gravity of its hemispherical base. being in the axis, tends to approach the horizontal plane as much as possible, and this can never be the case till the axis becomes perpendicular to the horizon; for the small figure above scarcely deranges it from its place, on account of the disproportion between its weight and that of its base.

In this manner, were constructed those small figures called Prussians, sold at Paris some years ago. They were formed into battalions, and being made to fall down by drawing a rod over them, they immediately started up again as soon as it was removed.

Screens of the same form have been since invented, which always rise up of themselves, when they happen to be pressed down.

PROBLEM XI.

If a rope AC B, to the extremities of which are affixed the given weights P and 2, be made to pass over two pulleys A and B; and if a weight R be suspended from the point C, by the cord RC; what position will be assumed by the three weights and the rope ACB? (fig. 11 pl. 2).

IN the line ab, perpendicular to the horizon, assume any part a c, and on that part as a base, describe the triangle a dc, in such a manner, that a c shall be to cd, as the weight R, to the weight P; and that ac shall be to a d, as R to Q; then through A, draw the indefinite line A C parallel to cd; and through B, draw B C, parallel to ad: the point C, where these two lines intersect each other,

will be the point required, and will give the position ACB of the rope.

For, if in R C continued we assume CD, equal to a c, and describe the parallelogram EDFC; it is evident that we shall have CF and C E, equal to cd and ad; and therefore the three lines EC, CD, and C F will be as the weights P, R and Q; consequently the two forces acting from C to F, and from C to E, or in the direction of the lines CA and C B, will be in equilibrio with the force which acts from C towards R.

REMARKS.

1st. If the ratio of the weights were such, that the point of intersection C should fall on the line A B, or above it, the problem in this case would be. impossible. The weight Q, or the weight P, would overcome the other two in such a manner, that the point C would fall in B or A; so that the rope would form no angle.

These weights also might be such that it would be impossible to construct the triangle ac d, as if one of them were equal to or greater than the other two taken together; for, to make a triangle of three lines, each of them must be less than the other two. In that case we ought to conclude that the weight equal or superior to the other two would overcome them both, so that no equilibrium could take place.

2d. If instead of a knot at C, we should suppose the weight R suspended from a pulley capable of rolling on the rope A CB, the solution would be still the same; for it is evident that, things being in the same state as in the first case, if a pulley were substituted for the knot C, the equilibrium

It

would not be destroyed. But there would be one limitation more than in the preceding case. would be necessary that the point of intersection, €, determined as above, should fall below the horizontal line, drawn through the point B; otherwise the pulley would roll to the point B, as if on an inclined plane.

PROBLEM XII.

Calculation of the time which Archimedes would have required to move the earth, with the machine of which he spoke to Hiero.

THE expression which Archimedes made use of to Hiero, king of Sicily, is well known, and particularly to mathematicians. "Give me a fixed point," said the philosopher," and I will move the earth from its place." This affords matter for a very curious calculation, viz. to determine how much time Archimedes would have required to move the earth only one inch, supposing his machine constructed and mathematically perfect; that is to say, without friction, without gravity, and in complete equilibrium.

For this purpose, we shall suppose the matter of which the earth is composed to weigh 300 pounds the cubic foot; being the mean weight nearly of stones mixed with metallic substances, such in all probability as those contained in the bowels of the earth. If the diameter of the earth be 7930 miles, the whole globe will be found to contain 261107411765 cubic miles, which make 1423499120882544640000 cubic yards, or 38434476263828705280000 cubic feet; and allowing 300 pounds to

each cubic foot, we shall have 11530342879148611584000000 for the weight of the earth in pounds.

Now, we know by the laws of mechanics that, whatever be the construction of a machine, the space paffed over by the weight, is to that passed over by the moving power, in the reciprocal ratio of the latter to the former. It is known also, that a man can act with an effort equally only to about 30 pounds for eight or ten hours, without intermission, and with a velocity of about 10000 feet per hour. If we suppose the machine of Archimedes then to be put in motion by means of a crank, and that the force continually applied to it is equal to 30 pounds, then with the velocity of 10000 feet per hour, to raise the earth one inch, the moving power must pass over the space of 384344762638287052800000 inches; and if this space be divided by 10000 feet, or 120000 inches, we shall have for quotient 3202873021985725440, which will be the number of hours required for this motion. But as a year contains 8766 hours, a century will contain 876600; and if we divide the above number of hours by the latter, the quotient, 3653745176803, will be the number of centuries during which it would be necessary to make the crank of the machine continually turn, in order to move the earth only one inch. We have omitted the fra tion of a century, as being of little consequence in. a calculation of this kind *.

The machine is here supposed to be constantly in action; but if it should be worked only 8 hours each day, the time required would be three times as long.

PROBLEM XIII.

With a very small quantity of wa'er, such as a fes pounds, to produce the effect of several thousands. (Plate 3 fig. 12).

PLACE a cask on one of its ends, and make a hole in the other end, capable of admitting a tube, an inch in diameter and from 12 to 15 feet in length; which must be fitted closely into the aperture by means of pitch or tow. Then load the upper end of the cask with several weights, so that it shall be sensibly bent downwards; and having filled the cask with water, continue to pour some in through the tube. The effort of this small cylinder of water will be so great, that not only the weights which pressed the upper end of the cask downwards will be raised up, but very often the end itself will be bent upwards, and form an arch in a contrary direction.

Care however must be taken that the lower end of the cask rest on the ground; otherwise the first effort of the water would be directed downwards, and the experiment might seem to fail.

By employing a longer tube, the upper end of the cask might certainly be made to burst.

The reason of this phenomenon may be easily deduced from a property peculiar to fluids, of which • it is an ocular demonstration, viz. that when they press upon a base they exercise on it an effort proportioned to the breadth of that base multiplied by the height. Thus, though the tube used in this experiment contains only about 150 or 180 cylindric inches of water, the effort is the same as if the tube