When a bar is written over a number, thus, 7︤ 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, 7︤ and 2︤ give 9︤, and let 7︤ with 2︤ subtracted give 5︤. But let the _addition_ of a number without the bar _diminish_ the negative number, and the _subtraction increase_ it. Thus, 7︤ and 4 are 3︤, 7︤ and 12 make 5, 7︤ with 8 subtracted is 1︦5. In fact, consider 1, 2, 3, &c., as if they were gains, and 1︤, 2︤, 3︤, 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 4︤ diminished by 1︦1 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 4︤ diminished by 2 is 6︤, 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 1︤; but ·018 and ·099 have 2︤; ·00017 has 4︤; ·000000001 has 9︤. 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 3︤; and 2 with 3︤ 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 3︤ we carry 1 before subtracting, and it then becomes 2︤, 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 3︤ diminished by 2 is 5︤; 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.