68 
Dr. Waring on 
1.3. The increment (P^) of the fpace divided by the velo- 
city V is ultimately as the increment of the time, and 
= the increment of the velocity (V) divided by the force 
2V 2 py . 
pv“ x sF 111 
VxPVxSP. 
lv 2 xPY ’ 
VxPVxSP 
~ 2 V 2 x PY 
the direction of the tangent, that is, ~ = , 
for Vp fubftitute — , and there refults 7 P x F( ? 
1 PY PY x V 
; and confequently 7 — X. ; but AAAA =: SY 
^ J SPxPV v 2PO 
— P, whence 
-p v 
P “V 7 
a 
and V = - , 
P 
where a is an. invariable 
quantity. v 
Cor. Since V x P, that is, SY the perpendicular multiplied 
into the velocity (which is ultimately as P^ the fpace defcribed 
in a given time) is ultimately as the area defcribed round the 
center S in a given time; but this redtangle —a, a given 
quantity; therefore the area, defcribed round the center of 
force S in a given time, will be a given quantity, and thence 
in unequal times will be proportional to the times. 
1.4. The fagitta QR is ultimately as the force, when the 
time is given; and when the time is not given, it will be as 
the force into the fquare of the time ; from which expreffion, 
by fubftituting for QR and the time their values, may be de- 
duced feveral others. 
Sir Isaac Newton has demonftrated this proportion with 
the greatefl fimplicity ; and this is given to fhew, that the 
fame proportion may be deduced from different principles. 
PROP. II. 
1. Fig. 3. Given the relation between SP 7 the diilance from, 
a point S, and SY 7 a perpendicular from the point S to P 7 Y, a 
line touching a curve in the point P 7 ; to fnd the relation 
3 between 
