When a bar is written over a number, thus, 7 let the number be called negative, and let it be thus used: Let it be augmented by additions of its own species, and diminished by subtractions; thus, 7 and 2 give 9 , and let 7 with 2 subtracted give 5 . But let the addition of a number without the bar diminish the negative number, and the subtraction increase it. Thus, 7 and 4 are 3 , 7 and 12 make 5, 7 with 8 subtracted is 15 . In fact, consider 1, 2, 3, &c., as if they were gains, and 1 , 2 , 3 , as if they were losses: let the addition of a gain or the removal of a loss be equivalent things, and also the removal of a gain and the addition of a loss. Thus, when we say that 4 diminished by 11 gives 7, we say that a loss of 4 incurred at the moment when a loss of 11 is removed, is, on the whole, equivalent to a gain of 7; and saying that 4 diminished by 2 is 6 , we say that a loss of 4, accompanied by the removal of a gain of 2, is altogether a loss of 6.

By the characteristic of a number understand as follows: When there are places before the decimal point, it is one less than the number of such places. Thus, 3·214, 1·0083, 8 (which is 8·00 ...) 9·999, all have 0 for their characteristics. But 17·32, 48, 93·116, all have 1; 126·03 and 126 have 2; 11937264·666 has 7. But when there are no places before the decimal point, look at the first decimal place which is significant, and make the characteristic negative accordingly. Thus, ·612, ·121, ·9004, in all of which significance begins in the first decimal place, have the characteristic 1 ; but ·018 and ·099 have 2 ; ·00017 has 4 ; ·000000001 has 9 .

To find the characteristic of a quotient, subtract the characteristic of the divisor from that of the dividend, carrying one before subtraction if the first significant figures of the divisor are greater than those of the dividend. For instance, in dividing 146·08 by ·00279. The characteristics are 2 and 3 ; and 2 with 3 removed would be 5. But on looking, we see that the first significant figures of the divisor, 27, taken by themselves, and without reference to their local value, mean a larger number than 14, the first two figures of the dividend. Consequently, to 3 we carry 1 before subtracting, and it then becomes 2 , which, taken from 2, gives 4. And this 4 is the characteristic of the quotient, so that the quotient has 5 places before the decimal point. Or, if abcdef be the first figures of the quotient, the decimal point must be thus placed, abcde·f. But if it had been to divide ·00279 by 146·08, no carriage would have been required; and 3 diminished by 2 is 5 ; that is, the first significant figure of the quotient is in the 5th place. The quotient, then, has ·0000 before any significant figure. A few applications of this rule will make it easy to do it in the head, and thus to assign the meaning of the first figure of the quotient even before it is found.

APPENDIX VI.
ON DECIMAL MONEY.

Of all the simplifications of commercial arithmetic, none is comparable to that of expressing shillings, pence, and farthings as decimals of a pound. The rules are thereby put almost upon as good a footing as if the country possessed the advantage of a real decimal coinage.

Any fraction of a pound sterling may be decimalised by rules which can be made to give the result at once.

Two shillings is£·100
One shilling is£·050
Sixpence is£·025
One farthing is£·00104⅙

Thus, every pair of shillings is a unit in the first decimal place; an odd shilling is a 50 in the second and third places; a farthing is so nearly the thousandth part of a pound, that to say one farthing is ·001, two farthings is ·002, &c., is so near the truth that it makes no error in the first three decimals till we arrive at sixpence, and then 24 farthings is exactly ·025 or 25 thousandths. But 25 farthings is ·026, 26 farthings is ·027, &c. Hence the rule for the first three places is

One in the first for every pair of shillings; 50 in the second and third for the odd shilling, if any; and 1 for every farthing additional, with 1 extra for sixpence.

Thus,0s.3½d. = £·014
0s.7¾d. = £·032
1s.2½d. = £·060
1s.11¼d. = £·096
2s.6d. = £·125
2s.9½d. = £·139
3s.2¾d. = £·161
13s.10¾d. = £·694