a Peripatetic philosopher, and native of Tarentum. He studied under Lampris, Xenophilius, and Aristotle; and, according to Staidas, was the author of 453 works on music, philosophy, and history. The only one of these that is now known is his *Elements of Harmonics*, in three books, of which Meibomius published an edition with Latin translation and notes in his *Antiquae Musicae Auctores*, Amsterdam, 1652. This is considered the best edition of Aristoxenus's work, although it is imperfect; for part of each book is wanting, and great confusion occurs from transposition of passages. The Abate J. Morelli, librarian of St Mark's library at Venice, published fragments of Aristoxenus's *Elements of Rhythm* from a MS. in that library, Venice, 1785.
By the term *diapodia*, the ancient Greeks did not mean *harmony* in the modern sense of that word as applied to music, but the system of sounds upon which *melody* for voice or instrument was founded. It is remarkable that Aristoxenus, after asserting that the ear is the sole judge of harmonic [melodic] intervals, should have plunged into a system of false calculations of these intervals. He, like the other Greek writers on music, has given us no information regarding the practical parts of the art, either in composition or performance.
**ARITHMETIC**
Is a science which explains the properties of numbers, and shows the method or art of computing by them.
I.—HISTORY OF ARITHMETIC.
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 hordes 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 numbers, at least, are expressed by combined strokes. 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 sets 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 | to signify one, another such || would express two, the junction of a third ||| three, and so repeatedly till the reckoner had reached ten. (See Plate LXXII.) 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 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 disperted semicircles divided by a diameter. This last form, by abbreviation on either side, gave two portions of I and IO to represent five hundred.
It was an easy process, therefore, to devise a universal Origin of 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 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 then most resembled them. (See Plate LXXII.) 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 resem- bling 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 cld, consisting of the letter I inclosed on both sides by C, and by the same letter reversed; a portion of this, again, or I, 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 Greek words ἑκατόν and ἑξήκοντα.
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, cclxxx was employed to represent ten thousand, and ccclxxx to signify an hundred thousand, the letter I, inclosed between the c, being, for the sake of greater distinctness, elongated. Again, each of these being divided, gives lxxx for five thousand, and lxxx 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 octodecim and novemdecim, the words for eighteen and nineteen, they frequently used duodecim et undecim, 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 philologists, 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 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.
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. 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 movetur.
As the Chinese constructed the Swan-pan on the principles of the Roman Abacus (See Abacus), so they 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 LXXII. 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. In addition to the figures in the Plate, we shall here give fac similis of a complete set of numerals, printed with metallic types in 1814, at Serampore, in the Elements of Chinese Grammar, by the Reverend Dr Marshman. In these characters, it will be perceived that each symbol has, for the sake of distinction, a small zero or º annexed to it.
| One, | Two, | Three, | Four, | Five, | Six, | Seven, | Eight, | Nine, | |------|------|--------|-------|-------|------|--------|-------|------| | Yi. | Irr. | San. | Se. | Ngco. | Lyèa.| Tihh. | Pih. | Kyèu.|
| Ten, | A Hundred, | A Thousand, | Ten Thousand, | A hundred Thousand, | A Million, | Ten Millions, | A hundred Millions, | |-----|------------|-------------|---------------|-------------------|-----------|--------------|-------------------| | Shih.| Pàh. | Ts'hyen. | Wan. | Eè. | Chao. | King. | Kyai. |
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 History. 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 outdoing 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 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.
| One. | Two. | Three. | Four. | Five. | Six. | Seven. | Eight. | Nine. | |------|------|--------|-------|-------|------|--------|--------|-------| | 一 | 二 | 三 | 四 | 五 | 六 | 七 | 八 | 九 |
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 pūh, 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.
