(B) Propositions 35 and 56 are so nearly alike in form that the difference between them should be carefully noted. it appears that the positions of the scales [T, U], [U, V] in Prop. 35 are interchanged in Prop. 56. Art. 214. NOTE 12. ON PROP. 58. The statement on lines 3-5 of Page 118 that two scales in which the first term is zero may be considered to be the same may present some difficulty, inasmuch as a relative multiple scale presupposes the existence of two magnitudes, and the relative multiple scale of zero and a magnitude has not been defined. Without going fully into the subject, which here touches upon the difficulties of the Infinitesimal Calculus, it may be sufficient to remark that since A Now imagine the integer n to increase without limit, then tends to the n 1 the limit zero, whilst at the same time the measure, viz. of the ratio of the two terms of the scale tends to the limit zero. 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 n 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. |