BOOK II Lj OF NATURAL PHILOSOPHY. 469 AE to AC. But, because £O is to SO as 3 to 1, and EG to XC in the same proportion, SX will be parallel to EB ; and, therefore, joining BX, the tri angle SEB will be equal to the triangle XEB. Wherefore if to the area ASEX.uA we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area ASBX,wA, equal to the area ASEX/^A. and therefore in proportion to the area ASCY//A as AE to AC. But the area ASBYwA is nearly equal to the area ASBX//A; and this area ASBY/zA is to the area ASCYwA as the time of description of the arc AB to the time of description of the whole arc AC ; and, therefore, AE is to AC nearly in the proportion of the times. Q.E.D. COR. When the point B falls upon the vertex \i of the parabola, AE is to AC accurately in the proportion of the times. SCHOLIUM. If we join //£ cutting AC in d, and in it take £//, in proportion to ^B as 27MI to 16Mf/, and draw B/?, this Bu will cut the chord AC, in the pro portion of the times, more accurately than before; but the point n is to be taken beyond or on this side the point £, according as the point B is more or less distant from the principal vertex of the parabola than the point p. LEMMA IX. AI2 The right lines Ip and /zM, and the length j~-, are equal among them selves. For 4.S/Z is the latus rectum of the parabola belonging to the vertex ft. LEMMA X. Produce Su to N and P, so as ^N may be one third of //I, and SP may be to SN as SN to S" ; and in the time that a comet would describe the arc AjuC. if it was supposed to move always forwards with the ve locity which it hath in a height equal to SP, it would describe a length equal to the chord AC. For if the comet with the velocity which it hath in \i was in the said time supposed to move uniformly forward in the right line which touches the parabola in p, the area which it would describe by a radius drawn to the point S would be equal to the parabolic area ASC/zA ; and therefore the space contained under the length described in the tangent and the length Su would be to the space contained under the lengths AC and SM as the