perpendicular on the axis; and drawing PH,
there will be
Whence . For write ;
Then dividing all the term by 3PO and multiplying them by 2AS,
we ſhall have to the area of to the area APS but GH was 3M and therefore is
Wherefore the area cut of APS is
area that was, to he cut of 4ASxM. Q. E. D.
Cor 1. Hence GH is to AS, as the time in
which the body deſcribed the arc AP to the
time in which the body deſcribed the arc between
the vertex A and the perpendicular
erected from the focus S upon the axis.
Cor 2. And ſuppoſe a circle ASP perpetually
to paſs through the moving body P, the
velocity of the point H, is to the velocity which
the body had in the vertex A. as 3 to 8; and
therefore in the ſame ratio is the line GH to the
right line which the body, in the time of its
moving from A to P, would deſcribe with that
velocity which it had in the vertex A.
Cor. 3. Hence alſo, on the other hand, the time
may be found, in which the body has deſcribed
any aſſigned arc AP. Join AP on its
middle point erect a perpendicular meeting the right
line GH in H.