we obtain successive dividends corresponding to quotients x − 200, x − 230 and x − 231. The original dividend is written as 0987063, since its initial figures are greater than those of the divisor; if the dividend had commenced with (e.g.) 3 ... it would not have been necessary to insert the initial 0. At each stage of the division the number of digits in the reduced dividend is decreased by one. The final dividend being 0000, we have x − 231 = 0, and therefore x = 231.
107. Methods of Division.—What are described as different methods of division (by a single divisor) are mainly different methods of writing the successive figures occurring in the process. In long division the divisor is put on the left of the dividend, and the quotient on the right; and each partial product, with the remainder after its subtraction, is shown in full. In short division the divisor and the quotient are placed respectively on the left of and below the dividend, and the partial products and remainders are not shown at all. The Austrian method (sometimes called in Great Britain the Italian method) differs from these in two respects. The first, and most important, is that the quotient is placed above the dividend. The second, which is not essential to the method, is that the remainders are shown, but not the partial products; the remainders being obtained by working from the right, and using complementary addition. It is doubtful whether the brevity of this latter process really compensates for its greater difficulty.
The advantage of the Austrian arrangement of the quotient lies in the indication it gives of the true value of each partial quotient. A modification of the method, corresponding with D of § 101, is shown in G; the fact that the partial product 08546 is followed by two blank spaces shows that the figure 2 represents a partial quotient 200. An alternative arrangement, corresponding to E of § 101, and suited for more advanced work, is shown in H.
108. Division with Remainder.—It has so far been assumed that the division can be performed exactly, i.e. without leaving an ultimate remainder. Where this is not the case, difficulties are apt to arise, which are mainly due to failure to distinguish between the two kinds of division. If we say that the division of 41d. by 12 gives quotient 3d. with remainder 5d., we are speaking loosely; for in fact we only distribute 36d. out of the 41d., the other 5d. remaining undistributed. It can only be distributed by a subdivision of the unit; i.e. the true result of the division is 3512d. On the other hand, we can quite well express the result of dividing 41d. by 1s (= 12d.) as 3 with 5d. (not “5”) over, for this is only stating that 41d. = 3s. 5d.; though the result might be more exactly expressed as 3512s.
Division with a remainder has thus a certain air of unreality, which is accentuated when the division is performed by means of factors (§ 42). If we have to divide 935 by 240, taking 12 and 20 as factors, the result will depend on the fact that, in the notation of § 17, 
In incomplete partition the quotient is 3, and the remainders 11 and 17 are in effect disregarded; if, after finding the quotient 3, we want to know what remainder would be produced by a direct division, the simplest method is to multiply 3 by 240 and subtract the result from 935. In complete partition the successive quotients are 771112 and 317111220 = 3215240. Division in the sense of measuring leads to such a result as 935d. = £3, 17s. 11d.; we may, if we please, express the 17s. 11d. as 215d., but there is no particular reason why we should do so.
109. Division by a Mixed Number.—To divide by a mixed number, when the quotient is seen to be large, it usually saves time to express the divisor as either a simple fraction or a decimal of a unit of one of the denominations. Exact division by a mixed number is not often required in real life; where approximate division is required (e.g. in determining the rate of a “dividend”), approximate expression of the divisor in terms of the largest unit is sufficient.
110. Calculation of Square Root.—The calculation of the square root of a number depends on the formula (iii) of § 60. To find the square root of N, we first find some number a whose square is less than N, and subtract a2 from N. If the complete square root is a + b, the remainder after subtracting a2 is (2a + b)b. We therefore guess b by dividing the remainder by 2a, and form the product (2a + b)b. If this is equal to the remainder, we have found the square root. If it exceeds the square root, we must alter the value of b, so as to get a product which does not exceed the remainder. If the product is less than the remainder, we get a new remainder, which is N − (a + b)2; we then assume the full square root to be c, so that the new remainder is equal to (2a + 2b + c)c, and try to find c in the same way as we tried to find b.
An analogous method of finding cube root, based on the formula for (a + b)3, used to be given in text-books, but it is of no practical use. To find a root other than a square root we can use logarithms, as explained in § 113.
(ii.) Approximate Calculation.
111. Multiplication.—When we have to multiply two numbers, and the product is only required, or can only be approximately correct, to a certain number of significant figures, we need only work to two or three more figures (§ 83), and then correct the final figure in the result by means of the superfluous figures.
A common method is to reverse the digits in one of the numbers; but this is only appropriate to the old-fashioned method of writing down products from the right. A better method is to ignore the positions of the decimal points, and multiply the numbers as if they were decimals between ·1 and 1·0. The method E of § 101 being adopted, the multiplicand and the multiplier are written with a space after as many digits (of each) as will be required in the product (on the principle explained in § 101); and the multiplication is performed from the left, two extra figures being kept in. Thus, to multiply 27·343 by 3·1415927 to one decimal place, we require 2 + 1 + 1 = 4 figures in the product. The result is 085·9 = 85·9, the position of the decimal point being determined by counting the figures before the decimal points in the original numbers.
112. Division.—In the same way, in performing approximate division, we can at a certain stage begin to abbreviate the divisor, taking off one figure (but with correction of the final figure of the partial product) at each stage. Thus, to divide 85·9 by 3·1415927 to two places of decimals, we in effect divide ·0859 by ·31415927 to four places of decimals. In the work, as here shown, a 0 is inserted in front of the 859, on the principle explained in § 106. The result of the division is 27·34.
113. Logarithms.—Multiplication, division, involution and evolution, when the results cannot be exact, are usually most simply performed, at any rate to a first approximation, by means of a table of logarithms. Thus, to find the square root of 2, we have log √2 = log (212) = 12 log 2. We take out log 2 from the table, halve it, and then find from the table the number of which this is the logarithm. (See Logarithm.) The slide-rule (see Calculating Machines) is a simple apparatus for the mechanical application of the methods of logarithms.
When a first approximation has been obtained in this way, further approximations can be obtained in various ways. Thus, having found √2 = 1·414 approximately, we write √2 = 1·414 + θ, whence 2 = (1·414)2 + (2·818)θ + θ2. Since θ2 is less than 14 of
