has given a short account of an almanack preserved in the library of Benet's College, Cambridge, and containing a table of eclipses for the cycle between 1330 to 1348. There is prefixed to it a very brief explication of the use of numerals, and the principles of the denary notation; from which we may see how imperfectly the practice of those ciphers was then understood. The figures are of the oldest form, but differ not materially from the present, except that the four has a looped shape, and the five and seven are turned about to the left and to the right. The one, two, three, and four, are likewise, perhaps for elucidation, represented by so many dots, thus, . . . . . : while five, six, seven, and eight, are signified by a semicircle or inverted O with the addition of corresponding dots—O O O O. Nine is denoted by o; ten by the same character, with a dash drawn across it; and twenty, thirty, or forty, by this last symbol repeated.

As a farther evidence of the inaccurate conceptions which prevailed respecting the use of the digits in the fourteenth century, we may refer to the mixture of Saxon and Arabic numerals which was copied from some French manuscripts by Mabillon, as exhibited in Plate XXVII. The Saxon r, signifying ten, is repeatedly combined with the ordinary figures; and , , , are immediately followed by 302, and 303, which must have been therefore intended to signify thirty-two and thirty-three, the force of the cipher not being still rightly understood. It should be observed, that the Greek epistemon or Fau, for the number six, had come to be represented by a character similar to G. The Saxon dates are taken from the Danish and Norwegian registers, preserved in Suhm's *Northern Collections*.

One of the oldest authentic dates in the numeral characters is that of the year 1375, which appears written by the hand of the famous Petrarchi on a copy of St Augustine, that had belonged to that distinguished Poet and Philosopher. The use of those characters had now begun to spread in Europe, but was still confined to men of learning. We have seen a short tract in the German language, entitled, *De Algorithmo*, and bearing the date 1390, which explained, with great brevity, the digital notation and the elementary rules of arithmetic. What is very remarkable, the characters, in their earliest form, are ranged thus, 0, 9, 8, 7, 6, 5, 4, 3, 2, 1, from right to left, the order which the Arabians would naturally follow. But it was not very easy to comprehend at first the precise force of the cipher, which, insignificant by itself, only serves to determine the rank and value of the other digits. The name, derived from an Arabic word signifying velocity, is sufficiently expressive; yet a sort of mystery, which has imprinted its trace on language, seemed to hang over the practice, for we still speak of deciphering, and of writing in cipher, in allusion to some dark or concealed art. After the digits had come to supply the place of the Roman numerals, a very considerable time probably elapsed before they were generally adopted in calculation. The modern practice of Arithmetic was unknown in England, till about the middle of the sixteenth century. But the lower orders, imitating the clerks of a former age, were still accustomed to reckon with their counters or avogram stones. In Shakespear's comedy of the *Winter's Tale*, written at the commencement of the seventeenth century, the clown, staggered with a very simple multiplication, exclaims that he will try it with counters.

Arithmetic was long considered in England as a higher branch of science, and therefore left, like Geometry, to be studied at the University. Most of the public or grammar schools of the South were, on the suppression of the monasteries, erected a little after the Reformation, during the short but auspicious reign of Edward VI. They were accordingly destined by their founders merely for teaching the dead languages; and the too exclusive pursuit of the same system is now one of the greatest defects in the English plan of liberal education.

It cannot be doubted that the calendars composed in France or Germany, and sent to the different religious houses, were the means of dispersing the knowledge of Arabic numerals over Europe. The library of the University of Edinburgh has a very curious almanack, presented to it, with a number of other valuable tracts, by the celebrated Drummond of Hawthornden, beautifully written on vellum, with most of the figures in vermilion. It is calculated especially for the year 1452, but contains the succession of lunar phases for three cycles, 1475, 1494; and 1513, with the visible eclipses of the sun and moon, from 1482 to 1530 inclusive. The date of this precious manuscript, which had once belonged to St Mary's Abbey at Cupar in Angus, is easily determined, and we have copied from it the oldest numerals exhibited in Plate XXVII. To these we have subjoined facsimiles from Caxton's *Mirror of the World*, and a wooden Cut from Shirwood's *Ludus Arithmomachia*, given in Dibdin's *Bibliotheca Spenceriana*.

The College accounts in the English Universities were generally kept in the Roman numerals, till the early part of the sixteenth century; nor in the parish registers were the Arabic characters adopted before the year 1600. The oldest date which we have met with in Scotland is that of 1490, which occurs in the rent-roll of the Diocese of St Andrew's, the change from Roman to Arabic numerals occurring, with a corresponding alteration in the form of the writing, near the end of the volume. The old characters in Plate XXVII. are copied from a manuscript history of the Scottish Bishoprics, apparently written about the year 1550, and now in the possession of Thomas Thomson, Esq. Advocate.*

---

* For want of attending to these facts, some learned Antiquaries have often suffered themselves to be grossly misled. Thus, Mr De Cardonnel, a respectable author, who has given views and short descriptions of the ancient edifices in Scotland, mentions, without marking the smallest doubt or surprise, that the date 1155 appears over the gateway of the ruins of the Castle of St Andrew's. But this front was built subsequent to the murder of the detested Cardinal Beaton, by Archbishop Hamilton, who likewise there affixed his Arms, but who long afterwards, on the capture of Dumbarton Castle, suffered an ignominious death, for his adherence to Queen Mary and the Popish faction. The real date was unquestionably 1555, only the second figure has been almost effaced by time and accident. Having endeavoured to trace the origin and introduction of our numeral characters, it only remains now to explain the operations of Figurate Arithmetic.

