Centre of. From Archimedes to Doctor Wallis, writers have assumed, that there belongs to every body an individual point, where, for most purposes, the whole weight may be conceived to reside; and they have called it the centre of gravity. The learned Englishman, however, believed himself the first to accomplish a demonstration from the nature of gravity. (Scholium to Prop. xv. De Centro Gravitatis.)

But the same point is admirably connected with other properties and contingencies of the body than its weight; and hence different names have been considered as preferable by many writers, such as centre of inertia, &c.

And from viewing a body as composed of physical points, it is found that the relation of this centre to their positions is fertile in some of the most beautiful theorems in the mathematics; whence it has properly received the geometrical designation of the centre of position.

We here introduce it in this geometrical character; and shall demonstrate very briefly, but we trust sufficiently, a collection of elegant and most important properties, of which some are virtually proved elsewhere, in a manner more suited to the direct object of their introduction.

If there be any multitude of distinct points in space, it is clear that a plane may move, parallel to any given plane, until the collective distances of the points on one side of it be equal to those on the other.

Let a plane thus situated, relatively to a system of points, be called a plane of equidistance. Suppose two planes of equidistance, A and B, perpendicular to each other. Then, first, we shall find that any plane C which passes through their common section or axis, is also a plane of equidistance to the same system of points.

For, let any plane, perpendicular to this axis in O, cut the planes A, B, C, in the straight lines Aa, Bb, Cc, respectively. Then Aa and Bb are at right angles. Let a perpendicular from any point of the system meet the plane ABC in p. Draw pq at right angles to Cc, and let it be positive when on the same side of the plane C as AO. Also draw pr perpendicular to Bb, and meeting Cc in s; and let pr and rs be positive when on the same sides of the planes B and C respectively as AO. The triangle pqrs being given in species, pq is as ps, that is, as pr + rs; so that the sum of the pq is as the sum of the pr together with that of the rs.

But the pr are the distances of the points of the system from the plane B, which is a plane of equidistance; and therefore their aggregate is nothing. And again, the rs are as the rO, which are the distances from the plane A; and, having different signs on the two sides, their aggregate is also nothing. Thus the sum of the pq is nothing; and the plane C is a plane of equidistance.

Now let O move along the axis to make ABC a plane of equidistance. Then shall any plane K which passes through O be such a plane.

For let the plane K cut ABC in the line Ce. The plane ABC, and the plane C, which cuts it perpendicularly in the line Ce, are planes of equidistance; therefore, by what we have just proved, the plane K which passes through their common axis Ce is also a plane of equidistance. Thus, to every system of points, there is a point through which every plane is a plane of equidistance. And there cannot be more than one. For if there were two, we might have through them two parallel planes of equidistance, which is manifestly absurd.

This single point has been called the centre of position.

Cor. 1. The difference of the centre of position from any given plane is an average of the distances of the several points of the system; the distances on different sides being distinguished by contrary signs.

For conceive the parallel plane of equidistance, at the distance D from the given plane. Let the distance of a point of the system from the given plane be denoted by D + d. Then d will vary its sign according as the point of the system and the given plane be on different sides of the plane of equidistance, or on the same side.

Now if there be p points in the system, the aggregate of their distances from the given plane will be pD, together with the aggregate of the d. But the collective d are nothing; so that it will be pD; or D is the average distance.

Cor. 2. If the average distances of the points of a system from three planes, perpendicular or inclined to each other, be given, the centre of position is given. For, by the distances of a point from three such planes, its position in space is given.

Cor. 3. If the centre of position, and the multitude of points in each of two systems be given, the centre of the united systems is found in the straight line joining the given centres at distances from these reciprocally proportional to the multitudes.

Let P and Q be the centres of position of two systems of p and q points respectively. Let perpendiculars from P and Q meet any plane of equidistance in a and b respectively.

Then \( p \cdot Pa + q \cdot Qb \) expresses the collective distances of the points of the united systems from the plane of equidistance, consequently the two terms are of opposite signs and equal magnitude.

And because Pa and Qb are perpendicular to the same plane, they are parallel; so that Pa, ab, bQ, are in one plane. Let O be the intersection of PQ and ab. Then PO and QO are as Pa and Qb; and since \( p \cdot Pa = q \cdot Qb \), we have also \( p \cdot PO = q \cdot QO \); or \( PO : QO :: q : p \).

Thus every plane of equidistance passes through the same point O, which is therefore the centre of position.

Scholium. Hence, if we have any points A, B, C, D, &c., bisecting AB in M, or making AM = 1/2AB; again, making MN = 1/2AC; and NO = 1/2OD, and so on: Then M is the centre of position of the two points A and B; N of the three points A, B, C; O of the four points A, B, C, D, and so on. And as we have proved that there cannot be more than one centre of position to the same system of points, the ultimate point must be the same in whatever order we take the given points into the operation.

