ARITHMETIC

History. IS a science which explains the properties of numbers, and shows the method or art of computing by them.

The various attempts of men in every state of society at representing numbers all spring from the same feeling, so strongly implanted in their breasts, which unceasingly prompts them to seek the applause of their fellows. It is the love of distinction that elevates the human character, infuses life and action through the mass, and 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 of the permanent amelioration of our species. The aspiring 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 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 rectilinear 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 $I$ 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 LXXVI. 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 $E$. 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 $\backslash Lambda$, or the upper half $V$, was employed to signify five. Next, the mark $E$ for an hundred, consisting of a triple stroke, was largely divided into $\backslash Gamma$ 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 $\backslash mathcal\{M\}$, frequently expressed thus $CIO$, by two departed 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 a 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 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 LXXVI.) 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 resen-

bling the divided symbol for fifty; while the entire symbol, or the tripled stroke, denoting an hundred, was exhibited by the hollow square $\backslash square$, 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 $\backslash text\{C\}\backslash square\backslash text\{C\}$, consisting of the letter I inclosed on both sides by C, and by the same letter reversed; a portion of this, again, or $\backslash text\{I\}\backslash square\backslash text\{I\}$, 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 $\backslash text\{ixatw\}$ and $\backslash text\{\chi ί\lambda \iota \alpha \}$.

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, $\backslash text\{cc\}\backslash square\backslash text\{c\}$ was employed to represent ten thousand, and $\backslash text\{cccc\}\backslash square\backslash text\{c\}$ to signify an hundred thousand, the letter I, inclosed between the $\backslash text\{c\}\backslash square$, being, for the sake of greater distinctness, elongated. Again, each of these being divided, gives $\backslash text\{Icc\}$ for five thousand, and $\backslash text\{Iccc\}$ 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 undeviginti, 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, $\backslash text\{VIII\}$, 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 $\backslash text\{XL\}$, and $\backslash text\{CD\}$, $\backslash text\{XC\}$, and $\backslash text\{CM\}$.

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 chronicleing 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 movebis.

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 LXXVI. 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 similes 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 ° annexed to it.

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 wan, 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, wan wan 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 wan wan far outdoing a thousand years, the measure of Spanish loyalty, is the usual shout of long live the emperor! The Chinese character chaò 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 fire, 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 püh, 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 here 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:—

零 萬 四 千 五
三 五 十 三 萬
千 七 百 四

Here 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.

ICCCCCCCC · CCCCCICCCCC ·
CCCCICCCCC · CCCCCICCCCC ·
CCCCICCCCC · CCCCCICCCC ·
CCCCICCC · CCCCCICCCC · CCCICCC ·
CCCIICC · CCCIICC · CCIIICC · ICC · III.

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

History. 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 LXXVI. 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 $\backslash angle$, 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.

History. minated 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 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.

Gen. 22. 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 Casket, 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 H of HIENTE marked five, the Δ of ΔΕΚΑ denoted ten, the Η of ΕΚΑΤΟΝ, anciently written ΗΕΚΑΤΟΝ, expressed an hundred, the Χ of ΧΙΑΙΑ a thousand, and the Μ of the word ΜΥΡΙΑ represented ten thousand. A simple and ingenious device was used for augmenting the powers of those symbols; a large Η placed over any letter made it signify five thousand times more. Thus, $\backslash overline\{\backslash Delta\}$ denoted fifty thousand, and $\backslash overline\{\backslash overline\{\backslash Delta\}\}$ five hundred thousand. See Plate LXXVI. 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 $\backslash epsilon$, called episemon, was inserted among the units, immediately after $\backslash iota$, the letter denoting five; and the koppa and sampi, represented by $\backslash zeta$, $\backslash eta$, or $\backslash theta$ ter-

History. 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 required for obtaining academical degrees, and in the legal age of majority.

Numerical fancies of the Chinese. 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 Yu. 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 LXXVI. 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 Hs-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.

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 myria, 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 Ψαμμης, 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 Mv, or the contraction for myria, ten thousand. Thus, to express thirty-four trillions, they wrote $\backslash lambda\backslash delta$ Mv. Mv. Mv. To signify units separately, it was customary with them to prefix the mark M^o, 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

History. 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 $\backslash mu$ were to be multiplied into $\backslash epsilon$, or forty into two hundred: Take the lower corresponding characters $\backslash delta$ and $\backslash beta$, or four and two, which were called $\backslash pi$ and $\backslash zeta$, or radicals, the one depressed ten times, and the other an hundred times, and multiply their product $\backslash pi$, or eight, successively by the ten and the hundred, or at once by a thousand, and the result is $\backslash eta$ 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.

In the first range, $\backslash phi$ multiplied into $\backslash omega$, being the same as the product of eight and five augmented ten thousand times, is consequently denoted by $\backslash mu$ or $\backslash mu$; $\backslash phi$ multiplied into $\backslash zeta$ gives the same result as five times six increased a thousand fold, and therefore expressed by $\backslash gamma$ or $\backslash gamma$; and $\backslash phi$ multiplied into $\backslash beta$ evidently makes a thousand, or $\backslash alpha$. In the second range, $\backslash pi$ multiplied into $\backslash omega$ gives the same product as eight repeated twice and then augmented a thousand times, or denoted by $\backslash alpha\; \backslash pi$; $\backslash pi$ multiplied into $\backslash zeta$ is equivalent to six repeated twice, and afterwards increased an hundred fold or, expressed by $\backslash alpha\; \backslash zeta$; and $\backslash pi$ multiplied by $\backslash beta$ gives forty, the value of $\backslash mu$. In the third range, $\backslash gamma$ multiplied into $\backslash omega$ produces twenty-four hundred, which is denoted by $\backslash beta\; \backslash nu$; $\backslash gamma$ multiplied into $\backslash zeta$ makes an hundred and eighty, or $\backslash epsilon\; \backslash pi$; and lastly, $\backslash gamma$ multiplied into $\backslash beta$ gives $\backslash varsigma$, 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 $\backslash mu\; \backslash epsilon\; \backslash omega\; \backslash pi\; \backslash varsigma$, 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 $\backslash delta$ signified one fourth, and $\backslash pi\; \backslash delta$ one twenty-fifth; but one half being of most frequent recurrence, was signified by a particular character, varying in its form, $C$, $\backslash angle$, $C\text{'}$, or $K$. 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, $\backslash beta^*$ intimated two-elevenths, and $\backslash pi\; \backslash alpha^\{**\}$ 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.

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 $\backslash beta^*$ and $\backslash varsigma^*$, while the product of the fraction itself gives eighty-one of an hundred and twenty-one parts, expressed by $\backslash pi\; \backslash alpha^\{**\}$.

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 calculated by 3 x

History. lation 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, 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:

Here in the first line, $\backslash lambda\backslash zeta$ multiplied into $\backslash lambda\backslash zeta$ in the place of units, gives $\backslash alpha\backslash tau\backslash zeta\backslash delta$ or thirteen hundred and sixty-nine degrees; $\backslash lambda\backslash zeta$ into $\backslash delta$ on the next bar, gives $\backslash epsilon\backslash mu\backslash eta$, or one hundred and forty-eight minutes; and $\backslash lambda\backslash zeta$ into $\backslash nu\backslash epsilon$ on the lowest bar gives $\backslash beta\backslash lambda\backslash epsilon$, or two thousand and thirty-five seconds. In the second line $\backslash delta$ multiplied into $\backslash lambda\backslash zeta$ gives the product $\backslash epsilon\backslash mu\backslash eta$ as before; $\backslash delta$ multiplied into $\backslash delta$, both of them on the bar of minutes, gives $\backslash iota$ or sixteen seconds; $\backslash delta$ into $\backslash nu\backslash epsilon$ gives $\backslash sigma\backslash kappa$, or two hundred and twenty thirds. Lastly, in the third line the $\backslash nu\backslash epsilon$ on the bar of seconds, multiplied successively into $\backslash lambda\backslash zeta$, and $\backslash delta$, produce, as before, $\backslash beta\backslash lambda\backslash epsilon$ and $\backslash sigma\backslash kappa$ on the bars of seconds and thirds; and $\backslash nu\backslash epsilon$ multiplied by itself gives $\backslash gamma\backslash kappa\backslash epsilon$, or three thousand and twenty-five fourths. These several products being reduced and collected together, formed the total amount of $\backslash alpha\backslash tau\backslash delta\; \backslash delta\; \backslash iota\backslash delta\; \backslash iota\; \backslash kappa\backslash epsilon$, 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 engrafted on the notation of the Greeks. The astronomers of Alexandria and Constanti-

nople 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 re-unite 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 Λογισμός, 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 indispensable 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, stooped 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, 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 compiling his 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 mould-

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 $\backslash circ$ to occupy the accidental blanks which occurred in the notation of sexagesimals. This letter was perhaps chosen by him, because immediately succeeding to $\backslash nu$, 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 classed 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 episemon, should denote the nine digits. The iota, 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 $\backslash circ$ 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 stopped 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 Siglae; 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 Golaila 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 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

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 Lilawati, 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 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 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 feeble genius of the Hindoos might just reach that desirable point, without diverging into an excessive 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 LXXVI. (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 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, wirey 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 em-

ployed 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 Persian, 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 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 Ινδική, or Υπηρεσία κατ' Ινδία, ή λογιστική μαγική, 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 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 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 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 Algorismus. 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 redoubtable acts of magic. But the writings of Bacon really discovered 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.¹

About the same period John of Halifax, named, in the quaint Latinity then used, Sacro-Bosco, who had likewise travelled, wrote his treatise De Sphæra, 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 Mabilon, 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 Archæologia, 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 $\backslash cup$ with the addition of corresponding dots—$\backslash cup$ . $\backslash cup$ : $\backslash cup$ : : . 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 Mabilon, as exhibited in Plate LXXVI. The Saxon $\backslash times$, signifying ten, is repeatedly combined with the ordinary figures; and $\backslash times\; \backslash times\; \backslash times$, $\backslash times\; \backslash times\; \backslash times\; \backslash times$, 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 episimon 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 Algorismo, and bearing the date 1390, which explained with great brevity the digital notation

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 salpetre 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 vacuity, 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 augrym stones. In Shakespeare'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 LXXVI. To these we have subjoined fac-similes 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 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 LXXVI. are copied from a manuscript his-

tory of the Scottish Bishoprics, apparently written about the year 1550, and now in the possession of Thomas Thomson, Esq. advocate.¹ (b.)

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 an 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 an 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.

¹ 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 significance 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.

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 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.¹

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.

346863
876734
123467
314213
712316
438987
279654
3092234
24433

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

¹ To abbreviate, it is sometimes convenient to indicate the four operations of arithmetic by the signs +, —, ×, ÷. The first, +, denotes addition; the second, —, indicates subtraction; the third, ×, multiplication; and the fourth, ÷, division. The sign = put between two quantities indicates that they are equal. Accordingly we may write $2\; +\; 3\; =\; 5$; $7\; -\; 3\; =\; 4$; $5\; \backslash times\; 3\; =\; 15$; $14\; \backslash div\; 2\; =\; 7$.

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. II. Avoirdupois Weight.

Scots money is divided in the same manner as sterling, and has one twelfth of its value. A pound Scots is equal to 1s. 8d. sterling, a shilling Scots to a penny sterling, and a penny Scots to a twelfth part of a penny sterling; a merk Scots is two thirds of a pound Scots, or 13$\backslash frac\{1\}\{3\}$d. sterling.

III. Troy Weight. IV. Apothecaries' Weight.

V. English Dry Measure. VI. Scots Dry Measure.

VII. English Land Measure. VIII. Scots Land Measure.

IX. Long Measure.

X. Time.

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 43, 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.

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.

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.

Examples.

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.

Note 5. As custom has established the method of placing the subtrahend under the minuend, we follow it when there is no reason for doing otherwise; the minuend may be placed under the subtrahend with equal propriety; and the learner should be able to work it either way with equal readiness, as this last is sometimes more convenient, of which instances will occur afterwards.

Note 6. The learner should also acquire the habit, when two numbers are marked down, of placing such a number under the lesser, that, when added together, the sum may be equal to the greater. The operation is the same as subtraction, though conceived in a different manner, and is useful in balancing accounts and on other occasions.

CHAP. IV.—MULTIPLICATION.

In multiplication two numbers are given, and it is required to find how much the first amounts to when reckoned as many times as there are units in the second. Thus, 8 multiplied by 5, or 5 times 8, is 40. The given numbers (8 and 5) are called factors, the first (8) the multiplicand, the second (5) the multiplier, and the amount (40) the product.

