CHAMBERS'S INFORMATION FOR CONDUCTED BY WILLIAM AND ROBERT CHAMBERS, EDITORS OF CHAMBERS'S Is the present and succeeding sheet, an attempt is made to convey to the comparatively unlearned mind some knowledge of Mathematical science, both as regards measurement by numbers (ARITHMETIC) and measurement of dimensions (GEOMETRY.) The sketch we offer of each is necessarily brief and imperfect; but our end will be gained if we afford that amount of information on the subject which is generally possessed by persons of moderately well-cultivated intellect.-Ed. A recognition of the value of numbers is coeval with the dawn of mental cultivation in every community; but considerable progress must be made before methods of reckoning are reduced to a regular system, and a notation adopted to express large or complex quantities. An inability to reckon beyond a few numbers is always a proof of mental obscurity; and in this state various savage nations have been discovered by travellers. Some are found to be able to count as far as five, the digits of the hand most likely familiarising them with that number; but any further quantity is either said to consist of so many fives, or is expressed by the more convenient phrase, "a great many." Among the North American Indians, any great number which the mind is incapable of distinctly recognising and naming is figuratively described by comparing it to the leaves of the forest; and in the same manner, the untutored Negro of Africa would define any quantity of vast amount by pointing to a handful of sand of the desert. On the first advance of any early people towards civilisation, it would be found impossible to give a separate name to each separate number which they had occasion to describe. It would therefore be necessary to consider large numbers as only multiplications of certain smaller ones, and to name them accordingly. This is, no doubt, what gave rise to classes of numbers, which are different in different countries. For instance, the Chinese count by twos; the ancient Mexicans reckoned by fours. Some counted by fives, a number which the fingers would always be ready to suggest. The Hebrews, from an early period, reckoned by lens, which would also be an obvious mode, from the number of the fingers of the two hands, as well as of the toes of the two feet. The Greeks adopted this plan; from the Greeks it came to the Romans, and by them was spread over a large part of the world. NOTATION. The representation of numbers by written signs is an art generally believed to have taken its rise after the formation of alphabets. One of the earliest sets of written signs of numbers of which we have any notice, is certainly the series of letters of the Hebrew alphabet which was used by that people-Aleph, beth, gimel, daleth, be, vau, zain, cheth, teth, standing respectively for the numbers one, two, three, four, five, six, seven, eight, nine. The Greeks directly adopted this plan PRICE lad. from the Hebrews, forming their numbers thus :1 alpha, 2 beta, 3 gamma, 4 delta, 5 epsilon-here, having no letter corresponding with the Hebrew vau, they put in the words sinμor Bau to denote six; after which they proceeded with 7 zeta, 8 eta, &c. Before adopting this plan, they had indicated one by iota, probably because it was the smallest of their letters, five by II (P) being the first letter of pente, five; ten by A (D) being the initial of deka, ten. After having for some time adopted the Hebrew plan, they divided their alphabet into three classes; the first ten letters expressing the numbers from one to ten, while twenty, thirty, forty, and so on up to a hundred, were signified by the next nine, ninety being expressed by a figure formed on purpose, and resembling the Arabic 3 inverted. The remaining seven letters expressed 200, 300, 400, 500, 600, 700, 800; and for 900 there was another inverted figure. Larger numbers were represented by letters accented in various ways. The Romans, from an early period, had a plan of expressing numbers, which seems to have been at first independent of the alphabet. The following clear account of it was given a few years ago by Professor Playfair :--" To denote one, a simple upright stroke was assumed; and the repetition of this expressed two, three, &c. Two cross strokes X marked the next step in the scale of numeration, or ten; and that symbol was repeated to signify twenty, thirty, &c. Three strokes, or an open square, were employed to denote the hundred, or the third stage of numeration; and four interwoven strokes M, sometimes incurved M, or even divided CIƆ, expressed a thousand. Such are all the characters absolutely required in a very limited system of numeration. The necessary repetition of them, however, as often occasionally as nine times, was soon found to be tedious and perplexing. Reduced or curtailed marks were therefore employed to express the intermediate multiples of five; and this improvement must have taken place at a very early period. Thus, five itself was denoted by the upper half