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Setting down the 1, we go to the 8, and finding it cannot ) arithmetic is erected, because these rules are founded
be taken from the 4 above it, we lend 10 to the 4, mak-in immutable truths. Mankind may change their de-
ing it 14, and then we say, 8 from 14, and 6 remains. nominations of money, weights, and measures, but they
In the same manner as before, adding the first figure can make no alteration in the doctrine of abstract
of the borrowed number (1) to the 6, we say, 7 from 8, numbers. That 2 and 2 are equal to 4, is a truth yes-
and I remains; thus the total remainder is found to terday, to-day, and for ever ; but as to how many pence
be 1617. From these explanations, which apply to all are in a shilling, or how many inches in a foot, these
calculations in subtraction, it will be observed, that are altogether matters of arbitrary arrangement, and
when the upper figure is less than the figure directly the treatment of them forms an inferior department of
under it, 10 is to be added, and for this one is carried arithmetical study, taking a different form in different
or added to the next under figure.

countries ; this local arithmetic, as we may call it, is
Subtraction is denoted by a small horizontal line, thus comprehended in the term
— between two figures; as, for example, 9 — 5 = 4,
which means, 5 subtracted from 9, and 4 remains.

COMPOUND NUMBERS OR QUANTITIES.
Again, 19 - Il = 8; that is, 11 taken from 19 leaves The calculation of the value of any number of ar-
8; or 19 diminished by ll, equal to 8.

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.
Division is that process by which we discover how vancing by tens--as, for example, 10 farthings 1 penny,
often one number may be contained in another, or by 10 pence 1 shilling, 10 shillings 1 pound. By making
which we divide a given number into any proposed num- both weights and measures on the same plan, as 10
ber of equal parts. By the aid of the Multiplication ounces 1 pound, 10 pounds 1 stone, 10 stones 1 hundred-
Table, we can ascertain without writing figures how weight; 10 inches 1 foot, 10 feet 1 yard, &c., ordinary
many times any number is contained in another, as far calculations would be rendered exceedingly easy. Thus
as 144, or 12 times 12 ; beyond this point notation is if an ounce cost ld., a pound would cost 18., and a
employed. There are two modes of working questions hundredweight would cost 100s. or £10; or, reversing
in division--one long, and the other short. Let the question, if we were asked £10 per hundredweight
it be required to divide 69 by 3: according to 3) 69 (23 for any article, we should know in an instant that it
the long inethod, we write the figures 69 as 6 was at the rate of ld. an ounce. In short, the greater
annexed, with a line at each side, and the

nuruber of arithmetical calculations would be accom-
divisor, or 3, on the left. The question is 9 plished by little more than a momentary reflection,
wrought out by examining how many times without the aid of pen or pencil.
3 is in 6, and finding it to be 2 times, we place 2 on This very convenient system of decimal arithmetic
the right side; then, placing 6 below 6, we draw a is established in France and Belgium, and it is there
line and bring down the 9, and proceed with it in carried to a most enviable degree of perfection : as, for
the same manner. The quotient is found to be 23. example, in money reckoning, the franc (equal to our
But we take a more difficult question--
the division of 7958 by 6. In commencing,

102.) is the standard coin of account, and is divided into
6)7958 (1326

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
out the l. The 1 being written, we bring 15 not on the same decimal scale, the advantage is of
down the 9 to it, and this makes 19. 12 comparatively small moment.
There being 3 times 6 in 18, we place the

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

,
trying how many sixes there are in 15, it appears there therefore, the French compute values in money by
are only 2. We place 2 to the quotient, and 12 below francs and centimes, and the Americans by dollars and
the 15. This leaves 3 over, and bringing down 8 to centimes, we compute by pounds, shillings, and pence;
the 3, we have 38, in which there are 6 sixes. Six and to ascertain the value of irregular quantities in
sixes make 36; therefore placing 6 to the quotient, these irregular denominations of money, there is a com,
and 36 below the 38, we find that there are 2 over. plex set of rules to be obeyed; indeed, it may be said
Here the account terminates, it being found that there that the principal part of the time usually spent by
are 1326 sixes in 7958, with a remainder of 2 over. In youth at school on arithmetic, is consumed in learning
this question, 6 is called the divisor ; the 7958 is the to work questions in this arbitrary and local depart-
dividend, and 1326 is the quotient.

ment of the science. We have only room to give a few
Skilful arithmeticians never adopt this long method examples in this species of computation.
of division ; they pursue a plan of working out part of L is the initial letter of the Latin word libra, a
the question in the mind, called Short Divi.

pound, and is used to denote pounds; s from the Latin
sion. They would, for example, treat the 6) 7958 word solidus, for shillings; and d from denarius, for
above question as here shown.