This operation is nothing else than addition of the same number several times repeated. If we mark 8 five times under each other, and add them, the sum is 40. But as this kind of addition is of frequent and extensive use, in order to shorten the operation, we mark down the number only once, and conceive it to be repeated as often as there are units in the multiplier.

For this purpose the learner must be thoroughly acquainted with the following multiplication table, which is composed by adding each digit twelve times.

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.

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 writ-

ing 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.

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

Multiplication. RULE.—Place the multiplier under the multiplicand, and multiply the latter successively by the significant figures of the former, placing the right-hand figure of each product under the figure of the multiplier from which it arises, then add the product.

A number which cannot be produced by the multiplication of two others is called a prime number; as 3, 5, 7, 11, and many others.

A number which may be produced by the multiplication of two or more smaller ones, is called a composite number; for example, 27, which arises from the multiplication of 9 by 3; and these numbers, 9 and 3, are called the component parts of 27.

Contractions and Varieties in Multiplication.

1st. If the multiplier be a composite number, we may multiply successively by the component parts.

Because the second product is equal to five times the first, and the first is equal to nine times the multiplicand, it is obvious that the second product must be five times nine, or forty-five times as great as the multiplicand.

2dly. If the multiplier be 5, which is the half of 10, we may annex a cipher and divide by 2. If it be 25, which is the fourth part of 100, we may annex two ciphers and divide by 4. Other contractions of the like kind will readily occur to the learner.

3dly. To multiply by 9, which is one less than 10, we may annex a cipher, and subtract the multiplicand from the number it composes. To multiply by 99999, or any number of 9's, annex as many ciphers, and subtract the multiplicand. The reason is obvious; and a like rule may be found though the unit place be different from 9.

4thly. Sometimes a line of the product is more easily obtained from a former line of the same than from the multiplicand.

In the first example, instead of multiplying by 8, we may multiply 5488 by 2; and, in the second, instead of multiplying by 3, we may divide 8088 by 2.

5thly. Sometimes the product of two or more figures may be obtained at once, from the product of a figure already found.

In the second example we multiply first by 4, then, because 12 times 4 is 48, we multiply the first line of the

product by 12, instead of multiplying separately by 8 and 4; lastly, because twice 48 is 96, we multiply the second line of the product by 2, instead of multiplying separately by 6 and 9.

When we follow this method, we must be careful to place the right-hand figure of each product under the right-hand figure of that part of the multiplier which it is derived from.

It would answer equally well in all cases to begin the work at the highest place of the multiplier; and contractions are sometimes obtained by following that order.

It is a matter of indifference which of the factors be used as the multiplier; for 4 multiplied by 3 gives the same product as 3 multiplied by 4; and the like holds universally true. To illustrate this, we may make three rows of points, four in each row, placing the . . . . rows under each other; and we shall have . . . . also four rows containing three points each, . . . . if we reckon the rows downwards.

Multiplication is proved by repeating the operation, using the multiplier for the multiplicand, and the multiplicand for the multiplier. It may also be proven by division, or by casting out the 9's, of which afterwards; and an account, wrought by any contraction, may be proven by performing the operation at large, or by a different contraction.

COMPOUND MULTIPLICATION.

RULE I.—If the multiplier do not exceed 12, the operation is performed at once, beginning at the lowest place, and carrying according to the value of the place.

EXAMPLES.

RULE II.—If the multiplier be a composite number, whose component parts do not exceed 12, multiply first by one of these parts, then multiply the product by the other. Proceed in the same manner if there be more than two.

Note 1. Although the component parts will answer in any order, it is best, when it can be done, to take them in such order as may clear off some of the lower places at the first multiplication, as is done in Ex. 2d.

Note 2. The operation may be proved by taking the

Reduce component parts in a different order, or dividing the multiplier in a different manner.

RULE III.—If the multiplier be a prime number, multiply first by the composite number next lower, then by the difference, and add the products.

$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}35\backslash \; 17\backslash \; 9\; \backslash \backslash \; \backslash times\; 8\; \backslash \backslash \; \backslash hline\; \backslash text\{\pounds \}287\backslash \; 2\; \backslash end\{array\}\; =\; 8\; \backslash text\{\; times.\}$
$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}2296\backslash \; 16\; \backslash \backslash \; \backslash times\; 3\; \backslash \backslash \; \backslash hline\; 107\backslash \; 13\backslash \; 3\; \backslash end\{array\}\; =\; 3\; \backslash text\{\; times.\}$
$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}2404\backslash \; 9\backslash \; 3\; \backslash \backslash \; \backslash times\; 1\; \backslash \backslash \; \backslash hline\; \backslash end\{array\}\; =\; 67\; \backslash text\{\; times.\}$

Here, because 8 times 8 is 64, we multiply twice by 8, which gives £2296. 16s. equal to 64 times the multiplicand: then we find the amount of 3 times the multiplicand, which is £107. 13s. 3d.; and it is evident that these added amount to 67, the multiplicand.

RULE IV.—If there be a composite number a little above the multiplier, we may multiply by that number, and by the difference, and subtract the second product from the first.

$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}17\backslash \; 4\backslash \; 5\; \backslash \backslash \; \backslash times\; 12\; \backslash \backslash \; \backslash hline\; \backslash text\{\pounds \}206\backslash \; 13\; \backslash end\{array\}\; =\; 108\; \backslash text\{\; times.\}$
$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}1859\backslash \; 17\; \backslash \backslash \; \backslash times\; 34\; \backslash \backslash \; \backslash hline\; \backslash text\{\pounds \}1825\backslash \; 8\; \backslash end\{array\}\; =\; 2\; \backslash text\{\; times.\}$

Here we multiply 108 by 12 and 9, the component parts of 108, and obtain a product of £1859. 17s. equal to 108 times the multiplicand; and, as this is twice oftener than was required, we subtract the multiplicand doubled, and the remainder is the number sought.

RULE V.—If the multiplier be large, multiply by 10, and multiply the product again by 10; by which means you obtain an hundred times the given number. If the multiplier exceed 1000, multiply by 10 again; and continue it further if the multiplier require it; then multiply the given number by the unit-place of the multiplier; the first product by the ten-place, the second product by the hundred-place, and so on. Add the products thus obtained together.

$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}34\backslash \; 8\backslash \; 2\backslash frac\{1\}\{2\}\; \backslash \backslash \; \backslash times\; 5\; \backslash \backslash \; \backslash hline\; \backslash end\{array\}\; =\; 5\; \backslash text\{\; times.\}$
$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}344\backslash \; 2\backslash \; 1\; \backslash \backslash \; \backslash times\; 6\; \backslash \backslash \; \backslash hline\; \backslash end\{array\}\; =\; 60\; \backslash text\{\; times.\}$
$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}3441\; \backslash \backslash \; \backslash times\; 4\; \backslash \backslash \; \backslash hline\; \backslash end\{array\}\; =\; 400\; \backslash text\{\; times.\}$
$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}34410\backslash \; 8\backslash \; 4\; \backslash \backslash \; \backslash times\; 3\; \backslash \backslash \; \backslash hline\; \backslash end\{array\}\; =\; 3000\; \backslash text\{\; times.\}$
$\backslash begin\{array\}\{r\}\; \backslash text\{\pounds \}119232\backslash \; 1\backslash \; 10\backslash frac\{1\}\{2\}\; \backslash \backslash \; \backslash times\; 3\; \backslash \backslash \; \backslash hline\; \backslash end\{array\}\; =\; 3465\; \backslash text\{\; times.\}$

The use of multiplication is to compute the amount of any number of equal articles, either in respect of measure, weight, value, or any other consideration. The multiplicand expresses how much is to be reckoned for each article, and the multiplier expresses how many times that is to be reckoned. As the multiplier points out the number of articles to be added, it is always an abstract number, and has no reference to any value or measure whatever. It is therefore quite improper to attempt the multiplication of shillings by shillings, or to consider the multiplier as expressive of any denomination.

REDUCTION.

Reduction is the computation for changing any sum of money, weight, or measure, into a different kind. When the quantity given is expressed in different denominations, we reduce the highest to the next lower, and add thereto the given number of that denomination; and proceed in like manner till we have reduced it to the lowest denomination.

Ex. To reduce £16. 13s. 8d. to farthings.

It is easy to take in or add the higher denomination at the same time we multiply the lower.

CHAP. V.—DIVISION.

In division two numbers are given, and it is required to find how often the former contains the latter. Thus, it may be asked how often 21 contains 7, and the answer is exactly 3 times. The former given number (21) is called the dividend, the latter (7) the divisor, and the number required (3) the quotient. It frequently happens that the division cannot be completed exactly without fractions. Thus it may be asked, how often 8 is contained in 19? the answer is twice, and the remainder of 3.

This operation consists in subtracting the divisor from the dividend, and again from the remainder, as often as it can be done, and reckoning the number of subtractions; as,

As this operation, performed at large, would be very tedious when the quotient is a high number, it is proper to shorten it by every convenient method; and for this purpose we may multiply the divisor by any number whose product is not greater than the dividend, and so subtract it twice or thrice, or oftener, at the same time. The best way is to multiply it by the greatest number that does not raise the product too high, and that number is also the quotient. For example, to divide 45 by 7, we inquire what is the greatest multiplier for 7 that does not give a product above 45, and we shall find that it is 6; and 6 times 7 is 42, which, subtracted from 45, leaves a remainder of 3. Therefore 7 may be subtracted 6 times from 45; or, which is the same thing, 45 divided by 7, gives a quotient of 6 and a remainder of three.

If the divisor do not exceed 12, we readily find the highest multiplier that can be used from the multiplication table. If it exceed 12, we may try any multiplier that we think will answer. If the product be greater than the dividend, the multiplier is too great; and if the remainder, after the product is subtracted from the dividend, be greater than the divisor, the multiplier is too small. In either of these cases we must try another. But the attentive learner, after some practice, will generally hit on the right multiplier at first.

If the divisor be contained oftener than ten times in the dividend, the operation requires as many steps as

Division. there are figures in the quotient. For instance, if the quotient be greater than 100, but less than 1000, it requires 3 steps. We first inquire how many hundred times the divisor is contained in the dividend, and subtract the amount of these hundreds. Then we inquire how often it is contained ten times in the remainder, and subtract the amount of these tens. Lastly, we inquire how many single times it is contained in the remainder. The method of proceeding will appear from the following example:

It is obvious, that as often as 8 is contained in 59, so many hundred times it will be contained in 5900, or in 5936; and as often as it is contained in 33, so many ten times it will be contained in 330, or in 336; and thus the higher places of the quotient will be obtained with equal ease as the lower. The operation might be performed by subtracting 8 continually from the dividend, which will lead to the same conclusion by a very tedious process. After 700 subtractions, the remainder would be 336; after 40 more it would be 16; and after 2 more the dividend would be entirely exhausted. In practice we omit the ciphers, and proceed by the following

RULES.—1. Assume as many figures on the left hand of the multiplier as contain the divisor once or oftener; find how many times they contain it, and place the answer as the highest figure of the quotient.

2. Multiply the divisor by the figure you have found, and place the product under the part of the dividend from which it is obtained.

3. Subtract the product from the figures above it.

4. Bring down the next figure of the dividend to the remainder, and divide the number it makes up as before.

The numbers which we divide, as 59, 33, and 16, in the first example, are called dividends.

It is usual to mark a point under the figures of the dividend as they are brought down, to prevent mistakes.

If there be a remainder, the division is completed by a vulgar fraction, whose numerator is the remainder, and its denominator the divisor. Thus, in Ex. 3 the quotient is 2671, and the remainder 17; and the quotient completed is 2671 $\backslash frac\{17\}\{365\}$.

A number which divides another without a remainder is said to measure it; and the several numbers which measure

another are called its aliquot parts. Thus 2, 4, 6, 8, and 12, are aliquot parts of 24. As it is often useful to discover numbers which measure others, we may observe,

1st, Every number ending with an even figure, that is, with 2, 4, 6, 8, or 0, is measured by 2.

2dly, Every number ending with 5 or 0 is measured by 5.

3dly, Every number whose figures, when added, amount to an even number of 3's or 9's, is measured by 3 or 9, respectively.

Contractions and Varieties in Division.

1st, When the divisor does not exceed 12, the whole computation may be performed without setting down any figures except the quotient.

2dly, When the divisor is a composite number, and one of the component parts also measures the dividend, we may divide successively by the component parts.

This method might be also used although the component parts of the divisor do not measure the dividend; but the management of the remainder requires the doctrine of vulgar fractions.

3dly, When there are ciphers annexed to the divisor, cut them off, and cut off an equal number of figures from the dividend; annex those figures to the remainder.

