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far as the (p+q)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 X 10m+n, rı being less than d, and n less than 10m. Then by (1) the quotient may be correctly found by dividing rı X 10m by d. Let d = 10 di tan; then ri X 10m ri X 10m

di 11

X 10m - 1 d


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

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 r2 X 10m-2

Agra by d, (if di = 10 X d, taz), the error being x 10m – 2. The first

da da figure of this quotient will be found by dividing r, by dg. 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 a, 12

am 'm
x 10m

da da

dm ay a, &c. being the digits of the divisor ; rar, &c. being the successive remainders ; d. &c. the successive divisors. Now d, containing ni ti digits, is greater than 10m, and less than 10m + 1, and d, d, &c., are respectively greater than 10 m - 1, 10m - 2 &c. and less than 10m, 10m — Also ay a, &c., are all less than 10; rır, &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 By taking note of the rejected digits of the divisor in forming the subtrahends, so as to obtain the first figures correct, the remainders r1, 72, &c. will be diminished, and the errors in consequence. Hence the Rule prescribed is evident.

-x 10m


- 1 dm

1 &c.




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, S + S2 +

(A 1+rT 1+r T2 i tot 2. At the equated time:

TS, (T, -t) r Si (t - T,) =


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

S, T, + S, T,


Sa+s, Secondly by considering the position of the debtor. 1. At the present time:

no S, T, q S, T2 r(S1 + S2)


1+r T 1+rT, 1+rt
wbich is obtainable from (A)
2. At the equated time:-


r S2 (T, -t)
r S(t – Ti) x

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

r S, T, r S, T,


=r(S1 + S,)t (E) 1+r (T, – t) 1+1(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 S+S, at time t makes no use of his acquired interest, then formulæ (D) and (E) become the

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-r11 + 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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