| Twenty. | Twenty-one. | Thirty. | Thirty-one. | Forty. | Forty-one. | Fifty. | Fifty-one. | Seventy-three. | |---------|-------------|---------|-------------|--------|------------|--------|------------|----------------| | 十 | 二十一 | 三十 | 三十一 | 四十 | 四十一 | 五十 | 五十一 | 七十三 |
A similar process extends to the notation of thousands, but for ten thousand the character wàn 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:
Here the first column on the right hand presents the marks for fifty and four, with the interjacent character wàn, 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 wàn, 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 17th century, the Jesuit missionaries Bouvet, Gerbillon, and others, then residing at the court of Pekin, and able mathematicians, appear to have still farther improved the numeral symbols of the Chinese traders, and reduced the whole system to a degree of simplicity and elegance of form scarcely inferior to that of our modern ciphers. With these abbreviated characters they printed, at the imperial press, Vlacq's Tables of Logarithms, extending to ten places of decimals, in a beautiful volume, of which a copy was presented by Father Gaubil, on his return to Europe, about the year 1750, to the Royal Society of London. From this very curious work, the marks in Plate LXXII. entitled Improved Chinese Numerals, were carefully copied. No more than nine characters, it will be seen, are wanted; the upright cross + for ten being a mere redundancy. The marks for one, two, and three, consist of parallel strokes as before; an oblique cross X denotes four; and a sort of bisected ten signifies five. This symbol again, being contracted into the angular mark Z, and combined with one, two, or three strokes drawn below it, represents six, seven, or eight; and still more abridged and annexed to the sign of four, it denotes nine. The distinction of units, tens, hundreds, &c. is indicated by giving the strokes alternately an horizontal and vertical position, while the blanks or vacant bars are expressed by placing small zeros. The very important collection of logarithmic tables just mentioned was printed by the command of the emperor Kang-hi, a man of enlarged views, who governed China with dignity and wisdom during a long course of years. This enlightened prince was much devoted to the learning of Europe, and is reported to have been so fond of calculation as to have those tables abridged and printed in a smaller character, which precious volume he carried constantly fastened to his girdle. The emperor Kien-long, who, after a beneficent reign of 60 years, in the decline of a protracted life spontaneously resigned the imperial office to his son, discovered a similar taste for the mathematical sciences.
The Greeks, after having communicated to the founders of Rome the elements of the numeral characters which are still preserved, again exercised their inventive genius in framing new systems of notation. Discarding the simple original strokes, they sought to draw materials of construction from their extended alphabet. They had no fewer than three different modes of proceeding. 1st, The letters of the alphabet, in their natural succession, were employed to signify the smaller ordinal numbers. In this way, for instance, the books of Homer's Iliad and Odyssey are usually marked. But the practice could scarcely be older than the time of Aristotle, who, it is well known, first collected and arranged those immortal poems, in the edition of the Codex, for the use of his illustrious pupil Alexander the Great. 2d, The first letters of the words for numerals were adopted as abbreviated symbols. Thus, employing capitals only, I, being retained as before, to denote one, the letter II of IIENTE marked five, the Δ of ΔEKA denoted ten, the Η of ΕKATON, anciently written HEKATON, expressed an hundred, the Χ of XIAIA a thousand, and the letter M of the word MYPIA represented ten thousand. A simple and ingenious device was used for augmenting the powers of those symbols; a large Π placed over any letter made it signify five thousand times more. Thus, [Δ] denoted fifty thousand, and [Π] five hundred thousand. See Plate LXXII. 3d, 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 ε, called epiešmon, was inserted among the units, immediately after ε, the letter denoting five; and the kappa and sanpi, represented by S, Λ, or Θ 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 dual, 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 dual 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 fourfold, 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 anyhow 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 *enniad*, 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 required 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 endowed 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 LXXII. 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.
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. The number 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 | Three Ten Four Nine |
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 *pegeia*, 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 notation properties of such progressions. In a curious tract, entitled *Yasurges* 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 Apollonius 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 Mo or the contraction for *pegeia*, ten thousand. Thus, to express thirty-four trillions, they wrote λδ Mo. Mo. Mo. To signify units separately, it was customary with them to prefix the mark M², or the abbreviation for *monad*.
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 the 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 \( \mu \) were to be multiplied into \( \sigma \), or forty into two hundred: Take the lower corresponding characters \( \delta \) and \( \beta \), or four and two, which were called \( \tau \omega \gamma \eta \zeta \varepsilon \nu \) or radicals, the one depressed ten times, and the other an hundred times, and multiply their product \( \pi \), or eight, successively by the ten and the hundred, or at once by a thousand, and the result is \( \eta \) or eight thousand.