The reason will appear by performing the operation at large, and comparing the steps.

To divide by 10, 100, 1000, or the like: Cut off as many figures on the right hand of the dividend as there are ciphers in the divisor. The figures which remain on the left hand compose the quotient, and the figures cut off compose the remainder.

Note.—Since 4 times 25 make 100, instead of dividing by 25, we may multiply the dividend by 4, and cut off the two last figures from the product. The figures left will be the quotient, and those cut off the remainder. In like manner, to divide by 125, which multiplied by 8 produces 1000, we may multiply the dividend by 8, and cut off three figures, which will be the remainder, and those left the quotient.

4thly, When the divisor consists of several figures, we may try them separately, by inquiring how often the first figure of the divisor is contained in the first figure of the dividend, and then considering whether the second and following figures of the divisor be contained as often in the corresponding ones of the dividend with the remainder (if any) prefixed. If not, we must begin again, and make trial of a lower number. When the remainder is nine or upwards, we may be sure the divisor will hold through the lower places; and it is unnecessary to continue the trial farther.

5thly, We may make a table of the products of the divisor, multiplied by the nine digits, in order to discover more readily how often it is contained in each dividend. This is convenient when the dividend is very long, or when it is required to divide frequently by the same divisor.

6thly, To divide by 9, 99, 999, or any number of 9's, transcribe under the dividend part of the same, shifting the highest figure as many places to the right hand as there are 9's in the divisor. Transcribe it again, with the like change of place, as often as the length of the dividend admits; add these together, and cut off as many figures from the right hand of the sum as there are 9's in the divisor. The figures which remain on the left hand compose the quotient, and those cut off the remainder.

If there be any carriage to the unit place of the quotient, add the number carried likewise to the remainder, as in Ex. 2; and if the figures cut off be all 9's, add 1 to the quotient, and there is no remainder.

To explain the reason of this, we must recollect, that whatever number of hundreds any dividend contains, it contains an equal number of 99's, together with an equal number of units. In Ex. 1, the dividend contains 3241 hundreds, and a remainder of 23. It therefore contains 3241 times 99, and also 3241, besides the remainder already mentioned.—Again, 3241 contains 32 hundreds and a remainder of 41. It therefore contains 32 99's, and also 32, besides the remainder of 41. Consequently the dividend contains 99, altogether, 3241 times and 32 times, that is, 3273 times, and the remainder consists of 23, 41, and 32, added, which make 96.

As multiplication supplies the place of frequent additions, and division of frequent subtractions, they are only repetitions and contractions of the simple rules; and, when compared together, their tendency is exactly opposite. As numbers, increased by addition, are diminished and brought back to their original quantity by subtraction; in like manner numbers compounded by multiplication are reduced by division to the parts from which they were compounded. The multiplier shows how many additions are necessary to produce the number; and the quotient shows how many subtractions are necessary to exhaust it. It follows that the product, divided by the multiplicand, will quote the multiplier; and because either factor may be assumed for the multiplicand, therefore the product divided by either factor quotes the other. It follows, also, that the dividend is equal to the product of the

divisor and quotient multiplied together; and hence these operations mutually prove each other.

To prove multiplication: Divide the product by either factor. If the operation be right, the quotient is the other factor, and there is no remainder.

To prove division: Multiply the divisor and quotient together; to the product add the remainder, if any; and, if the operation be right, it makes up the dividend. Otherwise divide the dividend (after subtracting the remainder, if any) by the quotient. If the operation be right, it will quote the divisor. The reason of all those rules may be collected from the last paragraph.

COMPOUND DIVISION.

RULE I.—When the dividend only consists of different denominations, divide the higher denomination, and reduce the remainder to the next lower, taking in (p. 547, Reduction) the given number of that denomination, and continue the division.

Examples.

RULE II.—When the divisor is in different denominations, reduce both divisor and dividend to the lowest denomination, and proceed as in simple division. The quotient is an abstract number.

It is best not to reduce the terms lower than is necessary to render them equal. For instance, if each of them consists of an even number of sixpences, fourpences, or the like, we reduce them to sixpences or fourpences, but not to pence.

The use of division is to find either of the factors by whose multiplication a given number is produced when the other factor is given, and therefore is of two kinds, since either the multiplier or the multiplicand may be

Division. given. If the former be given, it discovers what that number is which is contained so many times in another. If the latter be given, it discovers how many times one number is contained in another. Thus, it answers the questions of an opposite kind to those mentioned under Rule IV. p. 547, as, to find the quantity of a single parcel or share; to find the value, weight, or measure, of a single article; to find how much work is done, provisions consumed, interest incurred, or the like, in a single day, &c.

The last use of division is a kind of reduction exactly opposite to that described under Rule V. p. 547. The manner of conducting and arranging it, when there are several denominations in the question, will appear from the following examples.

1 To reduce 15783 pence to pounds, shillings, and pence.      2. To reduce 174865 grs. to lbs. oz. and dwt. Troy.

Answer, £65. 15s. 3d.

Ans. 30 lb. 4 oz. 6 dwt. 1 gr.

In the first example we reduce 15783 pence to shillings, by dividing by 12, and obtain 1315 shillings, and a remainder of 3 pence. Then we reduce 1315 shillings to pounds by dividing by 20, and obtain 65 pounds and a remainder of 15 shillings. The divisions might have been contracted.

In the practice of arithmetic questions often occur which require both multiplication and division to resolve. This happens in reduction, when the higher denomination does not contain an exact number of the lower.

RULE FOR MIXED REDUCTION.—Reduce the given denomination by multiplication to some lower one which is an aliquot part of both; then reduce that by division to the denomination required.

Ex.—Reduce £31742 to guineas.

Note 1. Guineas may be reduced to pounds by adding one twentieth part of the number.

2. Pounds may be reduced to merks by adding one half.

3. Merks may be reduced to pounds by subtracting one third.

Another case which requires both multiplication and division is, when the value, weight, measure, or duration of any quantity is given, and the value, &c. of a different quantity required, we first find the value, &c. of a single article by division, and then the value, &c. of the quantity required, by multiplication.

Ex. If 3 yards cost 15s. 9d. what will 7 yards cost at the same rate?

£1 16 9 price of 7 yards (by par. ult. p. 549, col. 2.)

Many other instances might be adduced where the operation and the reasons of it are equally obvious. These are generally, though unnecessarily, referred to the rule of proportion.

We shall now offer a general observation on all the operations in arithmetic. When a computation requires several steps, we obtain a just answer, whatever order we follow. Some arrangements may be preferable to others in point of ease, but all of them lead to the same conclusion. In addition or subtraction we may take the articles in any order, as is evident from the idea of number; or we may collect them into several sums, and add or subtract these either separately or together. When both the simple operations are required to be repeated, we may either complete one of them first, or may introduce them promiscuously; and the compound operations admit of the same variety. When several numbers are to be multiplied together, we may take the factors in any order, or we may arrange them into several classes, find the product of each class, and then multiply the products together. When a number is to be divided by several others we may take the divisors in any order, or we may multiply them into each other, and divide by the product; or we may multiply them into several parcels, and divide by the products successively. Lastly, When multiplication and division are both required, we may begin with either; and when both are repeatedly necessary, we may collect the multipliers into one product and the divisors into another, or we may collect them into parcels, or use them singly, and that in any order. Still we shall obtain the proper answer if none of the terms be neglected.

When both multiplication and division are necessary to obtain the answer of a question, it is generally best to begin with the multiplication, as this order keeps the account as clear as possible from fractions. The example last given may be wrought accordingly as follows:

Some accountants prove the operations of arithmetic by a method which they call casting out the 9's, depending on the following principles:

1st. If several numbers be divided by any divisor (the remainders being always added to the next number), the sum of the quotients, and the last remainder, will be the same as those obtained when the sum of the number is divided by the same divisor. Thus, 19, 15, and 23, contain together as many 5's, as many 7's, &c. as their sum 57 does, and the remainders are the same; and, in this way, addition may be proven by division. It is from the correspondence of the remainders that the proof by casting out the 9's is deduced.

2dly. If any figure with ciphers annexed be divided by 9, the quotient consists entirely of that figure; and the remainder is also the same. Thus, 40 divided by 9 quotes 4, remainder 4; and 400 divided by 9 quotes 44, remainder 4. The same holds with all the digits, and the reason will be easily understood: every digit, with a ci-

phier annexed, contains exactly so many tens; it must therefore contain an equal number of 9's, besides a remainder of an equal number of units.

3dly. If any number be divided by 9, the remainder is equal to the sum of the figures of the number, or to the remainder obtained, when that sum is divided by 9. For instance, 3765 divided by 9 leaves a remainder of 3; and the sum of 3, 7, 6, and 5, is 21, which divided by 9 leaves a remainder of 3. The reason of this will appear from the following illustration:

Again: 21 divided by 9 quotes 2, remainder 3; therefore, 3765 divided by 9 quotes 418, remainder 3; for the reason given. Hence we may collect the following rules for practice:

To cast the 9's out of any number, or to find what remainder will be left when any number is divided by 9: add the figures; and when the sum exceeds 9, add the figures which would express it. Pass by the 9's; and when the sum comes exactly to 9, neglect it, and begin anew. For example, if it be required to cast the 9's out of 3573294, we reckon thus: 3 and 5 are 8, and 7 is 15; 1 and 5 are 6, and 3 is 9, which we neglect; 2 and (passing by 9) 4 are 6; which is the remainder or RESULT. If the article out of which the 9's are to be cast contains more denominations than one, we cast the 9's out of the higher, and multiply the result by the value of the lower, and carry on the product (casting out the 9's, if necessary) to the lower.

To prove addition, cast the 9's out of the several articles, carrying the results to the following articles; cast them also out of the sum. If the operation be right, the results will agree.

To prove subtraction, cast the 9's out of the minuend; cast them also out of the subtrahend and remainder together; and if you obtain the same result, the operation is presumed right.

To prove multiplication, cast the 9's out of the multiplicand, and also out of the multiplier if above 9. Multiply the results together, and cast the 9's, if necessary, out of their product. Then cast the 9's out of the product, and observe if this result correspond with the former.

Ex. 1st. 9276 res. 6 $\backslash times$ 8 = 48 res. 3.

The reason of this will be evident, if we consider multiplication under the view of repeated addition. In the first example it is obviously the same. In the second, we may suppose the multiplicand repeated 48 times. If this be done, and the 9's cast out, the result, at the end of the 9th line, will be 0; for any number, repeated 9 times, and divided by 9, leaves no remainder. The same must happen at the end of the 18th, 27th, 36th, and 45th lines; and the last result will be the same as if the multiplicand had only been repeated 3 times. This is the reason for casting out the 9's from the multiplier as well as the multiplicand.

To prove division, cast the 9's out of the divisor, and

also out of the quotient; multiply the results, and cast the 9's out of the product. If there be any remainder, add to it the result, casting out the 9's if necessary. If the account be right, the last result will agree with that obtained from the dividend.

Ex. 42 2490 (59 res. 5 $\backslash times$ 6 = 30 res. 3.

And the result of the dividend is 6.

This depends on the same reason as the last; for the dividend is equal to the product of the divisor, and quotient added to the remainder.

We cannot recommend this method, as it lies under the following disadvantages.

1st. If an error of 9, or any of its multiples, be committed, the results will nevertheless agree; and so the error will remain undiscovered. And this will always be the case when a figure is placed or reckoned in a wrong column, which is one of the most frequent causes of error.

2dly. When it appears by the disagreement of the results that an error has been committed, the particular figure or figures in which the error lies are not pointed out, and consequently it is not easily corrected.

CHAP. VI.—RULE OF PROPORTION.

SECT. I.—SIMPLE PROPORTION.

Quantities are reckoned proportional to each other when they are connected in such a manner, that if one of them be increased or diminished, the other increases or diminishes at the same time, and the degree of the alteration on each is a like part of its original measure. Thus, four numbers are in the same proportion, the first to the second as the third to the fourth, when the first contains the second, or any part of it, as often as the third contains the fourth, or the like part of it. In either of these cases, the quotient of the first divided by the second is equal to that of the third divided by the fourth; and this quotient may be called the measure of the proportion.

Proportionals are marked down in the following manner:

The rule of Proportion directs us, when three numbers are given, how to find a fourth, to which the third may have the same proportion that the first has to the second. It is sometimes called the Rule of Three, from the three numbers given; and sometimes the Golden Rule, from its various and extensive utility.

RULE. Multiply the second and third terms together, and divide the product by the first.

Ex. To find a fourth proportional to 18, 27, and 34.

