88 The Three Sections, Tangencies, 



7/ A, B be two fixed points in the circumference of a circle, 

 and P any point whatever in the circumference, then will the 

 angle PA right to B be constant = 6 right. 



And I may remark tliat by adopting this property as the 

 fundamental definition of a circle, it would be easy to frame 

 an Element of Geometry in which we could give a direct 

 demonstration to the fact, that a point O, and but one, exists 

 from which lines draivn to the circumference of the circle (or 

 locus ofP) are equal to each other. 



This might be done in various ways ; but perhaps the most 

 simple, as well as the most natural,* would be to have a 

 knowledge of the Theory of Numbers, including proportion 

 in its most general form, precede the study of geometrical 

 science, so as to establish the relations of straight-lined 

 figures as early as possible. 



The demonstration might be made as follows : — 



Bisect AB in C and through C draw MM perpendicular to 

 AB ; find O in MM such that the angle OC right to B = 

 right ; then will O be the sought point. 



For let I be the other point in which the circle described 

 from C as centre with CA or CB as radius again cuts AP. 



* On careful examination it will be found that aU onr Geometry rests 

 ultimately on our power of conceiving the positions and motions of points, 

 lines, and surfaces, and on acquirements in the Theory of Numbers, whether 

 we depend on our intixitive ideas of these things, or that we acquire a more 

 extended systematic knowledge before commencing the study of a regular 

 course of geometry. And this should lead us to look on the Theory of 

 Numbers as pi-imary to geometry, inasmuch as without some knowledge of 

 numbers there can be no systematic treatise, and that the introduction of it 

 from the commencement confers simplicity and comprehensiveness on the 

 Elements. However, many are to be foiind who think they proceed with- 

 out the use of numbers or proportion, when they disguise their investigations 

 so as to mislead themselves : for wherever there is an idea of equahty there 

 must be an idea of proportion ; though not necessarily so refined or compre- 

 hensive a one as Euclid's. Others are to be found who would trammel the 

 natural action of the intellect, and compel the geometer to discard all con- 

 ceptions of motion, as foreign to the spirit of pure geometrical science. They 

 point confidingly to what works of the "Ancients" now remain in support of 

 their crotchet, as if the philosophy of the Greeks were perfect, and should 

 prescribe limits to the march of progress. But, though truth may be for a 

 f. tune suppressed, by the combined efforts of masters and professors it will yet 

 triumph ; and an axiomatic motional philosophy will correct the faulty links 

 and defective logic in the system of geometry which now imfortimately 

 prevails. Indeed, as in other sciences, it will be found, on careful examina- 

 tion, that geometry has inherent properties independent of man's endow- 

 ments, and exacts peculiar mental operations, from capable beings who fathom, 

 some of its various truths, which cannot be prescribed or varied at man's' 

 pleasm-e : for they are ordinances of the laws of nature, and can change 

 only at the Avill of Providence. 