V, and sometimes the under half_, of the character for ten; L, or the half of, the mark for a hundred, came to represent fifty; and the incurved symbol M, or CIO, for a thousand, was split into 15, to express five hundred. These important contractions having been adopted, another convenient abbreviation was introduced. To avoid the frequent repetition of a mark, it was prefixed to the principal character, and denoted the effect by counting backwards. Thus, instead of four strokes, it seemed preferable to write IV; for eight and nine, the symbols were XII and IX; and ninety was expressed by XC. This mode of reckoning by the defect was peculiar to the Romans, and has evidently affected the composition of their numerical terms. Instead of octodecem [eight and ten-for eighteen], and novemdecem [nine and ten-for nineteen], it was held more elegant, in the Latin language, to use undeviginti [one | designation of Sylvester II., travelled into Spain, and from twenty], and duodeviginti [two from twenty]. But the alphabetic characters now lent their aid to numeration. The uniform broad strokes were dismissed, and those letters which most resembled the several combinations were adopted in their place. The marks for one, five, ten, and fifty, were respectively supplied by the letters I, V, X, and L. The symbol for a hundred was aptly denoted by C, which had originally a square shape, and happened, besides, to be the initial of the very word centum. The letter D was very generally assumed as a near approximation to the symbol for five hundred; and M not only represented the angular character for a thousand, but was likewise, though perhaps accidentally, the first letter of the word mille."-Edin. Rev. xviii. 193. studied for several years the sciences there cultivated It would be impossible to calculate, even by their own transcendent powers, the service which the Arabic numerals have rendered to mankind. NUMERATION. The Arabic numerals take the following well-known forms:-1, 2, 3, 4, 5, 6, 7, 8, 9, 0. The first nine of these, called digits or digital numbers, represent, each, one of the numbers between one aud nine, and when thus employed to represent single numbers, they are considered as units. The last (0), called a nought, nothing, or cipher, is in reality, taken by itself, expres sive of an absence of number, or nothing; but, in cornexion with other numbers, it becomes expressive of number in a very remarkable manner. The Hebrew, improved Grecian, and Roman numerals, were perhaps sufficient to express any single number with tolerable precision; but it is easy to see that they must have been nearly unfitted for use in the processes of arithmetic. The Greeks certainly contrived to overcome many obstacles in the business of calculation, and even could express fractions-though, from a practice of adding from left to right, and ignorance of the plan of carrying tens to the higher places, their problems were at all times awkward and complicated. The Romans, however, careless of old inconveniences, were still more awkwardly situated than the Greeks. Let any reader just suppose, for instance, even so simple a question as the amount of XLVIII added to XXXIV! It is evident that placing the figures below each other, as we do with the Arabic numerals, would serve little to facilitate such a calculation. In fact, the Romans were obliged, where mental calculation would not serve, to resort to a mechanical process for performing problems in arithmetic. A box of pebbles called loculus, and a board called abacus, constituted their means of calculation; and of these every schoolboy, and many other persons, possessed a set. The word calculation claims no higher descent than from calculus, a stone or pebble. The board was divided from the right to the left hand by upright columns, on which the pebbles were placed, to denote units, tens, hundreds, thousands, &c. The labour The valuable peculiarity of the Arabic notation is the of counting and arranging the pebbles was afterwards enlargement and variety of values which can be given sensibly abridged by drawing across the board a hori- to the figures by associating them. The number ten is zontal line, above which each single pebble had the expressed by the 1 and 0 put together-thus, 10; and power of five. In the progress of luxury, tali, or dies all the numbers from this up to a hundred can be exmade of ivory, were used instead of pebbles; and after-pressed in like manner by the association of two figures wards the whole system was made more convenient by substituting beads strung on parallel threads, or pegs stuck along grooves; methods of calculation still used in Russia and China, and found convenient in certain departments of Roman Catholic devotion, and in several familiar games in more civilised countries. With such instruments, problems in addition and subtraction would not be very difficult; but those in multiplication and division, not to speak of the more compound rules, must have been extremely tedious and irksome. So