To explain the reason of this, we must observe, that if

The products 612, 918, and the quotients 34, 51, have therefore the same proportion to each other that 18 has to 27. In the course of this operation, the products of the first and third terms are divided by the first; therefore the quotient is equal to the third.

The first and second terms must always be of the same kind; that is, either both moneys, weights, measures, both abstract numbers, or the like. The fourth, or number sought, is of the same kind as the third.

When any of the terms is in more denominations than one, we may reduce them all to the lowest. But this is not always necessary. The first and second should not be reduced lower than directed, p. 549, col. 2, par. penult.; and when either the second or third is a simple number, the other, though in different denominations, may be multiplied without reduction.

The accountant must consider the nature of every question, and observe the circumstance which the proportion depends on; and common sense will direct him to this if the terms of the question be understood. It is evident that the value, weight, and measure of any commodity is proportioned to its quantity; that the amount of work or consumption is proportioned to the time; that gain, loss, or interest, when the rate and time are fixed, is proportioned to the capital sum from which it arises; and that the effect produced by any cause is proportioned to the extent of the cause. In these and many other cases the proportion is direct, and the number sought increases or diminishes along with the term from which it is derived.

In some questions, the number sought becomes less when the circumstances from which it is derived become greater. Thus, when the price of goods increases, the quantity which may be bought for a given sum is smaller; when the number of men employed at work is increased, the time in which they may complete it becomes shorter; and when the activity of any cause is increased, the quantity necessary to produce a given effect is diminished. In these and the like the proportion is said to be inverse.

GENERAL RULE for stating all questions, whether direct or inverse. Place that number for the third term which signifies the same kind of thing with what is sought, and consider whether the number sought will be greater or less. If greater, place the least of the other terms for the first; but if less, place the greatest for the first.

Ex. 1st. If 30 horses plough 12 acres, how many will 42 plough in the same time?

Here, because the thing sought is a number of acres, the place 12, the given number of acres, for the third term; and because 42 horses will plough more than 30, we make the lesser number, 30, the first term, and the greater number, 42, the second term.

Ex. 2d. If 40 horses be maintained for a certain sum on hay, at 5d. per stone, how many will be maintained on the same sum when the price of hay rises to 8d.

Here, because a number of horses is sought, we make the given number of horses, 40, the third term; and because fewer will be maintained for the same money when the price of hay is dearer, we make the greater price, 8d. the first term, and the lesser price, 5d. the second term.

The first of these examples is direct, the second inverse. Every question consists of a supposition and demand. In the first, the supposition is, that 30 horses plough 12 acres, and the demand, how many 42 will plough; and the first term of the proportion, 30, is found in the supposition, in this and every other direct question. In the second, the supposition is, that 40 horses are maintained on hay at 5d., and the demand, how many will be maintained on hay at 8d.? and the first term of the proportion, 8, is found in the demand, in this and every other inverse question.

When a proportion is stated, if the first and second terms, or first and third, be measured by the same number, we may divide them by that measure, and use the quotients in their stead.

SECT. II.—COMPOUND PROPORTION.

Sometimes the proportion depends upon several circumstances. Thus, it may be asked, if 18 men consume 6 bolts of corn in 28 days, how much will 24 men consume in 56 days? Here the quantity required depends partly on the number of men, partly on the time; and the question may be resolved into the two following ones:

1st. If 18 men consume 6 bolts in a certain time, how many will 24 men consume in the same time?

18) 144(8
2d. If a certain number of men consume 8 bolts in 28 days, how many will they consume in 56 days?

In the course of this operation, the original number of bolts, 6, is first multiplied into 24, then divided by 18, then multiplied into 56, then divided by 28. It would answer the same purpose to collect the multipliers into one product and the divisors into another, and then to multiply the given number of bolts by the former, and divide the product by the latter, p. 550, col. 2.

The above question may therefore be stated and wrought as follows:

In general, state the several particulars on which the question depends, as so many simple proportions, attending to the sense of the question to discover whether the proportions should be stated directly or inversely; then multiply all the terms in the first rank together, and all

those in the second rank together, and work with the product as directed in the simple rule (Sect. i. p. 551.)

Ex. If 100 men make 3 miles of road in 27 days, in how many days will 150 men make 5 miles?

Men 150 : 100 :: 27 days. Here the first stating is inverse, because more men will do it in fewer days; but the second is direct, because more miles will require more days.

The following contraction is often useful. After stating the proportion, if the same number occurs in both ranks, dash it out from both; or, if any term in the first rank and another in the second rank are measured by the same numbers, dash out the original terms, and use the quotients in their stead.

Ex. If 18 men consume £30 value of corn in 9 months when the price is 16s. per boll, how many will consume £54 value in 6 months when the price is 12s. per boll? In this question the proportion depends upon three particulars—the value of corn, the time, and price; the first of which is direct, because the greater the value of provisions is, the more time is required to consume them; but the second and third are inverse, for the greater the time and price are, fewer men will consume an equal value.

Value 30 : 54 :: 18 men.
Months 6 : 9. Here we observe 6 in the first rank measures 54 in the second; so we dash them out, and place the quotient 9 in the second rank. Next, because 30 and 9 are both measured by 3, we dash them out, and place down the quotients 10 and 3; then, because 12 and 16 are both measured by 4, we dash them out and place down the quotients 3 and 4. Lastly, because there is now 3 in both columns, we dash them out, and work with the remaining terms, according to the rule.

The moneys, weights, and measures of different countries may be reduced from the proportion which they bear to each other.

Ex. If 112 lb. avoirdupois make 104 lb. of Holland, and 100 lb. of Holland make 89 of Geneva, and 110 of Geneva make 117 of Seville, how many lbs. of Seville will make 100 lb. avoirdupois?

If it be required how many lb. avoirdupois will make 100 of Seville, then the terms must be placed in the different columns thus,—

SECT. III.—DISTRIBUTIVE PROPORTION.

If it be required to divide a number into parts which have the same proportion to each other that several other given numbers have, we add these numbers together, and state the following proportion: As the sum is to the particular numbers, so is the number required to be divided to the several parts sought.

Ex. 1. Four partners engage to trade in company: A's stock is £150, B's £320, C's £350, D's £500, and they gain £730: Required how much belongs to each if the gain be divided among them in proportion to their stocks?

Whole stock £1320 Proof £730

This account is proved by adding the gains of the partners, the sum of which will be equal to the whole gain if the operation be right; but if there be remainders, they must be added, their sum divided by the common divisor, and the quotient carried to the lowest place.

Ex. 2. A bankrupt owes A £146, B £170, C £45, D £480, and E £72; his whole effects are only £342. 7s. 6d. How much should each have?

This might also be calculated by finding what composition the bankrupt was able to pay per pound, which is obtained by dividing the amount of his effects by the amount of his debts, and comes to 7s. 6d., and then finding by the rules of practice how much each debt came to at that rate.

CHAP. VII.—RULES FOR PRACTICE.

The operations explained in the foregoing chapters comprehend the whole system of arithmetic, and are sufficient for every computation. In many cases, however, the work may be contracted by adverting to the particular circumstances of the question. We shall explain in this chapter the most useful methods which practice has suggested for rendering mercantile computations easy, in which the four elementary rules of arithmetic are sometimes jointly, sometimes separately employed.

SECT. I.—COMPUTATION OF PRICES.

The value of any number of articles, at a pound, a shilling, or a penny, is an equal number of pounds, shillings, or pence; and these two last are easily reduced to pounds. The value at any other rate may be calculated by easy methods, depending on some contraction already explained, or on one or more of the following principles.

1^d. If the rate be an aliquot part of a pound, a shilling, or a penny, then an exact number of articles may be bought for a pound, a shilling, or a penny; and the value is found by dividing the given number accordingly. Thus, to find the price of so many yards at 2s. 6d., which is the eighth part of a pound, we divide the quantity by eight, because every eight yards cost £1.

2^d. If the rate be equal to the sum of two other rates which are easily calculated, the value may be found by computing these separately, and adding the sums obtained. Thus, the price of so many yards at 9d. is found by adding their prices at 6d. and 3d. together.

3^d. If the rate be equal to the difference of two easy rates, they may be calculated separately, and the lesser subtracted from the greater. Thus, the value of so many articles at 11d. is found by subtracting their value at a penny from their value at a shilling. We may suppose that a shilling was paid for each article, and then a penny returned on each.

4^th. If the rate be a composite number, the value may be found by calculating what it comes to at one of the component parts, and multiplying the same by the other.

Case 1. When the rate is an aliquot part of a pound, divide the quantity by the number which may be bought for a pound.

Ex. 1st.] What is the value of 7463 yards, at 4s.?

2d.] What is the value of 1773 yards, at 3d.?

In the first example we divide by 5 because 4s. is $\backslash frac\{1\}\{5\}$ of a pound; the quotient 1492 shows how many pounds they amount to; besides which there remain three yards at 4s., and these come to 12s. In the second example we divide by 80, as directed, and the quotient gives £22, and the remainder 13 yards, which at 3d. come to 3s. 3d.

This method can only be used in calculating for the particular prices specified in the table. The following six cases comprehend all possible rates, and will therefore exhibit different methods of solving the foregoing questions.

CASE II. When the rate consists of shillings only, multiply the quantity by the number of shillings, and divide the product by 20: Or, if the number of shillings be even, multiply by half the number, and divide the product by 10.

It is easy to perceive that the method in which the second example is wrought must give the same answer as if the quantity had been multiplied by 14 and divided by 20; and as the division by 10 doubles the last figure for shillings, and continues all the rest unchanged for pounds, we may obtain the answer at once, by doubling the right-hand figure of the product before we set it down.

If the rate be the sum of two or more aliquot parts of a pound, we may calculate these as directed in Case I. and add them. If it be any odd number of shillings, we may calculate for the even number next lower, and add thereto the value of a shilling. If it be 19s. we may subtract the value at a shilling from the value at a pound.

Method 1. If the rate be an aliquot part of a shilling, divide the quantity accordingly, which gives the answer in shillings; if not, it may be divided into two or more aliquot parts: calculate these separately, and add the values; reduce the answer to pounds.

Here, because 4d. is one third of a shilling, we divide by 3, which gives the price in shillings, and reduce these by division to pounds.

It is sometimes easier to calculate at two rates whose difference is the rate required, and subtract the lesser value from the greater. Thus, the last example may be wrought by subtracting the value at a penny from the value at a shilling. The remainder must be the value at 11d.

Method 2. Multiply the quantity by the number of pence, the product is the answer in pence. Reduce it to pounds.

Method 3. Find the value at a penny by division, and multiply the same by the number of pence.

CASE IV. When the rate consists of farthings only, find the value in pence, and reduce it by division to pounds.

Here we suppose first 6d. and then 3d. to be paid for each article; half the quantity is the number of shillings they would cost at 6d. each, half of that is the cost at 3d., and these added and reduced give the answer.

Here we calculate what the articles would cost at 6d., at 3d., and at 2d., and add the values.

10d. may be wrought as the difference of 1s. and 2d.; and several other rates in like manner.

Method 2. Multiply the quantity by the number of pence, the product is the answer in pence. Reduce it to pounds.

Method 3. Find the value at a penny by division, and multiply the same by the number of pence.

CASE IV. When the rate consists of farthings only, find the value in pence, and reduce it by division to pounds.

Practice. It is sometimes best to join some of the pence with the farthings in the calculation. Thus, in Ex. 4, we reckon the value at 6d. and at 3 halfpence, which makes 7½d.

If the rate be 1½s., which is an eighth part of a shilling, the value is found in shillings by dividing the quantity by 8.

CASE VI. When the rate consists of shillings and lower denominations.

Method 1. Multiply the quantity by the shillings, and find the value of the pence and farthings, if any, from the proportion they bear to the shillings. Add and reduce.

Ex. 1st, 4258 at 17s. 3d.

Method 2. Divide the rate into aliquot parts of a pound; calculate the values corresponding to these, as directed in Case I, and add them.

Ex. 1st, 3894 at 17s. 6d. Ex. 2d, 1765 at 9s. 2d.

Sometimes part of the value is more readily obtained from a part already found; and sometimes it is easiest to calculate at a higher rate, and subtract the value at the difference.

Ex. 3d, 63790 at 5s. 4d. Ex. 4th, 3664 at 14s. 9d.

Method 3. If the price contain a composite number of pence, we may multiply the value at a penny by the component parts.

Ex. 5628 at 2s. 11d. or 35d.

CASE VII. When the rate consists of pounds and lower denominations.

Method 1. Multiply by the pounds, and find the value of the other denominations from the proportion which they bear to the pounds.

Ex. 1st, 3592 at £3. 12s. 8d.