But, on this branch of the subject, we need not dilate; since the common rules, with their various modifications, are given at some length in the Encyclopaedia. It will be more instructive to derive the practice of numbers from the principles already unfolded, in treating of Palpable Arithmetic. The same theory may likewise suggest other methods of varying and abridging the common operations. We shall follow the order observed under the first head, selecting as few examples as may be wanted for illustration. The Denary Scale, being the one generally received, will claim our chief attention; but we shall likewise compare its results with those of some other scales, particularly the Duodenary, which is partially adopted in commerce, and possesses certain peculiar advantages.

It would, in many cases, facilitate calculation, to have figures corresponding to the open counters. We have, therefore, to transform the ordinary characters into deficient digits, modifying their shape as much as to distinguish without entirely altering or disguising them. By help of such new figures, it will be easy to represent numbers by their defects as well as their excesses. This answers most conveniently in expressing the digits from 5 to 10. Thus 8 may be denoted by 42, meaning 40 with 2 abated: for the same reason, 829 may be written 1931, signifying that 1000 is to be diminished by 201.

For the operations with the Duodenary Scale, it becomes necessary to devise two additional characters for expressing ten and eleven. Not to seek far after such objects, we have contented ourselves with condensing the ordinary forms into 5 and 5, which are perhaps sufficiently distinct, while they shadow out the figures represented by them. We now proceed to explain the common operations.

**Addition.**

From the principle of numerical notation, it follows that addition is performed by collecting the digits of each bar or rank. Each class, whether it be units, hundreds or thousands, is treated in the same way. In adding two figures, it is only requisite to count forwards from one of them, as many steps as are signified by the other. Suppose 5 were to be joined to 8; reckoning onwards, we pass through 9, 10, 11, 12, to 13. This simple process may be more conveniently performed by counting over the fingers. But, for a learner, it is a preferable mode to frame a table of addition, which he may readily commit to memory. The construction and use of such a table are so very simple, as hardly to require any explanation. The one number occupies the horizontal row at the top, and the other the vertical row at the side. Thus, below the column of 7, and opposite to the horizontal range of 6, stands 13, the sum of these numbers. Such tables are found in the more ancient treatises of arithmetic; but they have been most injudiciously, as we think, omitted in the later systems of education.

Let it be sought to add these four numbers, 3709, 8540, 2618, and 706. Having set them in their ranks, the most natural way would be to write down the sum of each column. The first column on the right hand gives 23, the next 5, the third 25, and the last column 13. The same numbers, collected by a second summation, give for the final result 15573.

But the process would be rather shortened, by writing under each column the units of the sum, and below it the tens in a smaller character, which are to be joined to the figures of the next column in adding them. By a little practice, however, this precaution is rendered unnecessary, and the small subscribed figures are retained mentally, and carried to the successive higher columns.

This operation is somewhat easier with deficient figures. Thus, the numbers may be changed into others with the defects interspersed. In this mode, there being a sort of counterbalance, it will seldom be required to carry any to the higher columns.

Suppose those numbers were all transferred to the Duodenary Scale; they will stand thus: The figures of the first column give by summation twenty-one; that is 19 by the Duodenary Scale. The others in succession yield the several sums annexed.

If the condensed process be followed, the form will be a little different, and more distinct.

The operation will be somewhat shortened by introducing deficient figures; nor will there be any occasion for any carrying, the accumulated excess of the digits being partly counterbalanced by the intermixed defects.

To facilitate the working on the Duodenary Scale, it would be expedient to construct an Addition Table. By means of this, we may at once sum up a row of pence, and carry the excess to the place of shillings.

---

*These characters are larger, and differently shaped from what was intended; but is too late now to correct the oversight of the artist.* It is evident that the mode of operation will still be the same in the descending terms of any scale. The only thing needed is to preserve the respective ranks of the figures; to secure which, a point may be placed between the units and the fraction. Let the annexed example of integers and decimals be proposed. The fractional part of the sum thus corresponds to one-fourth.

Suppose deficient figures were employed. The several numbers might assume this form: As generally happens in such cases, no carrying whatever is here wanted.

The same numbers reduced to the Quaternary Scale will stand thus: Their addition is extremely simple, and gives of course the same result.

Subtraction. This operation having for its object to find the difference between two numbers, is precisely the reverse of addition. The same auxiliary table may hence answer for both. Thus, if 7 joined to 6 makes 13, it is equally clear, that 7 taken away from 13 must leave 6. For the sake of distinction, the greater of the two numbers is called the Minuend and the other one the Subtrahend.

The method of proceeding will be most clearly perceived from the inspection of an example. Let it be required to take 428053 from 702632. Here, beginning at the right hand, 3 cannot be taken from 2, but the effect will evidently be the same if ten were added to both the minuend and the subtrahend. Ten may therefore be joined to the 2, while one, as equivalent to it, is thrown to 5, which occupies the place higher. This addition of the 10 is called borrowing, and the countervailing addition of 1 in the next bar is called carrying. Take 3 then from 2, with the junction of 10 borrowed, or 12, and there remains 9. Now 5, with the addition of the unit carried, or 6, is to be taken from 3: To do this, 10 is again borrowed, making the figure of the minuend 13, which leaves 7; and the 1 carried to 0 in the higher place, is taken from 6, and therefore leaves 5. Again, 8 taken from 2 with the addition of 10, leaves 4; and the same addition of 10 is made to the subtrahend, by joining 1 to the 2 which stands a place higher. But 3 taken from 0 increased to 10, gives 7, for the remainder; and 1 carried to the 4, and subtracted from 7, leaves 2, as the last difference.

Let the same example be worked with deficient digits. In subtracting the lower number, it is only required to change the character of its digits, and then add them. The operation, therefore, needs no further explanation.

Next, suppose those numbers were converted into the Duodenary Scale. The subtraction will be performed thus: It is only to be observed, that when there is occasion for borrowing, twelve is joined to the digit of the minuend, and one is carried or annexed to the higher digit of the subtrahend.