disagreeable, indeed, was the whole labour, that the Romans generally left it to slaves and professional calculators. thus, twenty, 20; thirty, 30; eighty-five, 85; ninetynine, 99. These are called decimal numbers, from decem, Latin for ten. The numbers between a hundred and nine hundred and ninety-nine inclusive, are in like manner expressed by three figures-thus, a hundred, 100; five hundred, 500; eight hundred and eighty-five, 885; nine hundred and ninety-nine, 999. Four figures express thousands; five, tens of thousands; six, hundreds of thousands; seven, millions; and so forth. Each figure, in short, put to the left hand of another, or of several others, multiplies that one or more numbers by ten. Or if to any set of figures a nought (0) be added towards the The numerals now in use, with the mode of causing right hand, that addition multiplies the number by ten them by peculiar situation to express any number, and thus, 999, with 0 added, becomes 9990, nine thousand whereby the processes of arithmetic have been ren- nine hundred and ninety. Thus it will be seen that, in dered so highly convenient, have heretofore been sup-notation, the rank or place of any figure in a number posed to be of Indian origin, transmitted through the is what determines the value which it bears. The figure Persians to the Arabs, and by them introduced into third from the right hand is always one of the hundreds Europe in the tenth century, when the Moors invaded that which stands seventh always expresses millions and became masters of Spain. Such in reality ap- and so on. And whenever a new figure is added towards pears to have been in a great measure the true his 1, 2 3 4, 5 6 7,8 9 0 the right, each of the former tory of the transmission of these numerals; but as it has been lately found that the ancient hieroglyphical inscriptions of Egypt contain several of them, learned men are now agreed that they originated in that early seat of knowledge, between which and India there exist more points of resemblance, and more traces of intercourse, than is generally supposed. In the eleventh century, Gerbert, a Benedictine monk of Fleury, and who afterwards ascended the papal throne under the co Hundreds. Thousands. Tens of thousands. Hundreds of thousands. Tens of millions. Hundreds of millions. Thousands of millions. Tens. set obtains, as it were, a pro- The above number is therefore one thousand two | all simple numbers as far as 12 times 12, young per hundred and thirty-four millions, five hundred and sons commit to memory the following Multiplication sixty-seven thousands, eight hundred and ninety. Table, a knowledge of which is of great value, and saves Higher numbers are expressed differently in France much trouble in after life :and England. In the former country, the tenth figure expresses billions, from which there is an advance to tens of billions, hundreds of billions, trillions, &c. In our country, the eleventh figure expresses ten thousands of millions, the next hundreds of thousands of millions, the next billions, &c. plans will be clearly apprehended from the following 1 20 24 28 32 36 9 10 11 12 20 24 30 33 36 48 2 3 4 5 6 7 8 2 4 6 8 10 12 14 | 16 18 མ ཆ་རྩལ་ སེ ཆེ ༤།ཅེ༅ ༤ ཚེ རྩ 23888 60 50 30 36 42 48 54 60 66 72 70 77 84 96 Hundreds. Thousands. Tens of thousands. Hundreds of thousands. Tens of millions. Tens of billions. Tens of trillions. SIMPLE OR ABSTRACT NUMBERS. There are four elementary departments in arithmetic --Addition, Multiplication, Subtraction, and Division. Addition. 27 536 352 275 Addition is the adding or summing up of several numbers, for the purpose of finding their united amount. We add numbers together when we say, 1 and I make 2; 2 and 2 make 4; and so on. The method of writing numbers in addition, is to place the figures under one another, so that units will stand under units, tens under tens, hundreds under hundreds, &c. Suppose we wish to add together the following numbers-27, 5, 536, 352, and 275; we range them in columns one under the other, as in the margin, and draw a line under the whole. Beginning at the lowest figure of the right-hand column, we say 5 and 2 are 7-7 and 6 are 13-13 and 5 are 18-18 and 7 are 25; that is, 2 tens and 5 units. We now write the 5 below the line of units, and carry or add the 2 tens, or 20, to the lowest figure of the next column. In carrying this 20, we let the cipher go, it being implied by the position or rank of the first figure, and take only the 2; we therefore proceed thus-2 and 7 are 9-9 and 5 are 14-14 and 3 are 17-17 and 2 are 19. Writing down the 9, we proceed with the third column, carrying 1, thus-1 and 2 are 33 and 3 are 6-6 and 5 are 11. No more figures remaining to be added, both these figures are now put down, and the amount or sum of them all is found to be 1195. Following this plan, any quantity of numbers may be summed up. Should the amount of any column be in three figures, still, only the last or right-hand figure is to be put down, and the other two carried to the next column. For example, if the amount of a column be 127, put down the 7 and carry the other two figures, which are 12; if it be 234, put down the 4 and carry 23. 