460 THE MATHEMATICAL PRINCIPLES [BOOK IIL LEMMA IV That the comets are higher tliau tJie moon, and in the regions of the planets. As the comets were placed by astronomers above the moon, because they were found to have no diurnal parallax, so their annual parallax is a con vincing proof of their descending into the regions of the planets ; for all the comets which move in a direct course according to the order of the signs, about the end of their appearance become more than ordinarily slow or retrograde, if the earth is between them and the sun ; and more than ordinarily swift, if the earth is approaching to a heliocentric opposition with them ; whereas, on the other hand, those which move against the or der of the signs, towards the end of their appearance appear swifter than they ought to be, if the earth is between them and the sun ; and slower, and perhaps retrograde, if the earth is in the other side of its orbit. And these appearances proceed chiefly from the diverse situations which the earth acquires in the course of its motion, after the same manner as it hap pens to the planets, which appear sometimes retrograde, sometimes more slowly, and sometimes more swiftly, progressive, according as the motion of the earth falls in with that of the planet, or is directed the contrary wav. If the earth move the same way with the comet, but, by an angular motion about the sun, so much swifter that right lines drawn from the earth to the comet converge towards the parts beyond the comet, the comet seen from the earth, because of its slower motion, will appear retrograde ; and even if the earth is slower than the comet, the motion of the earth being subducted, the motion of the comet will at least appear retarded ; but if the earth tends the contrary way to that of the cornet, the motion of the comet will from thence appear accelerated; and from this apparent acceleration, or retardation, or regressive motion, the distance of the comet may be inF c B A ferred in this manner. Let TQA, TQ,B, TQ,C, be three observed lon gitudes of the comet about the time of its first appearing, and TQ,F its last observed longitude before its disappearing. Draw the right line ABC, whose parts AB, BC, intercepted between the right lines QA and Q.B, QB and Q.C, may be one to the other as the two times between the three first observations. Produce AC to G, so as AG may be to A B as the time between the first and last observation to the time between the first and second ; and join Q.G. Now if the comet did move uniformly in a right line, and the earth either stood still, or was likewise carried foru ards in a right line by an uniform motion, the angle TQG would be tht