[185] For it is easy to conceive a cylindric spiral described about a right-line, so as to preserve an equal distance from it in every part; and in this case the spiral and right-line will never coincide though infinitely produced. [186] [Illustration] As the conchoid is a curve but little known, I have subjoined the following account of its generation and principal property. In any given right line A P, call P the pole, A the vertex, and any intermediate point C the centre of the conchoid: likewise, conceive an infinite right line C H, which is called a rule, perpendicular to A P. Then, if the right line A _p_ continued at _p_ as much as is necessary, is conceived to be so turned about the abiding pole _p_, that the point C may perpetually remain in the right line C H, the point A will describe the curve A _o_, which the ancients called a conchoid. In this curve it is manifest (on account of the right line P O, cutting the rule in H that the point _o_ will never arrive at rule C H; but because _h_ O is perpetually equal to C A, and the angle of section is continually more acute, the distance of the point O from C H will at length be less than any given distance, and consequently the right line C H will be an asymptote to the curve A O. When the pole is at P, so that P C is equal to C A, the conchoid A O described by the revolution of P A, is called a primary conchoid, and those described from the poles _p_, and _π_, or the curves A _o_, A _ω_, secondary conchoids; and these are either contracted or protracted, as the eccentricity P C, is greater or less than the generative radius C A, which is called the altitude of the curve. Now, from the nature of the conchoid, it may be easily inferred, that not only the exterior conchoid A _ω_ will never coincide with the right line C H, but this is likewise true of the conchoids A O, A _o_; and by infinitely extending the right-line A _π_, an infinite number of conchoids may be described between the exterior conchoid A _ω_, and the line C H, no one of which shall ever coincide with the asymptote C H. And this paradoxical property of the conchoid which has not been observed by any mathematician, is a legitimate consequence of the infinite divisibility of quantity. Not, indeed, that quantity admits of an actual division in infinitum, for this is absurd and impossible; but it is endued with an unwearied capacity of division, and a power of being diffused into multitude, which can never be exhausted. And this infinite capacity which it possesses arises from its participation of the indefinite duad; the source of boundless diffusion, and innumerable multitude. [Illustration]