Since the terms of a descending scale are treated in the same way, it would, perhaps, be superfluous to take examples of the subtraction of decimal fractions.

Multiplication.

This operation, it was observed, is nothing but a repeated addition. The object which it seeks, is to add, for a Product, the Multiplicand, as often as there are units contained in the Multiplier. But such a process would have proved intolerably tedious, if the principle of numerical arrangement had not come to lend its aid. As in the Denary Scale, for instance, any digit is augmented tenfold at each shift to a higher place; so its product into the Multiplicand will give a similar increase, in the ascending progression, and must consequently maintain a corresponding value. Suppose it had been required to multiply 57 by 23; the most obvious way of proceeding would be this. First, the 57 is added three times, making 171. Next, the same number advanced a place higher, or having a zero annexed to it, is added twice, corresponding to the two of the multiplier, which, from its situation, has the value of 20. To this sum, or 1140, is joined the other, making in all 1311, for the compound product.

But the operation is shortened by performing mentally this repetition or summation, of each digit in the multiplicand. Thus, resuming the last example: 7 repeated three times makes 21, and 5 repeated as often on the advanced bar gives 15; again, 7 repeated twice, and moved a place higher, makes 14, and 5 likewise repeated twice, and shifted a place still higher, gives 10. The general amount is 1311, the same as before.

Such, we have seen, was the method practised by the Greeks; but in many cases this procedure becomes excessively tedious and perplexed. It is much simpler, by carrying, as in the process of Addition, to consolidate the figures at once on each bar, before they are written down. Instead of noting the 2 of the first product 21, it is joined immediately to the next product 15, and the sum 17 written down. Again, in the next row, the higher figure of the product 14 is combined at once with the lower figure of the next product 10, making 114. This now is the ordinary form of Compound Multiplication, and it seems scarcely to admit of any material improvement. But, to shorten the repeated summation of digits, it is expedient to construct a table, which must be engraved in the memory of the arithmetician. The mechanical method of multiplying digits, which has been already explained, may serve as an useful auxiliary, in fixing the recollection of the series of products. The table itself, though ascribed to Pythagoras, is most easily framed; but, notwithstanding, it has become now so very common, we make no hesitation in copying it, especially as we design to introduce another accommodated to the Duodenary Scale.

It may be observed, that the numbers 1, 4, 9, 16, 25, 36, 49, 64, and 81, which occupy the diagonal, are the second powers or squares of the successive digits. From the inspection of the table, we gather that one is the terminating figure in the three products 1, 21, and 81; that two terminates the six products 2, 12, 12, 32, 42, and 72; that three occurs as the terminating figure in only the two products 3 and 63; that four terminates the four products, 4, 14, 24, and 54; that five terminates likewise the five products 5, 15, 25, 35, and 45; that six is the terminating figure in the five products, 6, 16, 16, 36, and 56; that seven terminates only the two products 7 and 27; that eight terminates the five products, 8, 18, 28, and 48; and that nine occurs only twice as the terminating figure, in 9 and 9. It hence follows, that out of thirty-four chances, there are six that any composite number should end in 2; five chances that it should end in 3, 6, or 8; four chances that it should end in 4; three chances that it should end in 1; two chances that it should end in 3 or 7; and two chances likewise that the terminating figure should be 9. These very different proportions in the recurrence of the several digits at the end of a number, may be remarked in the large tables of products. It likewise appears, that the bulk of the prime numbers must terminate with 9, 3, or 7, and the rest with 1.

It may be instructive, to compare the operation of an example of compound multiplication in the ordinary way, with another performed by deficient figures.

In this instance, the working is evidently easier with the deficient figures, since lower digits are concerned in the multiplication. But it must be observed, that a deficient figure changes the character of all the digits which it multiplies. The restoration of the ordinary figures is better understood from being made, as here, by successive steps.

The same example treated after the method of the Greeks, where each product of the digits is set down separately, without any previous consolidation by carrying, will appear far more complex. This process may be conducted either from right to left, or from left to right, since each step is entirely independent of the rest. We have preferred the former mode, because it approaches nearer to the one generally practised.

The Persians, who probably communicated to the Persian Arabians the knowledge of the Indian numeral characters, but who had likewise carefully studied the Multiplication, Astronomy, and Arithmetic of the Greeks, made a capital step towards the improvement of the ancient mode of calculation. This will be readily understood from an example which we shall borrow from the judicious travels of Sir John Chardin. Suppose it were sought to multiply the number 36985 by 6428. The Persian Arithmeticians, having drawn a rhomboid, would, beginning at the top, write these numbers downwards along the upper sides, and then divide the figure into equilateral triangles, by combining oblique, with horizontal, lines.

Now, the multiplication is carried along the rows on the left side of the rhomboid: 6 into 3 gives 18, which is disposed in the uppermost triangle and the one below it; 6 into 6 gives 36, which is deposited in the two next triangles; and the same process is continued through the series. Again, 4 times 3 makes 12, which is placed in the two uppermost triangles of the next row. The rest of the operation of filling the triangles is easily understood. But to collect the products, the figures in each horizontal row, beginning at the bottom, are added up, and the tens carried to the one immediately above it. Thus, the zero at the point of the rhomboid remains unchanged; in the row above this, 4, 4, 0 make 8; in the next row, 2, 6, 6, 1, 0 make 15, and 5 being set... Thus, opposite to 8, the last digit of the multiplier, and proceeding from the right along the horizontal column, there occur these figures:

| 0, 4, and 4, or 8; 6, and 2, or 8; 7, and 8, or 15; and 1 carried to 4 | |---|---|---| | 237739580 |

and 4 makes 9; and lastly, the 2.

