888 Dr. Adamson on Geometrical Demonstrations. 



greater than wD; but (m— 1)C shall be not greater than nD, 

 whether the magnitudes be commensurable or not. 



Again, if there be two magnitudes D and F, and a third 

 magnitude C such that when equimultiples are taken of D and 

 F it is found that no multiple of C can exceed either of the 

 equimultiples wD or wF, without also exceeding the other, then 

 D is equal to F. For if either of them as F be the greater, or 

 F = D + Q, then substituting D + Q for F, and assuming n such 

 that nQ is greater than C, we must have mQ greater than 

 nD + C when mC is greater than nD, whatever m may be ; or in 

 all cases (m — 1)C must be greater than nD when mQ is greater 

 than nD, which is absurd. Hence these quantities are equal. 



If we now assume A, B, C, D, in conformity with Euclid's 

 definition, as our hypothesis, or have mK and mC such that in 

 regard to nB and nD each always exceeds its adjacent multiple, 

 when the other exceeds its adjacent multiple, then it is easy to 

 show that A : B = C : D, in the sense of the adjacent terms 

 forming equal fractions; for if a greater or smaller quantity 

 were substituted for any of them, such as D, so as to form a 

 proportion, then the application of the principle demonstrated 

 above would show that this involved a contradiction. 



This appears to be the natural and proper mode of making 

 Euclid's criterion to be a consequence of the principle now 

 adopted, as discriminating equality of ratios. It is evident that 

 no other relation of the multiples needs to be introduced into 

 the statement of the principle, except that of contemporaneous 

 excess in magnitude ; and it may therefore be expressed in the 

 simpler form, that if we take equimultiples of alternate magni- 

 tudes, and find that the multiples of the first and third must 

 each exceed its adjacent one, when the other exceeds its adja- 

 cent one, then these magnitudes are proportionals. 



It will be easily seen that the effect of the converse statement 

 is, as such ought always to be, restrictive or exclusive. It makes 

 the relation of the multiples to be a test determining what alone 

 can be proportionals, and renders unnecessary any other criterion 

 of non-proportionals. 



N.B. I may be in regard to this subject allowed to suggest 

 the following statement, as a useful epitome of various truths : — 

 " In a proportion the sums or the differences of adjacent terms 

 are as alternate terms, and the sums or the differences of alter- 

 nate terms are as adjacent terms.'' 



[To be continued,] 



