THE WEDGE.

The wedge has been juftly confidered as a species of inclined plane; for it confists of two inclined

have a common angle at B): fin. VOE, or VOC; for those angles being the complement of each other to two right angles, have the fame fine.

From this propofition the following corollaries are evidently deduced:

1. Since P: W:: fin. BAC: fin. VOC; it

weight W, and the inclination of the plane, or fin. BAC remain the same, the power muft increase or decrease inverfely as the fine of VOC; hence when the direction of the power is perpendicular to EC, or parallel to the plane AB, then the fine of VOC, being the fine of a right angle, is the greatest fine poffible, and, of courfe in that cafe the power P, which is required to fuftain the weight W, is the leaft poffible; or, which amounts to the fame thing, then the greatest weight may be fuftained by a given power. Alfo when the direction of the power coincides with OC, namely when the power acts in a direction perpendicular to the plane, then the angle VOC vanishes, and the power must be infinitely great.

2. If the direction of the power be parallel to, or coincide with the plane, then the equilibrium takes place when the power is to the weight :: OB: BC:: (Eucl. p. 8. B. VI.) BC BA; viz. as the elevation of the plane is to its length, or as the fine of its inclination is to radius.":

3. If the direction of the power be OR; that is, parallel to the horizon, then the equilibrium takes place when the

- power

inclined planes joined bafe to bafe, as fhewn in fig. 1, Plate VII. where AB or GC is the thicknefs of the wedge at its back, upon which the force or power is applied (be it the stroke of a mallet, or any other preffure); the middle line FD is the axis. or height of the wedge; DG and DC are the lengths of its flant fides; and OD is its edge, which is to be forced into the wood or other folid ; fince the use of this inftrument is for cleaving of wood, ftone, and other solid substances; or, in general, for feparating any two contiguous furfaces.

power is to the weight :: OR: CR :: (fince the triangles ORC and BAC are fimilar) BC: CA; viz. as the elevation of the plane is to its bafe.

4. The power must sustain the whole weight, when its direction is perpendicular to the horizon.

5. The power is to the preffure on the plane :: OV: OC :: fin. OCV : fin. OVC :: fin. BAC: fin. OVC.

6. The preceding analogy,by alternation, becomes P: fin. BAC:: preff.: fin. OVC, from which it appears that when the power and the inclination of the plane, or angle BAC, remain invariable, the preffure on the plane must increase or decrease according as the fine of OVC increases or decreases; therefore when the direction of the power is parallel to the base, and OVC becomes a right angle, whose fine is the greateft, then the preffure on the plane will likewife be greatest.

7. When the direction of the power is parallel to the plane, P: preffure :: OB: OC :: BC: AC.

8. When the direction of the power is parallel to the bafe AC, then P: preffure :: OR: OC:: BC: BA.

Hence its application is very extensive, and in fact, fciffars, knives, nails, chifels, hatchets, &c. are nothing but wedges under different fhapes.

Strictly speaking, in the geometrical language, a wedge may be called a triangular prifm; for it may be conceived to be generated by the motion of a plane triangle in a direction parallel to itself, as that of the triangle GCD, from GCD to ABO. And it is called an ifofceles or fcalene wedge, according as the generating triangle, or face, GCD, is ifofceles or scalene.

The action of the wedge, is evidently derived from that of the inclined plane; yet a variety of circumstances has rendered the investigation of the power of the wedge more perplexing than that of any other mechanical power *.

The most rational theory fhews, 1, That when the preffures on the fides of the ifofceles wedge are equal and act in directions perpendicular to thofe fides, the equilibrium takes place, when the force on the back of the wedge is to the fum of the preffures on the fides, as GF, viz. half the thickness of the back, is to either of its flant fides, GD, or CD. 2. That when the pref

fures

*The proportion between the power, which is applied to the back of the wedge, and the effect which is produced on the fides, has been ftated differently by different authors. Those who wish to examine the reafons of those different opinions, may consult Rowning's Comp. Syft. of Phil. P.I. chap. 10; and Ludlam's Mathem. Ellays.

fures are equal, but act in directions equally inclined to the fides of the ifofceles wedge, the equilibrium takes place when the force on the back is to the fum of the refiftances upon the fides, as the product of the fine of half the vertical angle GDC of the wedge, multiplied by the fine of the angle which the directions of the refiftances make with the fides, to the fquare of radius. And 3. that when in a scalene wedge three forces acting perpendicularly upon its three fides, keep each other in equilibrio, thofe three forces are refpectively proportional to the fides.

The three parts of this propofition will be found demonstrated in the note (2).

From

(2) In order to demonstrate the first part of the abovementioned propofition, let AKD, fig. 2, Plate VII. reprefent the face of an ifofceles wedge. B and E are two obftacles, which prefs upon its two fides in directions perpendicular to those fides. Suppose the wedge to be impelled downwards as far as the dotted reprefentation GLF, in confequence of which the obftacles B and E must be driven to the places O and M. Through O and M draw OI and MQ parallel to the middle line or axis CD of the wedge; which lines will meet those fides in two points I, Q. Join I, Q, as also O, M, with the lines IQ, OM. Then it is evident from the parallelism of the lines, that OM is equal to IQ; hence the part IQ of the wedge must have advanced as far as OM; therefore YN, or IO, or QM, represents the velocity of the wedge (that is of the power); whilst BO and EM represent the velocities of the obftacles..

From this it follows that by the addition of a little more force on the back of the wedge, than that which is fufficient to form the equilibrium, the refiftances will be overcome, &c.

It

Now the triangles IOB, and ACD are equiangular; (the angles at C and B being right, and the angle BIO equal to CDA; Eucl. p. 29. and 32. B. I.) and of courfe fimilar (Eucl. p. 4. B. VI.) therefore confidering half the wedge and one obftacle, OB:OI:: AC: AD; that is, the velocity of the obftacle B is to the velocity of the power, as half the thickness of the wedge is to its flant fide. Likewife for the fame reasons we say that the velocity of the preffing obftacle E is to that of the power, as half the thickness of the wedge is to its flant fide. Therefore, by adding those proportional quantities, we fay that the velocity of the obstacle B plus the velocity of the other obftacle E, is to the velocity of twice the half wedge, (viz. of the whole wedge) as the whole length AK of the back, is to the fum of the fides AD, DK; or as half the length of the back is to one fide.

But when oppofite powers, which act upon each other, are inversely as their velocities, they form an equilibrium ; therefore when the power on the back of the wedge is to the fum of the refiftances on the fides, as half the length of the back is to one flant fide, the wedge remains motionless, which is the first part of the propofition.

In order to prove the fecond part of the propofition; let ABC, fig. 3, Plate VII. be the face of an ifofceles wedge, HC its height or middle line, E and e two obftacles which prefs upon, or are to be removed, in the directions EF, ef equally inclined to the fides of the wedge, Let the force, represented