The other rows are easily formed in the same way. If the horizontal columns opposite to 8, 2, 4, and 6, were supposed to be detached and combined into an oblique group, the similarity to the Persian mode would be very striking.

But, without formally adopting either the figurate rods or the rhomboidal cells, it will sometimes be convenient, in very long multiplications, to form, by successive additions, an extemporaneous tablet of the digital products of the multiplicand. The application of this help is easily conceived.

It is evident, from the nature of notation, that, in the descending scale, the products corresponding to each figure of the multiplier, instead of being advanced, should be shifted backwards. Hence the common rule for the multiplication of decimal fractions—to cut off as many decimals as are found in both factors. But, since the remote decimals are of trifling import, a very commodious abbreviation is, to begin the process at the place of units. Passing to the 6 of the multiplier,

| 4236 | |---| | 1618 |

the last figure 6 of the multiplicand is struck off; but, as it would have given 36, the nearest whole number 4, expressing the tens, is carried to the product of 6 into 3, making 18. The same thing is repeated at each multiplication.

We shall now compare the ordinary method of multiplying numbers with the same process performed by deficient figures. By a little practice, the working with deficient figures would evidently become easier, and more expeditious than the common way.

In farther illustration of the process of multiplication, we shall perform the same examples on the Duodenary Scale. It will be convenient, however, though not quite essential, to construct previously a table of products.

**Multiplication Table.**

**Duodenary Scale.**

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 | Let it be required then to multiply the number 4819 by 378. Transformed into the Duodenary Scale, they will stand thus: The same operation is likewise here performed by deficient figures.

Suppose the large numbers 36985 and 6428 were now converted into the Duodenary Scale, their multiplication, in common, and in deficient figures, would proceed in this manner.

But this scale is of greater consequence, when viewed in the descending progression, since the duodecimal subdivision has, to a certain extent, obtained currency in the denomination of money, and of weights and measures. A few examples will explain the management of these fractions. Suppose it were sought to multiply L. 6, 15s. 3½d. by 53, the operation with Duodecimals will be performed thus:

The product 25½.6½.3 converted again into the Denary Scale, gives L. 358, 11s. 6½d. This result is more easily brought out than by Decimals, as appears by the comparison. If the Pound Sterling had been divided into 12 shillings, as the shilling is in 12 pence, the application of Duodecimals to accounts would have been extremely convenient.

Duodecimals are best adapted, however, to mensuration, where feet, inches, and their subordinate parts, enter into play. Thus, if it were required to find the solid contents of a log of timber, 2 feet 7½ inches square, and 27 feet 5½ inches long. The successive multiplications would be performed in this way, all the figures below the fourth place being excluded, as of little significance. The result is 190 cubic feet, with about 2½ twelfth parts, most inaccurately called, in this case, cubic inches.

But these fractions will likewise readily apply to the mensuration of round timber; for the relation of the circle to its circumscribing square, would be expressed in duodecimals by the number .9512. Suppose, for example, a cylinder 4 feet 2½ inches diameter, and 41 feet 10½ long.

The operation is thus performed: Four places of duodecimals only are retained, though in actual practice two places may generally be reckoned more than sufficient. The area of the circular base hence exceeds by a minute fraction 14 square feet, but the solid contents of the cylinder amounts extremely nearly to 700½ cubic feet. A similar mode of proceeding could easily be extended to the mensuration of cones and spheres.

Division.

This process, being exactly the reverse of Multiplication, consists in subtracting one number repeatedly from another. The former is called the Divisor, and the latter the Dividend, while the answer, signifying how often the subtraction needs to be made, is termed the Quotient. The principle of numerical arrangement suggests the means of abridging this operation. Suppose it were sought to divide 1554 by 37: Let 37 be subtracted in succession from 155, which, standing one place higher than the units, corresponds to tens; the several subtractions are marked by I, II, III, and IV, which belong to the place of tens, and from remainder 7 with 4 annexed to it, the divisor 37 is again subtracted twice. Whence the quotient is 42, or the number of times that 37 is contained in 1554, or must be subtracted before it exhausts this dividend.

But such an operation is evidently circuitous. The most obvious improvement is to frame, as in compound multiplication, a small tablet of the digital products of the divisor, and to subtract always the nearest less number from the successive terms of the dividend and the remainder. Let it be required to divide 22028148 by 423.

The tablet of products is formed by the successive addition of the divisor 423 and its multiples; of these, the number opposite to 5 comes nearest to the first four terms of the dividend; and the remainder 87, with the next figure annexed to it, is approached the nearest by 846, the next remainder 92 with the annexed 1 is less than the divisor, and, therefore, a zero is put in the quotient to preserve the place, and the following figure 4 is joined. The rest of the operation is easily conceived.

This method, however, is more tedious than needful, unless the quotient should consist of several figures. In other cases, a little practice will show how to choose the proper multiples of the divisor. Deficient figures may likewise be sometimes introduced with advantage. An example will explain this: dots placed under the figures of the dividend as fast as they are taken down, or annexed for a new division, point out the ranks of the divisor. With the deficient figures,

\[ \begin{array}{cccc} 472 & 1797848 & (3809) & 552 \\ & 18092 & 552 & (421) \end{array} \]

the operation is

\[ \begin{array}{cccc} 1316 & \cdots & 1928 & \cdots \\ 3818 & 1102 & & \\ 3776 & 1064 & & \\ 4248 & 1425 & & \\ 4248 & 532 & & \\ 0 & 1532 & & \\ 532 & & & \\ 0 & & & \\ \end{array} \]

In the first, a remainder 15, expressed in tenths, hundredths, thousands, &c. is divided by 64, and the quotient .234375 therefore expresses decimally the value of the fraction \( \frac{3}{15} \); in the second example, it is necessary to annex ciphers before the division begins, and the result .0368 consequently represents the fraction \( \frac{3}{15} \).

