Dr. Adamson on Geometrical Demonstrations. 337 



stated. It is therefore of importance that the latter be derived 

 from the former by rigid argument. The question being, 

 whether ratios are equal or not equal, there will be four state- 

 ments, by which relations between the two properties may be 

 indicated. 



1. In regard to four proportionals, equimultiples of alternate 

 terms, when compared adjacently, have symmetrical relations as 

 to magnitude. 



2. In regard to four non-proportionals, equimultiples of alter- 

 nate terms, when compared adjacently, have non-symmetrical 

 relations as to magnitude. 



3. Four magnitudes, of which the alternate equimultiples, 

 when compared adjacently, have symmetrical relations as to 

 magnitude, are proportionals. 



4. Four magnitudes, of which the alternate equimultiples, 

 when compared adjacently, have non-symmetrical relations as to 

 magnitude, are not proportionals. 



Of these four statements, two are enough to afford criteria 

 both of proportionals and of non-proportionals. But to serve 

 this end, we must, as has been already stated, take either, two 

 which are converse to each other, or else those which are both 

 affirmative, or else those which are both negative. To take the 

 second and third, or the first and fourth, will not suit. 



The character of a definition is such, that its direct and con- 

 verse statements are both applicable. Hence Euclid^s criterion 

 of proportion includes both the first and third. When, how- 

 ever, we treat all as theorems dependent on equality of fractions, 

 the case is different. If we so treat the first and fourth only, 

 we have a test of non-proportionals in the fourth, but we have 

 no test of proportionals at all. Neither can therefore be made 

 to suit the case — given — the relation of the equimultiples, to 

 infer — proportionality. The first is useless from the nature of 

 its hypothesis, and the last from the nature of its conclusion. 

 A misconception of this matter has prevailed generally, which it 

 is as well to remove. It is found in Peacock^s Algebra, vol. i. 

 p. 175 ; the demonstration of the fourth in that work, moreover, 

 is correct only for commensurable quantities ; and the attempt 

 to remedy this will require other provisions, introducing greater 

 complexity of reasoning. But the whole end is attained by 

 proving the first and third, which are converse to each other. 



From the equality of two fractions, to deduce the defining 

 property of Euclid, offers no difficulty. The converse statement 

 may be managed as follows : — If there be two homogeneous 

 magnitudes C and D, and we take of the latter any multiple, 

 such as wD, then may C -f C -f C, &c. be so taken that mC shall be 



