50. PROP. XVII. If a body balance itself upon a line in all positions, the Centre of Gravity of the body is in that line*.

Let the body AB balance itself in all positions upon the straight line CD, the Centre of C Gravity of the body shall be in CĎ.

For, if not, let a point G, without CD, be the Centre of Gravity; A and, first bringing CD into an horizontal position, turn the body round. CD until G is in the same horizon

H

B

tal plane with CD; then draw GH perpendicular to CD, meeting it in H, and GF vertical.

Now, since, by Definition, the body will balance itself on G in all positions, it will balance itself on G in this position, that is, the resultant of all the forces acting on the body passes through G; and since these forces (being the pressures exerted upon the several particles of the body by the force of gravity) are all parallel and vertical, the resultant will also be vertical, and equal to the sum of them, viz. the weight of the body. Replacing, then, all the forces acting on the body by their resultant, we have the case of a single force acting perpendicularly at G to turn the lever GH round the fulcrum H (CD, and therefore H, being supposed fixed in position); which force is not counteracted by any other, and therefore will turn the body round H, that is, round CD. But, by supposition, the body balances itself upon CD in all positions. Hence, the assumption that G lies anywhere without CD leads to an impossibility; and therefore G can only be in CD.

51. PROP. XVIII. To find the Centre of Gravity of two heavy points†, and to shew, that the pressure at the

* That is to say-If there be a line round which, as an axis, a body can be made to revolve, so that, when the line is held in any position, the body, after being made to revolve round it into any position, remains at rest, the Centre of Gravity of the body is in that line.-It is evident, that the line must pass through the body.

By "heavy points", in this Proposition and the next, are meant exceedingly small material bodies, and not geometrical points. For a geome

Centre of Gravity is equal to the sum of the weights in all positions.

Let P and Q be the weights of two heavy points A and B, supposed to be connected by a straight rigid rod AB without weight.

In AB take a point C, such

that BC AB: P: P+Q; then, M

dividendo,

B

N

Through C draw MCN horizontal; and through A and B draw the vertical lines PAM and QNB; these last are the lines in which the weights P and Q act.

Then, the angles at M and N being right angles, and the angle ACM being equal to the opposite angle BCN, the angle CAM is equal to the angle CBN, and the triangles ACM and BCN are equiangular, and... similar.

Now, P: Q: BC: AC,

:: CN: CM, by similar triangles;

therefore, if ABC be considered as a lever with fulcrum C, since P and Q are inversely as the perpendiculars drawn from the fulcrum to the lines in which the forces act, by the converse of Prop. vI. (Art. 26), P and Q will balance each other on C.

Also, if AB be turned round C into any other position, the same reasoning holds; and therefore A and B will balance on C in all positions of AB. Hence, by Definition, C is the Centre of Gravity of A and B.

Again, by what has been shewn, the weights P and Q will balance on C, when ACB is horizontal. But, in that case, by Axiom II. (Art. 19), the pressure on the fulcrum is equal to the sum of the weights. And in any other position of AB, as in the fig., P and Q will produce

trical point does not possess length, or breadth, or thickness, and consequently can be of no weight, since it can contain no matter.

the same effect as if they acted at M and N on the horizontal straight lever MCN, exerting a pressure on C equal to their sum. Therefore the pressure at the Centre of Gravity is equal to the sum of the weights in all positions of the system.

52. PROP. XIX. To find the Centre of Gravity of any number of heavy points; and to shew, that the pressure at the Centre of Gravity is equal to the sum of the weights in all positions.

Let A, B, C, three heavy points whose weights are P, Q, R, be connected together and placed

in any position.

Join AB, and take D a point in A

AB, such that

BD: AB:: P: P+Q;

B

.. BD: AB-BD, or AD, :: P: P+Q-P, or Q;

R

therefore, by Prop. XVIII., D is the Centre of Gravity of A and B; and the pressure produced by P and Q, in all positions of the system, is a pressure P+Q acting vertically at D.

Join DC, and in DC take a point E, such that

DE: DC: R: P+Q + R ;

.. DE: EC:: R: P+Q;

therefore E is the Centre of Gravity of the weight P+Q acting at D, and R acting at C; and if E be supported, those weights are supported in any position of the system. Since therefore the system will balance itself in all positions on E, that point is its Centre of Gravity;—and the Pressure on E is P+Q + R.

The construction here applied to a system of three bodies may be extended to a system of any number of bodies.

Wherefore the Centre of Gravity of any number of heavy points may always thus be found, and the pressure on the Centre of Gravity is equal to the sum of the weights in all positions.

53. By the definition given in Art. 49 of "the Centre of Gra vity" of a body, it will be understood, that to have a Centre of Gravity a body must have Weight. Now in the next two Propositions it is required to find the Centres of Gravity of a line, and of a plane, the former of which is defined by Euclid to have length merely, without either breadth or thickness; and the latter, though possessing length and breadth, is defined to be without thickness. A geometrical line, or plane, therefore, can have no weight; since there can be no weight where matter does not exist, and when matter exists under any form whatever, it is of three dimensions, or has length, breadth, and thickness. The line, therefore, and the plane, of which it is required to find the Centres of Gravity, are not the line and plane of Geometry.

But the line of which the Centre of Gravity is determined in the next Proposition is supposed to be formed of very small equal heavy bodies placed either in contact, or at equal distances, along the whole length of the line. And the plane triangle referred to in the next Proposition but one, is supposed to be made up of such lines, arranged parallel to any one of the sides of the triangle, and at equal distances from one another.

54. PROP. XX. To find the Centre of Gravity of a straight line.

Let AB be a straight line composed of small equal heavy bodies ranged either in contact, or at equal distances, along its whole length.

Bisect AB in C, and let P and Q be two of the small heavy bodies equally distant from C.

Then, by Axiom 1. Art. 19, and Prop. AP

VII., P and Q will balance in every

position on C.

9 B

And since the same is true of all such pairs of heavy bodies that are equidistant from C, the whole line will balance on C in every position, and therefore C is the centre of gravity of the line.

55. PROP. XXI. To find the Centre of Gravity of a triangle.

Let ABC be a triangle formed of lines ranged, at equal

Bisect AB in E, and BC in F; join AF, CE by lines intersecting in G;-G is the Centre of Gravity of the triangle.

For Af: fb

Abƒ, ABF'),

AFC, Afc).

AF: FB (from the equiangular triangles :: AF: FC, ·· BF=FC;

Af: fc (from the equiangular triangles

And since the first and third terms of this proportion are the same, fb is equal to fc; and therefore the straight line bc would balance in any position on f, by Prop. xx.

In the same manner all the other lines parallel to BC may be shewn to balance in any position on the points in which they are cut by AF; therefore the whole triangle will balance on AF in any position. Hence the Centre of Gravity of the triangle is in AF, by Prop. XVII.

Similarly, by supposing the triangle to be made up of lines parallel to AB, it may be shewn, that the Centre of Gravity of the triangle is in the line CE.

But AF and CE have only one point in common, that is, G, the intersection of AF and CE: therefore G is the Centre of Gravity of the triangle ABC.

56. PROP. XXII. When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line, drawn from its Centre of Gravity, falls within or

without its base.

[DEF. By "base" is here meant the area formed by drawing a string tightly round the lowest part of the body in contact with the plane. Thus, if the body be a three-legged stool, its 'base', for this purpose, will be a triangle—of a chair the 'base' will be a quadrilateral; and so on.-Again, the 'base' of a man, standing on a hori