Cor. 4. The difference of the squares of the distances of the centre of position from the extremities of any straight line is an average of the difference of the squares of the several points of a system from these extremities.

Let SO be the straight line; A a point of a system whose centre of position is G; and Aa and Ga perpendicular to the extension of SO. Then \( AS^2 - AO^2 = S^2 - O^2 = (aS - aO) \cdot OS \); making aS and aO negative when measured contrary to the tendency OS.

Hence \( \Sigma (AS^2 - AO^2) = \Sigma (aS \cdot OS + aO \cdot OS) \). But, if there be p points, \( \Sigma (aS) \) is pOS, and \( \Sigma (aO) \) is pOOS. Therefore, \( \Sigma (AS^2 - AO^2) = p \cdot (GS^2 - GO^2) \).

Cor. 5. The squares of the distances of any points in space from their centre of position are together less than from any point in the surface of a sphere about this centre, by the square of the radius multiplied by the number of points.

For when G coincides with O, g is also there; and we have \( \Sigma (AS^2 - AO^2) = p \cdot OS \cdot OS \), or \( p \cdot OS^2 \).

Cor. 6. If A, B, C, &c., express the number of points whose respective centres are in the positions A, B, C; and O be the centre of position of all the points united in one system; then, S being any other point,

\[ A \cdot SA^2 + B \cdot SB^2 + C \cdot SC^2 = A \cdot OA^2 + B \cdot OB^2 + C \cdot OC^2 + (A + B + C) \cdot OS^2. \]

For, by Cor. 4, \( A \cdot (SA^2 - OA^2) + B \cdot (SB^2 - OB^2) + C \cdot (SC^2 - OC^2) = (A + B + C) \cdot OS^2. \] Position. — \(OC^2\) expresses the excess of the squares of the distances of all the points from \(S\) over the squares of their distances from \(O\), which (by the last cor.) is equal to \((A + B + C)\).

Cor. 7. In any system of points, the motion of the centre of position, in any given direction, is an average of their several motions in that direction.

For assume a plane perpendicular to the given direction behind all their motions. Let there be \(p\) points, and the sum of their distances from the assumed plane, at the beginning of the motion \(pd\). Then the distance of the centre of position from this plane is \(d\).

At any instant afterwards, let their united distances from this plane be \(pD\), that of their centre of position being therefore \(D\).

The aggregate motion of the system in the given direction has been \(pD - pd\), or \(p(D - d)\), while that of the centre of position has been \(D - d\).

Cor. 8. If the several points of a system be uniform as to their motion, the centre of position either rests or moves uniformly.

If it do not rest, whatever be its first tendency, it must persevere in the same. For as it begins without any motion perpendicular to this direction, the aggregate of the motions of the system in any direction perpendicular to this must be nothing, and therefore will continue nothing; so that the centre will never be taken out of its first direction. And as the motion of each point, estimated in this direction, is the same during a given time, the average, or the motion of the centre is the same; so that it traces its path uniformly.

Otherwise.

Because the motion of any point of the system, estimated in three different directions not in the same plane, is uniform in each; that of the centre of position is uniform in these directions, or is nothing. Hence it can have no motion other than uniform in direction and velocity.

Cor. 9. If a body, or system of bodies, be put in motion, and subject to no other disturbance than the action of the parts or bodies upon each other; the centre of position will either rest or move uniformly in a straight line.

For every particle is at first determined to a certain direction and velocity, or put in a condition of uniform motion. And the actions of material particles or bodies upon each other, (whether by impulsion, attraction, impact, or leverage,) are equal and contrary; so that the aggregate motion, and consequently the average motion, is not affected by it.

Cor. 10. The aggregate momentum of a system, in any direction, is as if it were all collected into the centre of position.

The momentum in any direction is the united momenta of the several particles. Now, if there be \(p\) particles, and their several velocities in a given direction be denoted by \(v'\), \(v''\), \(v'''\), &c.; these also represent their several momenta; and if we put \(pv = v' + v'' + v''' + \ldots\); \(v\) is the velocity of the centre of position. Thus the aggregate momentum is as if every particle were in this centre.

Scholium. When the system is connected these parallel momenta compound a force in the given direction, which passes through the centre of position, or centre of gravity; and is equal to the sum of the several forces. So that the impact of the system is as if all were collected in the centre.

Cor. 11. If a body be left to itself in a state of rotation, the axis of rotation will pass through the centre of position.

For when there is rotation, no point out of the axis can either rest or move in a straight line. Therefore the centre of position will be in the axis.

Cor. 12. If attracting particles be left to their mutual attraction, they will coalesce about their centre of position. For their attractions do not affect the aggregate or average motion in any direction. Therefore the centre of position remains at rest while they are brought together around it.

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