Ex. 2d, 543 at £2. 5s. 10½d. Practice.

Method 2. Reduce the pounds to shillings, and proceed as in Case VI.

Ex. 1st, 3592 at £3. 12s. 8d. Ex. 2d, 3683 at £2. 4s. 11d.

We have hitherto explained the various methods of computation when the quantity is a whole number and in one denomination. It remains to give the proper directions when the quantity contains a fraction, or is expressed in several denominations.

When the quantity contains a fraction, work for the integers by the preceding rules, and for the fraction take proportional parts.

When the quantity is expressed by several denominations, and the rate given for the higher, calculate the higher, consider the lower one as fractions, and work by the last rule.

When the rate is given for the lower denomination, reduce the higher denomination to the lower, and calculate accordingly.

Note 1. 7 lb. 14 lb. and 21 lb. are aliquot parts of 1 qr.; and 16 lb. is ¼ of 1 cwt.; and are therefore easily calculated.

2. If the price of a dozen be so many shillings, that of an article is as many pence; and if the price of a gross be so many shillings, that of a dozen is as many pence.

3. If the price of a ton or score be so many pounds, that of 1 cwt. or a single article, is as many shillings.

4. Though a fraction less than a farthing is of no consequence, and may be rejected, the learner must be careful lest he lose more than a farthing, by rejecting several remainders in the same calculation.

SECT. II.—DEDUCTIONS ON WEIGHTS, &c.

The full weight of any merchandise, together with that of the cask, box, or other package, in which it is contained, is called the gross weight. From this we must make proper deductions in order to discover the quantity for which price or duty should be charged, which is called the net weight.

Tare is the allowance for the weight of the package; and this should be ascertained by weighing it before the goods are packed. Sometimes, however, particularly in payment of duty, it is customary to allow so much per cwt. or so much per 100 lb. in place of tare.

Tret is an allowance of 4 lb. on 104 granted on currants and other goods on which there is waste, in order that the weight may answer when the goods are retailed.

Cloff or Draught is a further allowance granted on some goods in London of 2 lb. on every 3 cwt. to turn the scale in favour of the purchaser. The method of calculating these, and the like, will appear from the following examples.

Commis-
sion, &c. Ex. 1st, What is the nett weight of 17 cwt. 2 q. 14 lb.,
tare 18 lb. per cwt.

In the first method we add the tare at 16 lb., which is $\backslash frac\{1\}\{4\}$ of the gross weight, to the tare at 2 lb., which is $\backslash frac\{1\}\{8\}$ of the former. In the second we multiply the gross weight by 18; the tare is 1 lb. for each cwt. of the product, and is reduced by division to higher denominations.

Ex. 2d, What is the tret of 158 cwt. 3 q. 4 lb.

3d, What is the cloff on 28 cwt. 2 q.?

3) 57 (19 lb.)

This allowance being 2 lb. on every 3 cwt., might be found by taking $\backslash frac\{1\}\{3\}$ of the number of cwts. and multiplying it by 2. It is better to begin with multiplication, for the reason given, p. 550, col. 2, par. 2.

SECT. III.—COMMISSION, &c.

It is frequently required to calculate allowances on sums of money, at the rate of so many per £100. Of this kind is COMMISSION, or the allowance due to a factor for buying or selling goods, or transacting any other business; PREMIUM OF INSURANCE, or allowance given for engaging to repay one's losses at sea or otherwise; EXCHANGE, or the allowance necessary to be added or subtracted for reducing the money of one place to that of another; PREMIUMS ON STOCK, or the allowance given for any share of a public stock above the original value. All these, and others of a like kind, are calculated by the following

RULE.—Multiply the sum by the rate, and divide the product by 100. If the rate contain a fraction, take proportional parts.

Ex. What is the commission on £728 at 2 $\backslash frac\{1\}\{2\}$ per cent.?

When the rate is given in guineas, which is common in cases of insurance, you may add a twentieth part to the sum before you calculate; or you may calculate at an equal number of pounds, and add a twentieth part to the answer.

When the given sum is an exact number of 10 pounds, the calculation may be done without setting down any figures. Every £10 at $\backslash frac\{1\}\{2\}$ per cent. is a shilling, and at other rates in proportion. Thus, £170 at $\backslash frac\{1\}\{2\}$ per cent. is 17s., and at $\backslash frac\{1\}\{4\}$ per cent. 8s. 6d.

SECT. IV.—INTEREST.

Interest is the allowance given for the use of money by the borrower to the lender. This is computed at so many pounds for each hundred lent for a year, and a like proportion for a greater or a less time. The highest rate is limited by our laws to 5 per cent. which is called the legal interest, and is due on all debts constituted by bond or bill, which are not paid at the proper term; and it is always understood when no other rate is mentioned.

The interest of any sum for a year, at any rate, is found by the method explained in the last section.

The interest of any number of pounds for a year at 5 per cent. is one twentieth part, or an equal number of shillings. Thus the interest of £34675 for a year is 34675 shillings.

The interest for a day is obtained by dividing the interest for a year by the number of days in a year. Thus the interest of £34675 for a day is found by dividing 34675 shillings by 365, and comes to 95 shillings.

The interest for any number of days is obtained by multiplying the daily interest by the number of days. Thus the interest of £34675 for 17 days is 17 times 95 shillings, or 1615 shillings; and this divided by 20, in order to reduce it, comes to £80. 15s.

It would have served the same purpose, and been easier, to multiply at first by 17, the number of days; and instead of dividing separately by 365, and by 20, to divide at once by 7300, the product of 365 multiplied by 20; and this division may be facilitated by the table inserted p. 549, col. 1.

The following practical rules may be inferred from the foregoing observations.

I. To calculate interest at 5 per cent.

Multiply the principal by the number of days, and divide the product by 7300.

II. To calculate interest at any other rate.

Find what it comes to at 5 per cent. and take a proper proportion of the same for the rate required.

Ex. 1st, Interest on £34675 for 17 days, at 5 per cent.

Ex. 2d, Interest on £304. 3s. 4d. for 8 days, at 4 per cent.

In order to understand the nature of vulgar fractions, we must suppose unity (or the number 1) divided into several equal parts. One or more of these parts is called a fraction, and is represented by placing one number in a small character above a line, and another under it: For example, two fifth parts is written thus, $\backslash frac\{2\}\{5\}$. The number under the line (5) shows how many parts unity is divided into, and is called the denominator. The number above the line (2) shows how many of these parts are represented, and is called the numerator.

It follows, from the manner of representing fractions, that when the numerator is increased, the value of the fraction becomes greater; but when the denominator is increased, the value becomes less. Hence we may infer, that if the numerator and denominator be both increased, or both diminished, in the same proportion, the value is not altered; and therefore, if we multiply both by any number whatever, or divide them by any number which measures both, we shall obtain other fractions of equal value. Thus, every fraction may be expressed in a variety of forms, which have all the same signification.

A fraction annexed to an integer or whole number makes a mixed number; for example, five and two third parts, or $5\backslash frac\{2\}\{3\}$. A fraction whose numerator is greater than its denominator is called an improper fraction; for example, seventeen third parts, or $\backslash frac\{17\}\{3\}$. Fractions of this kind are greater than unity. Mixed numbers may be represented in the form of improper fractions, and improper fractions may be reduced to mixed numbers, and sometimes to integers. As fractions, whether proper or improper, may be represented in different forms, we must explain the method of reducing them from one form to another before we consider the other operations.

PROBLEM I.—To reduce mixed numbers to improper fractions.

Multiply the integer by the denominator of the fraction, and to the product add the numerator. The sum is the numerator of the improper fraction sought, and is placed above the given denominator.

Because one is equal to two halves, or 3 third parts, or 4 quarters, and every integer is equal to twice as many halves, or four times as many quarters, and so on, therefore, every integer may be expressed in the form of an improper fraction, having an assigned denominator. The numerator is obtained by multiplying the integer into the

denominator. Hence the reason of the foregoing rule is evident: 5 reduced to an improper fraction whose denominator is 3, makes $\backslash frac\{5\}\{3\}$, and this added to $\backslash frac\{2\}\{3\}$ amounts to $\backslash frac\{7\}\{3\}$. Vulgar Fractions.

PROBLEM II.—To reduce improper fractions to whole or mixed numbers,

Divide the numerator by the denominator.

This problem is the converse of the former, and the reason may be illustrated in the same manner.

PROBLEM III.—To reduce fractions to lower terms.

Divide both numerator and denominator by any number which measures both, and place the quotients in the form of a fraction.

Here we observe that 135 and 360 are both measured by 5, and the quotients form $\backslash frac\{27\}\{72\}$, which is a fraction of the same value as $\backslash frac\{155\}\{60\}$ in lower terms. Again, 27 and 72 are both measured by 9, and the quotients form $\backslash frac\{3\}\{8\}$, which is still of equal value, and in lower terms.

It is generally sufficient, in practice, to divide by such measures as are found to answer on inspection, or by the rules given p. 547, col. 2. But if it be required to reduce a fraction to the lowest possible terms, we must divide the numerator and denominator by the greatest number which measures both. What number this is may not be obvious, but will always be found by the following rule.

To find the greatest common measure of two numbers, divide the greater by the lesser and the divisor by the remainder continually till nothing remains; the last divisor is the greatest common measure.

Ex. Required the greatest number which measures 475 and 589?

Here we divide 589 by 475, and the remainder is 114; then we divide 475 by 114, and the remainder is 19; then we divide 114 by 19, and there is no remainder; from which we infer that 19, the last divisor, is the greatest common measure.

To explain the reason of this, we must observe, that any number which measures two others will also measure their sum and their difference, and will measure any multiple of either. In the foregoing example, any number which measures 589 and 475 will measure their difference, 114, and will measure 456, which is a multiple of 114; and any number which measures 475 and 456 will also measure their difference, 19. Consequently, no number greater than 19 can measure 589 and 475. Again, 19 will measure them both, for it measures 114, and therefore measures 456, which is a multiple of 114 and 475, which is just 19 more than 456; and because it measures 475 and 114, it will measure their sum, 589. To reduce $\backslash frac\{589\}\{475\}$ to the lowest possible terms, we divide both numbers by 19, and it comes to $\backslash frac\{31\}\{25\}$.

If there be no common measure greater than 1, the fraction is already in the lowest terms.

If the greatest common measure of 3 numbers be required, we find the greatest measure of the two first, and then the greatest measure of that number and the third. If there be more numbers, we proceed in the same manner.

PROBLEM IV.—To reduce fractions to others of equal value that have the same denominator.

1st, Multiply the numerator of each fraction by all the deno-

minators except its own; the products are numerators to the respective fractions sought. 2d, Multiply all the denominators into each other; the product is the common denominator.

Here we multiply 4, the numerator of the first fraction, by 9 and 8, the denominators of the two others; and the product, 288, is the numerator of the fraction sought, equivalent to the first. The other numerators are found in like manner, and the common denominator, 360, is obtained by multiplying the given denominators 5, 9, 8, into each other. In the course of the whole operation, the numerators and denominators of each fraction are multiplied by the same number, and therefore their value is not altered.

The fractions thus obtained may be reduced to lower terms, if the several numerators and denominators have a common measure greater than unity. Or, after arranging the number for multiplication, as is done above, if the same number occur in each rank, we may dash them out and neglect them; and if numbers which have a common measure occur in each, we may dash them out and use the quotients in their stead; or any number which is a multiple of all the given denominators may be used as a common denominator. Sometimes a number of this kind will occur on inspection, and the new numerators are found by multiplying the given ones by the common denominator, and dividing the products by the respective given denominators.

If the articles given for any operation be mixed numbers, they are reduced to improper fractions by Problem I. If the answer obtained be an improper fraction, it is reduced to a mixed number by Problem II. And it is convenient to reduce fractions to lower terms, when it can be done, by Problem III. which makes their value better apprehended, and facilitates any following operation. The reduction of fractions to the same denominator by Problem IV. is necessary to prepare them for addition or subtraction, but not for multiplication or division.

SECT. I.—ADDITION OF VULGAR FRACTIONS.

RULE.—Reduce them, if necessary, to a common denominator; add the numerators, and place the sum above the denominator.

The numerators of fractions that have the same denominator signify like parts; and the reason for adding them is equally obvious as that for adding shillings or any other inferior denomination.

Mixed numbers may be added by annexing the sum of the fractions to the sum of the integers. If the former be a mixed number, its integer is added to the other integers.

SECT. II.—SUBTRACTION OF VULGAR FRACTIONS.

RULE.—Reduce the fractions to a common denominator; subtract the numerator of the subtrahend from the numerator of the minuend, and place the remainder above the denominator.