In both these examples, the operation terminates; but it will oftener happen, in the progress of the division, that the same remainder again emerges, after which the figures in the quotient must evidently maintain a perpetual recurrence. Thus, in the third example, the remainders of the division are successively 5, 11, 6, 8, 2, and again 7; from which point the series will recommence. The fraction \( \frac{1}{15} \) is therefore, when expanded into decimals, .588461, .588461, .588461, &c. continued in perpetual circulation. In the fourth example, the remainders are 19, 28, 37, 46, 55, 64, 73, and then 1 as at first; here consequently a recurrence takes place, and the value of the fraction \( \frac{1}{15} \) is expressed in these circulating decimals .012345679, .012345679, &c. In every case, the number of different remainders, and consequently the variety of changes, must obviously be less than the divisor. The last example is extremely remarkable, since it brings out all the digits in their natural succession, except 8. The reason of such a curious result, is that \( \frac{1}{15} \), being the square of \( \frac{1}{3} \), which, expanded into decimals, is .111111111, &c. will be expressed by this product: The value of the fraction \( \frac{1}{15} \) is therefore expressed, by the successive figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, &c. continued for ever, through all the descending places of the Denary Scale; but, at the junction of every ten, the order of the digits is partially disturbed; for the 11 behind the 10 changes it into 11, and this changes the 9 into 10, which again, by its influence, converts the 8 into 9.

Nothing more clearly discloses the properties of numbers, than the transferring of them to different scales of arrangement. Reckoning them, for instance, by successive braces, leashes, warps, &c. they are disposed on the Binary, Ternary, Quaternary, and succeeding scales. This process, requiring only the continued division by two, three, four, &c. is readily performed with figures. To procure greater variety in the results, we shall take a number that is rather large.

Let it be sought to exhibit the number 2138507 on successive scales. Its decomposition will be thus effected:

\[ \begin{array}{cccc} (1.) & 64 & 15.000000 & (.234375) \\ & 128 & \cdots & 1875 \\ & 220 & & 4250 \\ & 192 & & 3750 \\ & 280 & & 5000 \\ & 256 & & 5000 \\ & 240 & & 0 \\ & 192 & & \\ & 480 & & 81 \cdots \cdots \cdots \\ & 448 & & \\ & 320 & & 190 \\ & 320 & & 162 \\ & 0 & & 280 \\ & & & 243 \\ (2.) & 625 & 23.0000 & (.0368) \\ & 128 & \cdots & 1875 \\ & 220 & & 4250 \\ & 192 & & 3750 \\ & 280 & & 5000 \\ & 256 & & 5000 \\ & 240 & & 0 \\ & 192 & & \\ & 480 & & 81 \cdots \cdots \cdots \\ & 448 & & \\ & 320 & & 190 \\ & 320 & & 162 \\ & 0 & & 280 \\ & & & 243 \\ (3.) & 13 & 7.000000 & (.588461) \\ & 65 & \cdots & 370 \\ & 50 & & 324 \\ & 39 & & 460 \\ & 110 & & 405 \\ & 104 & & 550 \\ & 60 & & 486 \\ & 52 & & 640 \\ & 80 & & 567 \\ & 78 & & 730 \\ & 20 & & 729 \\ & 13 & & 1 \\ & 7 & & \\ \end{array} \]

* The formation of circulating decimals affords a fine illustration of that secret concatenation which binds the succession of physical events, and determines the various though lengthened Cycles of the returning seasons—a principle which the ancient Stoics, and some other Philosophers, have boldly extended to the moral world:

Alter est tua Tiphys, et altera qua vehat Argo Delectos heroas: erunt eiam altera bella, Atque iterum ad Trojam magnum mittetur Achilles.—See Monthly Review, vol. xviii. p. 14—17. Hence the same number 2138507 will be thus represented on the different scales:

| Binary | Ternary | Quaternary | Quinary | |--------------|-------------|--------------|------------| | 1000001010000110001011 | 11000122110222 | 20022012023 | 1021413012 |

The notation may be readily transferred to any higher scale which has, for its index, some power of that of the lower. If the number be distinguished into periods, consisting of two, three, or four places, the amount of these may be condensed into a single figure; and the corresponding index is the second, third, or fourth power of the primary index. Thus, let the expression 10,00,00,10,10,00,01,10,00,10,11 of the Binary Scale articulate at every alternate place from the right hand; the value of each period being substituted, or 1 for 01, 2 for 10, and 3 for 11, will transform the whole into this Quaternary arrangement 20022012023. If the same expression be divided into triplets 1,000,001,010,000,110,001,011, and each of these afterwards compressed into a single figure, assuming 1 for 001, 2 for 010, and 4 for 100, it will be changed into the Octary notation 10120613. In like manner, if the representation of the same number on the Ternary Scale, 11,00,01,22,11,02,22 be broken at every alternate place, and the values of those periods adopted, 1 being substituted for 01, and 3 for 10, the whole will be converted into the expression of the Nonary Scale, 4018428.

From the principle before investigated, we shall easily find the remainder of the division of the original number, by another number, which is one less than the index of any particular scale. Thus, to begin with the Ternary Scale, if the amount of all the figures be divided by two; or, what is the same thing, if every two be rejected and the other figures be successively added, retaining only at each step the excess above two; and at the end of the operation there will be left one. Hence it might be concluded that 2138507 is an odd number; a property indicated also by the character of its last digit. In the Quaternary Scale, by adding the figures together, and constantly throwing out the threes, there remains two; which shows that the division of the original number by three would leave two. In the Quinary Scale, the several figures being collected, omitting the fours as they arise, give three for the remainder of a division of 2138507 by four; which is indeed apparent from the inspection of its two last digits.