We shall take an example in multiplication, affording more variety than such as occur in Eutocius, 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.
\[ \begin{array}{cccc} \omega & \xi & \beta \\ \varphi & \times & \gamma \\ \end{array} \]
\[ \begin{array}{cccc} \mu & \gamma & \alpha \\ \alpha & \sigma & \mu \\ \beta & \upsilon & \epsilon \\ \end{array} \]
In the first range, \( \varphi \) multiplied into \( \omega \), being the same as the product of eight and five augmented ten thousand times, is consequently denoted by \( \mu \) or \( \frac{\mu}{10} \); \( \varphi \) multiplied into \( \xi \) gives the same result as five times six increased a thousand fold, and therefore expressed by \( \gamma \) or \( \frac{\gamma}{10} \); and \( \varphi \) multiplied into \( \beta \) evidently makes a thousand, or \( \alpha \). In the second range, \( \alpha \) multiplied into \( \omega \) gives the same product as eight repeated twice and then augmented a thousand times, or denoted by \( \alpha \); \( \alpha \) multiplied into \( \xi \) is equivalent to six repeated twice, and afterwards increased an hundred fold or, expressed by \( \epsilon \); and \( \alpha \) multiplied by \( \beta \) gives forty, the value of \( \mu \). In the third range, \( \gamma \) multiplied into \( \omega \) produces twenty-four hundred, which is denoted by \( \beta \); \( \gamma \) multiplied into \( \xi \) makes an hundred and eighty, or \( \epsilon \); and, lastly, \( \gamma \) multiplied into \( \beta \) gives \( \epsilon \), 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 \( \mu \times \alpha \times \epsilon \), 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, \( C \), \( \angle \), \( C \); 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, \( \beta^2 \) intimates two-elevenths, and \( \pi \alpha \xi \) 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.
\[ \begin{array}{cccc} \alpha & \omega & \lambda & \eta \\ \beta & \alpha & \omega & \lambda \\ \gamma & \eta & \zeta & \varepsilon \\ \end{array} \]
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 \( \beta^2 \) and \( \epsilon^2 \), while the product of the fraction itself gives eighty-one of an hundred and twenty-one parts, expressed by \( \pi \alpha \xi \).
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 calcu- 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*, vol. 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:
\[ \begin{array}{cccc} \lambda & \delta & \varepsilon \\ \lambda & \delta & \varepsilon \\ \alpha & \beta & \gamma \\ \alpha & \beta & \gamma \\ \end{array} \]
Here in the first line, \(\lambda\) multiplied into \(\delta\) in the place of units, gives \(\alpha\) or thirteen hundred and sixty-nine degrees; \(\lambda\) into \(\delta\) on the next bar, gives \(\beta\) or one hundred and forty-eight minutes; and \(\lambda\) into \(\varepsilon\) on the lowest bar gives \(\gamma\) or two thousand and thirty-five seconds. In the second line \(\delta\) multiplied into \(\lambda\) gives the product \(\alpha\) as before; \(\delta\) multiplied into \(\delta\), both of them on the bar of minutes, gives \(\beta\) or sixteen seconds; \(\delta\) into \(\varepsilon\) gives \(\gamma\) or two hundred and twenty thirds. Lastly, in the third line the \(\varepsilon\) on the bar of seconds, multiplied successively into \(\lambda\) and \(\delta\), produce, as before, \(\beta\) and \(\gamma\) on the bars of seconds and thirds; and \(\varepsilon\) multiplied by itself gives \(\gamma\) or three thousand and twenty-five fourths. These several products being reduced and collected together, formed the total amount of \(\alpha\) or \(\beta\) or \(\gamma\), 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. Wherefore 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 engraffed on the notation of the Greeks. The astronomers of Alexandria and Constanti- ed 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 ρ to occupy the accidental blanks which occurred in the notation of sexagesimals. This letter was perhaps chosen by him, because immediately succeeding to ρ, 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 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 epiesēmion, should denote the nine digits. The ῥότα, which signified ten, now losing its force, might have been employed as a convenient substitute for the cypher.
