fectly smooth and hard. It is represented by a line drawn in a vertical plane, and is supposed to pass through this line and to be perpendicular to the vertical plane. A vertical line is supposed to be drawn in the vertical plane from the upper extremity of the inclined plane; and both this vertical line, and the line which represents the inclined plane, are cut by a horizontal line or base, drawn in the same vertical plane. The portion of the inclined line and of the vertical line intercepted between the upper point of the plane and its horizontal base, are the length and the height of the inclined plane respectively.

*PROP. XX. The weight (W) being on an inclined plane, and the force (P) acting parallel to the plane, there is an equilibrium when P: W: the height of the plane its length.

F

Let AC be an inclined plane of which AC is the length, and let W be a weight on the inclined plane, supported by a force P, acting in the direction EF

parallel to AC.

The force of the weight DA

W

H

G

W acts in a vertical direction; draw EG vertical to represent this force. Also draw EH perpendicular and GH parallel to the plane AC.

The force EG is equivalent to the two forces EH, HG, (Prop. 13. Cor. 2); of these the force EH is balanced by the reaction of the plane AC, which will balance any force perpendicular to AC, by Axiom 13; and the weight W will be kept at rest if the force HG be counteracted by an equal

and opposite force P, acting in the direction EF. Therefore there will be equilibrium if P be represented by GH, when W is represented by EG; that is, P: W :: GH : EG.

But since EH is perpendicular and GH parallel to the plane AC, EHG is a right angle and therefore equal to ABC. Also the angle EGH is, by parallels, equal to GED, that is, to BFD, that is, to BCA. Therefore the two triangles ABC, EHG, have two angles equal, each to each, and are therefore equiangular, and therefore also similar. Hence GH EG :: BC AC, and therefore, by what has been proved already, P: W:: BC: AC, that is, P: W:: height of plane length of plane.

Q. E. D.

:

* DEF. If two points pass each through a certain space in the same time, the Velocities of the two points are to each other in the proportion of these two spaces.

* PROP. XXI. If P and W balance each other on the wheel and axle, and the whole be put in motion, P: W: W's velocity: P's velocity.

The construction being the same as in Prop. 16, let the machine turn round its axle CD through an angle ACa, or EDe; so that the radius of the wheel at which the acted, moves out of the position Ca into the position CA; and so that the radius of the axle at which the

power

H

E

K

e

power acted, moves out of the position De into the position DE. Then the string by which the power

P acts will be unwrapt from the portion a A of the circumference of the wheel, and therefore P will move through a space equal to a A. Also in the same time the string at which W acts will be wrapt upon the axle by a space equal to eE, and therefore W will move through a space equal to eE. Therefore by the definition of velocity a A, e E are as the velocities of P and W.

But since the wheel and axle is a rigid body, turning about the axis CD, all the parts move in planes perpendicular to the axis, and turn through the same angle; and since the plane of the wheel ACa, and of the axle EDe are both perpendicular to the axis, the angles ACa, EDe are the angles through which the radii CA, DE turn. Therefore the angles ACa, EDe, at the centers of the circles ACa, EDe are equal; and therefore, by the Lemma 3, DE : CA :: Ee Aa.

But by Prop. 16, DE : CA :: P: W; and by what has been just shewn, Ee: Aa :: W's velocity : P's velocity; therefore P: W:: W's velocity : P's velocity. Q. E. D.

* PROP. XXII. If P and W balance each other in the systems of pulleys described in Propositions 17, 18, and 19, and the whole be put in motion, P: W: W's velocity : P's velocity.

In Prop. 17, if W be raised through any space, as one inch, the string on each side of the pully A will be liberated for one inch, and therefore P will be at liberty to descend two inches: therefore W's

velocity: P's velocity :: 1: 2; and since by Prop. 17, Ꮲ : Ꮃ :: 1 : 2, P: W: W's velocity: P's

velocity.

::

In Prop. 18, if W be raised through any space, as one inch, each string at the lower block will be liberated one inch, and therefore as many inches of string will be liberated as there are strings at the lower block; and P will be at liberty to descend through a space equal to the whole of this. Therefore the space described by W: space described by P :: 1 : number of strings at the lower block; and hence by Prop. 18, and by the definition of velocity, P: WW's velocity: P's velocity.

In Prop. 19, if W be raised through any space, as one inch, each of the two strings at the lowest pully E will be liberated one inch; therefore the pully C will be liberated 2 inches, and will rise through 2 inches; therefore on each side the block C, 2 inches of string will be liberated; therefore the pully A will be liberated 2 x 2 inches; therefore the string on each side the pully A will be liberated 2 x 2 inches; therefore the string at which P acts will be liberated 2 x 2 x 2 inches, and since this happens at the same time that W is liberated one inch, W's velocity: P's velocity: 12 x 2 x 2. And it is clear that the last term is that power of 2 whose index is the number of moveable pulleys.

But by Prop. 19, P: W:: 1 : 2 × 2 × 2 as before; therefore, by what has been proved, P: W :: W's velocity: P's velocity.

* PROP. XXIII. If P support W on the inclined plane, acting parallel to the plane by means of a string of constant length, and if

the whole be put in motion, P: W :: W's velocity in the direction of of gravity P's velocity.

W

W

Let AC be the inclined plane, the weight W being supported by the force P acting parallel to the plane. Let W move to w, and P to p in the same time; and draw Wo horizontal and w v vertical. Then w v is the space through which W moves in the direction of gravity, while P moves through the space Pp, or Ww, which is equal to Pp, because the string w P is always of the same length. Therefore by the definition of velocity, W's velocity in the direction of gravity; P's velocity wv: Ww.

B

But since Wv is horizontal, or parallel to AB, and wv vertical, or parallel to CB, the triangle Wwv is similar to ACB. Therefore wv: Ww: BC : AC, that is, wv: Ww: height of the plane : length of the plane. But by Prop. 20, this proportion is that of P: W; therefore, by what has been proved P: W: W's velocity in the direction of gravity: P's velocity.

COR. If the string by which W is supported pass over a point C and hang vertically, as WC'Q, and if Q balance W, Q will descend through a space Qq equal to Ww, when W descends through a space Ww; and we may prove, as before, Q: W: W's velocity in the direction of gravity: P's velocity.

* DEF. The Center of Gravity of any body or system of bodies is the point about which the body or the system will balance itself in all positions.