To subtract a fraction from an integer, subtract the numerator from the denominator, and place the remain-

der above the denominator; prefix to this the integer diminished by unity.

To subtract mixed numbers, proceed with the fractions by the foregoing rule, and with the integers in the common method. If the numerator of the fraction in the subtrahend exceed that in the minuend, borrow the value of the denominator, and repay it by adding 1 to the unit place of the subtrahend.

Here, because 27, the numerator of the fraction in the minuend, is less than 35, the numerator of the subtrahend, we borrow 45, the denominator; 27 and 45 make 72, from which we subtract 35, and obtain 37 for the numerator of the fraction in the remainder; and we repay what was borrowed, by adding 1 to 5 in the unit place of the subtrahend.

The reason of the operations in adding or subtracting fractions will be fully understood if we place the numerators of the fractions in a column like a lower denomination, and add or subtract them as integers, carrying or borrowing according to the value of the high denomination.

SECT. III.—MULTIPLICATION OF VULGAR FRACTIONS.

RULE.—Multiply the numerators of the factors together for the numerator of the product, and the denominators together for the denominator of the product.

To multiply $\backslash frac\{2\}\{3\}$ by $\backslash frac\{3\}\{4\}$ is the same as to find what two third parts of $\backslash frac\{3\}\{4\}$ comes to. If one third part only had been required, it would have been obtained by multiplying the denominator 7 by 3, because the value of fractions is lessened when their denominators are increased; and this comes to $\backslash frac\{2\}\{7\}$; and, because two thirds were required, we must double that fraction, which is done by multiplying the numerator by 2, and comes to $\backslash frac\{4\}\{7\}$. Hence we infer that fractions of fractions, or compound fractions, such as $\backslash frac\{2\}\{3\}$ of $\backslash frac\{3\}\{4\}$, are reduced to simple ones by multiplication. The same method is followed when the compound fraction is expressed in three parts or more.

The foregoing rule extends to every case when there are fractions in either factor. For mixed numbers may be reduced to improper fractions, as is done in Ex. 2; and integers may be written, or understood to be written, in the form of fractions whose numerator is 1. It will be convenient, however, to give some further directions for proceeding when one of the factors is an integer, or when one or both are mixed numbers.

1st, To multiply an integer by a fraction, multiply it by the numerator, and divide the product by the denominator.

2d, To multiply an integer by a mixed number, we multiply it first by the integer and then by the fraction, and add the products.

Vulgar Fractions. 3d. To multiply a mixed number by a fraction, we may multiply the integer by the fraction, and the two fractions together, and add the products.

4th. When both factors are mixed numbers, we may multiply each part of the multiplicand, first by the integer of the multiplier, and then by the fraction, and add the four products.

SECT. IV.—DIVISION OF VULGAR FRACTIONS.

RULE I.—Multiply the numerator of the dividend by the denominator of the divisor. The product is the numerator of the quotient.

II.—Multiply the denominator of the dividend by the numerator of the divisor. The product is the denominator of the quotient.

To explain the reason of this operation, let us suppose it required to divide $\backslash frac\{3\}\{7\}$ by 7, or to take one seventh part of that fraction. This is obtained by multiplying the denominator by 7; for the value of fractions is diminished by increasing their denominators, and comes to $\backslash frac\{3\}\{49\}$. Again, because $\backslash frac\{3\}\{7\}$ is nine times less than seven, the quotient of any number divided by $\backslash frac\{3\}\{7\}$ will be nine times greater than the quotient of the same number divided by 7. Therefore we multiply $\backslash frac\{3\}\{7\}$ by 9, and obtain $\backslash frac\{27\}\{7\}$.

If the divisor and dividend have the same denominator, it is sufficient to divide the numerators.

Ex. $\backslash frac\{1\}\{2\}$ divided by $\backslash frac\{1\}\{3\}$ quotes 4.

The foregoing rule may be extended to every case by reducing integers and mixed numbers to the form of improper fractions. We shall add some directions for shortening the operation when integers and mixed numbers are concerned.

1st. When the dividend is an integer, multiply it by the denominator of the divisor, and divide the product by the numerator.

2d. When the divisor is an integer, and the dividend a fraction, multiply the denominator by the divisor, and place the product under the numerator.

3d. When the divisor is an integer, and the dividend a mixed number, divide the integer, and annex the fraction to the remainder; then reduce the mixed number thus formed to an improper fraction, and multiply its denominator by the divisor.

Hitherto we have considered the fractions as abstract numbers, and laid down the necessary rules accordingly. We now proceed to apply these to practice. Shillings and pence may be considered as fractions of pounds, and lower denominations of any kind as fractions of higher; and any operation, where different denominations occur, may be wrought by expressing the lower ones in one form of vulgar fractions, and proceeding by the following rules. For this purpose the two following problems are necessary.

PROBLEM V.—To reduce lower denominations to fractions of higher.

Place the given number for the numerator, and the value of the higher for the denominator.

Examples.

1. 1. Reduce 7d. to the fraction of a shilling. Ans. $\backslash frac\{7\}\{12\}$.
2. 2. Reduce 7d. to a fraction of a pound. Ans. $\backslash frac\{7\}\{240\}$.
3. 3. Reduce 15s. 7d. to a fraction of a pound. Ans. $\backslash frac\{387\}\{480\}$.

PROBLEM VI.—To value fractions of higher denominations.

Multiply the numerator by the value of the given denomination, and divide the product by the denominator; if there be a remainder, multiply it by the value of the next denomination, and continue the division.

In the first example we multiply the numerator 17 by 20, the number of shillings in a pound, and divide the product, 340, by 60, the denominator of the fraction, and obtain a quotient of 5 shillings; then we multiply the remainder, 40, by 12, the number of pence in a shilling, which produces 480, which, divided by 60, quotes 8d. without a remainder. In the second example we proceed in the same manner; but as there is a remainder, the quotient is completed by a fraction.

Sometimes the value of the fraction does not amount to an unit of the lowest denomination; but it may be reduced to a fraction of that or any other denomination by multiplying the numerator according to the value of the places. Thus $\backslash frac\{1\}\{240\}$ of a pound is equal to $\backslash frac\{1\}\{24\}$ of a shilling, or $\backslash frac\{1\}\{480\}$ of a penny, $\backslash frac\{1\}\{960\}$ of a farthing.

CHAP. IX.—DECIMAL FRACTIONS.

SECT. I.—NOTATION AND REDUCTION.

Decimal fractions are such as have 10, or some power of 10 (that is 100, 1000, &c.), for a denominator: such are these,—

They are more simply written thus:

the number of figures after the point being always the same as the number of ciphers in the denominators.

In decimal fractions, as thus written, the figure next

Decimal Fractions. the point, to the right, indicates so many tenths; the next so many hundredths, and so on. Thus, in the fraction $\backslash frac\{346\}\{1000\}$, the figure 3 expresses 3 tenths, 4 denotes 4 hundredths, and 6, 6 thousandths.

The use of ciphers in decimals, as well as in integers, is to bring the significant figures to their proper places, on which their value depends. As ciphers, when placed on the left hand of an integer, have no signification, but, when placed on the right hand, increase the value ten times each; so ciphers, when placed on the right hand of a decimal, have no signification, but when placed on the left hand, diminish the value ten times each.

The notation and numeration of decimals will be obvious from the following examples:

To reduce vulgar fractions to decimal ones. Annex a cipher to the numerator, and divide it by the denominator, annexing a cipher continually to the remainder.

The reason of this operation will be evident, if we consider that the numerator of a vulgar fraction is understood to be divided by the denominator; and this division is actually performed when it is reduced to a decimal.

In like manner, when there is a remainder left in division, we may extend the quotient to a decimal, instead of completing it by a vulgar fraction, as in the following example:

From the foregoing examples we may distinguish the several kinds of decimals. Some vulgar fractions may be reduced exactly to decimals, as Ex. 1st and 2d, and are

called terminate or finite decimals. Others cannot be exactly reduced, because the division always leaves a remainder; but, by continuing the division, we will perceive how the decimal may be extended to any length whatever. These are called infinite decimals. If the same figure continually returns, as in Ex. 3d and 4th, they are called repeaters. If two or more figures return in their order, they are called circulates. If this regular succession go on from the beginning, they are called pure repeaters, or circulates, as Ex. 3d and 5th. If otherwise, as Ex. 4th and 6th, they are mixed repeaters or circulates, and the figures prefixed to those in regular succession are called the finite part. Repeating figures are generally distinguished by a dash, and circulates by a comma or other mark, at the beginning and end of the circle; and the beginning of a repeater or circulate is pointed out in the division by an asterisk.

Lower denominations may be considered as fractions of higher ones, and reduced to decimals accordingly. We may proceed by the following rule, which is the same in effect as the former.

To reduce lower denominations to decimals of higher. Annex a cipher to the lower denomination, and divide it by the value of the higher. When there are several denominations, begin at the lowest, and reduce them in their order.

Here, in order to reduce 21 lb. to a decimal of 1 qr. we annex a cipher, and divide by 28, the value of 1 qr. This gives .75. Then we reduce 2.75 qrs. to a decimal of 1 cwt. by dividing by 4, the value of 1 cwt., and it comes to .6875. Lastly, 5.6875 cwt. is reduced to a decimal of a ton by dividing by 20, and comes to .284375.

To value a decimal fraction. Multiply it by the value of the denomination, and cut off as many decimal places from the product as there are in the multiplicand. The rest are integers of the lower denomination.

Example. What is the value of .425 of £1?

SECT. II.—ARITHMETIC OF TERMINATE DECIMALS.

The value of decimal places decreases like that of integers, ten of the lower place in either being equal to one of the next higher; and the same holds in passing from decimals to integers. Therefore, all the operations are performed in the same way with decimals, whether placed by themselves or annexed to integers, as with pure integers. The only peculiarity lies in the arrangement and pointing of the decimals.

In addition and subtraction, Arrange units under units, tenth parts under tenth parts, and proceed as in integers.

In multiplication, Allow as many decimal places in the product as there are in both factors. If the product has not so many places, supply them by prefixing ciphers on the left hand.

The reason of this rule may be explained, by observing, that the value of the product depends on the value of the factors; and since each decimal place in either factor diminishes its value ten times, it must equally diminish the value of the product.

To multiply decimals by 10, move the decimal point one place to the right; to multiply by 100, 1000, or the like, move it as many places to the right as there are ciphers in the multiplier.

In division, Point the quotient so that there may be an equal number of decimal places in the dividend as in the divisor and quotient together.

Therefore, if there be the same number of decimal places in the divisor and dividend, there will be none in the quotient.

If there be more in the dividend, the quotient will have as many as the dividend has more than the divisor.

If there be more in the divisor, we must annex (or suppose annexed) as many ciphers to the dividend as may complete the number in the divisor, and all the figures of the quotient are integers.

If the division leave a remainder, the quotient may be extended to more decimal places; but these are not regarded in fixing the decimal point.

The reason for fixing the decimal point as directed may be inferred from the rule followed in multiplication. The quotient multiplied by the divisor produces the dividend; and therefore the number of decimal places in the dividend is equal to those in the divisor and quotient together.

The first figure of the quotient is always at the same distance from the decimal point, and on the same side, as the figure of the dividend which stands above the unit place of the first product. This also takes place in integers; and the reason is the same in both.

Multiplication by fractions corresponds with division by integers, and division by fractions with multiplication by integers; when we multiply by $\backslash frac\{1\}\{2\}$ or .5, we obtain the same answer as when we divide by 2, and every integer has a correspondent decimal, which may be called its reciprocal. Multiplication by that decimal supplies the place of division by the integer, and division supplies the place of multiplication.

To find the reciprocal of any number, divide 1 with ciphers annexed by that number.

Ex. Required the reciprocal of 625.

The product of any number multiplied by .0016 is the same as the quotient divided by 625.

Because .0016 is $\backslash frac\{1\}\{625\}$ of unity, any number multiplied by that fraction will be diminished 625 times. For a like reason, the quotient of any number divided by .0016 will be equal to the product of the same multiplied by 625.

SECT. III.—APPROXIMATE DECIMALS.

It has been shown that some decimals, though extended to any length, are never complete; and others, which terminate at last, sometimes consist of so many places that it would be difficult in practice to extend them fully. In these cases, we may extend the decimal to three, four, or more places, according to the nature of the articles and the degree of accuracy required, and reject the rest of it as inconsiderable. In this manner we may perform any operation with ease by the common rules, and the answers we obtain are sufficiently exact for any purpose in business. Decimals thus restricted are called approximates.

