each figure of the multiplier is placed under the first figure of the multiplicand which is used with it in forming the first _four_ places of decimals. And here observe, that the units’ figure in the multiplier 51·30742, viz. 1, comes under 4, the _fourth_ decimal place in the multiplicand. If there had been no carrying from the right of the vertical line, the rule would have been: Reverse the multiplier, and place it under the multiplicand, so that the figure which was the units’ figure in the multiplier may stand under the last place of decimals in the multiplicand which is to be preserved; place ciphers over those figures of the multiplier which have none of the multiplicand above them, if there be any: proceed to multiply in the usual way, but begin each figure of the multiplier with the figure of the multiplicand which comes above it, taking no account of those on the right: place the first figures of all the lines under one another. To correct this rule, so as to allow for what is carried from the right of the vertical line, observe that this consists of two parts, 1st, what is carried directly in the formation of the different lines, and 2dly, what is carried from the addition of the first column on the right. The first of these may be taken into account by beginning each figure of the multiplier with the one which comes on its right in the multiplicand, and carrying the tens to the next figure as usual, but without writing down the units. But both may be allowed for at once, with sufficient correctness, on the principle of (151), by carrying 1 from 5 up to 15, 2 from 15 up to 25, &c.; that is, by carrying the nearest ten. Thus, for 37, 4 would be carried, 37 being nearer to 40 than to 30. This will not always give the last place quite correctly, but the error may be avoided by setting out so as to keep one more place of decimals in the product than is absolutely required to be correct. The rule, then, is as follows: [20] The 1′ here means that the 1 is in the multiplier. 154. To multiply two decimals together, retaining only _n_ decimal places. I. Reverse the multiplier, strike out the decimal points, and place the multiplier under the multiplicand, so that what was its units’ figure shall fall under the _n_ᵗʰ decimal place of the multiplicand, placing ciphers, if necessary, so that every place of the multiplier shall have a figure or cipher above it. II. Proceed to multiply as usual, beginning each figure of the multiplier with the one which is in the place to its right in the multiplicand: do not set down this first figure, but carry its _nearest_ ten to the next, and proceed. III. Place the first figures of all the lines under one another; add as usual; and mark off _n_ places from the right for decimals. It is required to multiply 136·4072 by 1·30609, retaining 7 decimal places. 1364072000 906031 ---------- 1364072000 409221600 8184432 122766 ----------- 178·1600798