16 24 32 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 This table is so well known, that it is almost superfluous to explain that, when any number in the top row is multiplied by any number in the left-hand side row, the amount is found in the compartment or square beneath the one and opposite the other. Thus, 2 times 2 are 4; 5 times 6 are 30; 12 times 12 are 144. The multiplying of numbers beyond 12 times 12 is usually effected by a process of calculation in written figures. The rule is to write down the number to be multiplied, called the multiplicand; then place under it, on the right-hand side, the number which is to be the multiplier, and draw a line under them. For example, to find the amount of 9 times 27, we set down the figures thus one place farther to the left. A line is 76843 then drawn, and the two products added 4563 together, bringing out the result of 185742. 230,529 We may, in this manner, multiply by three, 4,610,58 four, five, or any number of figures, always 38,421,5 placing the product of one figure below the other, but shifting a place farther to the left in each line. An example is here given in the multiplying of 76843 by 4563. Multiplication is denoted by a cross of this shape x : thus, 3 x 824, signifies, that by multiplying 8 by 3, the product is 24. A number which is produced by the multiplication of two other numbers, as 30 by 5 and 6, leaving nothing over, is called a composite number. The 5 and 6, called the factors (that is, workers or agents), are said to be the component parts of 30, and 30 is also said to be a multiple of either of these numbers. The equal parts into which a number can be reduced, as the twos in thirty, are called its aliquot parts. A number which cannot be produced by the multiplication of two other numbers, is called a prime number. When the multiplicand and multiplier are the same, that is, when a number is multiplied by itself once, the product is called the square of that number: 144 is the square of 12. Subtraction. 537 Subtraction is the deducting of a smaller number from a greater, to find what remains, or the difference between them. We subtract when we say, take 3 from 5, and 2 remains. To ascertain what remains, after taking 325 from 537, we proceed by writing the one under the other, as here indicated, and then subtracting. Commencing at 5, the right-hand figure of the lower and smaller number, we say, 5 from 7, and 2 remains; setting down the 2, we say next, 2 from 3, and 1 remains; and setting down the 1, we say, 3 from 5, and 2 remains; total remainder, 212. 325 212 8432 1617 To subtract a number of a higher value, involving the carrying of figures and supplying of tens, we proceed as in the margin. Commencing as before, we find that 5 cannot be subtracted from 2, and therefore supply or lend 10 to the 2, making it 12; then we say, 5 from 12, and 7 remains. Setting down the 7, we take 1, being the decimal figure of the number which was borrowed, and give it to the 1, making it 2, and taking 2 from 3, we find that 1 remains. Setting down the 1, we go to the 8, and finding it cannot be taken from the 4 above it, we lend 10 to the 4, making it 14, and then we say, 8 from 14, and 6 remains. In the same manner as before, adding the first figure of the borrowed number (1) to the 6, we say, 7 from 8, and 1 remains; thus the total remainder is found to be 1617. From these explanations, which apply to all calculations in subtraction, it will be observed, that when the upper figure is less than the figure directly under it, 10 is to be added, and for this one is carried or added to the next under figure. Subtraction is denoted by a small horizontal line, thus -between two figures; as, for example, 95 = which means, 5 subtracted from 9, and 4 remains, Division. Skilful arithmeticians never adopt this long method of division; they pursue a plan of working out part of the question in the mind, called short divi6) 7958 sion. They would, for example, treat the 1326-2 above question as here shown. The over number of 1 from the 7 is carried in the mind to the 9, making 19; the 1 from 19 is in the same manner carried to the 5; and the 3 from it is carried to the 8, leaving the overplus of 2. Division is denoted by the following character; thus, 7525, signifies that 75 is to be divided by 25. These explanations conclude the subject of simple or abstract numbers. On the substructure of the few rules in Addition, Multiplication, Subtraction, and Division, which we have given, whether in reference to whole numbers or fractions, every kind of conventiona! arithmetic is erected, because these rules are founded in immutable truths. Mankind may change their denominations of money, weights, and measures, but they can make no alteration in the doctrine of abstract numbers. That 2 and 2 are equal to 4, is a truth yesterday, to-day, and for ever; but as to how many pence are in a shilling, or how many inches in a foot, these are altogether matters of arbitrary arrangement, and the treatment of them forms an inferior department of arithme. tical study, taking a different form in different countries: this local arithmetic, as we may call it, is comprehended in the term COMPOUND NUMBERS OR QUANTITIES. 