15G Dr. Day on the Theory of Parallels. 



a very elegant explauation of the uses of the different parts of 

 the retina, examined in its radial section after the manner of 

 M. H. Miiller, and of those of the choroid coat, — structures 

 which are without meaning on the ordinary theory of vision. 



Yours truly. 

 University, New York, J. W. Deaper. 



Jauuarj- 26, 185/. 



XXIV. Remarks on the Theory of Parallels. 

 By Alfred Day, LL.D* 



THE appearance of several short articles on the doctrine of 

 parallels in recent Numbers of the Philosophical Maga- 

 zine, recalls my attention to a subject which I have long and 

 attentively considered. While agreeing in general in the views 

 of Drs. "\Tliewell, Whatelj', Lardner, and Messrs. Mill and Hen- 

 nessy, on the nature of geometrical definition, that we do not 

 conclude from mere names, but that there lies at the basis of the 

 reasoning a something assumed which is not susceptible of proof 

 in its simplest form, call it axiom or conception, or what you will, 

 it appears to me that this something, to be of any avail, must be 

 self-evident, universally and invariably commanding assent, and 

 incapable of being disputed, however its nature may be expressed 

 in different verbal propositions. The question for consideration 

 is, do the definitions of straight line, circle, or parallel line, con- 

 tain distinct and independent assumptions ? and if so, are these 

 all seen to be equally uecessaiy, or is there but one and the same 

 assumption inevitable in the present constitution of the human 

 mind, and embracing and implying these various modes of ex- 

 pression, lying at the ground of our deductions ? Is not the 

 conception of space common to all men, aud does it not contain 

 in it all that is necessary for a science of geometry ? If so, there 

 should be nothing disputable or open to cavil in the minds of 

 competent thinkers. An axiom, to be valid, must express some 

 ultimate truth involved in this conception, and should be inca- 

 pable of further proof, otherwise it is superfluous. If, after 

 repeated attempts to analyse a proposition already so far element- 

 ary that we are intuitively confident of its truth, we find our- 

 selves di'iven to modes of explauation which arc in themselves 

 more difficult than the deductive processes to which geometi'y is 

 applied, such, for instance, as the use of limits or arguments from 

 continuity or iuconceivableness, we may be pretty sure that we 

 have passed from the region of deduction and proof into that of 

 fundamental conception. There is an endless variety of such 



* Communicated by the Author. 



