9 Setting down the 1, we go to the 8, and finding it cannot ) arithmetic is erected, because these rules are founded countries ; this local arithmetic, as we may call it, is COMPOUND NUMBERS OR QUANTITIES. ticles, or a summation of values, in relation to money, would be comparatively simple if the scale of money Division. were constructed on a principle of decimals, or ad. nuruber of arithmetical calculations would be accom- 102.) is the standard coin of account, and is divided into 100 parts called centimes. There is an equal simplicity 6 we find that there is only one 6 in 7, and in the money reckoning of the North American Union, I over; we therefore place the 6 below 19 in which the dollar (equal to our 48. 3d.) is divided the 7, and subtract it, in order to bring 18 into 100 centimes; but as weights and measures are 38 In the United Kingdom, the pound or sovereign is 3 to the product (which in division is 36 the standard in money. It consists of a series of inferior called the quotient— literally, How many 2 coins, advancing irregularly from a farthing upwards ; times ?), and 18 below the 19, leaving 1 as 2 farthings I halfpenny, 2 halfpence or 4 farthings over as before. To this l we bring down the 5, and 1 penny, 12 pence 1 shilling, 20 shillings 1 pound. While , ment of the science. We have only room to give a few pound, and is used to denote pounds; s from the Latin The over 1326-2 pence: £ s. d. are therefore respectively placed over number of 1 from the 7 is carried in the columns of pounds, shillings, and pence. The mark mind to the 9, making 19; the 1 from 19 is in the for a halfpenny is , for a farthing !, and for three same manner carried to the 5; and the 3 from it is farthings . To find the number of farthings, carried to the 8, leaving the overplus of 2. £5 pence, and shillings, in any number of pounds, Division is denoted by the following character ; ; 20 we multiply by 20, which brings the pounds thus 75 = 25, signifies that 75 is to be divided by 25. 100 into shillings; next by 12, which brings the For example, 240 = 8 = 30. 12 shillings into pence; and lastly, by 4, which By employing the signs or abbreviations above ad- 1200 brings the pence into farthings; as, for example , verted to, any arithmetical operation may be very 4 Required the number of farthings in £5—we briefly stated ; thus 8 + 6 - 4 x 3 + 5 = 6. 4800 proceed as in the margin. The result is observed to be 4800 farthings. Compound Addition. , the addition s. we can manner, $. an account of sums to be reckoned up. The first thing and 3 from 3, nothing remains. Total sum remaining, Compound Division. viding 87 by 7, as in Simple Division, penny over. We set down for the 57 12 35 £ d. the answer is 12, and 3 remain--that halfpenny, and carry the 2 to the £320 12 7317)87 14 9 is, 3 pounds are over. We set down the pence column ; this being added, we 12 10 87 12, and taking the 3 which is over, we find there are 31 pence, which make 2 shillings and 7 reduce it to its equivalent in shillings, pence. We write down the 7, and carry the 2 shillings to that is 60; we then add the 60 to 14, making 74, which the shillings column; adding them to the under figure being divided by 7 gives 10 shillings, and 4 shillings at the right-hand side, we reckon up thus-2 and 2 are 4, over. Setting down the 10, we carry forward the 4; and 5 are 9, and 7 are 16, and 4 are 20, and 2 are 22; we 4 shillings are 48 pence, which, added to 9, makes 57. put down 2 aside, and carrying 2 to the second row of This divided by 7 gives 8 and 1 penny over; a penny the shillings column, we find on summing it up, that it is 4 farthings; add to these the 3 in the dividend, amounts to 7; this 7 and the 2 set aside make 72 shil- thus making 7; 7 divided by 7 gives 1, that is d. lings--that is, £3, 128.; 12, therefore, is written down The sum desired, then, is £12, 10s. 8.4d. under the shillings column, and the 3 pounds are If the divisor is a composite number the product of carried to the pounds column, which is added up as two numbers individually not exceeding 12– in Simple Addition, making 320. Thus the sum-total is divide first by one and then by the other, £320, 12s. 7d. All accounts in Compound Addition £ d. as follows: Divide £376, lls. 1}d. referring to British money are performed in the same 7)376 11_1} by 63: 63 is a composite number; its We recommend young persons to acquire 9)53 15 104 component parts are 7 and 9 (seven facility in adding; and it will save much time if they 5 1964 nines are 63). The given amount, learn to sum up the columns by a glance of the eye, therefore, is first divided by 7, and the without naming the numbers; for instance, instead of quotient, £53, 155. 