332 Renew of the Principia of Newton, 
10. The velocity of a body moving in a parabola, is to the 
velocity moving in a circle at the same distance in the ratio 
stance. 
12. The velocity of a body revolving in any conic section 
is to the velocity of a body revolving in a circle at the same 
distance, as a mean proportional between that common dis- 
tance, and half the principal /atws rectum of the section to the 
perpendicular let fall from the common focus upon the tan- 
gent of the section, 
- ~ From the foregoing principles, and those previously estab- 
lished, it follows that bodies moving in any of the conic sec- 
tions, or in a straight line, by virtue of a centripetal force, 
will have their positions after any lapse of time, in the same 
ordinate, perpendicular to the common axis; for the areas cut 
off by a line drawn from the focus to the perimeter of any of 
the conic sections, must be proportional to the times of their 
description, and in different sections, having one common 
focus and vertex, they will be to. one another as their ordi- 
nates: but those conditions cannot be fulfilled, when the times 
are the same, except the line be drawn to the points of inter- 
section of one common ordinate, with the perimetres of the 
different sections. 
To show the place of a body moving in a rectilinear di- 
rection towards the centre of force, it is only necessary te 
find the corresponding position of it at any time moving in 
an ellipsis, parabola, or hyperbola, and letting fall a perpen- 
dicular ordinate en the axis of the curve. This, however, 
will obtain only in the. cases most important, viz. when the 
ree is such as to cause the bodies to move in conic sections, 
about the focus, or about the centre of the figure. The si- 
multaneous positions of a body moving in orbicalar or trajec- 
tory motion, in a curve designated by the ordinate on is 
of the curve, furnish a very ready method of ascertaining the 
times of bodies ending to the centre of force, supposing 
2 ne force) to vary as before, andi to, be in the focus; for 
— will be the. same as that of a semi-revolution in a 
whose radius is half the distance fallen through, The 
