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- fractions taken together. Now if all the fractions have the same denominator, they indicate that the products of the quantity, upon which the operations are supposed to be performed, by each numerator, are all to be divided by the same number, and added; but the sum of the quotients of several quantities is equal to the quotient of the sum (Prop. 12); therefore the result may be obtained by adding the products, and dividing the sum by the denominator; or-since the sum of the products by several numbers is equal to the product of their sum (Prop. 8)—by multiplying by the sum of the numerators, and dividing by the denominator. Hence the sum of several abstract fractions with the same denominator is equivalent to the fraction, whose numerator is the sum of the numerators, and denominator the common denominator. Therefore if fractions be brought to a common denominator, they may be readily added.

In a similar manner the method of Subtraction of fractions may be explained.


Note to Prop. 41.- To explain the abbreviated method of

Multiplication of decimals. If it be required to find the product of two decimals correct only to a given number of figures, (for instance 6,) evidently there is no need to trouble ourselves with any more than 6 places of decimals in the partial products, if we can find these correctly, provided that we also know what is to be carried from the sum of the figures in the 7th place.

Now a figure in the 6th place of decimals in any product may arise from the multiplication of a number of units, tens, &c. by a number in the 6th, 7th, &c. places respectively, or of a number in the 1st, 2nd, &c. places by a number in the 5th, 4th, &c. places respectively, these products being increased by the numbers carried from previous multiplications. Hence if units, tens, &c. tenths, hundredths, &c. in the multiplicand be placed over figures in the 6th, 7th, &c. 5th, 4th, &c. places in the multiplier, each figure in the multiplier will stand under the first figure of the multiplicand, whose complete product is to form part of the total. But this arrangement exactly reverses the order of the figures of the multiplier, and places its units' figure under the last place of decimals of the multiplicand which is to be preserved, viz. the 6th. Hence the rule prescribes that this be done. This being done, the several products may be formed by multiplying by each figure of the multiplier, the figures of the multiplicand, as far as that which is over the particular multiplier being used. The first figures of these products being all in the 6th place of decimals (or in the last which is to be reserved), must all be ranged under each other for addition. But in order to obtain these first figures at all correctly, it will be necessary to commence each multiplication mentally with the figure to the right of the multiplier, so to find the number to be carried. At the same time an allowance may be made for the carriage from the sum of the figures in the 7th place by carrying not the actual number, which appears due, but the nearest number of tens; i.e. if the product be between 5 and 15 carry 1, if between 15 and 25 carry 2, and so on.

As an example, let the number 999.9 be multiplied by 99.9, according to the above Rule, so as to retain six places of decimals :999.99999999


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In this case, (which is an extreme one, all the figures being 9’s) it appears that not one out of the six decimal figures is correct; but at the same time it is seen that the product obtained differs from the true product by a number less than .00001; so that even in this extreme case the error cannot be very great; and the error may be lessened by increasing the number of decimal places reserved. Thus if 6 decimal figures are required, and it be attempted to obtain 7, the error will be less than .00000). Of course the larger the quantities involved in a question, which are affected by such products, the greater number of places should be reserved.

Note to Prop. 42.-To explain the abbreviated method of

Division of decimals.

1. Let d be a divisor, containing m + 1 digits, D a dividend: let D=N X 10m + n, n being the number composed of the last m digits in D: then

D N X 10m n.


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and, as n is less than 10m, and d is not less than 10m, is a proper fraction ; therefore the integral quotient of D by d is the same as that of N X 10m


by d.

2. If a divisor contain a decimal places, and p be required in the quotient, the dividend must be made to have p + q decimal places; and the figures of the quotient will then be found by dividing the dividend, so far as the (p + g)th decimal place, by the divisor, considered as whole numbers.

3. Let now d be the number composed of the figures of a divisor containing 9, D that composed of those of a dividend containing p + 4, decimal places ; let there be m + 1 digits in all in d: let D be divided by d till there remain to be found m figures of the quotient; then the remainder, from which these m figures have to be obtained, will be equal to rı X10m +n, rı being less than d, and n less than 10m. Then by (1) the quotient may be correctly found by dividing ri X 10m by d.

