80 81 82 83 84 ---- + ---- + ---- + ---- + ---- + &c. = 9·88286. 81 82 83 84 85 151. We now enter upon methods by which unnecessary trouble is saved in the computation of decimal quantities. And first, suppose a number of miles has been measured, and found to be 17·846217 miles. If you were asked how many miles there are in this distance, and a rough answer were required which should give miles only, and not parts of miles, you would probably say 17. But this, though the number of whole miles contained in the distance, is not the nearest number of miles; for, since the distance is more than 17 miles and 8 tenths, and therefore more than 17 miles and a half, it is nearer the truth to say, it is 18 miles. This, though too great, is not so much too great as the other was too little, and the error is not so great as half a mile. Again, if the same were required within a tenth of a mile, the correct answer is 17·8; for though this is too little by ·046217, yet it is not so much too little as 17·9 is too great; and the error is less than half a tenth, or ¹/₂₀. Again, the same distance, within a hundredth of a mile, is more correctly 17·85 than 17·84, since the last is too little by ·006217, which is greater than the half of ·01; and therefore 17·84 + ·01 is nearer the truth than 17·84. Hence this general rule: When a certain number of the decimals given is sufficiently accurate for the purpose, strike off the rest from the right hand, observing, if the first figure struck off be equal to or greater than 5, to increase the last remaining figure by 1. The following are examples of a decimal abbreviated by one place at a time. 3·14159, 3·1416, 3·142, 3·14, 3·1, 3·0 2·7182818, 2·718282, 2·71828, 2·7183, 2·718, 2·72, 2·7, 3·0 1·9919, 1·992, 1·99, 2·00, 2·0