Review ofT. P. Thompson s Geometry without A.rioms. 09 



are prone to disguise tlieir incapacity to prove what they ought to prove, 

 under the specious name of axioms, which they allege to be self-evident 

 truths. 



A very slight reflection must show every man who opens a geometrical 

 treatise, how many propositions are demonstrated by geometers, which are 

 yet to him quite as self-evident as those which they style axioms : there- 

 fore he will at least expect some reason assigned, why some more than 

 others should be thus designated. But it needs no great insight into the 

 matter, to perceive furtlier, that the setting up of axioms is a mere after- 

 thought : that is, no geometer calls a proposition self-evident, because he 

 intuitively discerned it ; but because, after trying, he failed to prove it. 

 Further : it is easy to see, that of half a dozen propositions equally self- 

 evident, the geometer selects arbitrarily which shall be his axiom : one will 

 do as well as another ; but one or other he must have. This appears ex- 

 ceedingly like a petitio principii. If done openly, with an avowal that it 

 should not be so, the student has fair warning : but when all is smuggled 

 under fine words, axioms, postulates, self-evident truths, &c. a mist is 

 thrown over the principles of abstract science, and false philosophy is so 

 far inculcated. 



The only circumstance under which there appears the least pretence for 

 axioms, is, when a definition is notoriously in defect. We all know that 

 there must be some stop to defining. If we be asked to define the terms 

 of our definitions, the matter may be pushed back and back, till we are in 

 inextricable confusion. We are then to expect terms which involve ideas 

 so easy and so primary, as that he who alleges them to be incapable of 

 definition, will say nothing unplausible. We would give in illustration the 

 words Quantity and Equal. If a mathematician finds it necessary to reject 

 ail the definitions that are offered of these words, on the ground that the 

 words themselves recur in the very definitions alleged ; he may think 

 himself forced into another track, viz. to lay down a set of propositions 

 concerning the words in question, [as, quantities which are equal to the 

 same are equal to one anotiier, &c.] which propositions, taken together, 

 shall adequately restrict and settle the meaning of the words. In this view, 

 a set of axioms are substituted for a definition, precisely as a set of func- 

 tional equations for the explicit declaration of a function. Thus, if the 

 defining of Napier's logarithms were attended with any difficulty, the two 

 following fljfjows would amount to a definition. 



log ,r -f- log y ■=. log (a;i]J . . . . for all values oi .r Si. y. . and 

 log. 27182818 = 1. 



But of course we are here beset by the difficulty of determining, u priori, 

 whether we have given too much data for the result j or in algebraic lan- 

 guage, whether our equations have a possible solution. And this objection 

 renders it (juitc unprofitable to adopt axioms concerning any thing that is 