10:1d. is divided by 9. The result saying 2 and 2 are 4, and 5 are I, and 7 are 16, and is the same as if the original sum had been divided 4 are 20, and 2 are 22, acquire the knack of summing by 63. £5, 19s. 6td. is the quotient. the figures in the mind, thus__2, 4, 5, 9, 16, 20, 22. When the divisor is a prime d. number above 12, the work is in Compound Multiplication. 73)484 1976 every respect similar to the former; Questions in Compound Multiplication are determined 438 but it is performed by long diviin the following manner :-Having written down the 46 sion, as in the annexed example: number to be multiplied, place the multiplier under the 20 Divide £484, 198. 740. by 73. The lowest denomination, and proceed as in this example. 73939(12 amount being written down as in We wish to multiply the sum of £37, 16s. 8 d. by 6. 73 long division of simple numbers, the We begin by multiplying the farthings pounds are first divided by 73; the by the 6; this makes 18 farthings, or £37 16 83 209 answer is 6. The remainder 46 is 146 4£d. Setting down the d, we carry the reduced to shillings by multiplying 4 to the pence, saying 6 times 8 are £227 0 45 by 20, and the 198. in the sum 48, and 4 are 52, which is equal to 4 12 we are dividing being taken in, shillings and 4 pence. Setting down the 4 pence, we 73)763(10 makes together 9398., which, dicarry the 4 shillings onward, and multiplying 16 by 6 73 vided by 73, gives 12, and 63 of find 96, which, added to the 4 shillings, gives 100. This 33 & remainder. These 63 shillings is equal to £5, so we set down (, and carry the 5 to are now reduced to pence by being the 37. The amount is 227. The answer of the ques. multiplied by 12, and the 7 being tion is therefore £227, 0s. 4d. 73) 135(1 taken in, makes 763; this, divided 73 by 73, gives 10, and 33 over, which, Compound Subtraction. 62 being reduced to farthings by being Compound Subtraction is performed as in the fol multiplied by 4, and the three lowing question :-If we take £27, 179. 8 d. from taken in, makes 135; and this, divided by 73, gives 1, £36, 148. 54d., how much remains? The first thing and 62 over. The whole answer is £6, 125. 10 d. and we are called on to do, is to take 3 farthings from a fraction {i over. 2 farthings, or }d., and as this cannot be done, we borrow a penny, or 4 farthings, and British Weights and Measures. adding these to the 2 farthings, we £36 14 52 The working of accounts in weights and measures, have 6. We now take 3 from 6, and 27_17_81 as respects addition, multiplication, subtraction, or find that 3 remains, which is therefore £8 16 8 division, proceeds on principles similar to those which written down. It is now necessary to bave now been explained. The only real difference is account for the borrowed penny, and a means of doing that, for example, in reduction, instead of multiplying this would be to consider the pence of the upper line by 20, by 12, and by 4, to reduce a sum to farthings, of figures as so much lese, or 4d. instead of 5d. It if the question refer to ordinary weights we multiply is found, however, to be the most convenient plan to to bring out the number of hundredweights, 112 for add 1 to the pence of the lower line, which comes to pounds, and 16 for ounces. Suppose we wish to know the same thing. Adding 1 to 8, in this case, we have | how many ounces are in 15 tons: we multiply 15 by 9 to subtract from 5. As this cannot be done, we 20, and the result is 300-that is, 300 hundredweights; borrow 19., which is 12 pence, and adding that 12 to a hundredweight is 112 pounds, so we now multiply the 5 makes 17, from which taking 9, there will re- } 300 by 112, and the result is 33,600 pounds; this mulmain 8, which is placed under the pence. The bor- tiplied by 16, the number of ounces in a pound, gives lowed ls. is also repaid by adding 1 to the 17, making 537,600 ounces. In the addition of quantities, each thus 18 to be taken from 14 ; but as we cannot do denomination is set down in its own column, as in this either, we borrow £1, which is 20s. Adding 20s. money, and the summation is made also as we add to 14 makes 34; then 18 from 34 leaves 16. This is money, with the difference, that we carry forward placed under the shillings, and 1 is carried to the lower ounces, pounds, or whatever it may be, to the next amount of pounds, which are then subtracted as in column. The mark cwt, is usually employed to indiSimple Subtraction; thus, 1 to 7 is 8, 8 from 6, cannot, cate hundredweights, 11. pounds, and oz. ounces. but *8 from 16, there remains 8; carry 1 to 2 is 3, 1 The following are the principal tables of weights and 6 63 4 ༧ 3 5 1.MEASURE OF LENGTH. 12 36 198 1 Foot. 660 5280 220 1760 40 320 8 . II.-MEASURE OF SURFACE. 144 9 ... 1210 .. 40 160 4 measures established by law in the United Kingdom, ( pipe, &c. ; but these are now to be considered rather as follows :-Pipe of port, 115 imp. galls. ; pipe of Lisbon, 117 do.; pipe of Cape or Madeira, 92 do.; pipe of Teneriffe, 100 do.; butt of sherry, 108 do.; hogshead of claret, 46 do.; aum of hock, 30 do. Herrings are measured by the barrel of 263, or cran 1 Furlong. of 374 gallons. 