FUNCTIONS AND VALUE OF THE SYLLOGISM. 125 



amples of the rudest and most spon- 

 taneous form of the operations of 

 superior minds. It is a defect in 

 them, and often a source of errors, 

 not to have generalised as they went 

 on ; but generalisation, though a help, 

 the most important indeed of all helps, 

 is not an essential. 



Even the scientifically instructed, 

 who possess, in the form of general 

 propositions, a systematic record of 

 the results of the experience of man- 

 kind, need not always revert to those 

 general propositions in order to apply 

 that experience to a new case. It is 

 justly remarked by Dugald Stewart, 

 that though the reasonings in mathe- 

 matics depend entirely on the axioms, 

 it is by no means necessary to our 

 seeing the conclu.siveness of the proof 

 that the axioms should be expressly 

 adverted to. When it is inferred that 

 A B is equal to C D because each of 

 them is equal to E F, the most un- 

 cultivated understanding, as soon as 

 the propositions were understood, 

 would assent to the inference, without 

 having ever heard of the general truth 

 that " things which are equal to the 

 same thing are equal to one another." 

 This remark of Stewart, consistently 

 followed out, goes to the root, as I 

 conceive, of the philosophy of ratio- 

 cination ; and it is to be regretted 

 that he himself stopt 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 pro- 

 bative 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 com- 

 position of forces in dynamics, the 

 equal mobility of fluids in hydro- 

 statics, 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 demon- 

 stration, 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 generalisation, 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 example, of the circle 

 is to the properties of the circle what 

 the laws of equilibrium and of the 

 pressure of the atmosphere are to the 

 rise of the mercury in the Torricellian 

 tube. Yet all that he had asserted 

 respecting the function to which the 

 axioms are confined in the demon- 

 strations of geometry holds equally 

 true of the definitions. Every demon- 

 stration in Euclid might be carried on 

 without them. This is apparent from 

 the ordinary process of proving a pro- 

 position of geometry by means of a 

 diagram. What assumption, in fact, 

 do we set out from to demonstrate by 

 a diagram any of the pi'operties 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 general ; 

 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 

 singular proposition, combined with 

 other propositions of a similar kind, 

 some of which when generalised are 

 called definitions, and others axioms, 

 we prove that a certain conclusion is 

 true, not of all circles, but of the 

 particular 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 