Shillings, pence, and farthings, may be easily reduced to decimals of three places by the following rule: Take half the shillings for the first decimal place, and the number of farthings increased by one, if it amount to 24 or upwards; by two, if it amount to 48 or upwards; and by three, if it amount to 72 or upwards, for the two next places.

The reason of this is, that 20 shillings make a pound, two shillings is the tenth part of a pound, and therefore half the number of shillings makes the first decimal place. If there were 50 farthings in a shilling, or 1000 in a pound, the units of the farthings in the remainder would be thousandth parts, and the tens would be hundredth parts, and so would give the two next decimal places; but because there are only 48 farthings in a shilling, or 960 in a pound, every farthing is a little more than the thousandth part of a pound; and since 24 farthings make 25 thousandth parts, allowance is made for that excess by adding 1 for every 24 farthings, as directed.

If the number of farthings be 24, 48, or 72, and consequently the second and third decimal places 25, 50, and 75, they are exactly right; otherwise they are not quite complete, since there should be an allowance of $\backslash frac\{1\}\{24\}$, not only for 24, 48, and 72 farthings, but for every other single farthing. They may be completed by the following rule: multiply the second and third decimal places, or their excess above 25, 50, 75, by 4. If the product amount to 24 or upwards, add 1; if 48, add 2; if 72, add 3. By this operation we obtain two decimal places more; and by continuing the same operation, we may extend the decimal till it terminate in 25, 50, 75, or in a repeater.

Decimals of sterling money of three places may easily be reduced to shillings, pence, and farthings, by the following rule: Double the first decimal place, and if the second be 5 or upwards, add 1 thereto for shillings. Then divide the second and third decimal places, or their excess above 50, by 4, first deducting 1, if it amount to 25

Decimal or upwards; the quotient is pence, and the remainder farthings.

As this rule is the converse of the former one, the reason of the one may be inferred from that of the other. The value obtained by it, unless the decimal terminate in 25, 50, or 75, is a little more than the true value; for there should be a deduction, not only of 1 for 25, but a little deduction of $\backslash frac\{1\}\{2\}$ on the remaining figures of these places.

We proceed to give some examples of the arithmetic of approximates, and subjoin any necessary observations.

If we value the sum of the approximates, it will fall a little short of the sum of the articles, because the decimals are not complete.

It is proper to add 1 to the last decimal place of the approximate, when the following figure would have been 5 or upwards. Thus the full decimal of 3 qrs. 22 lb. is .946,428571, and therefore .94643 is nearer to it than .94642. Approximates thus regulated will give exact answers, sometimes above the true one and sometimes below it.

The mark + signifies that the approximate is less than the exact decimal, or requires something to be added. The mark — signifies that it is greater, or requires something to be subtracted.

MULTIPLICATION.

Here the last four places are quite uncertain. The right-hand figure of each particular product is obtained by multiplying 8 into the figures of the multiplier; but if the multiplicand had been extended, the carriage from the right-hand place would have been taken in; consequently the right-hand place of each particular product, and the four places of the total product, which depend on these, are quite uncertain. Since part of the operation therefore is useless, we may omit it; and for this purpose it will be convenient to begin (as in p. 547, col. 1, fifth variety) at the highest place of the multiplier. We may perceive that all the figures on the right hand of the line in Meth. 2 serve no purpose, and may be left out if we only multiply the figures on the multiplicand, whose products are placed on the left hand of the line. This is readily done by inverting the multiplier in Meth. 3, and beginning each product with the multiplication of that figure which stands above the figure of the multiplier that produces it, and including the carriage from the right-hand place.

If both factors be approximates, there are at least as many uncertain places in the product as in the longest factor. If only one be an approximate, there are as many uncertain places as there are figures in that factor, and sometimes a place or two more, which might be affected by the carriage. Hence we may infer how far it is necessary to extend the approximates in order to obtain the requisite number of certain places in the product.

DIVISION.

Here all the figures on the right hand of the line are uncertain, for the right-hand figure of the first product 7448 might be altered by the carriage if the divisor were extended; and all the remainders and dividends that follow are thereby rendered uncertain. We may omit these useless figures, for which purpose we dash a figure on the right hand of the divisor at each step, and neglect it when we multiply by the figure of the quotient next obtained; but we include the carriage. The operation, and the reason of it, will appear clear, by comparing the operation at large, and contracted, in the above example.

CHAP. X.—INTERMEDIATE DECIMALS.

SECT. 1.—REDUCTION OF INTERMEDIATE DECIMALS.

We have seen that some vulgar fractions admit of being converted into exact decimal fractions, while others have not that property, but proceed interminably, the numerator being either the same figure, or else a combination of figures always repeated. The fraction $\backslash frac\{3\}\{8\}$ is of the first mentioned kind, its decimal value being $\backslash frac\{375\}\{1000\}$, that is .375; again, $\backslash frac\{1\}\{2\}\; =\; .333$, &c. and $\backslash frac\{1\}\{3\}\; =\; .714285,\; 714285$, &c. are examples of the second kind. Let us suppose a fraction reduced to its lowest terms, $\backslash frac\{3\}\{8\}$, for example. Now, to convert this into a decimal, we annex ciphers to the numerator (that is, we multiply it by some power of 10), and divide by the denominator, the decimal denominator being always that power of ten by which the numerator was multiplied. In the case of $\backslash frac\{3\}\{8\}$, the numerator is multiplied by 1000, which is exactly divisible by 8; for when the numbers are expressed by the product of their simple factors, we have $1000\; =\; 2\; \backslash times\; 2\; \backslash times\; 2\; \backslash times\; 5\; \backslash times\; 5\; \backslash times\; 5$ and $8\; =\; 2\; \backslash times\; 2\; \backslash times\; 2$, and $\backslash frac\{1000\}\{8\}\; =\; 5\; \backslash times\; 5\; \backslash times\; 5\; =\; 125$, therefore the decimal value of $\backslash frac\{3\}\{8\}$ is $\backslash frac\{375\}\{1000\}$, that is, .375. Here it appears that the decimal fraction has a finite form, because the divisors of the denominator of the vulgar fraction are also divisors of a power of 10.

A like remark may be made concerning the fraction $\backslash frac\{1\}\{25\}\; =\; \backslash frac\{1\}\{5\; \backslash times\; 5\}$, the two equal divisors 5 being divisors of 10, and therefore their product a divisor of 100, and the fraction = .28.

In general, it appears that if the simple factors of the denominators of a vulgar fraction be all either 2 or 5, its decimal value will be finite, otherwise not.

The fraction $\backslash frac\{1\}\{3\}$ is therefore, from its nature, incapable of being converted into a finite decimal; but this fraction, when converted into a decimal, being .1111, &c. ad infinitum, therefore $\backslash frac\{2\}\{3\}$ will be .222, &c. and $\backslash frac\{1\}\{2\}\; =\; .333$, &c. and so on to $\backslash frac\{8\}\{9\}$, which will be .888, &c. Hence it is manifest that every repeating decimal is equivalent to a vulgar fraction whose denominator is 9.

Again, since $\backslash frac\{1\}\{9\}\; =\; .010101$, &c. ad infinitum, therefore $\backslash frac\{2\}\{9\}\; =\; .0202$, &c.; and in like manner $\backslash frac\{1\}\{7\}\; =\; .1717$, &c. and so on, from $\backslash frac\{1\}\{7\}$ to $\backslash frac\{6\}\{7\}$, which is equal to .89, 89, 89, &c. Therefore every circulating decimal of two figures is convertible into a vulgar fraction whose denominator is 99.

And since $\backslash frac\{1\}\{99\}\; =\; .001001001$, &c., therefore any fraction whose numerator consists of three figures, and deno-

Decimal numerator 999, must produce a circulate of three figures; for example, $\backslash frac\{355\}\{999\}\; =\; .355355355$, &c.; and in like manner, since $\backslash frac\{999\}\{999\}\; =\; .00010001$, &c., therefore every fraction whose numerator consists of not more than four figures, and which has 9999 for a denominator, must produce a circulate of four figures, and so on.

It is now sufficiently manifest that all circulating decimals may be generated from vulgar fractions whose denominators are formed by a repetition of the figure 9; and hence it follows that every circulate may be converted into a vulgar fraction of that form.

RULES for reducing interminate decimals to vulgar fractions.

I. If the decimal be a pure repeater, place the repeating figure for the numerator, and 9 for the denominator.

II. If the decimal be a pure circulate, place the circulating figures for the numerator, and as many 9's as there are places in the circle for the denominator.

III. If there be ciphers prefixed to the repeating or circulating figures, annex a like number to the 9's in the denominator.

IV. If the decimal be mixed, subtract the finite part from the whole decimal. The remainder is the numerator, and the denominator consists of as many 9's as there are places in the circle, together with as many ciphers as there are finite places before the circle.

Thus, $.235,62\; =\; \backslash frac\{235,62\}\{99999\}$
From the whole decimal      23562
We subtract the finite part      235

and the remainder      23327 is the numerator.

The reason may be illustrated by dividing the decimal into two parts, whereof one is finite, and the other a pure repeater or circulate, with ciphers prefixed. The sum of the vulgar fractions corresponding to these will be the value of the decimal sought.

$.235,62$ may be divided into $.235\; =\; \backslash frac\{235\}\{1000\}$ by Rule I.
and $.000,62\; =\; \backslash frac\{62\}\{99999\}$ by Rules II. III.

In order to add these vulgar fractions, we reduce them to a common denominator; and for that purpose we multiply both terms of the former by 99, which gives $\backslash frac\{23265\}\{99\}$; then we add the numerators.

The value of circulating decimals is not altered though one or more places be separated from the circle and considered as a finite part, providing the circle be completed. For example, $.27$ may be written $.2,72$, $\backslash frac\{272\}\{99\}$, or $\backslash frac\{27\}\{99\}$, which is also the value of $.27$; and if two or more circles be joined, the value of the decimal is still the same. Thus, $.2727\; =\; \backslash frac\{2727\}\{9999\}$, which is reduced by dividing the terms by 101 to $\backslash frac\{27\}\{99\}$.

All circulating decimals may be reduced to a similar form, having a like number both of finite and circulating places. For this purpose, we extend the finite part of each as far as the longest, and then extend all the circles to so many places as may be a multiple of the number of places in each.

Ex. $.34,725$, extended, $.34,725725725725$
$.1,4562$,      $.14,562456245624$

Here the finite part of both is extended to two places, and the circle to 12 places, which is the least multiple for circles of 3 and 4 places.

SECT. II.—ADDITION AND SUBTRACTION OF INTERMINATE DECIMALS.

To add repeating decimals. Extend the repeating figures one place beyond the longest finite ones, and when you add the right-hand column, carry to the next by 9.

To subtract repeating decimals. Extend them as directed for addition, and borrow at the right-hand place, if necessary, by 9.

The reason of these rules will be obvious, if we recollect that repeating figures signify ninth parts. If the right-hand figure of the sum or remainder be 0, the decimal obtained is finite; otherwise it is a repeater.

To add circulating decimals. Extend them till they become similar (p. 563, col. 1, par. ult. &c.); and when you add the right-hand column, include the figure which would have been carried if the circle had been extended farther.

Note 1. Repeaters mixed with circulates are extended and added as circulates.

Note 2. Sometimes it is necessary to inspect two or more columns for ascertaining the carriage; because the carriage from a lower column will sometimes raise the sum of the higher, so as to alter the carriage from it to a new circle. This occurs in Ex. 2.

Note 3. The sum of the circles must be considered as a similar circle. If it consist entirely of ciphers, the amount is terminate. If all the figures be the same, the amount is a repeater. If they can be divided into parts exactly alike, the amount is a circle of fewer places; but, for the most part, the circle of the sum is similar to the extended circles.

To subtract circulating decimals. Extend them till they become similar; and when you subtract the right-hand figure, consider whether 1 would have been borrowed if the circles had been extended farther, and make allowance accordingly.

SECT. III.—MULTIPLICATION OF INTERMINATE DECIMALS.

CASE I.—When the multiplier is finite and the multiplicand repeats, carry by 9 when you multiply the repeating figure; the right-hand figure of each line of the product is a repeater, and they must be extended and added accordingly.

If the sum of the right-hand column be an even number of 9's, the product is finite; otherwise it is a repeater.

CASE II.—When the multiplier is finite, and the multiplicand circulates, add to each product of the right-hand figure the carriage which would have been brought to it if the circle had been extended. Each line of the product is a circle similar to the multiplicand, and therefore they must be extended and added accordingly.

The product is commonly a circulate similar to the

Decimal multiplicand; sometimes it circulates fewer places, re-
Fractions. peats or becomes finite; it never circulates more places.