63,360 1 Mile. Apothecaries' Weight.—20 troy grains make 1 scruple, The hand – 4 inches; the English ell – 45 inches; the pace 3 scruples make 1 drachm, and 8 drachms make 1 ounce. - 5 feet; and the fathom – 6 feet. The geographical degrec The ounce and pound are the same as in troy weight. 20 nautical leagues, or 69:121 miles. In land measure, the This weight is used in medical prescriptions only. The fluid measures used by apothecaries are—60 mi- 20 fluid ounces 1 pint, 8 pints 1 gallon. (See No. 48.) The Scottish acre, formerly a standard in Scotland, 1 Sq. Foot. and now abolished, consisted of 54937:15 imperial 1,296 1 sq. Yard. 301 square feet : 23 Scots acres were very nearly equal to 1 Rood. 29 imperial acres. The Scottish choppin was very 6,272,640 13,560 4840 1 Acre nearly the English wine quart, and the mutchkin was And 640 acres make 1 square mile. rather more than the English pint. Previous to the Union with England, the Scottish money pound was equal to one shilling and eightpence English. Hence £100 Scots was equal to the sum of £8, 6s. 8d. in our present money. 1 cubic yard. min. 1 minuto 2. Measure for Liquids, Corn, and Dry Goods. 60 min. - 1 hour 1 hr. 60 1 day, 24 hr. = l day 1 day 86400 = 1440 -- 24 wk. Cubic Wat. 7 days - 1 week I wk. 604800 - 10080 168 - 7.1 Inches. lbs. av. 365 days, or 52 weeks 1 day - 1 common year. 366 days, or 52 weeks 2 days = 1 leap year. 365 days 6 hours - 1 Julian year. 365 days 5 hours 48 minutes 49 seconds - ) solar year. 1 Quart. 277.274 1 Gal. French Weights and Measures. 554.518 20 1 Peck. As French weights and measures are now frequently 2218-19 64 32 17745.5 640 2018 256 64 32 referred to in literature, we think it proper to offer the following particulars on the subject :- The French system of weights and measures is estab- application, the latter can never be anything but local. The French unity of length and weight is based on an According to the system introduced January 1, 1826, invariable dimension of the terrestrial globe, which is the imperial gallon contains 277,274 cubic inches, or recognisable in all countries. It is independent of all 10 lbs. avoirdupois of distilled water at 62° Fahrenheit, extrinsic notions, such as gravity and the arbitrary subthe barometer being at 30 inches. Heaped measures divisions of duration, an advantage which the length of (formerly used for dry goods) were abolished in 1835. a seconds pendulum certainly does not present. The admeasurement of a fourth of the earth's meridian-an ideal circle going round the globe from pole to pole at right angles with the equator-constitutes the basis of the French system. The length of this fourth of the meridian is divided into 10,000,000 parts; a single ten- millionth part is the metre, or the unity of long measure. 224 (A metre is equal to 39British inches.) i Qr. A square, measuring on each side 16 metres, forms 781000 28,672 1,792 1 Cwt. the are, or the unity of the mensuration of surface. 15680000 573,440 35,840 160 1 Ton. (40$ ares are nearly equal to one British acre.) FLOUR WEIGHT.- peck 14 pounds; 1 boll 140 pounds; A cube, measuring on each of its sides 1 metre, 1 sack -- 280 pounds, or 2 cwt. ; I barrel - 196 lbs. constitutes the stère: used for dry measure. A cube, measuring on each of its sides the tenth part 2. Troy Weight. of a metre, is the unity of volume. A vessel gauging such a cube, is the unity of liquid measures, and is called the litre. (A litre is equal to about a pint and 1 Pound. three-quarters, or nearly a quart British measure.). The weight of a cube of water, measuring on each of These unities being often too great or too small for In Wine Measure, besides the gallon and its sub-unities are formed from Greek and Latin words. If to divisions, various denominations are used, as the butt, I express multiplication of the original unity, Greek is 598 It on. used ; if to express division of the original unity, Latin decimal notation, for it allows of adding up columns of is the decimal;or, what is the same thing, 75-hundredth SERIES AND RATIOS OF NUMBERS. gramme is the weight of a litre of pure water at its A series of numbers is a succession of numbers that maximum density. increase or decrease according to some law. Of the The currency of the country being assimilated by two kinds of series usually treated of in arithmetic, decimal reckoning to the weights and measures, it may the simpler is one whose terms increase or decrease be safely averred that the whole world cannot produce by some constant number called the common diffea more simple and immutable plan of calculation than rence. This common difference or rate of increase that now in use in France and in Belgium. is only one, when we say 4, 5, 6, 7, 8; it is two when we say 7, 9, 11, 13; and four when we say FRACTIONS. 