though they always diminish the interval, never coincide. And they mutually converge, indeed, but never perfectly nod to each other; which is indeed a theorem in geometry especially admirable, exhibiting certain lines endued with a non affenting nod. But the right-lines, which are always diftant by an equal interval, and which never dimi

nish

When the pole is at P, fo that PC is equal to CA, the choncoid A O defcribed by the revolution of PA, is called a primary conchoid, and thofe defcribed from the poles p, and x, or the curves Ao, Aw, fecondary conchoids; and there are either contracted or protracted. as the excentricity P' C, is greater or lefs 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 will never coincide with the right line C H, but this is likewife true of the conchoids A O, Ao; and by infinitely extending the right-line A, an infinite number of conchoids may be defcribed between the exterior conchoid Aw, and the line C H, no one of which fhall ever coincide with the affymptote C H. And this paradoxical property of the conchoid which has not been obferved by any mathematician, is a legitimate confequence of the infinite divifibility of quantity. Not, indeed, that quantity admits of an actual divifion in infinitum, for this is abfurd and impoffible; but it is endued with an unwearied capacity of divifion, and a power of being diffused into multitude, which can never be exhausted. And this infinite capacity which it poffeffes arifes from its participation of the indefinite duad; the fource of boundlefs diffufion, and innumerable multitude.

But this fingular property is not confined to the chencoid, but is found in the following curve.. Conceive that the right line AC which is perpendicular to the indefinite line X Y, is

equal to the quadrantal arch H.D, defcribed from the centre C, with the radius CD: then from the fame centre C, with the feveral diftances C E, CF, CG, defcribe the arches E 1, F", Gp, each of which must be conceived equal to the first arch H D, and fo on infinitely. Now, if the points H, k, l, n, p, be joined, they will form a curve line, approaching continually nearer to the right-line A B (parallel to C Y) but never effecting a perfect coincidence. This will be evident from confidering that each of the fines of the arches H D, E, F, &c. being less than its refpective arch, muft alfo be less than the right-line AC, and confequently can never coincide with the right-line A B.

n

B

Y

nifh the space placed between them in one plane, are parallel lines. And thus much we have extracted from the ftudies of the elegant Geminus, for the purpose of explaining the prefent definition.

But if other arches Di, Em, Fo, &c. each of them equal to the right-line A C, and defcribed from one centre, tongents to the former arches HD, IE, n F, &c. be supposed; it is evident that the points H, i, m, o, &c. being joined, will form a curve line, which shall pafs beyond the former curve, and converge ftill nearer to the line A B, without a poffibility of ever becoming coincident: for fince the arches Di, Em, Fo, &c. have lefs curvature than the former arches, but are equal to them in length, it is evident that they will be fubtended by longer fines, and yet can never touch the right-line A B. In like manner, if other tangent arches be drawn to the former, and fo on infinitely, with the fame conditions, an infinite number of curve-lines will be formed, each of them paling between Hp and A B, and continually diverging from the latter, without a poffibility of ever coinciding with the former. This curve, which I invented fome years fince, I fufpect to be a parabola; but I have not yet had opportunity to determine it with ceainty.

END OF THE FIRST VOLUME.