DEMONSTKATION, AND NECESSARY TRUTHS. 151 



Those who say that the premisses of geometry are hypotheses, arc not 

 bound to maintain them to be hypotlieses which have no r^jlation what- 

 ever to fact. Since an hypothesis framed for the purpose of scientific 

 inquiry must relate to something whicli has real existence (for there 

 can be no science respecting non-entities), it folliows that any hypothe- 

 sis we make respecting an object, to faciUtatc onr study of it, must, not 

 involve anything which is distinctly false, and repugnant to its real 

 nature : we must not ascrilx) to the thing any property which it has 

 not ; ^our liberty extends only to suppressing some of those which it 

 has, under the indispensable obligation of restoring them whenever, 

 and in as far as, their presence or absence would make any material 

 diilerence in the truth of our conclusions. Of this nature, accordingly, 

 are the fir^it principles involved in the definitions of geometry. In 

 their positive part they are observed facts ; it is only in their negative 

 part that they are hypothetical. That the hypotheses should be of 

 this particular character, is, however, no further necessary, than inas- 

 much as no others could enable us to deduce conclusions which, with 

 due corrections, would be true of real objects : and in fact, when our 

 aim is only to illusti'ate truths and not to investigate them, we are not 

 under any such restriction. We might suppose an imaginary animal, 

 and work out by deduction, from the known laws of physiology, its 

 natural history; or an imaginary commonwenlth, and from the elements 

 composing it, might argue what would be its fate. And the conclu- 

 sions which we might thus draw from purely arbitrary hypotheses, 

 might form a highly useful intellectual exercise : but as they could only 

 teach us what would be the properties of objects which do not really 

 exist, they would not constitute any addition to our knowledge : while 

 on the contrary, if the hypothesis merely divests a real object of some 

 portion of its properties, \Vithout clothing it in false ones, the conclu- 

 sions will always express, under known liability to correction, actual 

 truth. 



§ 3. But although Mr. Wliewell has not shaken Stewart's doctrine 

 as to the hypothetical chai'acter of that portion of the first principles of 

 geometry which are involved in the so-called definitions, he has, I con- 

 ceive, gi'esLtly the advantage of Stewart on another important point 

 in the theory of geometrical reasoning; the necessity of admitting, 

 among those first jn-inciples, axioms as well as definitions. Some of 

 the axioms of Euclid might, no doubt, be exhibited in the form of defi- 

 nitions, or might be deduced, by reasoning, fi-om propositions similar to 

 ivhat 9,re so called. Thus, if instead of the axiom. Magnitudes which 

 can be made to coincide are equal, we introduce a definition, " Equal 

 magnitudes arc those which may be -so applied to one another as to 

 coincide ;" the three axioms which follow, (Magnitudes which are equal 

 to the same are equal to one another — If equals are added to equals 

 the sums are equal — ^If equals are taken from equals the remainders 

 are equal,) may be proved by, an imaginai'y superposition, resembling 

 that by which the fourth proposition of the first book of Euclid is de- 

 monstrated. But although these and several others may be struck out 

 of the list of first princi])les, because, though not requiring demon- 

 stration, they are susceptible of it ; there will be found in the list of 

 axioms two or three fundamental trutJis, not capable of being demon- 

 strated : among which I agree with Mr. WTiewell in placing the prop- 