6, 10, 14, 18, and so on. Every advancement of this Hitherto we have spoken only of whole numbers, nature, by which the same number is added at every which in arithmetic are called integers. We have step, is called arithmetical progression. There is a now to treat of fractions, or the parts into which different species of advancement, by which the last integers may be broken. The more ordinary fractions number is always multiplied by a given number, thus of any single article or number are a half, third, quar- causing the series to mount rapidly up. Suppose 4 ter, &c.; but a number admits of being divided into is the multiplier, and we begin at 2, the progression any quantity of equal parts. All such fractions are will be as follows:-2, 8, 32, 128, 512, 2048, and so called vulgar fractions, from their being common. It is here observed, that multiplying the 2 by is the practice to write vulgar fractions with two or 4, we have 8; multiplying the 8 by 4, we have 32; more small figures, one above the other, with a line and multiplying the 32 by 4, we have 128, &c. till at between, as follows:- (one-half), } (one-third), 1 (one- the fifth remove we attain 2048. This kind of advancefourth or quarter), }(one-eighth), 4 (four-fifths), i ment of numbers is called geometrical progression. (nine-tenths), and so on. In these and all other in- (The very great difference between the two kinds of stances, the upper number is called the numerator, the progression is exemplified in the following two lines, lower the denominator. Thus in the fraction is, 9 is the number 3 being added in the one case, and being the numerator, and 10 the denominator the latter used as the multiplier in the other :denominating or showing into how many parts the 5, 8, 11, 14, 17-Arithmetical Progression. original unit is supposed to be divided, and the former 5, 15, 45, 135, 405–Geometrical Progression, pointing out how many such parts are in the fraction spoken of In the case of arithmetical progression, as above or in It may happen that it is necessary to add together any other manner exemplified, it may be noticed that different fractions to make up whole numbers. In the amount of the first and last term is always the working all such questions, we must, in the first place, same as twice the amount of the middle term; thus bring all the fractions into one kind: if we have to add | 5 and 17 being 22, are equal to twice 11, or 22. The 9, 4, and together, we make all into eighths, and see cause of this is, that as the numbers increase or dehow many eighths we have got; thus is g; then is crease in equal degrees, the last number is just as much I, that is 2 and 4, which make 6, and makes a total more as the first is less than the number in the middle; of 1. The same plan is to be pursued in the subtrac- and the two being added, the amount must consequently tion of vulgar fractions. be double the central number. The same rule holds It is sometimes necessary to speak of the tenths, good with respect to any two numbers at equal dishundredths, or thousandths of a number, and for this tances from the number in the middle. If the series arithmetic has provided a system of decimal fractions. be an even number, and do not possess a middle term, Where great exactness of expression is required, deci- then the two terms nearest the middle (called the mean mals are indispensable. It has been already shown terms) must be added together; thus in the natural that in writing common numbers, the value of a figure series from 1 to 24, 12 and 13 are the two nearest the increases by ten times as we proceed from right to left; middle, and one being added to the other makes 25, the in other words, we ascend by tens. Now there is no- sum of the first and last term, thing to prevent us in the same manner descending by In geometrical progression, each term is a factor of tens from unity. This is done by decimal fractions, all the numbers or terms that follow, and a product of We place a dot after unity, or the unit figure, all that go before, so that there is a harmonious ratio which dot cuts off the whole number from its frac- pervading the whole. Each term bears an exact protional tenths ; thus, 120-3 means 120 and 3-tenths of a { portion to its predecessor, because the multiplier is the whole; if we write 120-31, the meaning is 120 and 31. same. Supposing, as above, the multiplier to be 3, the hundredths of a whole-that is, 31 parts in 100 into term 15 is proportionally greater than 5, as 45 is which a whole is supposed to be divided. If we go on greater than 15. In the technical language of arithadding a figure to the right, we make the fraction into metic, as 15 is to 5, so is 45 to 