(iv) Cut off from the sum of these partial products the required number of decimal places.

For greater accuracy it is preferable to calculate one figure more than necessary, and to cut it off from the

answer.

Thus, to multiply 3·14159 by 19·14383 correctly to three places of decimals,

3.141,59

383 419 1

31 416

28 274

314

126

12

60.142

The correct answer is 60·1420648897.

When any number occurs frequently, it is convenient to make once for all a table of its multiples from 1 to 9, by adding the number to itself eight times. These multiples have then to be simply added, the decimal point being moved as required.

Thus, to find the circumference of a circle in metres the diameter of which is one yard (=91438 metre),

(i) A number is divisible by 2, 4, or 8, when the number formed by the last one, two, or three figures on the right is divisible by 2, 4, or 8 respectively.

(ii) Every number ending with 5 or 0 is divisible by 5 or 10 respectively.

(iii) Every number is divisible by 3 or 9, when the sum of the digits is divisible by 3 or 9 respectively.

(iv) A number is divisible by 11, when the difference between the sum of the figures in the odd places, and the sum of those in the even places is divisible by 11, or is zero.

The division of decimals may be shortened as follows:

Make the number of decimal places equal in the divisor and in the dividend by annexing ciphers or the next figures of a recurring period. Divide in the ordinary way, but instead of bringing down ciphers cut off a figure from the divisor, remembering to carry from the multiple of the figure cut off.

The simple form of proportion usually required in chemical sums, is treated of under the name of 'Rule of Three,' and can be dealt with without making use of the signs of proportion.

Thus if 100 lbs. of mercury form 108 lbs. of mercuric oxide, how much oxide will be formed by 80 lbs. of mercury? Since 100 lbs. of mercury form 108 lbs. of mercuric oxide,

*In comparing two quantities of the same kind of thing expressed in the same denomination, the number of times that one is of the other is called the ratio of that one to that

When two ratios are equal the four numbers form a proportion, which is written a : b :: c: d, or a : b = c : d.

In any proportion the product of the extremes is equal to the product of the means ad = bc. Hence if three terms are given in a proportion the fourth can be found,

Thus, if 100 lbs. of mercury form 108 lbs. of mercuric oxide, how much oxide will be formed by 80 lbs. of mercury?

(i) Commencing with the unit figure cut off periods of two figures each in both directions. 365-3650.

(ii) The first figure of the required root will be the square root of the left-hand period, or of the greatest square contained in it.

(iii) Subtract the square of this figure from the first period; and to the remainder annex the next period for dividend; and for part of the divisor double the portion of the root already obtained.

(iv) Try how often this part of the divisor is contained in the dividend wanting the last figure, and annex the figure thus found to the parts of the root and divisor already found.

(v) Then multiply and subtract as in division, to the remainder bring down the next period and proceed as before.

(vi) When all the whole numbers have been used, proceed to find decimals, the period consisting of two decimal figures or of two ciphers.