mucT 20.

LOGARITHMS.

309

bers to unity, the ratio of equality, or of unity to unity, mustbe represented by 0; because that ratio has properly no mag-nitude, since, when it is added to, or subtracted from, anyother ratio, it neither increases nor diminishes it.

The second observation that occurred to them was, thatany number whatever might be chosen at pleasure for therepresentative of the ratio of any given natural number tounity; but that, when once such choice was made, all theother representative numbers would be thereby determined,because they must be greater or less than that first represen-tative number, in the same proportions in which the ratiosrepresented by them, or the ratios of the corresponding na-tural numbers to unity, were greater or less than the ratio ofthe said given natural number to unity. Tlius, either 1, or2, or 3, &c, might be chosen for the representative of theratio of 10 to 1. But, if 1 be chosen for it, the representa-tives of the ratios of 100 to 1 and 1000 to 1, which are doubleand triple of the ratio of 10 to 1, must be 2 and 3, and can-not be any other numbers; and, if 2 be chosen for it, the re-presentatives of the ratios of 100 to 1 and 1000 to l, will be4 and 6, and cannot be any other numbers; and, if 3 be cho-sen for it, the representatives of the ratios of 100 to 1 and1000 to 1, will be 6 and 9, and cannot be any other numbers;and so on.

The third observation that occurred to them was, that, asthese artificial numbers were representatives of, or propor-tional to, ratios of the natural numbers to unity, they mustbe expressions of the numbers of some smaller equal ratiosthat are contained in the said ratios. Thus, if 1 be taken forthe representative of the ratio of 10 to 1, then 3, which is therepresentative of the ratio of 1000 to 1, will express the num-ber of ratios of 10 to 1 that are contained in the ratio of 1000to 1. And if, instead of 1, we make 10,000,000, or ten mil-lions, the representative of the ratio of 10 to 1, (in which case1 will be the representative of a very small ratio, or ratiuncula ,which is only the ten-millionth part of the ratio of 10 to 1,or will be the representative of the 10,000,000th root of 10,