[ 5^9 ] 
of the greater axis of the earth’s orbit, of the ratio 
of the cofine of the angle, which the orbit of the 
planet makes with the plane of the ecliptic, to the 
radius, and of the ratio of a line drawn in any angle 
from the center of the fun to the tangent of the or- 
bit of the earth at the point, wherein the earth is, to 
a line drawn in the fame angle from the fun to the 
tangent of the orbit of the planet projected upon the 
plane of the ecliptic at the place of the planet in the 
ecliptic. 
Let A be the fun, B C the orbit of any planet, 
D E the fame projected on the plane of the ecliptic, 
F G being the line of the nodes, B the place of the 
planet in its orbit, D its projected place : then the 
plane through B and D, which fhall be perpendi- 
cular to both the planes B C and D E, interfe&ing 
thofe planes in BH, DH, the lines BH, DH will 
be both perpendicular to the line of the nodes, and 
the angle BHD the inclination of the orbit to the 
plane of the ecliptic. But tangents drawn to B C 
and DE at the points B and D refpedtively will 
meet the line of the nodes, and each other in the 
fame point I, and the velocity of the planet in lon- 
gitude will be to its velocity in the orbit B C, as D I 
to BI. 
Now from the point A let A K fall perpendicular 
on B I, and A L be perpendicular to D I : then 
the ratio of D I to IB will be compounded of the 
ratio of D I to D H, or of A I to A L, of the ratio 
of DH to BH, and of that of BH to B I, that is, 
of AK to A I. But DH is to BH as the cofine 
of the inclination of the orbit to the radius, and the 
two ratios, that of A I to AL, and that of AK to 
A I, compound the ratio of AK to A L : therefore 
X x x 2 the 