In the Senary Scale, omitting all the fives, there is left only two; a remainder which might indeed be inferred, from the circumstance that the last digit of the original number was seven. In the Septenary Scale, collecting the figures, and rejecting the sixes, there is an excess of five; the same which is left by the division of the original number by six. In the Octary Scale, when the sevens are thrown out, nothing remains; which shows that the number was divisible by seven. In the Nonary Scale, rejecting the eights, there is left three; but this remainder of the division of the original number by eight, might be inferred from the simple inspection of the three last digits 507, since all the rest, being thousands, are evidently divisible by eight. In the Denary Scale, by adding the figures, and separating the nines as fast as they arise, there is still a surplus of eight; being the remainder of the division of 2138507 by nine. It thus appears, that the common operation of casting out the nines is only the application of a general principle to the received system of notation. But to pursue the illustrations,—in the Undenary Scale, by casting out the tens, there is left seven or the digit of the original number, or the remainder of its division by ten. Finally, in the Duodenary Scale, by rejecting the elevens from the collected figures, there remains eight, the last figure in the preceding scale, or the residue of the division of the original number by eleven. In general, we may perceive the terminating figure of the expression on any scale is the same, as the remainder of the division, by its index, on the next higher scale.

But the quotients of those divisions are, likewise, directly discovered from what was unfolded under Palpable Arithmetic. Not to multiply examples, let us begin with the Senary Scale. If the figures... be repeated on all the lower places, their summation will appear thus: The excesses corresponding to the several rows, amount to 22, which leaves 2, and gives 4 to the next column. The sum of this again, with the 6 borrowed, is 21, which contains 6 three times, with a remainder of 3. The 3 is then carried to the next column, and the operation repeated; the result is 13100083, which, being transferred from the Senary to the Denary Scale, gives 427701 for the quotient of the number 2138507 by 5.

The expressions on some of the other scales may be treated more concisely.

| OCTARY SCALE | NONARY SCALE | |--------------|-------------| | 11111111 | 44444444 | | 111111 | 111111 | | 22992 | 88888 | | 666 | 444 | | 11 | 22 | | 3 | 8 | | 1124533 | 4466143 |

Which, transferred to the Denary Scale, gives 305501 for the quotient of the original number by seven.

| DENARY SCALE | |--------------| | 00000000 | | 111111 | | 333333 | | 88888 | | 555 | | 7 | | 237611 |

Which expresses the quotient of 2138507 by nine.

But the extension which was made of the same principle to open counters, may be applied to discover the quotient and remainder of any number divided by the index of the next higher scale. Not to dwell on this subject, we shall take only three of the last examples to illustrate the mode of operation. Beginning at the left hand, each figure must be repeated through all the succeeding places, alternately as an excess and a defect, and the balance of addition taken as the result. The same as 720053; which being transformed to the Denary Scale, gives 237611, with a remainder of 8 for the quotient of 2138507 by nine, the index of the scale augmented by one.

The same as 194409; which is the quotient of the original number by eleven, the remainder of the division being 8.

| DUODENARY SCALE | |-----------------| | 88888 | | 77777 | | 111 | | 66 | | 8 | | 94609 |

Which, converted into a Denary expression, is 194409, the quotient of the original number by eleven.

It is of more consequence, however, in such divisions, to discover the remainder than the quotient. Nines, If the process be confined to the Denary Scale, the number eleven will appear to have properties analogous to those of nine. But to find the remainder of the division of any number by eleven, or, in the vulgar phrase, to cast out the elevens, will require attention to the alternate character of the ciphers, fluctuating in succession from excess to defect. The easiest mode is, beginning at the right hand to mark the alternate figures; and, from the amount of these, augmented by eleven, if necessary, take that of the rest, and the difference is the remainder sought. Thus, resuming the original number 2138507', the sum of the marked figures is 17, and that of the rest only 9; wherefore, if divided by eleven, it would leave an excess of 8. Again, taking the number 52904682', the marked figures together make only 10, while the others amount to 22; the former must consequently be augmented repeatedly by the addition of eleven, till the sum 32 comes to exceed 22. The remainder of this division would therefore be 10.

It hence follows, that, as a number is divided by nine when the amount of its figures is any multiple of nine; so a number is divisible by eleven, when the sums of the alternate figures are either equal or differ by eleven or its multiples. This proposition leads to some curious results; but we shall notice only the more striking and simple. It is an obvious consequence, that the difference between any number and its reverse is divisible by nine: Thus, the number 2138507 being reversed into 7058312, gives the difference 4919805, divisible by 9. The reason is plain, since this number and its reverse are expressed by identical figures, they are both multiples of 9 with the same excess, and consequently their difference must only be some multiple of 9. Again, the difference between a number and its reverse is likewise divisible by eleven, if it consists of odd figures: Thus, the last difference 4919805' is divisible by 11, for the sums of the alternate figures are each 18. But the sum of a number and its reverse is divisible by eleven when it consists of even figures. Thus, the number 52904682 has 28640925 for its reverse, Figurate and their sum is $31'54'56'07'$; which is evidently divisible by 11, for each set of alternate figures amounts to 18. It is not difficult to perceive the reason of these properties of eleven; when the number consists of odd figures, they preserve the same character of abundant or defective in its reverse, and consequently the subtraction of the opposite numbers will destroy whatever inequality there had before existed; but when the number proposed consists of even figures, the abundant and defective by reverting change mutual places, and hence the sum of the number and its reverse will extinguish any original inequality between these, balancing any surplus of the one set by the equal deficiency of the other. Thus, in the first example, the number $2'13'85'07'$ exceeds by 8 a multiple of 11, and so does its reverse $7'05'83'12'$, the marked figures being the same in both; consequently their difference is a multiple of 11. In the second example, the number $52'90'46'82'$, is greater than a multiple of 11 by 10, and its reverse $28'64'09'25'$ is greater than another multiple of 11 by 1, or is less than the next lower multiple by 10: Wherefore these opposite numbers added together will produce a mutual balance accurately divisible by 11.

