Let us now examine how much more weight B will require than A, to balance. As the radii of circles are in proportion to their circumferences, they are also pro portionate to similar parts of them; there fore, as the arches, AD, CB, are similar, the radius, or arm, DE, bears the same proportion to EC. that the arch AD beat's to CB. But the arches AD and CB repre sent the velocities of the ends of the lever, because they are the spaces which they moved over in the same time ; therefore the arms DE and EC may also represent these velocities. Hence, an equilibrium will take place, when the length of the arm AE, multiplied into the power A, shall equal EB, multiplied into the weigh t 13; and, consequently, that the shorter EB is, the greater must be the weight B ; that is, the power and the weight must be to each other inversely, as their dis tances from the fulcrum. Thus, suppose AE, the distance of the power from the prop, to be twenty inches, and Ell, the distance of the weight from the prop, to be eight incises, also the weight to be raised at B to be five pounds ; then the power to be applied at A, must be two pounds ; because the distance of the weight from the fulcrum eight, multipli ed into the weight five, makes forty ; therefore twenty, the distance of the power from the prop, must be multiplied by two, to get an equal product ; which will produce an equilibrium.
The second kind of lever, when the weight is between the fulcrum and the power, is represented by fig. 3, in which A is the fulcrum, B the weight, and C the power. The advantage gained by this lever, as in the first, is as great as the dis tance of the power from the prop ex ceeds the distance of the weight from it. Thus, if the point cc, on which the power acts, be seven times as far from A as the point b, on which the weight acts, then one pound applied at C will raise seven pounds at B. This lever shews the rea son why two men carrying a burden upon a stick between them, bear shares of the burden, which are to one another in the inverse proportion of their distances from it.
It is likewise applicable to the case of two horses of unequal strength to be so yoked, as that each horse may draw a part proportionable to his strength ; which is done by so dividing the beam they pull, that the point of traction may be as much nearer to the stronger horse than to the weaker, as the strength of the former exceeds that of the latter. To this kind of lever may be reduced rud ders of ships, doors turning upon hinges, &c. The hinges being the centre of motion, the hand applied to the lock is the power, while the door is the weight to be moved.
If in this lever we suppose the power and weight to change places, so that the power may be between the weight and the prop, it will become a lever of the third kind ; in which, that there may a balance between the power and the weight, the intensity of the power must exceed the intensity of the weight just as much as the distance of the weight from the prop exceeds the distance of the power. Thus, let E, fig 4, be the prop of the lever EF, and W a weight of one pound, placed three times as far from the prop as the power P acts at F, by the cord going over the fixed pulley D ; in this case, the power must be equal to three pounds, in order to support the weight of one pound. To this sort of lever are generally referred the bones of a man's arm ; for when he lifts a weight by the hand, the muscle that exerts its force to raise that weight, is fixed to the bone about one tenth part as far below the elbow as the hand is. And the elbow being the centre round which the lower part of the arm turns, the muscle must therefore exert a force ten times as great as the weight that is raised. As this kind of lever is a disadvantage to the moving power, it is used as little as possible ; but in some cases it cannot be avoided ; such as that of a ladder, which being fixed at one end, is by the strength of a man's arms reared against a wall.
What is called the hammer-lever, dif fers in nothing but its form from a lever of the first kind. Its name is derived from its use, that of drawing a nail out of wood by a hammer. Suppose the shaft of a hammer to be five times as long as the iron part which draws the nail, the lower part resting on the board as a ful crum ; then, by pulling backwards the end of the shaft, a man will draw a nail with one-fifth part Of the power that Ise must use to pull it out with a pair of pin cers ; in which case, the nail would move as fast as his hand; but with the hammer, the hand moves five times as much as the nail by the time that the nail is drawn out. Hence it is evident, that in ever' species of lever there will be an equilibrium, when the power is to the weight as the distance of the weight from the fulcrum is to the distance of the power from the fulcrum. In experiments with the lever we take care that the parts are perfectly balanced before the weights and powers are applied. The bar, therefore, has the short end so much thicker than the long arm, as will be sufficient to balance it on the prop. If several levers be combined together in such a manner, as that a weight being appended to the first lever may be sup ported by a power applied to the last, as in fig. 5, which consists of three levers of the first kind, and is so contrived, that a power applied at the point L of the lever C, may sustain a weight at the point S of the lever A, the power must here be to the weight, in a ratio, or proportion, compounded of the several ratios, which those powers that can sustain the weight by the help of each lever, when used singly and apart from the rest, have to the weight. For instance, if the power which can sustain the weight W by the help of the lever A, be to the weight as 1 to 5 ; and if the power which can sus tain the same weight, by the lever B alone, be to the weight as]. to 4; and if the power which could sustain the same weight by the lever C, he to the weight as 1 to 5 ; then the power which will sus tain the weight by help of the three le vers joined together, will be to the weight in a proportion consisting of the several proportions multiplied together, of 1 to 5, 1 to 4, and 1 to 5; that is as 1 : 5 X4 X 5, or of 1 : 100. For since, in the lever A, a power equal to one-fifth of the weight W, pressing down the lever at L, is sufficient to balance the weight, and since it is the same thing whether that power be applied to the lever A at L, or the lever B at S, the point S bearing on the point L, a power equal to one-fifth of the weight P, being applied to the point S of the lever B, will support the weight ; but one-fourth of the same power being applied to the point L of the lever B, and pushing the same upward, will as ef fectually depress the point S of the same lever, as if the whole power were ap plied at S ; consequently a power equal to one-fourth of one-fifth, that is one-twen tieth of the weight P, being applied to the point L of the lever B, and pushing up the same, will support the weight : in like manner, it matters not whether that force be applied to the point L of the le ver B, or to the point S of the lever C, since, if S be raised, L, which rests on it, must be raised also; but one-fifth of the power applied at the point L of the lever C, and pressing it downwards, will as ef fectually raise the point S of the same lever, as if the whole power were appli ed at S, and pushed up the same; conse quently a power equal to one-fifth of one twentieth, that is, one hundredth part of the weight P, being applied to the point L of the lever C, will balance the weight at the point S of the lever A. This me thod of combining levers is frequently used in machines and instruments, and is of great service, either in obtaining a greater power, or in applying it with more convenience.