By using the qualities of conic sections and proportions, Newton brings the system of two bodies with masses m_{1} and m_{2}, that are acting on each other to the system of one body in free motion. In the beginning, instead of attraction between them, comes attraction to their general center of gravity (Theorem XXIV), then the nonmoving point transfers to "the center of gravity of the largest and most internal body" m_{1} (reduction and Theorem XXIV), and orbits of (points) with masses m_{1} + m_{2} and m_{2}(?) are built (Theorems XX, XXII, XXIII). Then, the imovable point is transferred to the general "center of gravity of the two internal bodies," and we obtain the orbit of a point with a mass μ. The orbit of the μpoint, an ellipse, (with time) becomes a circle with (the) radius of the larger semiaxis, so the motion of the μpoint will be without acceleration. For this reason, Newton introduced projective coordinates, i.e., {they transform a metric to a straight interconnected body}.
Page 160 From (3.1), by using proportions
By using r_{1}+r_{1} = r_{12} , v_{1}+v_{1} = v_{12} we obtain
Substituting (3.3) into (3.1) gives
which formally defines the system of two bodies of equal mass
b. The Problem of n Bodies
Let us look on the problem of n bodies. For its solution it is
necessary and sufficient to transform, by using Newton's laws, a series of
n bodies acting on each other, into a system of a finite (number) of
Newton's subsystems, each one with two bodies. This is done by
using the {main axes of quadratic form}  a metric that is induced in
R^{3} by a system of n bodies, that at the moment of
time t_{0} are polyhedra, located at the n centers of
gravity of the bodies.
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where
are second powers of radius vectors r_{i}.
 their scalar products, which are proportional to the projections of the
ith vector on the jth, m_{i}  masses that
normalize the values of the vectors r_{i}.
The zero components of the quadratic forms, (are) transferred to the main
axes of the centers of gravity of (n1) Newtonian subsystems of
n bodies.
From this we obtain the first equation of the center of gravity of the first Newtonian subsystem of two bodies  a body with mass m and the general center of gravity for n1 bodies, located at the distance from the general center of gravity for system C_{n} .