The casting out of the nines and elevens, furnishes a ready, though but a negative proof of the accuracy of arithmetical operations. In addition, for instance, by casting the nines or elevens out of each number, and collecting those excesses, the result of another ejection should be the same as from the sum. This principle applies equally to subtraction; but the mode of proceeding in multiplication and division will require some investigation. By casting the nines out of a number, it is converted into the Nonary Scale, of which the excess would occupy the place of units; if another number, therefore, were treated in the same way, and their product represented on that scale, that part of it formed by multiplying the excesses would still retain the rank of units, and exhibit the remainder of the division by nine. The same principle, it is obvious, will extend to the casting out of eleven.

Thus, in the first example of compound multiplication, the nines, being cast out of the multiplier, and multiplicand, 472 and 3809, leave 4 and 2, which produce 8, the remainder of the division of the product 1797848 by 9. In like manner, on casting eleven out of the former numbers, there are left 10 and 3, which being multiplied yield 30, or an excess of 8; but the product 1797848 divided, by 11, leaves also 8. Again in the next example, if nine be cast out of the multiplier and multiplicand, 6428 and 36985, there remains 2 and 4 as formerly, and their product 8 is excess of 287739580 above the nearest multiple of nine. With eleven likewise the remainders are 4 and 3, whose product 12 gives an excess of 1 for the remainder of the division of the result by eleven.

In the Duodenary Scale, the corresponding property must belong to the numbers eleven and thirteen. Thus, in the first example, casting the elevens out of 276 and 2957, the excesses are 4, and 1; but their product 7350156 being divided by eleven, leaves 4. Casting thirteen out of the same factors, there remain 1 and 9, and the product, divided by thirteen, leaves 9. Again, the second example, casting out the elevens from 3878 and 19421, there are left 4 and 3; which being multiplied give 12, or 1 for the excess of the product 67750938 above the nearest multiple of eleven; but casting thirteen out of the same numbers, the remainders are 6 and 9, which give 54 or 10 for the excess of the product.—The agreement in these examples affords a strong presumption of the correctness of the operations; as any discordance would be an absolute proof of inaccuracy.

To transfer any numeral expression from one scale to another, the most obvious way, is, to decompound ring of it and then dispose it again on the new scale. Suppose it were required to convert the binary expression 1000011001 into another on the Quinary Scale.

Beginning at the left hand, and constantly doubling the terms, shifting them every time a place lower, the successive results are 2, 4, 8, 16, 33, 67, 134, 268 and 537. This number again, if decomposed by a series of pentads, gives 4122 for its Quinary expression. But the conversion might be performed directly with more expedition, by dividing the original expression successively by the root of the new scale.

The operation would be conducted thus:

| BINARY SCALE | |--------------| | 101 | | 101 | | 101 | | 101 | | 101 | | 101 | | 101 | | 101 | | 101 | | 101 |

The divisor 5 being denoted on the Binary Scale by 101, the repeated analysis is pursued on the same scale; the series of inverted remainders is 100, 1, 10, and 10, giving, as before, the final result 4122 on the Quinary Scale.

Let it be sought to convert this last expression to the Ternary Scale, the division will be performed in this manner:

| TERNARY SCALE | |---------------| | 34122 | | 12040 | | 2142 | | 342 | | 111 | | 20 |

The Ternary expression is hence 201220.

To reduce vulgar fractions to any scale, we have only to multiply the numerator by the root of that scale, and to divide the product by the denominator, and to repeat this process till it either terminates or glides into a circulation. The series of quotients exhibits the value developed on the descending bars. But this decomposition has been already anticipated. The same method applies likewise to the conversion of mixed fractions. Suppose it were sought to express fifteen shillings and three pence three farthings on the Denary Scale of pounds. Multiply this sum repeatedly by ten, and the integral results deferred to the lower bars, must evidently express the same value. The corresponding decimal is hence .765625, as formerly employed.

This example, however, is better adapted to the Duodenary Scale, the multiplication being performed by successive twelves. The duodecimal expression for the same sum is therefore .923, consisting of only three figures.

These reductions might be performed differently, by ascending gradually to the higher denominations; the farthings being changed into parts of a penny, the pence into parts of a shilling, and the shillings into parts of a pound.

This process requires no farther explanation; but, in the Duodenary Scale, it becomes expedient to represent the 12 and 20, by 10 and 18.

To transfer the expression of a fraction from one scale to another, it should be multiplied successively by the root of this, and the integral results retained. Thus, to convert the decimal fraction 7854, which expresses the relation of the circle to its circumscribing square, to the Duodenary Scale: Multiplying the whole by twelve, the product would represent on the next bar of this scale the same value; again, the fractional surplus .4248 being multiplied by twelve, gives another residue for a lower bar; and this process being continued, the fraction proposed is at last changed into the duodecimal .9512, being the number already employed in the mensuration of circular timber.

Again, suppose it were required to transfer the expression .2132032, from the Quaternary to the Ternary Scale: This operation will be performed by continually multiplying the fractional part by three, and retaining only the integral results. The value of the expression is, therefore, on the Ternary Scale, denoted by .12121122, &c.

The finding of roots is a branch of Arithmetic, to which the system of numerical arrangement is well adapted. The method of proceeding here, is to approach by successive steps to the final result, descending from the highest to the lowest point of the Root.