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 ρ 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 Alshadhī, a learned Arabian doctor, boasted of three very different inventions—the composition of the *Golaila Wadamma*, 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 Universal Diplomacy*, maintains that our ciphers were only prior 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 Mussulmans 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 History. 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 Lilavati, of such inestimable 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 Lilavati 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 Dera-nagari 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 Hindoos 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 Hindoos 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 LXXII. (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 Dera-nagari characters appears very striking. From these, again, the common Hindoo and the vulgar Bengalee digits are evidently moulded, with only slight alterations of figure. The Birman numerals, which we have copied from Symes's Embassy to the Kingdom of Ava, are manifestly of the same origin; only they have a thin, wiry body, being generally written on the palmrya-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 Gentooes, 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 farther 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 Λογιστική Ἰνδῶν, or Υπεροχὴ τῶν Ἰνδῶν, ἡ λογιστικὴ μηχανῆ, 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.
2. 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 250 years later.
While the thickest darkness brooded over the Christina 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 flamed 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 bishoprics 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 acts 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 Toulouse, 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.
In the tenth volume of the Archaeologia, the Rev. Mr North has given a short account of an almanack preserved in the library of Bennet College, Cambridge, and containing a table of eclipses for the cycle between 1330 and 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 LXXII. The Saxon X, signifying ten, is repeatedly combined with the ordinary figures; and XXX, XXXI., 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 episonom 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 Petrarch 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.
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1 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. History, 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 executio, 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 oemery stones. In Shakspeare'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 1482, 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 LXXII. To these we have subjoined fac-similes from Caxton's Mirror of the World, and a wooden cut from Shirwood's Ludus Arithmomachiae, 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 Andrews; 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 LXXII. are copied from a manuscript history of the Scottish Bishoprics, apparently written about the year 1550.
II.—OPERATIONS OF ARITHMETIC.
Chap. I.—Notation and Numeration.
The first elements of arithmetic are acquired during our infancy. The idea of one is the simplest of any, and is suggested by every single object. Two is formed by placing one object near another; three, four, and every higher number, by adding one continually to the former collection. As we thus advance from lower numbers to higher, we soon perceive that there is no limit to this increasing operation; and that, whatever number of objects be collected together, more may be added, at least in imagination; so that we can never reach the highest possible number, nor approach near it. The idea of numbers, which is first acquired by the observation of sensible objects, is afterwards extended to measures of space and time, affections of the mind, and other immaterial qualities.
Small numbers are most easily apprehended: a child soon knows what two and what three is, but has not any distinct notion of seventeen. Experience removes this difficulty in some degree: as we become accustomed to handle larger collections, we apprehend clearly the number of a dozen or a score; but perhaps could hardly advance to a hundred without the aid of systematic arrangement, which is the art of forming so many units into a class, and so many of these classes into one of a higher kind, and thus advancing through as many ranks of classes as occasion requires. If a boy arrange a hundred stones in one row, he would be tired before he could reckon them; but if he place them in ten rows of ten stones each, he will reckon a hundred with ease; and if he collect ten such parcels, he will reckon a thousand. In this case, ten is the lowest class, a hundred is a class of the second rank, and a thousand is a class of the third rank.
There does not seem to be any number naturally adapted for constituting a class of the lowest or any higher rank, to the exclusion of others. However, as ten has been universally used for this purpose by the Hebrews, Greeks, Romans, and Arabians, and by all nations who have cultivated this science, it is probably the most convenient for general use; but other scales may be assumed, perhaps, on some occasions with superior advantage; and the principles of arithmetic will appear in their full extent, if the student can adapt them to any scale whatever. Thus, if eight were the scale, 6 times 3 would be two classes and two units, and the number 18 would then be represented by 22. If 12 were the scale, 5 times 9 would be three classes and nine units, and 45 would be represented by 39.
Whatever number of units constitutes a class of the lower rank, the same number of each class should make one of the next higher. This is observed in our arithmetic, ten being the universal scale; but it is not regarded in the various kinds of monies, weights, and the like, which do not advance by any universal measure; and much of the difficulty in the practice of arithmetic arises from that irregularity.
