192

HISTORICAL VIEW OF THE

whose common multiplier is likewise 10, it is evidentthat the second term will he 100, or 10 in the powerwhose index is 2 ; the third will be 1000, or 10in the power of 3, &c. But since in algebra thenotion of powers includes those whose indices are anynumbers, whole or fractional, there may he terms inter-mediate to these, which shall be powers of 10whose indices are fractions, or mixed numbers interme-diate to the whole indices. Upon this principle weshould have, for example, 40, as a term in the seriesbetween 10 and 100, which is a power of 10 whoseindex is the mixed number 1-602. Again, we maycarry the series backward to numbers below 10, andwe find 5, a power of 10 whose index is the frac-tion 0-6989 ; and 1 is, upon a well known algebra-ical principle, the power of 10 whose index is 0.

In a word, all the natural numbers may find theirplaces, by interpolation somewhere among the terms ofsuch a series, and the corresponding indices are calledtheir logarithms.* Different systems of logarithmswill be formed, according as different geometrical seriesare assumed. To find the means of assigning suchseries, and to calculate the indices belonging to thenatural numbers, or, conversely, the numbers from theindices, has been the subject of the profound labours ofmodern analysts, who have devised various methodsfor the purpose: but the undivided honour of havingfirst accomplished such a work belongs exclusively toNapier.

It has been said, that a hint of some method of thesame kind, ascribed to Longomontanus, had been givento Napier; but, from the total absence of all traces ofsuch a principle in the writings of the former, and evenof any claim or pretence to it, we can attach no im-portance to the story. Napier associated in his labourshis friend Mr. Briggs, who conversed freely with himon the improvements of which his system was sus-ceptible. Napier, however, published his “ Canon* A oyot, ratio, xaidpos, number; the numbers or measures of ratios.