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.
These explanations conclude the subject of simple
or abstract numbers. On the substructure of the few

Compound Addition.
rules in Addition, Multiplication, Subtraction, and In ordinary transactions of business, and making up
Division, which we have given, whether in reference to of accounts, Compound Addition, that is

, the addition
whole numbers or fractions, every kind of conventional of monies--is principally required. In the margin is

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an account of sums to be reckoned up. The first thing and 3 from 3, nothing remains. Total sum remaining,
done is to add together the halfpence and farthings £8, 168. 8 d.
in the right-hand side; and in doing
50, we throw all into farthings. Thus, £31 12 71

Compound Division.
2 and I are 3, and 3 are 6, and 2 are 73 14 88 Compound Division is performed as follows:-We
8, and 2 are 10. Ten farthings are 2 69 17 5 wish to divide £87, 14s. 94d. into 7 equal parts. Di.
pence, and 2 farthings, or one half- 87 15 6

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

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1.MEASURE OF LENGTH.

12 36 198

1 Foot.
3

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660 5280

220 1760

40 320

8

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II.-MEASURE OF SURFACE.

144

9

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1210

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40 160

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measures established by law in the United Kingdom, ( pipe, &c. ; but these are now to be considered rather
and hence called imperial. That which is entitled as the names of casks than as expressing any definite
avoirdupois weight is the table in use for all ordinary number of gallons. The standard gauges in trade are
purposes :-

as follows :-Pipe of port, 115 imp. galls. ; pipe of

Lisbon, 117 do.; pipe of Cape or Madeira, 92 do.;
Inches.

pipe of Teneriffe, 100 do.; butt of sherry, 108 do.;

hogshead of claret, 46 do.; aum of hock, 30 do.
1 Yard.
161 5! 1 Pole or Perch.

Herrings are measured by the barrel of 263, or cran
7,920

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.
chain of 100 links - 66 feet; a link is equal to 7-92 inches.

The fluid measures used by apothecaries are—60 mi-
nims make 1 fluid drachm, 8 drachms 1 fluid ounce,

20 fluid ounces 1 pint, 8 pints 1 gallon. (See No. 48.)
Sq. Inches.

The Scottish acre, formerly a standard in Scotland, 1 Sq. Foot.

and now abolished, consisted of 54937:15 imperial 1,296

1 sq. Yard.
39,204 272

301
1 Sg. Pole.

square feet : 23 Scots acres were very nearly equal to
1,568,160 10,890

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
III. MEASURES OF CAPACITY.

equal to one shilling and eightpence English. Hence
1. Solidity.

£100 Scots was equal to the sum of £8, 6s. 8d. in our
1728 cubic inches
I cubic foot.

present money.
27 cubic feet

1 cubic yard.
The ton measurement – 8 barrel bulk, or 40 cubic feet. A second

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.
1 Gill.

365 days 6 hours - 1 Julian year.
13
1 Pint.

365 days 5 hours 48 minutes 49 seconds - ) solar year.
25

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 :-
4 pecks - 1 bushel; 8 bushels - 1 quarter.

The French system of weights and measures is estab-
The peck, bushel, and quarter, are used for dry goods only. lished on a principle much more simple and unerring
There are, besides, the coom - 4 bushels; the wey or load 5 than that in use in England—the former is of universal
quarters; and the last - 2 loads or 10 quarters.

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
Troy

ideal circle going round the globe from pole to pole at
Grains.
1. Avoirdupois.

right angles with the equator-constitutes the basis of

the French system. The length of this fourth of the
27-34375 1 Drachm.

meridian is divided into 10,000,000 parts; a single ten-
1 Ounce.
1 lb.

millionth part is the metre, or the unity of long measure.
98000 3,584

224
1 Stone.

(A metre is equal to 39British inches.)
196000 7,168

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
Grains.

such a cube, is the unity of liquid measures, and is
1 Pennyweight.
1 Ounce.

called the litre. (A litre is equal to about a pint and
5760

1 Pound.

three-quarters, or nearly a quart British measure.).

The weight of a cube of water, measuring on each of
This weight is used in weighing gold, silver, pla- its sides the 100th part of a metre, is the unity of
tina, &c. The fineness of gold is expressed in carats weight, and is called the gramme. A thousand grammes
and grains, the pound or other weight being divided of pure water at its greatest density (about 40 degrees
into 24 carats, and the carat into 4 grains. Thus the of Fahrenheit's thermometer), are of course equivalent
carat pound is the 24th part of the troy pound, or 10 to the litre. (A thousand grammes, forming 1 kilo.
dwts., and the carat grain the 96th part of the troy gramme, weigh about 24 pounds British.)
pound, or 60 troy grains.