For want of attending to these facts, some learned antiquaries have often suffered themselves to be grossly misled. Thus, Mr De Cardonnell, 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 Andrews. But this front was built subsequent to the murder of the detested Cardinal Beaton, by Archibald 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. As higher numbers are somewhat difficult to apprehend, we naturally fall on contrivances to fix them in our minds, and render them familiar; but notwithstanding all the expedients we can fall upon, our ideas of high numbers are still imperfect, and generally far short of the reality; and though we can perform any computation with exactness, the answer we obtain is often incompletely apprehended.
It may not be amiss to illustrate, by a few examples, the extent of numbers which are frequently named without being attended to. If a person employed in telling money reckon an hundred pieces in a minute, and continue at work ten hours each day, he will take seventeen days to reckon a million; a thousand men would take 45 years to reckon a billion. If we supposed the whole earth to be as well peopled as Britain, and to have been so from the creation, and that the whole race of mankind had constantly spent their time in telling from a heap consisting of a quadrillion of pieces, they would hardly have yet reckoned the thousandth part of that quantity.
All numbers are represented by the ten following characters:
1 2 3 4 5 6 7 8 9 0
One, two, three, four, five, six, seven, eight, nine, cipher.
The nine first are called significant figures, or digits; and sometimes represent units, sometimes tens, hundreds, or higher classes. When placed singly, they denote the simple numbers subjoined to the characters; when several are placed together, the first or right-hand figure only is to be taken for its simple value; the second signifies so many tens, the third so many hundreds, and the others so many higher classes, according to the order they stand in. And as it may sometimes be required to express a number consisting of tens, hundreds, or higher classes, without any units or classes of a lower rank annexed, and as this can only be done by figures standing in the second, third, or higher places, while there are none to fill up the lower ones; therefore an additional character or cipher (0) is necessary, which has no signification when placed by itself, but serves to supply the vacant places, and bring the figures to their proper station.
The following table shows the names and divisions of the classes.
| Trillions | Billions | Millions | Thousands | Hundreds | Tens | Units | |-----------|----------|----------|-----------|---------|------|-------| | 8 4 3 7 9 8 | 2 5 6 4 7 3 | 8 9 7 2 6 4 5 |
The first six figures from the right hand are called the unit period, the next six the million period, after which the trillion, quadrillion, quintillion, sextillion, septillion, octillion, and nonillion periods follow in their order.
It is proper to divide any number, before we reckon it, into periods and half periods, by different marks. We then begin at the left-hand, and read the figures in their order, with the names of their places, from the table. In writing any number, we must be careful to mark the figures Addition in their proper places, and supply the vacant places with ciphers.
As there are no possible ways of changing numbers, except by enlarging or diminishing them according to some given rule, it follows that the whole art of arithmetic is comprehended in two operations, Addition and Subtraction. However, as it is frequently required to add several equal numbers together, or to subtract several equal ones from a greater, till it be exhausted, proper methods have been invented for facilitating the operation in these cases, and distinguished by the names of Multiplication and Division; and these four rules are the foundation of all arithmetical operations whatever.
As the idea of number is acquired by observing several objects collected, so is that of fractions by observing an object divided into several parts. As we sometimes meet with objects broken into two, three, or more parts, we may consider any or all of these divisions promiscuously, which is done in the doctrine of vulgar fractions. However, since the practice of collecting units into parcels of tens has prevailed universally, it has been found convenient to follow a like method in the consideration of fractions, by dividing each unit into ten equal parts, and each of these into ten smaller parts; and so on. Numbers divided in this manner are called Decimal Fractions.
**Chap. II.—Addition.**
Addition is that operation by which we find the amount of two or more numbers. The method of doing this in simple cases is obvious, as soon as the meaning of number is known, and admits of no illustration. A young learner will begin at one of the numbers, and reckon up as many units separately as there are in the other, and practice will enable him to do it at once. It is impossible, strictly speaking, to add more than two numbers at a time. We must first find the sum of the first and second, then we add the third to that number, and so on. However, as the several sums obtained are easily retained in the memory, it is neither necessary nor usual to mark them down. When the numbers consist of more figures than one, we add the units together, the tens together, and so on. But if the sum of the units exceed ten, or contain ten several times, we add the number of tens it contains to the next column, and only set down the number of units that are over. In like manner we carry the tens of every column to the next higher. And the reason of this is obvious from the value of the places; since an unit, in any higher place, signifies the same thing as ten in the place immediately lower.
