But this singular property is not confined to the conchoid, but is found in the following curve. Conceive that the right line A C which is perpendicular to the indefinite line X Y, is equal to the quadrantal arch H D, described from the centre C, with the radius C D: then from the same centre C, with the several distances C E, C F, C G, describe the arches E _l_, F _n_, G _p_, each of which must be conceived equal to the first arch H D, and so 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 considering that each of the sines of the arches H D, _l_ E, _n_ F, &c. being less than its respective arch, must also be less than the right-line A C, and consequently can never coincide with the right-line A B. But if other arches D _i_, E _m_, F _o_, &c. each of them equal to the right-line A C, and described from one centre, tangents to the former arches H D, _l_ E, _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 pass beyond the former curve, and converge still nearer to the line A B, without a possibility of ever becoming coincident: for since the arches D _i_, E _m_, F _o_, &c. have less curvature than the former arches, but are equal to them in length, it is evident that they will be subtended by longer lines, and yet can never touch the right-line A B. In like manner, if other tangent arches be drawn to the former, and so on infinitely, with the same conditions, an infinite number of curve-lines will be formed, each of them passing between H _p_ and A B, and continually diverging from the latter, without a possibility of ever coinciding with the former. This curve, which I invented some years since, I suspect to be a parabola; but I have not yet had opportunity to determine it with certainty. Transcriber’s Notes: 1. Obvious printers’, punctuation and spelling errors have been corrected silently. 2. Where hyphenation is in doubt, it has been retained as in the original. 3. Some hyphenated and non-hyphenated versions of the same words have been retained as in the original. 4. The errata have been soilently corrected. 5. Italics are shown as _xxx_.