MATHEMATICAL

AND

PHILOSOPHICAL

RECREATIONS.

PART THIRD.

Containing various Problems in Mechanics.

AFTER arithmetic and geometry, mechanics is the next of the physico-mathematical sciences having their certainty resting on the simplest foundations. it is a science also the principles of which, when combined with geometry, are the most fertile and of the most general use in the other parts of the mixed mathematics. All those mathematicians therefore who have traced out the development of mathematical knowledge, place mechanics immediately after the pure mathematics, and this method we shall here adopt also.

We suppose, as in every other part of the mathe

matics introduced into this work, that the reader is acquainted with the first principles of the science of which we treat.--Thus, in regard to mechanics, we suppose him acquainted with the principles of equilibrium and of hydrostatics; with the chief laws of motion, &c. For it is not our intention to teach these principles; but only to present a few of the most curious and remarkable problems, which arise from them.

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PROBLEM I,

To cause a ball to proceed in a retrograde direction, though it meets with no apparent obstacle.

PLACE an ivory ball on a billiard table, and give it a stroke on the side or back part, with the edge of the open hand, in a direction perpendicular to the table, or downward.-It will then be seen to proceed a few inches forward, or towards the side where the blow ought to carry it; after which it will roll in a retrograde direction, as it were of itself, and without having met with any obstacle.

REMARK.

This effect is not contrary to the well known principle in mechanics, that a body once put in motion, in any direction, will continue to move in that direction until some foreign cause oppose and prevent or turn it. For, in the present case, the blow given to the ball, communicates to it two kinds of motion; one of rotation about its own centre, and the other direct, by which its centre moves parallel to the table, as impelled by the blow, The latter motion,

on account of the friction of the ball on the table, is soon annihilated; but the rotary motion about the centre continues, and, when the former has ceased, the latter makes the ball roll in the retrograde direction. In this effect, therefore, there is nothing contrary to the well known laws of me chanics.

PROBLEM II.

To make a false ball, for playing at nine pins.

MAKE a hole in a common ball used for playing at the above game; but in such a manner as not to proceed entirely to the centre; then put some lead into it, and close it with a piece of wood, so that the joining may not be easily perceived. When this ball is rolled towards the pins, it will not fail to turn aside from the proper direction, unless thrown by chance or dexterity in such a manner, that the lead shall turn exactly at the top and bottom while the ball is rolling.

REMARK.

The fault of all balls used for billiards depends on this principle. For, as they are all made of ivory, and as, in every mass of that substance, there are always some parts more solid than others, there is not a single ball perhaps which has the centre of gravity exactly in the centre of the figure. On this account every ball deviates more or less from the line in which it is impelled, when a slight motion is communicated to it, in order to make it proceed towards the other side of the billiard table,

unless the heaviest part be placed at the top or bottom. We have heard an eminent maker of these balls declare, that he would give two guineas for a ball that should be uniform throughout; but that he had never been able to find one perfectly free from the abovementioned fault.

Hence it happens, that when a player strikes the ball gently, he often imagines that he has struck it unskillfully, or played badly; while his want of success is entirely the consequence of a fault in the ball. A good billiard player, before he engages to play for a large sum, ought carefully to try the ball in order to discover the heaviest and lightest parts. This precaution was communicated to us by a first rate player.

PROBLEM III.

How to construct a balance, which shall appear just when not loaded, as well as when loaded with unequal weights.

WE certainly do not here intend to teach people how to commit a fraud, which ought always to be condemned; but merely to shew that they should be on their guard against false balances, which often appear to be exact; and that in purchasing valuable articles, if they are not well acquainted with the vender, it is necessary to examine the balance, and to subject it to trial. It is possible indeed to make one, which when unloaded shall be in perfect equilibrium, but which shall nevertheless be false. The method is as follows.

Let A and B be the two scales of a balance, and let A be heavier than B: if the arms of the balance

be made of unequal lengths, in the same ratio as the weights of the two scales, and if the heavier scale A be suspended from the shorter arm, and the lighter scale B from the longer, these scales when empty will be in equilibrium. They will be in equilibrium also when they contain weights which are to each other in the same ratio as the scales. A person therefore unacquainted with this artifice will imagine the weights to be equal; and by these means may be imposed on.

Thus, for example, if one of the scales weighs 15, and the other 16; and if the arms of the balance from which they are suspended be, the one 16 and the other 15 inches in length; the scales when empty will be in equilibrium, and they will remain so when loaded with weights which are to each other in the ratio of 15 to 16, the heaviest being put into the heaviest scale. It will even be difficult to observe this inequality in the arms of the balance. Every time therefore that goods are weighed with such a balance, by putting the weight into the heavier scale and the merchandise into the other, the purchaser would be cheated of a sixteenth part, or an ounce in every pound.

But, this deception may be easily detected by transposing the weights; for if they are not then in equilibrium, it is a proof that the balance is not just.

And indeed in this way the true weight of any thing may be discovered, even by such a false balance, namely by first weighing the thing in the one scale, and then in the other scale; for a mean proportional between the two weights, will be the true quantity; that is, multiply the numbers of these two weights together, and take the square