These unities being often too great or too small for
In Beer Measure, the barrel contains 4 firkins or 36 common use, they constitute the basis of new unities on
gallons; and the hogshead ld barrel or 54 gallons. the simple decimal principle. The names of these new

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

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used ; if to express division of the original unity, Latin decimal notation, for it allows of adding up columns of
is used, or words slightly modified from it. The Greek decimals all of the same denomination. Their great
words are, deka, for ten, hecto, a hundred, kilo, a thou- excellence, indeed, consists in the uniformity which
sand, and myria, ten thousand. The Latin words are, they give to calculation, and the easy methods which,
decem, for ten, centum, a hundred, and mille, a thousand. by these means, they present of pursuing fractional
These various words are placed before, or prefixed to, numbers to any degree of minuteness.
the principal unity. Thus the decametre is equal to The method of reducing a vulgar to a decimal frac-
ten metres, and the decimetre is the tenth part of a tion is a simple question in Division. For instance,
metre; the hectolitre is equal to 100 litres, and the cen- to reduce 1 to a decimal, we take the 3, and putting two
telitre is the hundredth part of a litre; the kilogramme ciphers after it, divide by 4, thus *.; therefore 75
is equal to a thousand grammes, and the milligramme
is the thousandth part of a gramme.

is the decimal;or, what is the same thing, 75-hundredth
The connection between these weights and measures parts of a whole are equal to three-quarters.
will now be clearly seen. The are is the square deca-
metre; the litre is the cubic decimetre; and the kilo-

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

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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
Taking the four regularly-advancing terms, 15, 45, 405,

work in 8 days, how long will 4 men
and 1215, he proceeds--Suppose that we had only the

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
second terms, in order to do which, as before shown,

both, include fractional parts, they
we divide the larger by the smaller, and this gives us must be reduced to the denomination of the fractions
the ratio 3, with which, by multiplying the third term, before working: thus if one be reduced to shillings,
we produce the fourth ; or let the three terms be the other must be made shillings also; if to pence,
these, 405, 1215, 5, and let it be our wish to find a both must be pence; and so on. If the middle term
fourth which shall bear the same relation to the 15 as be also a compound quantity, it may either be reduced
1215 does to 405. We divide and multiply as before, to its lowest term, before multiplying and dividing by
and the fourth term is produced. And in this manner, the other terms, or you may multiply and divide by
having two numbers or two quantities of any kind, Compound Division and Multiplication.
bearing a certain proportion towards each other, and a If the middle term be reduced to its lowest term,
third, to which we would find a number or quantity the answer will be in that denomination to which it
that should bear a like proportion, in this manner do was reduced; thus if it were brought to farthings, the
we proceed, and thus easily may we find the number answer would be in farthings; if to ounces, the answer
we require.'

would be in ounces.
Referring to the discovered ratio of 45 to 15 to be Example. If 2 cwts. I qr. 7 lbs, sugar cost £8, 148, 4d.,
3, or the fifteenth part—Now,' continues this author, what will 14 cwts. 3 qrs. cost ?
• what would have been the consequence had we multi-

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
à fifteenth part of the second? The consequence would

4
20

4
have been, that we should have had a term or number

9
174

59
fifteen times larger than that required. But this would

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
15. And this is the rule, when the terms are properly simple number, it is

3304
placed-Multiplying the second and third terms together, reduced to its lowest 14868
and dividing the product by the first; this avoids all term-namely, lbs. The 33040
the difficulties arising from the occurrence of fractions third term is therefore
in the course of the process, and gives us in all cases reduced to lbs. also,

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
constitute a proportion. Thus the ratio of 8 to 6 is the The answer is £55, lls.

70
same as that of 12 to 9, and the equality of these two 11 d.
ratios is represented thus:-

The following is a kind of question which often occurs
8:6 12:9, or 8:6:;12:9.

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
second does to the first. Suppose the question be this:
If 3 lbs. of tea cost 9s., how many

800

answer, is to work out 9:3::21

each creditor's share pounds may be purchased for 21s. ?

3

as a distinct account.
-state the terms, with the larger

1000)16000(16
eum last.
Should the question,
9)63

Thus, first, as to A's
16000

share :however, be the reverse - If 7 lbs.

The answer is £199, 168. Following a similar calcu600

7 lbs.

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