**Rule.—Write the numbers distinctly, units under units, tens under tens, and so on. Then reckon the amount of the right-hand column. If it be under ten, mark it down. If it exceed ten, mark the units only, and carry the tens to the next place. In like manner, carry the tens of each column to the next, and mark down the full sum of the left-hand column.**
Example: \[ \begin{array}{cccc} \text{Units} & \text{Tens} & \text{Hundreds} & \text{Thousands} \\ 876734 & 123467 & 314213 & 712316 \\ & & 438987 & 279654 \\ & & & 3092234 \\ & & & 24433 \\ \end{array} \]
As it is of great consequence in business to perform addition readily and exactly, the learner ought to practise it till it become quite familiar. If the learner can readily add any two digits, he will soon add a digit to a higher number with equal ease. It is only to add the unit place.
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1 To abbreviate, it is sometimes convenient to indicate the four operations of arithmetic by the signs \(+\), \(-\), \(\times\), \(\div\). The first, \(+\), denotes addition; the second, \(-\), indicates subtraction; the third, \(\times\), multiplication; and the fourth, \(\div\), division. The sign = put between two quantities indicates that they are equal. Accordingly we may write \(2 + 3 = 5\); \(7 - 3 = 4\); \(5 \times 3 = 15\); \(14 \div 2 = 7\). Addition of that number to the digit; and if it exceed ten, it raises the amount accordingly. Thus, because 8 and 6 are 14, 48 and 6 are 54. It will be proper to mark down under the sums of each column, in a small hand, the figure that is carried to the next column. This prevents the trouble of going over the whole operation again, in case of interruption or mistake. If you wish to keep the account clean, mark down the sum and figure you carry on a separate paper, and after revising them, transcribe the sum only. After some practice, we ought to acquire the habit of adding two or more figures at one glance. This is particularly useful when two figures which amount to 10, as 6 and 4, or 7 and 3, stand together in the column.
Every operation in arithmetic ought to be revised, to prevent mistakes; and as one is apt to fall into the same mistake, if he revise it in the same manner he performed it, it is proper either to alter the order, or else to trace back the steps by which the operation advanced, which will lead us at last to the number we began with. Every method of proving accounts may be referred to one or other of these heads.
1st. Addition may be proved by any of the following methods: Repeat the operation, beginning at the top of the column, if you began at the foot when you wrought it.
2d. Divide the account into several parts; add these separately, and then add the sums together. If their amount correspond with the sum of the account when added at once, it may be presumed right. This method is particularly proper when you want to know the sums of the parts as well as that of the whole.
3d. Subtract the numbers successively from the sum; if the account be right, you will exhaust it exactly, and find no remainder.
When the given number consists of articles of different value, as pounds, shillings, and pence, or the like, which are called different denominations, the operations in arithmetic must be regulated by the value of the articles. We shall give here a few of the most useful tables.
I. Sterling Money.
| Sterling Money | Avoidupois Weight | |----------------|-------------------| | £ | t. cwt. gr. lb. | | 127 | 13 3 | | 43 | 5 10½ | | 806 | 18 7 | | 190 | 2 5¾ | | 214 | 0 3 | | 85 | 15 4½ | | 1467 | 15 9½ |
II. Troy Weight.
| Troy Weight | |-------------| | 20 mites = 1 grain, gr. | | 24 grains = 1 pennyw, dwt. | | 20 pennyw = 1 ounce, oz. | | 12 ounces = 1 pound, lib. |
III. English Dry Measure.
| English Dry Measure | |---------------------| | 2 pints = 1 quart | | 4 quarts = 1 gallon | | 2 gallons = 1 peck | | 4 pecks = 1 bushel | | 8 bushels = 1 quarter |
IV. Apothecaries' Weight.
| Apothecaries' Weight | |----------------------| | 20 grains = 1 scruple, 9 | | 3 scruples = 1 dram, 3 | | 8 drams = 1 ounce, 3 | | 12 ounces = 1 pound, lb. |
