294 Central Forces. 
. Et 
Vu yu ays, 
BD=",- Xt (25) or because >= const. ¢ is as the area BD, or 
supposing the particle to fall from A, the time from A to B is to the 
time from C to Bas the area of the semicircle ADB to the area BD. 
In like manner (23) gives his case 2. (see his fig. 2.) the axis AB 
of his rectangular hyperbola =2a, CB=r, AC=2a-+-r, and 
Cot. o/ =*/ — = tan. ang. CBD, a oe po’ = CBD,. hence as 
a ies : 
before the area BD= = xt (26), and the time from C to B isas 
the area BD. Also (24) gives his case 3. supposing that p= the 
semiparameter of his’ parabola, (see his fig. 3.) r=CB, then 
ay eo ak ' P * 
a cot. o/ = / P= tan. ang. CBD .".5 —9”=CBD, hence as before 
% 
ng 
: ve . 
_ Ihave the area BD — Xt (27), and the time of motion of the 
= particle from C to B is as the area BD. Again, (25), (26), (27 ) 
agree with Newton’s conclusions, (Prin. Vol. I. Sec. 7, prop: 35,) 
the areas BD in (25), (26) are equal to a 
dius of the circle described by the particle at the distance a, from 
the centre of force = aot ‘oe of the circle, or rectangular hy- 
perbola, and V’t= the arc of the circle described by the: particle 
» in which a= the 1a- 
, eee : ; ‘ 
in the time ¢, -“.—g—= the area described by the radius vector, " 
in the circle (rad. =a,) whence his first case is evident; also his se- 
cond case follows in the same manner from (27), the radius of the 
circle in this case =p= half the latus rectum of the parabola. 4° 
Newton’s constructions of props. 36 and.37, are evident from. what 
has been done, his 37th being equivalent to supposing that Viet 
in (7) and (8) are given to find a 3 whence by squaring those gpm: 
tions and writing them in the form of proportions, his proportions 
will be obtained for finding a, &c. but as these proportions are very 
simple, I shall here leave them. ‘ 
a, 
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