AE to AC. But, because *ξ*O 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 ASEX*μ*A we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area ASBX*μ*A, equal to the area ASEX*μ*A, and therefore in proportion to the area ASCY*μ*A as AE to AC. But the area ASBY*μ*A is nearly equal to the area ASBX*μ*A; and this area ASBY*μ*A is to the area ASCY*μ*A 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 *ξ*n in proportion to *μ*B as 27MI to 16M*μ*, and draw B*n*, this B*n* 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 I*μ *and* μ*M, and the length , are equal among themselves.*

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

LEMMA X.

*Produce*S*μ to*N*and*P*, so as μ*N*may be one third of μ*I*, and*SP*may be to*SN*as*SN*to*S*μ; and in the time that a comet would describe the arc*A*μ*C*, 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 ASC*μ*A; and therefore the space contained under the length described in the tangent and the length S*μ* would be to the space contained under the lengths AC and SM as the