Now imagine the integer n to increase without limit, then 1 the limit zero, whilst at the same time the measure, viz. two terms of the scale tends to the limit zero. of the ratio of the Now when the terms of a scale are given, they determine a ratio, and also its measure. Conversely, when the measure of a ratio is given, the corresponding scale is determined. If then the measure of a certain ratio is zero, it is possible to say either that there is no corresponding scale, or that there is one and only one corresponding scale. The latter alternative is the one implied in the text. In this connection the following proposition is of interest. If the scale of A, B is the same as that of C, D; if A can be made as small as we please, and if B and D be fixed magnitudes, then C can be made smaller than any magnitude E, however small E may be. It is possible by Archimedes' Axiom to choose ʼn so that So that when the first term of the scale of A, B tends to zero, so also does the first term of the scale of C, D. Another proposition of a similar kind is this: If the scale of A, B is the same as that of C, D; if A can be made as small as we please, if C and D be fixed magnitudes, then B can be made smaller than any magnitude E, however small E may be. It is possible by Archimedes' Axiom to choose n so that nC > D. |