V. Scots Dry Measure.
| Scots Dry Measure | |-------------------| | 4 lippies = 1 peck | | 4 pecks = 1 firlo | | 4 firlots = 1 boll | | 16 bolls = 1 calder |
VI. English Land Measure.
| English Land Measure | |----------------------| | 30½ square yards = 1 pole | | or perch | | 40 poles = 1 rood | | 4 roods = 1 acre |
VII. Scots Land Measure.
| Scots Land Measure | |--------------------| | 36 square ells = 1 fall | | 40 falls = 1 rood | | 4 roods = 1 acre |
IX. Long Measure.
| Long Measure | |--------------| | 12 inches = 1 foot | | 3 feet = 1 yard | | 5½ yards = 1 pole | | 40 poles = 1 furlong | | 8 furlongs = 1 mile | | 3 miles = 1 league |
X. Time.
| Time | |------| | 60 seconds = 1 minute | | 60 minutes = 1 hour | | 24 hours = 1 day | | 7 days = 1 week | | 365 days = 1 year | | 52 weeks and 1 day = 1 year |
Rule for Compound Addition.—Arrange like quantities under like, and carry according to the value of the higher place.
Note 1. When you add a denomination which contains more columns than one, and from which you carry to the higher by 20, 30, or any even number of tens, first add the units of that column, and mark down their sum, carrying the tens to the next column; then add the tens, and carry to the higher denomination, by the number of tens that it contains of the lower. For example, in adding shillings, carry by 10 from the units to the tens, and by 2 from the tens to the pounds.
Note 2. If you do not carry by an even number of tens, first find the complete sum of the lower denomination, then inquire how many of the higher that sum contains, and carry accordingly, and mark the remainder, if any, under the column. For example, if the sum of a column of pence be 48, which is three shillings and sevenpence, mark 7 under the pence column, and carry 3 to that of the shillings.
Note 3. Some add the lower denominations after the following method: when they have reckoned as many as amounts to one of the higher denomination, or upwards, they mark a dot, and begin again with the excess of the number reckoned above the value of the denomination. The number of dots shows how many are carried, and the last reckoned number is placed under the column.
Examples.
| Sterling Money | Avoidupois Weight | |----------------|-------------------| | £ | t. cwt. gr. lb. | | 127 | 13 3 | | 43 | 5 10½ | | 806 | 18 7 | | 190 | 2 5¾ | | 214 | 0 3 | | 85 | 15 4½ | | 1467 | 15 9½ |
CHAP. III.—Subtraction.
Subtraction is the operation by which we take a lesser number from a greater, and find their difference. It is exactly opposite to addition, and is performed in a like manner, beginning at the greater, and reckoning downwards the units of the lesser. The greater is called the minuend, and the lesser the subtrahend.
If any figure of the subtrahend be greater than the corresponding figure of the minuend, we add ten to that of the minuend, and having found and marked the difference, we add one to the next place of the subtrahend. This is called borrowing ten. The reason will appear, if we consider that, when two numbers are equally increased by adding the same to both, their difference will not be altered. When we proceed as directed above, we add ten to the minuend, and we likewise add one to the higher place of the subtrahend, which is equal to ten of the lower place.
Rule.—Subtract units from units, tens from tens, and so on. If any figure of the subtrahend be greater than the corresponding one of the minuend, borrow ten.
Example.
| Minuend | Subtrahend | |---------|------------| | 173694 | 21453 | | Remainder | 152241 | | | 359406 | To prove subtraction, add the subtrahend and remainder together; if their sum be equal to the minuend, the account is right.
Or subtract the remainder from the minuend. If the difference be equal to the subtrahend, the account is right.
**Rule for Compound Subtraction.—Place like denominations under like; and borrow, when necessary, according to the value of the higher place.**
| £ | s. d. | Cwt. gr. lbs. | A. R. F. E. | |---|------|--------------|------------| | 146 | 3 3 | 12 3 19 | 15 2 24 18 | | 58 | 7 6 | 4 3 24 | 12 2 36 7 | | 87 | 15 9 | 7 3 23 | 2 3 28 11 |
**Note 1.** The reason for borrowing is the same as in simple subtraction. Thus, in subtracting pence, we add 12 pence when necessary to the minuend, and at the next step we add one shilling to the subtrahend.