CASE III.—When the multiplier repeats or circulates, find the product as in infinite multipliers, and place under it the products which would have arisen from the repeating or circulating figures if extended.

It is evident, that if a repeating multiplier be extended to any length, the product arising from each figure will be the same as the first, and each will stand one place to the right hand of the former. In like manner, if a circulating multiplier be extended, the product arising from each circle will be alike, and will stand as many places to the right hand of the former as there are figures in the circle. In the foregoing examples there are as many of these products repeated as is necessary for finding the total product. If we place down more, or extend them farther, it will only give a continuation of the repeaters or circulates.

The multiplication of interminate decimals may be often facilitated by reducing the multiplier to a vulgar fraction, and proceeding as directed p. 558, col. 2, par 8. Thus,

Therefore, in order to multiply by $\backslash frac\{1\}\{3\}$, we take one third part of the multiplier; and, to multiply by $\backslash frac\{2\}\{3\}$, we take two thirds of the same. Thus,

As the denominator of the vulgar fractions always consists of 8's or of 9's with ciphers annexed, we may use the contraction explained p. 549, col. 1, par 1, &c.; and this will lead us exactly to the same operation which was explained p. 564, col. 1, par. 1, &c. on the principles of decimal arithmetic.

When the multiplier is a mixed repeater or circulate, we may proceed as in Ex. 4th and 7th; or we may divide the

multiplier into two parts, of which the first is finite, and the second a pure repeater or circulate, with ciphers prefixed, and multiply separately by these, and add the products.

In the following examples the multiplicand is a repeater, and therefore the multiplication by the numerator of the vulgar fraction is performed as directed p. 564, col. 1.

In the following examples the multiplicand is a circulate, and therefore the multiplication by the numerator is performed as directed, p. 563, col. 2, par ult.

In Ex. 12 we have omitted the products of the divisor, and only marked down the remainders. These are found

Decimal by adding the left-hand figure of the dividend to the remaining figures of the same. Thus, 363 is the first dividend, and 3, the left hand figure, added to 63, the remaining figures, gives 66 for the first remainder; and the second dividend, 666, is completed by annexing the circulating figure 6, the reason of which may be explained as follows. The highest place of each dividend shows, in this example, how many hundreds it contains; and as it must contain an equal number of ninety-nines, and also an equal number of units, it follows that these units, added to the lower places, must show how far the dividend exceeds that number of ninety-nines. The figure of the quotient is generally the same as the first place of the dividend, sometimes one more. This happens in the last step of the foregoing example, and is discovered when the remainder found, as here directed, would amount to 99 or upwards; and the excess above 99 only must in that case be taken to complete the next dividend.

The number of places in the circle of the product is sometimes very great, though there be few places in the factors; but it never exceeds the product of the denominator of the multiplier, multiplied by the number of places in the circle of the multiplicand. Therefore, if the multiplier be 3 or 6, the product may circulate three times as many places as the multiplicand; if the multiplier be any other repeater, nine times as many; if the multiplier be a circulate of two places, ninety-nine times as many: thus, in the last example, .01, a circulate of two places, multiplied by .01, a circulate of two places, produces a circulate of twice 99, or 198 places. And the reason of this limit may be inferred from the nature of the operation; for the greatest possible number of remainders, including 0, is equal to the divisor 99; and each remainder may afford two dividends, if both the circulating figures, 3 and 6, occur to be annexed to it. If the multiplier circulate three places, the circle of the product, for a like reason, may extend 999 times as far as that of the multiplicand. But the number of places is often much less.

CASE I.—When the dividend only is interminate, proceed as in common arithmetic; but when the figures of the dividend are exhausted, annex the repeating figure, or the circulating figures in their order, instead of ciphers, to the remainder.

In these accounts the quotient is never finite. It may repeat if the dividend repeat; or, if the dividend circulate, it may circulate an equal number of places, often more, and never fewer. The greatest possible extent of the circle is found by multiplying the divisor into the number of places in the circle of the dividend. Thus, a circulate of 3 places, divided by 3, quotes a circulate of 3 times 3 or 9 places.

CASE II.—When the divisor is interminate, the multiplications and subtractions must be performed according to the directions given for repeating and circulating decimals.

Ex. 1st, Divide .37845 by 5.

2d, Divide .245892 by 2.18

The foregoing method is the only one which properly depends on the principles of decimal arithmetic; but it is generally shorter to proceed by the following rule.

Reduce the divisor to a vulgar fraction, multiply the dividend by the denominator, and divide the product by the numerator.

Ex. 1st, Divide .37845 by $\backslash frac\{5\}\{9\}$.

2d, Divide .37845 by $\backslash frac\{2\}\{3\}$.

Note 1. Division by 3 triples the dividend, and division by 6 increases the dividend one half.

Note 2. When the divisor circulates, the denominator of the vulgar fraction consists of 9's, and the multiplication is sooner performed by the contraction explained p.546, col.1. It may be wrought in the same way when the divisor repeats, and the denominator of consequence is 9.

Note 3. If a repeating dividend be divided by a repeating or circulating divisor; or, if a circulating dividend be divided by a similar circulating divisor; or, if the number of places in the circle of the divisor be a multiple of the number in the dividend; then the product of the dividend multiplied by the denominator of the divisor will be terminate, since like figures are subtracted from like in the contracted multiplication, and consequently no remainder left.

Note 4. In other cases the original and multiplied dividend are similar, and the form of the quotient is the same as in the case of a finite divisor.

Note 5. If the terms be similar, or extended till they become so, the quotient is the same as if they were finite, and the operation may be conducted accordingly; for the quotient of vulgar fractions that have the same denominator is equal to the quotient of their numerators.

The origin of powers of involution has already been explained under the article ALGEBRA. There now remains, therefore, only to give the most expeditious methods of extracting the square and cube roots; the reasons of which will readily appear from what is said under that article. As for all powers above the cube, unless such as are multiples of either the square or cube, the extraction of their roots admits of no deviation from the algebraic canon, which must be always constructed on purpose for them.

If the root of any power not exceeding the seventh power be a single digit, it may be obtained by inspection from the following Table of powers.

RULE I.—Divide the given number into periods of two figures, beginning at the right hand in integers, and pointing toward the left. But in decimals begin at the place of hundreds, and point toward the right. Every period will give one figure in the root.

II.—Find by the table of powers, or by trial, the nearest lesser root of the left-hand period; place the figure so found in the quot; subtract its square from the said period, and to the remainder bring down the next period for a dividend or resolvend.

III.—Double the quot for the first part of the divisor; inquire how often this first part is contained in the whole resolvend, excluding the unit's place; and place the figure denoting the answer both in the quot and on the right of the first part; and you have the divisor complete.

IV.—Multiply the divisor, thus completed, by the figure put in the quot, subtract the product from the resolvend, and to the remainder bring down the following period for a new resolvend, and then proceed as before.

Note 1st, If the first part of the divisor, with unity supposed to be annexed to it, happen to be greater than the resolvend, in this case place 0 in the quot, and also on the right of the partial divisor; to the resolvend bring down another period; and proceed to divide as before.

Note 2d, If the product of the quotient figure into the divisor happen to be greater than the resolvend, you must go back and give a lesser figure to the quot.

Note 3d, If, after every period of the given number is brought down, there happen at last to be a remainder,

you may continue the operation by annexing periods, or Extra pairs of ciphers, till there be no remainder, or till the decimal part of the quot repeat or circulate, or till you think proper to limit it.

Ex. 1st, Required the square root of 133225.

2d, Required the square root of 72, to eight decimal places.

3d, Required the square root of 2916.

If the square root of a vulgar fraction be required, find the root of the given numerator for a new numerator, and find the root of the given denominator for a new denominator. Thus the square root of $\backslash frac\{3\}\{4\}$ is $\backslash frac\{1\}\{2\}$, and the root of $\backslash frac\{1\}\{2\}$ is $\backslash frac\{1\}\{2\}$; and thus the root of $\backslash frac\{3\}\{2\}$ ($=\; 6\backslash frac\{1\}\{2\}$) is $\backslash frac\{1\}\{2\}\; =\; 2\backslash frac\{1\}\{2\}$.

But if the root of either the numerator or denominator cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root, as in Ex. 3d, above.

RULE I.—Divide the given number into periods of three figures, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of thousands, and point toward the right. The number of periods shows the number of figures in the root.

II.—Find by the table of powers, or by trial, the nearest lesser root of the left-hand period; place the figure so found in the quot; subtract its cube from the said period; and to the remainder bring down the next period for a dividend or resolvend.

The divisor consists of three parts, which may be found as follows:

III. The first part of the divisor is found thus: Multiply the square of the quot by 3, and to the product annex two ciphers; then inquire how often this first part of the divisor is contained in the resolvend, and place the figure denoting the answer in the quot.

IV. Multiply the former quot by 3, and the product by the figure now put in the quot; to this last product annex a cipher, and you have the second part of the divisor. Again, square the figure now put in the quot for the third part of the divisor; place these three parts under one another as in addition, and their sum will be the divisor complete.

V. Multiply the divisor thus completed by the figure last put in the quot, subtract the product from the resolvend, and to the remainder bring down the following period for a new resolvend, and then proceed as before.

Note 1. If the first part of the divisor happen to be equal to or greater than the resolvend, in this case place 0 in the quot, annex two ciphers to the said first part of the divisor, to the resolvend bring down another period, and proceed to divide as before.

Note 2. If the product of the quotient figure into the divisor happen to be greater than the resolvend, you must go back, and give a lesser figure to the quot.

Note 3. If, after every period of the given number is brought down, there happen at last to be a remainder, you may continue the operation by annexing periods of three ciphers till there be no remainder, or till you have as many decimal places in the root as you judge necessary.

Ex. 1st, Required the cube root of 12812904.

Proof.

Ex. 2d, Required the cube root of 284.

97384 remainder.

Proof.

28.750000 cube.

If the cube root of a vulgar fraction be required, find the cube root of the given numerator for a new numerator, and the cube root of the given denominator for a new denominator. But if the root of either cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root.

ARIUS, a divine of the fourth century, the head and founder of the Arians, a sect which denied the eternal divinity and substantiality of the Word, was born in Libya. Eusebius, bishop of Nicomedia, a great favourite of Constantia, sister of the emperor Constantine, and wife of Licinius, became a zealous promoter of Arianism. He took Arius under his protection, and introduced him to Constantia; so that the sect increased, and several bishops embraced it openly. In the cities, however, disputes arose so high, that the emperor was obliged to assemble a council—that of Nice—where, in the year 325, the doctrine of Arius was condemned, he himself banished by the emperor, all his books ordered to be burnt, and capital punishment denounced against whoever dared to keep them. After five years' banishment, he was recalled to Constantinople, where he presented the emperor with a confession of his faith, drawn up so artfully, that it fully satisfied him. Notwithstanding which, Athanasius, now advanced to the see of Alexandria, refused to admit him and his followers to communion. This so enraged them, that by their interest at court they procured the deposition and banishment of that prelate. But the church of Alexandria still refusing to admit Arius into their communion, the emperor sent for him to Constantinople, where, upon delivering in a fresh confession of his faith in terms less offensive, the emperor commanded Alexander, the bishop of that church, to receive him the next day into his communion; but that very evening Arius died suddenly. The heresy, however, did not ex-

pire with the heresiarch: his party continued still in great credit at court. Athanasius, indeed, was soon recalled from banishment, but as soon removed again; for the Arians, under the countenance of government, made and deposed bishops as it best served their purposes. In short, this sect shone with great lustre above 300 years. It was the reigning religion of Spain for above two centuries; it was on the throne both in the East and West; it prevailed in Italy, France, Pannonia, and Africa; and was not extirpated till about the end of the 8th century. It was afterwards set on foot in the West by Servetus. Erasmus seems to have aimed at reviving it in his Commentaries on the New Testament; and the learned Grotius appears to favour the doctrine.

ARK, NOAH'S, a floating vessel built by Noah, for the preservation of his family, and the several species of animals, during the deluge. The ark has afforded several points of curious inquiry relating to its form, capacity, materials, &c. The wood of which it was built is called in the Hebrew gopher wood, and in the Septuagint square timbers. Some translate the original cedar, others pine, others box, &c. Pelletier prefers cedar, on account of its incorruptibility and the great plenty of it in Asia. Fuller and Bochart contend that it was built of what the Greeks call καραβας, or the cypress tree; for, taking away the termination, kypar and gopher differ very little in sound. In what place Noah built and finished his ark is no less a matter of disputation; but the most general opinion is, that it was built in Chaldea, in the territories