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, 9 - 5 = 4, which means, 5 subtracted from 9, and 4 remains. Again, 19 118; that is, 11 taken from 19 leaves 8; or 19 diminished by 11, equal to 8.

Division.

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

19

18

15 12

38

6 9

it be required to divide 69 by 3: according to 3) 69 (23 the long method, we write the figures 69 as annexed, with a line at each side, and the divisor, or 3, on the left. The question is 9 wrought out by examining how many times 3 is in 6, and finding it to be 2 times, we place 2 on the right side; then, placing 6 below 6, we draw a line and bring down the 9, and proceed with it in the same manner. The quotient is found to be 23. But we take a more difficult questionthe division of 7958 by 6. In commencing, 6)7958 (1326 6 we find that there is only one 6 in 7, and 1 over; we therefore place the 6 below the 7, and subtract it, in order to bring out the 1. The 1 being written, we bring down the 9 to it, and this makes 19. There being 3 times 6 in 18, we place the 3 to the product (which in division is called the quotient-literally, How many times?), and 18 below the 19, leaving 1 over as before. To this 1 we bring down the 5, and trying how many sixes there are in 15, it appears there are only 2. We place 2 to the quotient, and 12 below the 15. This leaves 3 over, and bringing down 8 to the 3, we have 38, in which there are 6 sixes. Six sixes make 36; therefore placing 6 to the quotient, and 36 below the 38, we find that there are 2 over. Here the account terminates, it being found that there are 1326 sixes in 7958, with a remainder of 2 over. In this question, 6 is called the divisor; the 7958 is the dividend, and 1326 is the quotient.

36

2

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 Division. They would, for example, treat the 6)7958 above question as here shown.

The over

1326-2

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. For example, 240÷8=30.

By employing the signs or abbreviations above adverted to, any arithmetical operation may be very briefly stated; thus 8+ 6-4 x 3 ÷ 5 = 6.

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 conventional

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 arithmetical 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.

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, 10 pence 1 shilling, 10 shillings 1 pound. By making both weights and measures on the same plan, as 10 ounces 1 pound, 10 pounds 1 stone, 10 stones 1 hundredweight; 10 inches I foot, 10 feet 1 yard, &c., ordinary calculations would be rendered exceedingly easy. Thus if an ounce cost 1d., a pound would cost ls., 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.

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 comparatively small moment.

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 1 halfpenny, 2 halfpence or 4 farthings 1 penny, 12 pence 1 shilling, 20 shillings 1 pound. While, 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 local 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 libra, a 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 columns of pounds, shillings, and pence. The mark for a halfpenny is, for a farthing, and for three farthings. To find the number of farthings, pence, and shillings, in any number of pounds, we multiply by 20, which brings the pounds into shillings; next by 12, which brings the shillings into pence; and lastly, by 4, which brings the pence into farthings; as, for example, Required the number of farthings in £5-we proceed as in the margin. The result is observed to be 4800 farthings.

£5 20

100

12

1200

4

4800

Compound Addition.

In ordinary transactions of business, and making up of accounts, Compound Addition-that is, the addition of monies-is principally required. In the margin is

£31 12 7

73 14 84 69 17 5 87 15 64 57 12 3 £320 12 7

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 halfWe set down for the penny over. halfpenny, and carry the 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, 128.; 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. 7 d. 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 Division.

Compound Division is performed as follows:-We wish to divide £87, 14s. 9 d. into 7 equal parts. Dividing 87 by 7, as in Simple Division,

£ S. 7)87 14 12 10

d.

93

the answer is 12, and 3 remain-that is, 3 pounds are over. We set down the 8 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 id. The sum desired, then, is £12, 10s. 84d.

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, £ S. d. as follows: Divide £376, 11s. 1d. 7)376 11 1 by 63: 63 is a composite number; its 9)53 15 10 component parts are 7 and 9 (seven 5 19 6 nines are 63). The given amount, therefore, is first divided by 7, and the quotient, £53, 15s. 104d. is divided by 9. The result is the same as if the original sum had been divided by 63. £5, 19s. 6d. is the quotient.

£ S. d. 73)484 19 7(6 438

46

20

Compound Subtraction.

73

£37 16 83

209

6

146

£227 0 44

63 12

73)763(10

73

Compound Subtraction is performed as in the following question:If we take £27, 178. 8d. from £36, 14s. 54d., how much remains? The first thing we are called on to do, is to take 3 farthings from 2 farthings, or d., and as this cannot be done, we borrow a peuny, or 4 farthings, and adding these to the 2 farthings, we £36 14 51 have 6. We now take 3 from 6, and 27 17 83 find that 3 remains, which is therefore £8 16 8 written down. It is now necessary to account for the borrowed penny, and a means of doing this would be to consider the pence of the upper line 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 ls. 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,

When the divisor is a prime number above 12, the work is in every respect similar to the former; but it is performed by long division, as in the annexed example :Divide £484, 19s. 7 d. 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. 10 d. and a fraction over.

33

4

73)135(1 73 62

British Weights and Measures.