10 The calculation of the value of any number of articles, or a summation of values, in relation to money, would be comparatively simple, if the scale of money were constructed on a principle of decimals, or advancing by tens-as, for example, 10 farthings 1 penny, 4,10 pence 1 shilling, 10 shillings 1 pound. By making both weights and measures on the same plan, as, ounces 1 pound, 10 pounds 1 stone, 10 stones 1 hundredweight; 10 inches 1 foot, 10 feet 1 yard, &c., ordinary calculations would be rendered exceedingly easy. Thus, if an ounce cost 1d., a pound would cost 1s., and a hundredweight would cost 100s. or £10; or, reversing the question, if we were asked £10 per hundredweight for any article, we should know in an instant that it was at the rate of 1d. an ounce. In short, the greater number of arithmetical calculations would be accomplished by little more than a momentary reflection, without the aid of pen or pencil. Division is that process by which we discover how often one number may be contained in another, or by which we divide a given number into any proposed number of equal parts. By the aid of the Multiplication Table, we can ascertain without writing figures how many times any number is contained in another, as far as 144, or 12 times 12; beyond this point notation is employed. There are two modes of working questions in division, one long and the other short. Let 6 19 18 15 12 6 9 9 This very convenient system of decimal arithmetic is established in France and Belgium, and it is there carried to a most enviable degree of perfection: as, for example, in money reckoning, the franc (equal to our 10d.) is the standard coin of account, and is divided into 100 parts called centimes. There is an equal simplicity in the money reckoning of the North American Union, in which the dollar (equal to our 4s. 3d.) is divided into 100 centimes; but as weights and measures are not on the same decimal scale, the advantage is of compara tively small moment. it be required to divide 69 by 3: according to 3) 69 (23 38 36 2 In the United Kingdom, the pound or sovereign is the standard in money. It consists of a series of inferior coins, advancing irregularly from a farthing upwards; as, 2 farthings I halfpenny, 2 halfpence or 4 farthings 1 penny, 12 pence 1 shilling, 20 shillings 1 pound. White, therefore, the French compute values in money by francs and centimes, and the Americans by dollars and centimes, we compute by pounds, shillings, and pence; and to ascertain the value of irregular quantities in these irregular denominations of money, there is a complex set of rules to be obeyed; indeed, it may be said that the principal part of the time usually spent by youth at school on arithmetic, is consumed in learning to work questions in this arbitrary and lecal department of the science. We have only room to give a few examples in this species of computation. L is the initial letter of the Latin word libro, pound, and is used to denote pounds; s from the Latin word solidus, for shillings; and d from denarius, for pence: £ s. d. are therefore respectively placed over £5 TOO columns of pounds, shillings, and pence. The mark Compound Addition. 12 1200 4 4800 £31 12 7 In ordinary transactions of business, and making up of accounts, Compound Addition, that is, the addition of moneys, is principally required. In the margin is an account of sums to be reckoned up. The first thing done is to add together the halfpence and farthings in the right-hand side; and in doing so, we throw all into farthings. Thus, 2 and 1 are 3, and 3 are 6, and 2 are 8, and 2 are 10. Ten farthings are 2 pence, and 2 farthings, or one halfpenny, over. We set down for the halfpenny, and carry the 73 14 8 2 to the pence column; this being added, we find there are 31 pence, which make 2 shillings and 7 pence. We write down the 7, and carry the 2 shillings to the shillings column; adding them to the under figure at the right-hand side, we reckon up thus-2 and 2 are 4, and 5 are 9, and 7 are 16, and 4 are 20, and 2 are 22; we put down 2 aside, and carrying 2 to the second row of the shillings column, we find, on summing it up, that it amounts to 7; this 7 and the 2 set aside make 72 shillings, that is, £3, 12s.; 12, therefore, is written down under the shillings column, and the 3 pounds are carried to the pounds column, which is added up as in Simple Addition, making 320. Thus, the sum-total is £320, 12s. 74d. All accounts in Compound Addition, referring to British money, are performed in the same manner. We recommend young persons to acquire facility in adding; and it will save much time if they learn to sum up the columns by a glance of the eye, without