units numeral-wheel to the sight or reading line of the machine; we would then strike the 4 key of the same series, which would bring 13 to the reading line, the ,3 being on the units numeral-wheel and the 1 on the tens numeral-wheel. By the operation of the Felt subtraction cut-off, this error of 1 on the tens numeral-wheel will be prevented, and only the true remainder, 3, will appear on the reading line' of the machine.
Thé cause of this error is manifest when we look into the nature of the operation of additive subtraction. We have, as illustrated in the foregoing example, performed subtraction by adding the complement of the subtrahend, 6, which is 10 — 6, or 4, to the minuend, 9, making the result 13, or 10 in excess of the true remainder. In this operation we have committed the obvious error of increasing the minuend by 4 instead of decreasing it by 6, and the measure of this error is plainly the sum of 6 and 4, or 10. In other words, the measure of the error is the power of ten from which the subtrahend is subtracted to produce the complement. It follows that the error in the remainder in any given example of additive subtraction is 10, or 100, or 1000, or 10,000, according to the power of 10 from which the subtrahend may be subtracted, and, since naughts count for nothing, that this error will- be represented by-1 on the left of the true remainder.
In referring to this feature of the Felt machine, the patent says:
“Another object of my invention is to prevent the carrying of tens from any column to the next higher whenever a subtraction is made by means of adding a complementary number, as hereinafter explained.”
We have here the underlying conception of the Felt subtraction cut-off, which was so to organize his machine as to prevent the opera-’ tion of the carrying, mechanism whenever it is necessary to prevent it in performing additive subtraction. That the error in' question is caused by the carrying mechanism, and may be prevented by preventing the operation of a particular carry, clearly appears from the following considerations:
As the tens are carried mentally in arithmetical addition, so an adding machine in its normal operation carries the tens mechanically by carrying 1 to the numeral wheel of the next higher order, as from the units wheel to the tens wheel, and from the tens wheel to the hundreds wheel, and so on throughout the series of wheels. Now, it is apparent that this error of 1 on the left of the true remainder is due in each example to the particular carry of 1 either from the units column to the tens column, or from the tens column to the hundreds column, or from the hundreds column to the thousands column, dependent in each case upon the power of 10 from which the complement is subtracted.
For instance, in the example already given of subtracting 6 from *9 by additive subtraction, the error of 1 in the tens column was caused by the carrying of 1 from the units column; and, if this carry had not taken-place, the result would have been the true remainder, 3, on the reading line of the machine. So, if the example had been to subtract 66 from 99, the error of 1 in the hundreds column would have been caused by the carrying of 1 from thé tens column to the hundreds