It also appears that the fmaller the angle GDC is, the lefs force will be required to drive the wedge: into any folid fubftance.

We

represented by the line EF, be refolved into two other forces; viz. FD parallel, and DE perpendicular, to AC; then the former of those forces, being parallel to the fide of the wedge, cannot have any power upon it; therefore the original force EF will have juft the fame effect upon the wedge as the leffer perpendicular force DE; the former being to the latter as radius to the fine of the inclination of the force EF to the fide AC. But, by the first part of this propofition, this perpendicular force DE is to the power on the back of the wedge which balances it, as AC is to AH, or as radius to the fine of the angle ACH, (viz. half the angle at the vertex of the wedge) therefore, by compounding those ratios, EF XED power on the back x ED:: force EF: power on the back square of radius ; fine of half the vertical angle x fine of the inclination of the refiftance.

The oblique force ef on the other fide of the wedge, being equal to EF, will require another power equal to the former on the back of the wedge, to balance it; therefore the fum of the refiftances on the fides of the wedge is to the whole power on the back, as the square of radius is to the product of the fine of half the vertical angle multiplied by the fine of the inclination of the refiftances to the fides of the wedge.

In order to prove the last part of the propofition, let GD, GF, and GE, fig. 4, Plate VII. reprefent the directions of the three forces perpendicular to the fides of the scalene wedge A B C. Produce EG ftraight towards O, and through F draw FO parallel to DG. Then fince those three forces balance each other, they must be (by prop. IV.

We fhall laftly obferve, that when the wood fplits below the edge of the wedge, as is fhewn by fig. 5, Plate VII. which is generally the cafe; then the fide of the wedge must be confidered as equal to either fide of the cleft; for in fact if we fuppofe that the wedge is lengthened downwards to the very apex of the cleft; the effect will be the fame.

THE SCREW.

The screw is the laft mechanical power that remains to be defcribed. This is likewife confidered as a fpecies of inclined plane; it being in fact no

thing

of chap. VIII.) refpectively proportional to the three fides of the triangle, GOF; but this triangle GOF is equiangular, and therefore (Eucl. p. 4. B. VI.) fimilar, to the triangle ABC; therefore the three forces are likewise respectively proportional to the three fides of the triangle or wedge ABC.

The triangles GOF and ABC are equiangular; for the four angles of the quatrilateral figure AEGF are equal to four right angles (Eucl. p. 32. B. I.) and fince the angles at E and F are right, the two angles FAE and FGE muft be equal to two right angles; that is, equal to FGE plus FGO, Therefore taking away the common angle FGE, there remains the angle FAE equal to FGO, Alfo in the like manner is proved that the angle DBE is equal to the angle OGD, and likewise equal (Eucl. p. 29. B. I.) to FOG. And fince the two angles at O and G of the tri-. angle FGO are respectively equal to the two angles at A and B of the triangle ABC; the third angle of the former. must be equal to the third angle of the latter.

thing more than an inclined plane coiled round a cylinder; and the nut or perforated body which moves up or down a fcrew, moves up or down an inclined plane in a circular, inftead of a rectilinear, direction.

Either the screw A may be moved forwards and backwards in a fixed nut as in fig. 6, Plate VII. or the fcrew A remains fixt, and the nut BC, or perforated piece, is made to move upon the fcrew as in fig. 7.-By way of diftinction A is called the male. Screw, and the nut B with its perforation shaped. like an hollow fcrew, is called the female ferew.The spiral projections e, f, g, &c, are called the threads of the screw.

The power which moves this most useful, and most powerful engine, is applied either to one end of the fcrew, which is generally furnished with a fort of head or projection, or to the end of a lever which is fixed either in the head of the fcrew as in fig. 6. or in the nut BC, as in fig. 7. And then indeed it may with more propriety be called an engine compounded of a fcrew and a lever.

In all cafes the equilibrium takes place between the effect which is produced at the end of the fcrew or at the nut, and the power, when the former is to the latter as the circumference defcribed by the power in one revolution, is to the diftance between two contiguous threads of the fcrew. Thus fup

pofing that, the distance between the

threads be

half an inch and the length of the lever CD be 12

inches;

inches; the circle defcribed by the end D of the lever will be about 76 inches, or 152 times the diftance between two contiguous threads; therefore if the power at the end D of the lever be equivalent to one pound, it will balance a preffure of 152 pounds acting against the end of the fcrew in fig. 6; or it will fupport a weight of 152 pounds on the board. B, fig. 7, &c.

The reafon of this is fo evidently dependent on the properties of the inclined plane, that nothing more needs here be faid about it.

The leaft reflection on the preceding explanation of the nature and properties of the mechanical powers will fufficiently prove that, strictly speaking, the real and original mechanical powers are not more than two in number; namely, the lever and the inclined plane; fo that all the others are only fpecies of thofe two; the balance, the wheel and axle, and the pulley, being fpecies of lever; and the wedge with the fcrew being fpecies of inclined plane. It is however immaterial whether those powers be reckoned all primitive and diftinct from each other, or not; for the theory remains always true and the fame. The only advantage which might be derived from the idea of the original mechanical powers being only two, is that their properties might, in that cafe, be explained in a much more concife manner; yet it is to be obferved that, after a certain limit, theories became obfcure and perplexing,

in proportion as they are rendered more concife and comprehenfive.

Before we quit the prefent chapter it will be proper to make the following remark, the object of which is to prevent the establishment of wrong notions in the mind of the reader, with refpect to the powers of the above-mentioned engines.

Beginners in this branch of natural philofophy frequently imagine that by means of the mechanical powers, a real increase of power is obtained; whereas this is not true. For instance, if a man be just able to convey 100 weight from the bottom to the top of his house in one minute's time, no mechanical engine will enable him to convey 300 weight to the fame height in the fame time; but the engine will enable him to convey the 300 weight in three minutes; which amounts to the fame thing as to fay that the man could, without the engine, carry the 300 weight by going three times to the top of the houfe, and carrying 100 weight at a time, provided the load admitted of its being fo divided. Therefore the engine increases the effect of a given force by lengthening the time of the operation; or (fince uniform velocity is proportional to the time) by increafing the velocity of that force or power.

Thus again, if any active force is able to raise a weight of 10 pounds with a given velocity, it will be found impoffible, by the use of any instrument,

to