15. To save words, such thousands ; as, for instance, 120-315, which signifies a proposition is written with signs or abbreviations 120 and 315 out of a thousand parts. thus--15:5:: 45:15. The two dots mean is to, and Tables of specific gravities, population, mortality, and four dots mean so is. The same formula is applicable many matters of statistics, are greatly made up of to any series of proportional terms, though not in condecimal fractions, and therefore it is proper that all tinued proportion to each other. should comprehend the principle on which they are In order to discover the ratio between any two terms, designed. In many cases, it would answer the purpose we divide the largest by the least, and the quotient is to write the fractions as vulgar fractions ; but there is the ratio : 45 divided by 15 gives 3 as the ratio. By a great advantage in reducing all broken parts to the thus ascertaining the ratio of two terms, we are fur 1 £ 8. d. nished with the means of arriving at the ratio of other of tea cost 21s., how much may be terms. We cannot do better than explain the method | 21:7 :: 3 purchased for 3s.!-then the sum of working out this principle in the ratio of numbers, 7 to be ascertained is less, and is put by giving the following passages from the admirable 21)21 last. Suppose another plain example: • Lessons on Arithmetic,' by Mr T. Smith of Liverpool. 1 lb. If 10 men can execute a piece of work in 8 days, how long will 4 men take to do the same? 4:8:: 10 first three, and that it were our wish to find the fourth, Such is the principle of working 8 which term bears the same proportion to the third as Rule of Three questions, whatever the second does to the first. The thing we have first 4)80 be their apparent complexity. If to do is, to discover the ratio between the first and 20 days. either the first or third term, or both, include fractional parts, they would be in ounces. cwt. qr. lb. cwt. gr. lb. plied the third term (405) by the whole, instead of by 2 1 7 8 14 4 :: 14 3 0 4 4 9 59 28 12 be a matter of no difficulty; for it would be set right 28 at once, and our purpose gained, by dividing the over 259 2092 472 large product by 15. Let us write this process down : 118 405 x 45 18225, and 18225 + 15 = 1215,—which Here, in order to 1652 1215 bears the same proportion to 405 as does 45 to make the first term a 2092 3304 259)3455984(12)13343 259 any proportional terms we may require.? that both may be alike. 20)1111-11 The second or middle 865 £55 11 11 Rule of Three. term is reduced to 777 On the principle now explained, we can, in any affairs its lowest term, pence. 889 of business, ascertain the amount of an unknown quan- After multiplying and 777 tity, by knowing the amount of other three quantities, dividing, according to 1128 which, with the unknown quantity, bear a proportional rule, the quotient is 1036 relation. The word quantity is here used, but any sum 13343 pence, which are 924 of money is also meant. brought to shillings and Let it be remembered that the ratio of one number pounds. The remain 777 to another is the number of times that the former con- der, 147, being farther 147 tains the latter; for example, the ratio of 6 to 3 is 2, reduced to farthings, 4 that of 12 to 4 is 3, and that of 8 to 12 is g. When and divided by the first 259)58862 two numbers have the same ratio as other two, they term, gives 2 farthings. 518 70 The following is a kind of question which often occurs in business:--A person is unable to pay his debts. He The following is the rule for stating and working owes to A £540, to B £260, to c £200, being in all questions:--Make that term which is of the same kind € € £1000. On examinas the answer sought, the second or middle term. Con- 1000 : 370 :: 540 ing his affairs, it is sider, from the nature of the question, whether the 370 found that he posanswer should be more or less than this term ; if 37800 sesses property only more, make the smaller of the other two terms the first , 1620 to the value of £370. and the greater the third; if the answer should be less £ than the middle term, make the greater of the two How, then, is this to 1000)199800(199 16 be divided propor. terms the first, and the smaller the third; then mul 1000 tionally among the tiply the second and third terms together, and divide 9980 three creditors, so the result by the first term. The quotient found will 9000 that each may receive be the answer to the question, and it will be found to 9800 a fair share? The bear the same proportion to the third term as the 9000 way to arrive at an 800 answer, is to work out 9:3::21 each creditor's share pounds may be purchased for 21s. ? 3 as a distinct account. 1000)16000(16 Thus, first, as to A's share :however, be the reverse - If 7 lbs. The answer is £199, 168. Following a similar calcu600 7 lbs. 20 |