# Page:Newton's Principia (1846).djvu/475

Book III.]
469
of natural philosophy.

AE to AC. But, because ξO is to SO as 3 to 1, and EO to XO in the same proportion, SX will be parallel to EB; and, therefore, joining BX, the triangle SEB will be equal to the triangle XEB. Wherefore if to the area ASEXμA we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area ASBXμA, equal to the area ASEXμA, and therefore in proportion to the area ASCYμA as AE to AC. But the area ASBYμA is nearly equal to the area ASBXμA; and this area ASBYμA is to the area ASCYμA 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 μ of the parabola, AE is to AC accurately in the proportion of the times.

SCHOLIUM.

If we join μξ cutting AC in δ, and in it take ξn in proportion to μB as 27MI to 16Mμ, and draw Bn, this Bn will cut the chord AC, in the proportion of the times, more accurately than before; but the point n is to be taken beyond or on this side the point ξ, according as the point B is more or less distant from the principal vertex of the parabola than the point μ.

LEMMA IX.

The right lines Iμ and μM, and the length ${\displaystyle \scriptstyle {\frac {AI^{2}}{4S\mu }}}$, are equal among themselves.

For 4Sμ is the latus rectum of the parabola belonging to the vertex μ.

LEMMA X.

Produce Sμ to N and P, so as μN may be one third of μI, and SP may be to SN as SN to Sμ; and in the time that a comet would describe the arc AμC, if it was supposed to move always forwards with the velocity 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 μ was in the said time supposed to move uniformly forward in the right line which touches the parabola in μ, the area which it would describe by a radius drawn to the point's would be equal to the parabolic area ASCμA; and therefore the space contained under the length described in the tangent and the length Sμ would be to the space contained under the lengths AC and SM as the