[156] Because -ẋ ⁄ √(1 - xx) and -Ẋ ⁄ √(1 - XX) are known to express the fluxions of the circular arcs whose co-sines are x and X, it is evident, if those arcs be supposed in any constant ratio of 1 to n, that nẋ ⁄ √(1 - xx) = Ẋ ⁄ √(1 - XX), and consequently that nẋ ⁄ √(xx - 1) (= nẋ ⁄ √-1 × √(1 - xx) = Ẋ ⁄ √-1 × √(1 - XX)) = Ẋ ⁄ √(XX - 1). From whence, by taking the fluents, n × Log. (x + √xx - 1) (or Log. (x + √xx - 1)ⁿ) = Log. X + √XX - 1; and consequently (x + √xx - 1)ⁿ = X + √XX - 1: whence also, seeing x - √xx - 1 is the reciprocal of x + √xx - 1, and X - √XX - 1 of X + √XX - 1, it is likewise evident, that (x - √xx - 1)ⁿ = X - √XX - 1. Hence, not only the truth of the above assumption, but what has been advanced in relation to the roots of the equation zⁿ - 1 = 0, will appear manifest. For if x ± √xx - 1 be put = z, then will zⁿ (= (x ± √xx - 1)ⁿ) = X ± √XX - 1: where, assuming X = 1 = co-s. 0 = co-s. 360° = co-s. 2 × 360° = co-s. 3 × 360°, &c. the equation will become zⁿ = 1, or zⁿ - 1 = 0; and the different values of x, in the expression (x ± √xx - 1) for the root z, will consequently be the co-sines of the arcs, 0 ⁄ n, 360° ⁄ n, 2 × 360° ⁄ n, &c. these arcs being the corresponding submultiples of those above, answering to the co-sine X (= 1).——In the same manner, if X be taken = -1 = co-s. 180° = co-s. 3 × 180° = co-s. 5 × 180°, &c. then will zⁿ = -1, or zⁿ + 1 = 0; and the values of x will, in this case, be the co-sines of 180° ⁄ n, 3 × 180° ⁄ n, 5 × 180° ⁄ n, &c.
[157] Avellana purgatrix; in French, medicinier.
[158] This refers to Mr. Baker’s having supposed, that old iron and old brass may be mixt sometimes, and melted down together.
[159] Vide Wilkins’s real Character, p. 131. Bellon. aquat. p. 330.
[160] Some of the Pour-contrel kind have but one row of suckers on the arms: such an one I have seen, whose arms were thirty inches long.
[161] Of this I gave an account some years ago, in my attempt towards a Natural History of the Polype, chap. v.
[162] See Plate [xxxi]. Fig. 1.
[163] De Num. quibusd. Sam. et Phœn. &c. Dissert. p. 56-59. & Tab. II. Oxon. 1750.
[164] Marm. Palmyren. a Cl. Dawk. edit. pass.
[165] Vid. Hadr. Reland. Palæst. Illustrat. p. 1014. Traject. Batavor. 1714. Erasm. Frœl. ad Annal. Compendiar. Reg. & Rer. Syr. Tab. VIII. &c. Viennæ, 1754.