PHYSICAL AND MATHEMATICAL SCIENCES. 191
The first suggestion of the principle may be thusexplained: — If we suppose a series of numbers ingeometrical progression, any one tolerably conversantwith arithmetic will see that, if any two terms insuch a progression are multiplied together, the productwill also be a term in the same series, and will befound by inspection, if the series has in the first instancebeen carried far enough. And we can always tell atwhat term it will be found : it will be that term whosenumber, reckoning from the commencement, is thesum of those of the two terms which are to be multi-plied. Thus, to find the product of the third and seventhterms, we have only to take that number which formsthe tenth in the series. It is also evident, that thesenumbers of the terms are also the indices of the powersof the common multiplier, which enter into each termrespectively. This is the leading idea which may besupposed suggested by a passage in Archimedes .
Thus, if our computations always involved no othernumbers than such as are terms in a geometrical pro-gression, we should only have to add the indices, andthus be led directly to the product; or, conversely, tosubtract them, and find the term which is the quotient.Again, by doubling, tripling, &c. the indices, weshould find their products indicating a term whichwould be respectively the square, cube, &c.of the originalterm ; and, conversely, we should have the square, cube,&c. roots. Thus far all was sufficiently clear and simple.
But here arose the main difficulty: this would applyonly to a very few limited systems of numbers, andcould not be of any general practical utility. Thegrand discovery of Napier, therefore, amounted to this:—that a geometrical progression may be found in whichall the natural numbers are terms.
Of the methods by which he arrived at this conclu-sion, or of the general principle on which such a seriescan be assigned, we cannot here say much ; but wemay sufficiently illustrate it by an example. If wesuppdse a geometrical series whose first term is 10 and