Dr. Adainson on Geometrical Demonstrations. 333 



that whicli is only auxiliary in one science,, may be auxiliary in 

 others also. Axiom is the title for such auxiliary truths, and it 

 ought to be restricted to them alone. 



With a correct nomenclature, therefore, the question whether 

 axioms in geometry are essential as the foundations of its theo- 

 rems, is easily settled. They enter into no hypothesis, and are 

 therefore not essential, but are auxiliary only. The relations 

 expressed by them necessarily occur, but the expressions of them 

 in general terms, which make them to be axioms, are non-essen- 

 tial. In fact those truths to which the name is generally ap- 

 plied, are corollaries from definitions of general terms. Their 

 existence as axioms depends therefore on the existence of such 

 general terms. There are tongues in which these are scanty. 

 There may be tongues in which there are no such words as equal, 

 whole, part, &c., and yet thought would follow its laws of ana- 

 lysis or proof; and demonstrations, though less or more cum- 

 brous, would be as strict and as conclusive as they are in lan- 

 guages more prolific in terms. 



Their auxiliary character, and their exclusion from being hy- 

 potheses, ought to be the only circumstances determining the 

 character of axioms. To describe them as being " self-evi- 

 dent truths,^^ is indefinite, and useless. Whether their truth be 

 more or less easily perceived, depends not on their character, 

 but on the mind, and on its consciousness of the signification of 

 terms. Controversies, therefore, as to whether one proposition 

 assumed to be an axiom be more or less clearly evident than an- 

 other, are futile. All truths of this auxiliary character have a 

 common nature, and deserve a common name. This common 

 nature is most fitly indicated by the term aasiomic. The propo- 

 sition that " half the sum added to half the difference makes the 

 greater of two magnitudes," does not differ in kind from this 

 other — that " if equals be added to equals the sums are equal.^' 

 With these also the following are identical in kind, notwith- 

 standing their greater complexity — " If there be three magni- 

 tudes such that any two are together greater than the third, then 

 half the sum of the three is greater than any one of them,^' — 

 or " if, in a series of continued proportionals, any term is equal 

 to the difference of the two which precede it, then aJso any term 

 is equal to the sum of the two which follow it :" and so in re- 

 gard to multitudes of others. All are of the same order, and 

 ought to come under the same name. 



It is evident that whatever may be the subject of a science, 

 truths of this class may become auxiliary in reasonings relating 

 to these subjects. They are therefore not concrete, but abstract. 

 Hence axioms in geometry are algebraic truths, or are true of 

 any concrete magnitudes, as space, force, money, sound, &c. 