Extraction of Square of given scale, by discovering the series of additions required to complete the number sought. The extraction of the square root, is the easiest and the most important. To investigate the principle which should direct this operation, we must examine what takes place in the process of compound multiplication. If a number composed of two parts be multiplied into itself, the result will evidently consist of the squares of those parts, together with twice their mutual product. Now, taking away the square of one of these parts, suppose the greater, there must remain the square of the other, joined to their double product; or, what is the same thing, this residue will be the product of the smaller, into a number formed by annexing it to double the greater. Consequently, to discover the secondary or additive portion of the root, we have only, after the square of the principal part has been separated, to divide what is left, by twice its root, annexing always the quotient to this divisor, in closing the process of division. The same operation, descending successively to lower terms, must be repeated, till the number proposed for extraction be entirely exhausted. It is only requisite to observe the rank and number of the figures which the root should contain. But, for this purpose, since every compound number will evidently by squaring have its places of figures doubled, we need only distinguish each pair in the number whose root is sought.

An example will best explain the whole procedure. Suppose it were required to find the square root of 107584. Beginning at the right hand, and marking every second figure, it is divided into three periods; which shows that the root must consist of hundreds, tens, and units. To the first period 10, the nearest square is 9, whose root 3 must occupy the place of hundreds.

Subtracting and taking down the next period, the residue 175 comes to be decomposed; doubling, therefore, the root, we have 6 in the place of hundreds, to which the quotient 2, as denoting twenty, is annexed, and the product 124 set down for subtraction. The remainder, with the last period, making 5184, is finally to be analysed. Twice the root already found, amounting to 640, with the quotient 8 itself, forms the new divisor, and the product extinguishes what was left of the proposed number. The root is therefore 300, with the successive additions of 20 and of 8.

It will sometimes be convenient, in performing this operation, to employ deficient figures, especially as they will rectify the oversight, in case too large a quotient may have been assumed.

We subjoin an example of the The same mode of proceeding will obviously extend to the descending terms of any scale, a pair of zeros being annexed to the remainder after each successive division. As an example, we shall select the calculation of the greater segment of a line, divided by extreme and mean ratio, which is denoted by $\sqrt{\frac{1}{2}}$, the whole being unit. The radical is thus found on three different scales.

**Quaternary.**

| 110(1.013208) | 1 | |---------------|---| | 201(1000) | 21 | | 202(13300) | 221 | | 203(12901) | 222 | | 204(103300) | 2228 | | 205(10300) | 22282 | | 206(13300000)| 223603 |

**Denary.**

| 1.25(1.118034) | 1 | |----------------|---| | 21 | 21 | | 221 | 211 | | 222 | 211 | | 2228 | 17900 | | 22282 | 17824 | | 223603 | 760000 |

**Duodecenary.**

| 1.30(1.145537) | 1 | |----------------|---| | 21 | 30 | | 224 | 300 | | 228 | 894 | | 2280 | 894 | | 22800 | 894 | | 228000 | 894 |

It hence appears, that the greater segment of a line, divided by extreme and mean ratio, is expressed in duodecimals by .745537, or extremely nearly by .75; and, therefore, that it consists of 89 parts, of which the whole contains 144. The very same result is obtained from the recurring series, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, &c., which continually approximates to the required value.

The main advantage of this scale, consists in its fitness to denote fractional parts. Its root has indeed no fewer than four factors — 2, 3, 4, and 6; while ten is divisible only by 2 and 5. Several attempts, accordingly, have, at different times, been made to carry the Duodecenary Scale into actual practice. It is a singular fact, that the famous Charles XII. of Sweden, whose views, though often disturbed by the wildness of heroism, were on the whole beneficent, seriously deliberated on a scheme of introducing this system of numeration into his dominions, a very short time before his death, while lying in the trenches, during the depth of winter, before the towering Norwegian fortress of Fredericksball.

With respect to square numbers, it may be remarked, that they terminate only in these five digits,—1, 4, 5, 6, and 9,—which lie equally distant on each side of the middle. When a number ends in 1, its square root must end in 1 or 9; when it ends in 4, the root ends 2 or 8; when it ends in 5, the root will likewise end in 5; when it ends in 6, the root will end in 4 or 6; and when it ends in 9, the root will end in 3 or 7. Unless, therefore, in the case of 5, there are always two corresponding terminations of the root, making together the number ten.

Cube numbers, or those denoting the third powers, and produced by a triple multiplication, are not affected by such ambiguity. They have for their terminations indiscriminately all the nine digits. If any cube should terminate in one of the fore-mentioned figures, 1, 4, 5, 6, or 9, its root will end in the same figure; but if it terminate in any of the remaining digits, 2, 3, 7, or 8, the corresponding root will end in 8, 7, 3, or 2, that is, in the difference of each from ten.

But we have already far exceeded our limits, and must abruptly conclude. Still we should commemorate, though not in the proper place, one of the earliest and ablest writers in England on Arithmetical Science—Robert Recorde, Fellow of all Souls, Oxford, and Doctor in Physic, published in 1540 a treatise of Arithmetic, in the form of a dialogue.*

He afterwards proceeded farther; and, in his Whetstone of Witte, containing the Extraction of Rootes and Cosike practise, he introduced the first rudiments of Algebra. But this benefactor of his country, who seems to have devoted himself to the teaching of Mathematics in London, was treated with ingratitude, and, to the disgrace of the age, left to struggle with chilling poverty, and constrained to end his days in the gloom of a prison.

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* We subjoin another specimen of the Roman or Saxon numerals till then used: In the accounts of the Scottish Exchequer for the year 1331, the sum of L.6896, 5s. 5d. stated as paid to the King of England, is thus marked, *V*10, viii, iii, xvj, h, v, s, v, d. It may be observed, that, in Scotland, the contraction 3m for one thousand, is still used in the dates of charters, and other legal instruments.