THE MATHEMATICAL PRINCIPLES [BOOK III. Prop. XL, Book III ; and tV a perpendicular upon the chord TT. In the mean observed longitude tfB take at pleasure the point B, for the place of the comet in the plane of the ecliptic ; and from thence, towards the sun S, draw the line BE, which may be to the perpendicular /V as the content under SB and St2 to the cube of the hypothenuse of the right angled tri angle, whose sides are SB, and the tangent of the latitude of the comet in the second observation to the radius ^B. And through the point E (by Lemma VII) draw the right line AEC, whose parts AE and EC, terminat ing in the right lines TA and rC. may be one to the other as the times V and \V : then A and C will be nearly the places of the comet in the plane of the ecliptic in the first and third observations, if B was its place rightly assumed in the second. Upon AC, bisected in I, erect the perpendicular li. Through B draw the obscure line Ei parallel to AC. Join the obscure line Si, cutting AC in A, and complete the parallelogram il AJU. Take \o equal to 3IA ; and through the sun S draw the obscure line<0£ equal to 3So -f 3 fa. Then, cancelling the letters A, E, C, I, from the point B towards the point £, draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity Sju + 1 fa. And through the point E draw again the right line AEC by the same rule as before ; that is, so as its parts AE and EC may be one to the other as the times V and W between the observations. Thus A and C will be the places of the comet more accurately. Upon AC, bisected in I, erect the perpendiculars AM, CN, IO, of which AM and CN may be the tangents of the latitudes in the first and third ob servations, to the radii TA and TC. Join MN, cutting IO in O. Draw the rectangular parallelogram zlAjt/, as before. In I A produced take ID equal to Sfi + f fa. Then in MN, towards N, take MP, which may be to the above found length X in the subduplicate proportion of the mean distance of the earth from the sun (or of the semi-diameter of the orbis 'tnagnus] to the distance OD. If the point P fall upon the point N; A, B, and C, , equal respectively to CG, eg, KJ ; through the points P, f, and 0, draw the circumference of a circle Vf, cutting the right line AT in X ; and the point X will be another place of the comet in the plane of