of eight or twelve days, and the longitudes taken when the comet moves with the greatest velocity; for thus the errors of the observations will bear a less proportion to the differences of the longitudes.
LEMMA IV.
To find the longitudes of a comet to any given times.
It is done by taking in the line FG the distances Rr, Rρ, proportional to the times, and drawing the lines Tr, Tρ. The way of working by trigonometry is manifest.
LEMMA V.
To find the latitudes.
On TF, TR, TG, as radiuses, at right angles erect Ff, RP, Gg, tangents of the observed latitudes; and parallel to fg draw PH. The perpendiculars rp, ρῶ, meeting PH, will be the tangents of the sought latitudes to Tr and Tρ as radiuses.
PROBLEM I.
From the assumed ratio of the velocity to determine the trajectory of a comet.
Let S represent the sun; t, T, τ, three places of the earth in its orbit at equal distances; p, P, ῶ, as many corresponding places of the comet in
its trajectory, so as the distances interposed betwixt place and place may answer to the motion of one hour; pr, PR, ῶρ, perpendiculars let fall on the plane of the ecliptic, and rRp the vestige of the trajectory in this plane. Join Sp, SP, Sῶ, SR, ST, tr, TR, τρ, TP, and let tr, τρ, meet in O, TR will nearly converge to the same point O, or the error will be in considerable. By the premised lemmas the angles rOR, ROρ, are given, as well as the ratios pr to tr, PR to TR, and ῶρ to τρ. The figure tTτO