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, if the question refer to ordinary weights we multiply to bring out the number of hundredweights, 112 for pounds, and 16 for ounces. Suppose we wish to know how many ounces are in 15 tons: we multiply 15 by 20, and the result is 300-that is, 300 hundredweights; a hundredweight is 112 pounds, so we now multiply 300 by 112, and the result is 33,600 pounds; this multiplied by 16, the number of ounces in a pound, gives 537,600 ounces. In the addition of quantities, each denomination is set down in its own column, as in money, and the summation is made also as we add money, with the difference, that we carry forward ounces, pounds, or whatever it may be, to the next column. The mark cwt. is usually employed to indicate hundredweights, lb. pounds, and oz. ounces.

The following are the principal tables of weights and

The connection between these weights and measures will now be clearly seen. The are is the square decametre; the litre is the cubic decimetre; and the kilogramme is the weight of a litre of pure water at its maximum density.

The currency of the country being assimilated by decimal reckoning to the weights and measures, it may be safely averred that the whole world cannot produce a more simple and immutable plan of calculation than that now in use in France and in Belgium.

FRACTIONS.

The method of reducing a vulgar to a decimal fraction is a simple question in Division. For instance, to reduce to a decimal, we take the 3, and putting two 4)300 ; therefore 75 ⚫75 is the decimal; or, what is the same thing, 75-hundredth parts of a whole are equal to three-quarters.

ciphers after it, divide by 4, thus

Hitherto we have spoken only of whole numbers, which in arithmetic are called integers. We have now to treat of fractions, or the parts into which integers may be broken. The more ordinary fractions of any single article or number are a half, third, quarter, &c.; but a number admits of being divided into any quantity of equal parts. All such fractions are called vulgar fractions, from their being common. is the practice to write vulgar fractions with two or more small figures, one above the other, with a line between, as follows:- (one-half), (one-third), (onefourth or quarter), (one-eighth), (four-fifths), (nine-tenths), and so on. In these and all other instances, the upper number is called the numerator, the lower the denominator. Thus in the fraction 5,9 is the numerator, and 10 the denominator-the latter denominating or showing into how many parts the original unit is supposed to be divided, and the former pointing out how many such parts are in the fraction spoken of.

It may happen that it is necessary to add together different fractions to make up whole numbers. In working all such questions, we must, in the first place, bring all the fractions into one kind: if we have to add 4, 4, and together, we make all into eighths, and see how many eighths we have got; thus is; then is , that is 2 and 4, which make 6, and makes a total of. The same plan is to be pursued in the subtraction of vulgar fractions.

It is sometimes necessary to speak of the tenths, hundredths, or thousandths of a number, and for this arithmetic has provided a system of decimal fractions. Where great exactness of expression is required, decimals are indispensable. It has been already shown that in writing common numbers, the value of a figure increases by ten times as we proceed from right to left; in other words, we ascend by tens. Now there is nothing to prevent us in the same manner descending by tens from unity. This is done by decimal fractions. We place a dot after unity, or the unit figure, which dot cuts off the whole number from its fractional tenths; thus, 120-3 means 120 and 3-tenths of a whole; if we write 120-31, the meaning is 120 and 31hundredths of a whole-that is, 31 parts in 100 into which a whole is supposed to be divided. If we go on adding a figure to the right, we make the fraction into thousands; as, for instance, 120-315, which signifies 120 and 315 out of a thousand parts.

Tables of specific gravities, population, mortality, and many matters of statistics, are greatly made up of decimal fractions, and therefore it is proper that all should comprehend the principle on which they are designed. În many cases, it would answer the purpose to write the fractions as vulgar fractions; but there is a great advantage in reducing all broken parts to the

It is here observed, that multiplying the 2 by 4, we have 8; multiplying the 8 by 4, we have 32; and multiplying the 32 by 4, we have 128, &c. till at the fifth remove we attain 2048. This kind of advancement of numbers is called geometrical progression. The very great difference between the two kinds of progression is exemplified in the following two lines, the number 3 being added in the one case, and being used as the multiplier in the other:—

5, 8, 11, 14, 17-Arithmetical Progression. 5, 15, 45, 135, 405-Geometrical Progression. In the case of arithmetical progression, as above or in any other manner exemplified, it may be noticed that the amount of the first and last term is always the same as twice the amount of the middle term; thus 5 and 17 being 22, are equal to twice 11, or 22. The cause of this is, that as the numbers increase or decrease in equal degrees, the last number is just as much more as the first is less than the number in the middle; and the two being added, the amount must consequently be double the central number. The same rule holds good with respect to any two numbers at equal distances from the number in the middle. If the series be an even number, and do not possess a middle term, then the two terms nearest the middle (called the mean terms) must be added together; thus in the natural series from 1 to 24, 12 and 13 are the two nearest the middle, and one being added to the other makes 25, the sum of the first and last term.

In geometrical progression, each term is a factor of all the numbers or terms that follow, and a product of all that go before, so that there is a harmonious ratio pervading the whole. Each term bears an exact proportion to its predecessor, because the multiplier is the same. Supposing, as above, the multiplier to be 3, the term 15 is proportionally greater than 5, as 45 is greater than 15. In the technical language of arithmetic, as 15 is to 5, so is 45 to 15. To save words, such a proposition is written with signs or abbreviations thus-15:5: 45:15. The two dots mean is to, and four dots mean so is. The same formula is applicable to any series of proportional terms, though not in continued proportion to each other.