**Note 2.** When there are two places in the same denomination, if the next higher contain exactly so many tens, it is best to subtract the units first, borrowing ten when necessary; and then subtract the tens, borrowing, if there is occasion, according to the number of tens in the higher denomination.
**Note 3.** If the value of the higher denomination be not an even number of tens, subtract the units and tens at once, borrowing according to the value of the higher denomination.
**Note 4.** Some choose to subtract the place in the subtrahend, when it exceeds that of the minuend, from the value of the higher denomination, and add the minuend to the difference. This is only a different order of proceeding, and gives the same answer.
| Twice | Thrice | Four times | Five times | Six times | Seven times | Eight times | Nine times | Ten times | Eleven times | Twelve times | |-------|--------|------------|------------|-----------|-------------|-------------|------------|-----------|-------------|-------------| | 1 is 2 | 1 is 3 | 1 is 4 | 1 is 5 | 1 is 6 | 1 is 7 | 1 is 8 | 1 is 9 | 1 is 10 | 1 is 11 | 1 is 12 | | 2 | 4 | 6 | 8 | 10 | 12 | 14 | 16 | 18 | 20 | 22 | 24 | | 3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | | 4 | 8 | 12 | 16 | 20 | 24 | 28 | 32 | 36 | 40 | 44 | 48 | | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | | 6 | 12 | 18 | 24 | 30 | 36 | 42 | 48 | 54 | 60 | 66 | 72 | | 7 | 14 | 21 | 28 | 35 | 42 | 49 | 56 | 63 | 70 | 77 | 84 | | 8 | 16 | 24 | 32 | 40 | 48 | 56 | 64 | 72 | 80 | 88 | 96 | | 9 | 18 | 27 | 36 | 45 | 54 | 63 | 72 | 81 | 90 | 99 | 108 | | 10 | 20 | 30 | 40 | 50 | 60 | 70 | 80 | 90 | 100 | 110 | 120 | | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 121 | 132 | | 12 | 24 | 36 | 48 | 60 | 72 | 84 | 96 | 108 | 120 | 132 | 144 |
If both factors be under 12, the table exhibits the product at once. If the multiplier only be under 12, we begin at the unit place and multiply the figures in their order, carrying the tens to the higher place, as in addition.
**Ex.** 76859 multiplied by 4, or 76859 added 4 times.
\[ \begin{array}{c} 4 \\ \times 76859 \\ \hline 307436 \\ \end{array} \]
If the multiplier be 10, we annex a cipher to the multiplicand; if the multiplier be 100, we annex two ciphers, and so on. The reason is obvious, from the use of ciphers in notation.
If the multiplier be any digit, with one or more ciphers on the right hand, we multiply by the figure, and annex an equal number of ciphers to the product. Thus, if it be required to multiply by 50, we first multiply by 5, and then annex a cipher. It is the same thing as to add the multiplicand 50 times; and this might be done by writing the account at large, dividing the column into 10 parts of 5 lines, finding the sum of each part, and adding these ten sums together.
If the multiplier consists of several significant figures, we multiply separately by each, and add the products. It is the same as if we divided a long account of addition into parts corresponding to the figures of the multiplier.
**Example.** To multiply 7329 by 365.
\[ \begin{array}{cccc} 7329 & 7329 & 7329 & 36645 = 5 \text{ times.} \\ \times 365 & \times 365 & \times 365 & \times 365 \\ \hline 5 & 60 & 300 & 439740 = 60 \text{ times.} \\ \times 36645 & \times 36645 & \times 36645 & \times 36645 \\ \hline 2198700 & 2198700 & 2198700 & 2198700 \\ \hline 2675085 = 365 \text{ times.} \\ \end{array} \]
It is obvious that 5 times the multiplicand added to 60 times, and to 300 times the same, must amount to the product required. In practice we place the products at once under each other; and as the ciphers arising from the higher places of the multiplier are lost in the addition, we omit them. Hence may be inferred the following...