^{page 263 note *} See Note E.

^{page 272 note *} See, for other forms, Note A, at the end of this paper.

^{page 274 note *} It is altogether unnecessary to prove, that, for in these and most of the other results of this paper, we may substitute (2n + 1), where n is any whole number positive or negative.

^{page 285 note *} It is indeterminate, because the equation (11) is fulfilled by the co-efficients, which should determine the circle,—not from a particular value assumed by the variable θ, which would merely indicate an extraneous factor, that might be eliminated by differentiation.

^{page 296 note *} “Centre of the plane” is used to signify the point where the plane of projection is cut by the axis of the sphere.

^{page 298 note *} These two equations are foreign to the inquiry. The manner of their appearance here is easily shewn; but the extent to which this paper runs, forbids my enlarging upon this topic at present.

^{page 301 note *} Certain precautions, which are not very prominently brought forward in plano, are necessary in the use of this and of all other forms of spherical equations. We shall explain at a future time.

^{page 301 note †} Pappus, Coll. Math. lib. iv. prop. 30.

^{page 302 note *} In these figures, we suppose the sphere orthographically projected on the plane of the meridian PEP'Q. As this will require us to represent both the hemispheres on one plane, we shall distinguish that which is between us and the meridian by the name of convex and the other by the name concave hemisphere. We shall trace in full line the parts of the locus that lie on the convex hemisphere, and in dots those which lie on the concave. The same letters are used for corresponding points on both hemispheres; but those belonging to the concave are accentuated.

^{page 306 note *} See his Works, collected by Cramer, vol. ii. p. 512.

^{page 306 note †} See Art. XX. Indeed this has been done by Bernoulli himself, very simply and elegantly, Op. tom. ii. p. 744.

^{page 306 note ‡} Bern. Op. ii. p. 513.

^{page 308 note *} For MPQ = NOQ = θ; or θ is the same both on the spherical equator and its projection on the plane MHG. It may also be remarked, that there is no essential difference between Bernoulli's own proof and the one above given, except the notation.

^{page 309 note *} Mém. de l'Institut. tom. ii. p. 228.

^{page 309 note †} Hist. des Math. tom. ii. p. 94.

^{page 309 note ‡} Leybourn's Mathematical Repos. vol. i. pt. ii. p. 1. New Series. See also Woodhouse in Phil. Trans. 1801, p. 153. Other works are referred to by historical writers: but as additional detail would be inconsistent with my plan, I shall not consider further examination to be at present necessary.

^{page 311 note *} Some preliminary considerations intimately connected with this subject, are given in Note (B) at the end.

^{page 312 note *} Acta Petrop. tom, v.—Dr Brewster's Translation of Legendre's Geometry, p. 266, and other places.

^{page 315 note *} See also Note (C) at tha end.

^{page 318 note *} In the Mathematical Repository, vol. v. p. 240, pt. 1, are solutions of this case, which had heen proposed some years before by Professor Wallace. How far that gentleman had carried his inquiries into this subject, or whether he had systematically entered upon it at all, does not appear from the Repository; nor have I other means of ascertaining.

^{page 320 note *} See Note (A) at the end.

^{page 324 note *} There is a copy of Hermann's dissertation inserted in John Bernoulli's works, vol. iii. p. 211.

^{page 324 note †} Page 240; also his works, vol. iii. p. 220.

^{page 324 note ‡} Mém. de l'Acad. 1732, p. 243. Opera Omnia, tom. iii. p. 223.

^{page 324 note §} Maupertius, p. 255: Nicole, p. 271: Clairault, p. 289. It is rather singular that Dr Young, in his Catalogue, refers to the paper of Maupertius, but does not mention those of Nicole and Clairault, which follow them in the same volume. Dr Young mentions a paper on Spherical Epicycloids, by Lexell, in the third volume of the Petersburg Acts; but I have no means of consulting that series of Mémoires, there not being a copy in this city.

^{page 325 note *} Perhaps it would be more accurate to say the one should be a multiple of the other by a constant factor: but it will be found that this introduces a complexity into the expression, which is fatal to the subsequent rationalization of the general formula for the arc. The result of such a supposition is an imaginary quadratic. However, as upon this hinges the only doubt respecting the general conclusion in the subsequent investigation, I shall enter upon it with all requisite detail on some future occasion.

^{page 329 note *} Vide Bern. Op. om. iii. p. 233.; or Mém. de l'Acad. 1732, p. 234. The statement above of the property is not exactly in form the same as that given by Bernoulli. He calls the ‘abscissa’ the versed sine of the arc of the rolling circle which has already been in contact. This versed sine is in the plane of the circle itself, and therefore is to the sine of the declination as sine declination is to radius. Making these transformations, the result I have given ahove will be found identical with Bernoulli's, though obtained by so completely different a process.

^{page 330 note *} Mém. de l'Acad. des Scien. 1732, pp. 293-4.

^{page 332 note *} Mém. de l'Acad. 1732, p. 245; or Opera Omnia, tom. iii. p. 226.

^{page 332 note †} Mém. de l'Acad. 1732, pp. 257-8.

^{page 333 note *} Mem. 1704, p. 315.

^{page 333 note †}’‘ 1730, p. 27.

^{page 337 note *} The common form of the symbol of infinity is open to some objections. Baron Fourier, in his Treatise on Heat, and in his posthumous work on the solution of Equations, has employed , and certainly it has the advantage of expressing distinctly the arithmetical origin and signification of that quantity. It has too, some collateral advantages, which, however, it would be foreign to our subject to discuss.

^{page 340 note *} Prefixed to Robertson's Navigation. The remark is in the foot-note at p. xv, xvi, of Wales's edition (or fourth) of that work.

^{page 341 note *} This is the general equation of Cotes's spirals: but as it does not admit of the values , the sixth species cannot be formed from the rhumb line by projection on a plane at right angles to the axis of projection. All the others can. See also Hymers's Geometry of Three Dimensions, p. 136.

^{page 342 note *} Phil. Trans. 1696. Abt. vol. i. p. 577., or New Abt. vol. iv. p. 68.

^{page 342 note †} Jac. Bern. Op. Omn. Ed. Cramer, tom. ii. p. 491.

^{page 343 note *} See the two editions of this tract, Basle, 1536, p. 280, edited by Ziegler, and the other edited by Commandine, Ven. 1558, p. 30. I have given some account of these in the Mathematical Repository, vol. vi. pt. ii. p. 42, in treating of the History of the Stereographic Projection of the Sphere.

^{page 344 note *} See note (D).

^{page 344 note †} This very appropriate epithet seems to have been introduced into the science by Mr Gompertz.—See his second tract on Imaginary Quantities, p. x. Some authors have been so far from understanding the nature of “Mercator's Projection,” that they have designated it as made “upon a plane at an infinite distance” !—Dealtry's Fluxions, p. 427. This singular oversight originated, doubtless, in the want of a proper verbal distinction being made between the radial projection of a figure upon a given surface, and the figure generated upon that surface, by taking as co-ordinates some function of the curve said to be “projected.”

^{page 346 note *} Page 66. and 69.

^{page 346 note †} Vide Professor Leybourn's edition of this curious, and in many respects valuable, Miscellany, vol. ii. p. 249. We might, in a manner very similar to that just employed, determine the development of the Stereographic projection of the Loxodrome upon the equatorial cylinder, and should find, as is done in the Gentleman's Diary for 1820, p. 43. that it is the logarithmic curve.

^{page 349 note *} See Figure on Page 350.

^{page 353 note *} Vide Note B.