128 REASONING. 



from which all the othei* truths of the science are synthetically deduced 

 (as the laws of motion and of the composition of forces in mechanics, 

 the equal mobility of fluids in hydrostatics, the laws of refle.ction and 

 refraction in optics, are the first princijiles of those sciences) ; but are 

 merely necessary assumptions, self-evident indeed, and the denial of 

 which would annihilate all demonstration, but firom which, as premisses, 

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

 stances, 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 pit of darkness, and not seeing that this is equally ti'ue 

 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 defi- 

 nition, 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 ai"e confined in 

 the- demonstrations of geometry, holds equally true of the definitions. 

 Every demonstration in Euclid might be carried on without them. 

 This is apparent from the ordinary process of proving a proposition of 

 geometry by means of a diagi'am. What assumption, in fact, do we 

 set out fi'om, to demonsti'ate by a diagi'am 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 general; but it is only 

 necessary that you should grant the assumption in the case of the par- 

 ticular 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 others 

 axiqms, 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 general theorem which stands at the head of the 

 demonstration, is not the proposition 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 other instances ; in every 

 instance which conforms to certain conditions. The contrivance of 

 general language furnishing us with terms which connote these con- 

 ditions, 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 alphabet, we might prove the 

 general theorem directly, that is, we might demonstrate all the cases 

 at once ; and to do this we must, of course, employ as our premisses, 

 the axioms and definitions in their general form. ]3ut this only means, 

 that if we can prove an individual conclusion by assuming an individual 

 fact, then in Avhatever case we are waiTanted in making an exactly 

 similar assumption, we may draw an exactly similar conclusion. The 

 definition is a sort of notice to ourselves and others, what assumptions 

 we think ourselves entitled to make. And so in all cases, the general 

 propositions, whether called definitions, axioms, or laws of nature, 



