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XVII. — On Superposition. By the Rev. Philip Kelland, M.A., Professor of 

 Mathematics in the University of Edinburgh. (With a Plate.) 



(Read 19th February 1855.) 



The subject which I propose to discuss in this paper is the value of the method 

 of demonstration by superposition. I am satisfied that it has been much under- 

 rated, and in some cases misunderstood. It may be stated, that the essential 

 characteristic of this method of demonstration, is the mental comparison of two 

 magnitudes, by placing one of them upon the other. Euclid's axiom of equality 

 (which, perhaps, is rather a definition) is this : " Magnitudes which coincide with 

 one another, that is, which exactly fill the same space, are equal." Accordingly, 

 in his first four books, Euclid never regards two magnitudes as equal, except under 

 circumstances wherein it can be shown that this condition is satisfied. Only in 

 one proposition has he avoided the labour which a strict attention to this re- 

 quirement necessarily imposes ; and perhaps, even in that case, it is hypercritical 

 to object to what he has done. All that he assumes is this : it being admitted 

 that when A fills the same space as B, A is equal to B ; it must therefore be ad- 

 mitted, that when A and C together fill the same space as B and C together, A 

 is also equal to B. 



In the 6th book, as depending on the 5th, Euclid makes another step in the 

 assumptions necessary, and, in the 2d proposition, he admits the test of equality 

 to be this : When two magnitudes can be multiplied equally to any extent, and 

 when it can be shown, that on every occasion in which one exceeds or falls short 

 of a magnitude, the other does the same exactly ; then the two cannot fill other 

 than the same space. 



For the next step in this line of argument, we must advance to Newton's 

 Principia. In the 7th Lemma will be found a beautiful process, by which it is 

 shown that two inconceivably small magnitudes are, as they get smaller, ap- 

 proaching to a ratio of equality. The process is simply that of applying a mi- 

 nute magnifying power, so that one of the things to be compared shall always 

 be magnified up to a fixed standard. 



This is the method of demonstration in geometry. It is not too much to say 

 that geometry owes nine-tenths of its value, as an educational agent, to its being 

 a consistent and systematic exemplification of this method. The treatises of 

 Legendre, in French, and Sir John Leslie, in English, in which method is set at 

 defiance, however valuable as introductory to other branches of science, are com- 

 vol. xxi. part ii. 4 D 



