Review of the Principia of Newton. “8S 
nearly in curvature to them, that it may safely be assumed 
for an orbit of that kind, in the part of it near the vertex. 
On account of the greater facility of calculating the angular 
motion in a parabola, than in a very eccentric ellipse, Astro- 
-nomers have generally considered comets as moving in thdt 
curve. Our author, for that reason, has, with his usual sa- 
gacity, given a geometrical solution of the angular motion 
rabola. Dr. Halley, and others since his time, have pro- 
duced analytical solutions better adapted to practice. They 
had their prototype in our author, and have only rendered 
the solution easier, but not more elegant. The assumption 
of a periodical time infinite, since it has been found to agree 
with the motions of comets in the lower parts of their orbits, 
shows how very eccentric they must be, and that their peri- 
odical times cannot be determined from observations of their 
motions within the regions of the planets. 
The next in order of the grand problems of our author, is 
that celebrated one of Kepler, which, since his time, has been 
considered as the foundation of all true Astronomy. As- 
suming the principles established by Newton, it is reduced to 
one nearly mathematical, viz. to cut an ellipsis by a line 
drawn from its focus to its perimeter, so that the area in- 
cluded between that line and any other line drawn to the = 
rimeter from the focus, may include a given 
will amount to the same thing, the equable description of 
areas, which corresponds with the equable angular motion of 
a body ina circle, being given, to determine from thence the 
angular motion of a body about the focus of re foe Si a 
describing the same areas. pe this 
roblem have been attempted, first Ragder himself b an 
indirect method, next by Bishop Ward fiyporteticaliy. by. 
lialdus, who te ater Ward’s hypothesis, by Cassini and 
—. But a direct geometrical or analytical solution was 
r, I believe, given before those of our author, as ex- 
hibited j in the 6th section of the work before us. His ana- 
lytical solution has been much improved, and accommodated 
to practice, by Dr. Keil, but as a speculative problem, in 
which the powers of genius are displayed, nothing can ex- 
ceed the solution given by Newton, if we except some of his 
own in his 2d Book. 
In a preliminary article, the author proves, that in no oval 
figure, can an area cut off at = iit from a given point 
VOL. XII.—No. 1. 