In order to discover the ratio between any two terms, we divide the largest by the least, and the quotient is the ratio: 45 divided by 15 gives 3 as the ratio. By thus ascertaining the ratio of two terms, we are fur

nished with the means of arriving at the ratio of other terms. We cannot do better than explain the method of working out this principle in the ratio of numbers, by giving the following passages from the admirable Lessons on Arithmetic,' by Mr T. Smith of Liverpool. Taking the four regularly-advancing terms, 15, 45, 405, and 1215, he proceeds- Suppose that we had only the first three, and that it were our wish to find the fourth, which term bears the same proportion to the third as the second does to the first. The thing we have first to do is, to discover the ratio between the first and second terms, in order to do which, as before shown, we divide the larger by the smaller, and this gives us the ratio 3, with which, by multiplying the third term, we produce the fourth; or let the three terms be these, 405, 1215, 5, and let it be our wish to find a fourth which shall bear the same relation to the 15 as 1215 does to 405. We divide and multiply as before, and the fourth term is produced. And in this manner, having two numbers or two quantities of any kind, bearing a certain proportion towards each other, and a third, to which we would find a number or quantity that should bear a like proportion, in this manner do we proceed, and thus easily may we find the number we require.'

Referring to the discovered ratio of 45 to 15 to be 3, or the fifteenth part-Now,' continues this author, 'what would have been the consequence had we multiplied the third term (405) by the whole, instead of by a fifteenth part of the second? The consequence would have been, that we should have had a term or number fifteen times larger than that required. But this would be a matter of no difficulty; for it would be set right at once, and our purpose gained, by dividing the overlarge product by 15. Let us write this process down: 405 × 45 = 18225, and 18225 ÷ 15 = : 1215,-which 1215 bears the same proportion to 405 as does 45 to 15. And this is the rule, when the terms are properly placed-Multiplying the second and third terms together, and dividing the product by the first; this avoids all the difficulties arising from the occurrence of fractions in the course of the process, and gives us in all cases any proportional ternis we may require.'

Rule of Three.

On the principle now explained, we can, in any affairs of business, ascertain the amount of an unknown quantity, by knowing the amount of other three quantities, which, with the unknown quantity, bear a proportional relation. The word quantity is here used, but any sum of money is also meant.

Let it be remembered that the ratio of one number to another is the number of times that the former contains the latter; for example, the ratio of 6 to 3 is 2, that of 12 to 4 is 3, and that of 8 to 12 is . When two numbers have the same ratio as other two, they constitute a proportion. Thus the ratio of 8 to 6 is the same as that of 12 to 9, and the equality of these two ratios is represented thus:

Con

The following is the rule for stating and working questions:-Make that term which is of the same kind as the answer sought, the second or middle term. sider, from the nature of the question, whether the answer should be more or less than this term; if more, make the smaller of the other two terms the first, and the greater the third; if the answer should be less than the middle term, make the greater of the two terms the first, and the smaller the third; then multiply the second and third terms together, and divide the result by the first term. The quotient found will be the answer to the question, and it will be found to bear the same proportion to the third term as the second does to the first. Suppose the question be this: If 3 lbs. of tea cost 9s., how many 9: 3:21 pounds may be purchased for 21s.? -state the terms, with the larger sum last. Should the question, however, be the reverse-If 7 lbs.

3

9)63

7 lbs.

21:7:3

7

21)21 1 lb.

4:8:10

8

4)80

20 days.

of tea cost 21s., how much may be purchased for 3s.then the sum to be ascertained is less, and is put last. Suppose another plain example: If 10 men can execute a piece of work in 8 days, how long will 4 men take to do the same?

Such is the principle of working Rule of Three questions, whatever be their apparent complexity. If either the first or third term, or both, include fractional parts, they must be reduced to the denomination of the fractions before working: thus if one be reduced to shillings, the other must be made shillings also; if to pence, both must be pence; and so on. If the middle term be also a compound quantity, it may either be reduced to its lowest term, before multiplying and dividing by the other terms, or you may multiply and divide by Compound Division and Multiplication.

If the middle term be reduced to its lowest term, the answer will be in that denomination to which it was reduced; thus if it were brought to farthings, the answer would be in farthings; if to ounces, the answer would be in ounces.

Example.-If 2 cwts. 1 qr. 7 lbs. sugar cost £8, 14s. 4d., what will 14 cwts. 3 qrs. cost?

1652

2092

3304

14868

Here, in order to make the first term a simple number, it is reduced to its lowest term-namely, lbs. The third term is therefore reduced to lbs. also, that both may be alike. The second or middle term is reduced to its lowest term, pence. After multiplying and dividing, according to rule, the quotient is 13343 pence, which are brought to shillings and pounds. The remainder, 147, being farther reduced to farthings, and divided by the first term, gives 2 farthings. The answer is £55, 11s. 11d.