JOHN NAPIER. 91
by 2 it gives 4 ; which divided by 1 still remains 4 ; while, in the arithmetical proportion, 1 is to 2 as 2 is to 3 ; if 2 be added to 2 it gives 4 ; from which if 1 be subtracted, there remains the fourth term 3. It is plain, therefore, that, especially where large numbers are concerned, operations by arithmetical must be much more easily performed than operations by geometrical proportion ; for, in the one case you have only to add and subtract, while in the other you have to go through the greatly more laborious processes of multiplication and division.
" Now, it occurred to Napier, reflecting upon this important distinction, that a method of abbreviating the calculation of a geometrical proportion might per- haps be found, by substituting, upon certain fixed principles, for its known terms, others in arithmetical proportion, and then finding, in the quantity which should result from the addition and subtraction of these last, an indication of that which should have resulted from the multiplication and division of the original figures. It had been remarked before this, by more than one writer, that if the series of numbers 1, 2, 4, 8, &c., that proceed in geometrical pro- gression, that is, by a continuation of geometrical ratios, were placed under or along side of the series 0, 1, 2, 3, &c., which are in arithmetical progression, the addition of any two terms of the latter series would give a sum, which would stand opposite to a number in the former series indicating the product of the two terms in that series, which corresponded in place to the two in the arith- metical series first taken. Thus, in the two lines,
1, 2, 4, 8, 16, 32, 64, 128, 256, 0, 1, 2, 3, 4, 5, 6, 7, 8,
the first of which consists of numbers in geometrical, and the second of numbers in arithmetical progression, if any two terms, such as 2 and 4, be taken from the latter, their sum 6, in the same line, will stand opposite to 64 in the other, which is the product of 4 multiplied by 16, the two terms of the geometrical series which stand opposite to the 2 and 4 of the arithmetical. It is also true, and follows directly from this, that if any three terms, as, for instance, 2, 4, 6, be taken in the arithmetical series, the sum of the second and third, diminished by the subtraction of the first, which makes 8, will stand opposite to a number (256) in the geometrical series which is equal to the product of 16 and 64 (the opposites of 4 and 6), divided by 4 (the opposite of 2).
" Here, then, is, to a certain extent, exactly such an arrangement or table as Napier wanted. Having any geometrical proportion to calculate, the known terms of which were to be found in the first line or its continuation, he could substitute for them at once, by reference to such a table, the terms of an arith- metical proportion, which, wrought in the usual simple manner, would give him a result that would point out or indicate the unknown term of the geometrical proportion. But, unfortunately, there were many numbers which did not occur in the upper line at all, as it here appears. Thus, there were not to be found in it either 3, or 5, or 6, or 7, or 9, or 10, or any other numbers, indeed, ex- cept the few that happen to result from the multiplication of any of its terms by two. Between 128 and 256, for example, there were 127 numbers wanting, and between 256 and the next term (512) there would be 255 not to be found.
" We cannot here attempt to explain the methods by which Napier's ingenuity succeeded in filling up these chasms, but must refer the reader, for full informa- tion upon this subject, to the professedly scientific works which treat of the his- tory and construction of logarithms. Suffice it to say, that he devised a mode by which he could calculate the proper number to be placed in the table over against any number whatever, whether integral or fractional The new numeri- cal expressions thus found, he called Logarithms, a term of Greek etymology,