FUNCTIONS AND VALUE OF THE SYLLOGISM. 145 



root, as I conceive, of the philosophy of ratiocination ; and it is to be re- 

 gretted that he himself stopped short at 'a much more limited application 

 of it. He saw that the general propositions on which a reasoning is said 

 to depend, may, in certain cases, be altogether omitted, without impairing 

 its probative force. But he imagined this to be a peculiarity belonging 

 to axioms ; and argued from it, that axioms are not the foundations or first 

 principles of geometry, from which all the other truths of the science are 

 synthetically deduced (as the laws of motion and of the composition of 

 forces in dynamics, the equal mobility of fluids in hydrostatics, the laws of 

 reflection and refraction in optics, are the first principles of those sciences) ; 

 but are merely necessary assumptions, self-evident indeed, and the denial 

 of which would annihilate all demonstration, but from which, as premises, 

 nothing can be demonstrated. In the present, as in many other instances, 

 this thoughtful and elegant writer has perceived an important truth, but 

 only by halves. Finding, in the case of geometrical axioms, that general 

 names have not any talismanic virtue for conjuring new truths out of the 

 well where they lie hid, and not seeing that this is equally true in every 

 other case of generalization, he contended that axioms are in their nature 

 barren of consequences, and that the really fruitful truths, the real first 

 principles of geometry, are the definitions; that the definition, for exam- 

 ple, of the circle is to the properties of the circle, what the laws of equi- 

 librium and of the pressure of the atmosphere are to the rise of the mer- 

 cury in the Torricellian tube. Yet all that he had asserted respecting the 

 function to which the axioms are confined in the demonstrations of geome- 

 try, holds equally true of the definitions. Every demonstration in Euclid 

 might be crrried on without them. This is apparent from the ordinary 

 process of proving a proposition of geometry by means of a diagram. 

 What assumption, in fact, do we set out from, to demonstrate by a dia- 

 gram any of the properties of the circle ? Not that in all circles the radii 

 are equal, but only that they are so in the circle ABC. As our warrant 

 for assuming this, we appeal, it is true, to the definition of a circle in gen- 

 eral ; but it is only necessary that the assumption be granted in the case of 

 the particular circle supposed. From this, which is not a general but a sin- 

 gular proposition, combined with other propositions of a similar kind, some 

 of which when generalized are called definitions, and other axioms, we 

 prove that a certain conclusion is true, not of all circles, but of the partic- 

 ular circle ABC ; or at least would be so, if the facts precisely accorded 

 with our assumptions. The enunciation, as it is called, that is, the gener- 

 al theorem which stands at the head of the demonstration, is not the propo- 

 sition actually demonstrated. One instance only is demonstrated : but the 

 process by which this is done, is a process which, when we consider its 

 nature, we perceive might be exactly copied in an indefinite number of oth- 

 er instances ; in every instance which conforms to certain conditions. The 

 contrivance of general language furnishing us with terms which connote 

 these conditions, we are able to assert this indefinite multitude of truths 

 in a single expression, and this expression is the general theorem. By 

 dropping the use of diagrams, and substituting, in the demonstrations, 

 general phrases for the letters of the alphapet, we might prove the general 

 t,heorem directly, that is, we might demonstrate all the cases at once ; and 

 :o do this we must, of course, employ as our premises, the axioms and 

 lefinitions in their general form. But this only means, that if we can 

 jrove an individual conclusion by assuming an individual fact, then in 

 vhatever case we are warranted in making an exactly similar assumption, 



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