naming the numbers; for instance, instead of saying 2 and 2 are 4, and 5 are 9, and 7 are 16, and 4 are 20, and 2 are 22, acquire the knack of summing the figures in the mind, thus-2, 4, 5, 9, 16, 20, 22. Compound Multiplication. £37 16 8 of figures as so much less, or 4d. instead of 5d. It is found, however, to be the most convenient plan to add 1 to the pence of the lower line, which comes to the same thing. Adding 1 to 8, in this case, we have 9 to subtract from 5. As this cannot be done, we borrow 1s., which is 12 pence, and adding that 12 to the 5 makes 17, from which taking 9, there will remain 8, which is placed under the pence. The borrowed Is. is also repaid by adding 1 to the 17, making thus 18 to be taken from 14; but as we cannot do this either, we borrow £1, which is 20s. Adding 20s. to 14 makes 34; then 18 from 34 leaves 16. This is placed under the shillings, and 1 is carried to the lower amount of pounds, which are then subtracted as in Simple Subtraction; thus, 1 to 7 is 8; 8 from 6, cannot, but 8 from 16, there remains 8; carry 1 to 2 is 3, and 3 from 3, nothing remains. Total sum remaining, £8, 16s. 8d. £ s. d. 7) 87 14 93 Compound Division. Compound Division is performed as follows:-We wish to divide £87, 14s. 93d. 8 into 7 equal parts. Dividing 87 by 7, as in Simple Division, the answer is 12, and 3 remain, that is, 3 pounds are over. We set down the 12, and taking the 3 which is over, we reduce it to its equivalent in shillings, that is 60; we then add the 60 to 14, making 74, which being divided by 7 gives 10 shillings, and 4 shillings over. Setting down the 10, we carry forward the 4; 4 shillings are 48 pence, which, added to 9, makes 57. This divided by 7 gives 8 and 1 penny over; a penny is 4 farthings; add to these the 3 in the dividend, thus making 7; 7 divided by 7 gives 1, that is d. The sum desired, then, is £12, 10s. 84d. £ S. d. 7) 376 11 14 9) 53 15 10 If the divisor is a composite number-the product of two numbers individually not exceeding 12-we can divide first by one and then by the other, as follows:Divide £376, 11s. 14d. by 63: 63 is a 5 19 6 composite number; its component parts are 7 and 9 (seven nines are 63). The given amount, therefore, is first divided by 7, and the quotient, £53, 15s. 101d., is divided by 9. The result is the same as if the original sum had been divided by 63. £5, 193. 64d. is the quotient. 438 46 20 73) 939 (12 73 209 6 146 63 12 Questions in Compound Multiplication are determined in the following manner-Having written down the number to be multiplied, place the multiplier under the lowest denomination, and proceed as in this example. We wish to multiply the sum of £37, 16s. 8d. by 6. We begin by multiplying the farthings by the 6; this makes 18 farthings, or 4d. Setting down the 4, we carry the 4 to the pence, saying 6 times 8 are 48, and 4 are 52, which is equal to 4 shillings and 4 pence. Setting down the 4 pence, we carry the 4 shillings onward, and multiplying 16 by 6 find 96, which, added to the 4 shillings, gives 100. This is equal to £5, so we set down 0, and carry the 5 to the 37. The amount is 227. The answer of the question is therefore £227, 0s. 41d. Compound Subtraction. £227 0 4 Compound Subtraction is performed as in the following question:If we take £27, 17s. 8d. from 36, 148, 5d., how much remains? The first thing e are called on to do, is to take 3 farthings from 2 farthings or d., and as this cannot be done, we borrow a penny, or 4 farthings, and adding these to the 2 farthings, we have 6. We now take 3 from 6, and 27 17 83 find that 3 remains, which is therefore written down. It is now necessary to £8 16 84 thunt for the borrowed penny, and a means of doing this would be to consider the pence of the upper line 261 £36 14 5 d. When the divisor is a prime £ S. number above 12, the work is in 73) 484 19 72(6 every respect similar to the former; but it is performed by long division, as in the annexed example:-Divide £484, 19s. 73d. by 73. The amount being written down as in long division of simple numbers, the pounds are first divided by 73; the answer is 6. The remainder 46 is reduced to shillings by multiplying by 20, and the 19s. in the sum we are dividing being taken in, makes together 939s., which, divided by 73, gives 12, and 63 of a remainder. These 63 shillings are now reduced to pence by being multiplied by 12, and the 7 being taken in, makes 763; this, divided by 73, gives 10, and 33 over, which, being reduced to farthings by being multiplied by 4, and the three taken in, makes 135; and this, divided by 73, gives 1, and 62 over. The whole answer is £6, 12s. 104d. and a fraction over. 73) 763 (10 73 4 73) 135 (1 62 British Weights and Measurės. The working of accounts in weights and measures, as respects addition, multiplication, subtraction, or division, proceeds on principles similar to those which have now been explained. The only real difference is that, for example, in reduction, instead of multiplying by 20, by 12, and by 4, to reduce a sum to farthings, |