Let d = 10 di tau; then

ni X 10m

riX 10m

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X 10m - 1


12 X 10m


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d di or the quotient may be found by dividing rı X 10m - 1 by dı, the error being airi

x 10m – 1. The first figure of this quotient will be obtained from d di the division of r, by dı; let it be found, and let r, be the remainder: then

1 has to be divided by dı : but, as before, it may be shown that the quotient of r2 X 10m - 1 by d, may be found by dividing r, X 10m—2 by d, (if di = 10 X d, taz), the error being x 10m — 2. The first

dy do figure of this quotient will be found by dividing r, by dz. In a similar manner it may be shown that all the figures of the quotient may be obtained by rejecting, one by one, the figures of the divisor, and dividing the remainders by the other figures. The sum of the errors committed in this process is a1 11 d2 T 2

am rem
x 10m

? + +
d dı
da da

da a, a, &c. being the digits of the divisor; rir, &c. being the successive remainders; d, d, &c. the successive divisors. Now d, containing n + 1 digits, is greater than 10m, and less than 10m + 1, and dy d2 &c., are respectively greater than 10 m - 1, 10m 2 &c. and less than 10m, 10m - 1 &c. Also a, a, &c., are all less than 10; rı r, &c., being respectively less than d, d, &c. are less than 10m +1, 10m &c. therefore each term of the above sum is less than 102 ; and the whole less than m X 102. Consequently the

m X 102 error in the quotient is less than

10p 10P By taking note of the rejected digits of the divisor in forming the subtrahends, so as to obtain the first figures correct, the remainders r 1, 72, &c. will be diminished, and the errors in consequence. Hence the Rule prescribed is evident.


x 10m


- 1 dm

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Note to Prop. 84.




The only true principle, on which the equated time should be calculated, is, that the interest of the creditor and debtor shall appear both unaffected, at whatever time they be considered. But if Simple Interest only be reckoned, we obtain a different value for the equated time, for every different time, at which we view the subject, and different also according as we consider the position of the creditor or of the debtor, except we regard them at the present time. The formulæ obtained are the following:

First by considering the position of the creditor. 1. At the present time :


S, Si + S2 +

(A) 1+rT 1+r T2 1 trt 2. At the equated time:

rS, (T, -t) r Si (t - T1) =


1+r(T, -t) 3. At the end of the transactions :

Si Ti + S, T, t =


Si + S2
Secondly by considering the position of the debtor.
1. At the present time:-
no S, T, q S, T,

r (S, +S2) t
1+r Ti

which is obtainable from (A)
2. At the equated time:

1-5 Ti r S, (T, – t) r Si (t - Tı) x

(D) 1-pt 1+r(T, -t) which is nearly the same as (B). 3. At the end of the transactions:

r(t – T1) r Si Ti


=r(Si +52) + (E) 1+(T,

1+r(T, - t) These formulæ are calculated on the supposition that the debtor makes the best use of the money, the payment of which he is allowed to defer, putting it out to interest till it is required for payment, and then putting the acquired interest out again. Thus in fact Compound Interest is allowed to the debtor, while the creditor has only Simple Interest. If we suppose that the debtor, after having paid S, at time T1, or Si + S, at time t makes no use of his acquired interest, then formulæ (D) and (E) become the

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same as (B) and (C). Still we shall have 3 different values for the equated time; and yet the principle on which each has been obtained is the same.

If, however, we allow Compound Interest, we shall find but one value, at whatever time we consider the subject, and whether we regard the position of creditor or debtor. The formula obtained is

S, e-rTi + S, e-rT2 = (S, +52) e-rt e being the base of the Napierian system of logarithms. If the second and higher powers of r be neglected, this formula, as all the others, gives that on which the rule is founded.

The End.


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