Popular scientific lectures

elated to it. The latter principle, however, is by no means new, for in the province of mechanics it has controlled for centuries t

g.

ica, Tom. IV, De statica, (Leyden, 1605, p. 34), tre

ented in cross-section in Figure 41. Since we can imagine the lower symmetrical part of the cord ABC taken away, Stevinus concludes that the four balls on AB hold in equilibrium the two ball

he eight globes to the left would again be heavier than the six to the right, and therefore those eight would sink a second ti

iple the laws of equilibrium on the inclined

up the following principle: "Aquam datam, datum sibi intra aquam locum

g.

demonstrated as fo

D. This being posited, the water which succeeds A will, for the same reason, also flow down to D; A will be forced out of its place in

with the help of that remark. If we were to reproduce Stevinus's demonstration to-day, we should have to change it slightly. We find no difficulty in imagining the cord on the prism possessed of unending uniform motion if all hindrances are thought away, but we should protest against the assumption of an accelerated motion or even against that of a uniform motion, if

part, that a body in virtue of the velocity acquired in its descent can rise exactly as high as it fell. This principle, which appears frequently and

y the weights of the water which flowed out of a small orifice in a large vessel. In this experiment he assumes as a fundamental principle, that the velocity acquired in descent down an inclined plane always corresponds to the vertical height descended through, a conclusion which for him is the immediate outcome of the fact that a body which ha

a body in descent down planes of different inclinations

alviati say in

his experiment and repeated it several times, let us drive in the wall, in the projection of the vertical AB, as at E or at F, a nail five or six inches long, so that the thread AC, carrying as before the ball through the arc CB, at the moment it reaches the position AB, shall strike the nail E, and the ball be thus compelled to move up the arc BG described about E as centre. Then we shall see what the same impetus will here accomplish, acquired now as before at the same point B, which then drove the same moving body through the arc BD to the height of the horizontal CD. Now gentlemen, you will be pleased to see the ball rise to the horizontal line at the point G, and the same thing also happen if the nail be placed lower as at F, in which case the ball would describe the arc BJ, always terminating its ascent precisely at the line CD. If the nail be placed so low that the length of thread below it does not reach to the height of CD (which would happen if F were nearer B than to the intersecti

g.

applied to the inclined plane and leads to

red, and by its nature immutably impressed, provided all causes of new acceleration or retardation are taken away: I say acceleration, having in view its possible further progress along the plane extended; retardation, in view of the p

g.

eights of ascent which was so fruitful in Galileo's hands. He employs the latter principle in the solution of the problem of the centre of oscillation an

en occur (Hugenii, Horologium oscil

ons of bodies, a body would keep up forever the motion once imp

orologium de centro o

the common centre of gravity of the weights as a whole cannot possib

ve upwards.-And truly if the devisers of the new machines who make such futile attempts to construct a perpetual motion would acquaint themselves wi

e words "mechanical means." One might be led to believe from them

rinciple is still more clearly pu

the individual weights, with their common connexions dissolved, change their acquired velocities upwards and ascend as far as they can, the co

lation. Lagrange characterises this principle as precarious and is rejoiced at James Bernoulli's successful attempt, in 1681, to reduce the theory of the centre of oscillation to the laws of the lever, which appeared to him clearer. All the great inquirers of the seventeenth and eig

of living force," as that was enunciated by John and Daniel Bernoulli and employed with such signal success by the latter in

r principle. Torricelli assumed that the liquid which flows out of the basal orifice of a ves

or virtual velocities. This principle was not first enunciated, as is usually stated, and as Lagrange also a

s axiom of static

of the body acted upon, so is the power of the bo

le in the consideration of the simple machines, and also

resented by weights, is that the common centre of gravity of the weights shall not sink. Conversely, if the centre of gravity cannot sink equilibrium obtains, because heavy

rces of the system, passes a cord through these pulleys, and appends to its free extremity a weight which is a common measure of all the forces of the system. With no difficulty, now, the number of elements of each pulley may be so chosen that the forces in question shall be replaced by them. It is then clear that if the weight at the extremity cannot sink, equilibrium subsists, because heavy bodies cannot of themselves move upwards. If we do not

extraneous elements and fully satisfactory, but without

of his progress, feels the uncomfortableness of this state of affairs; every one wishes it removed; but seldom is the difficulty stated in words. Accordingly, the zealous pupil of the science is highly r

the demonstration of the principle of virtual velocities. But that quest brought back all the difficulties that we had overcome by the principle itself. That law so general, wherein are mingled the vague and unfamiliar ideas of infinitely small movements and of perturbations of equili

pon a different basis: for the demonstration of a law which embraces a whole science is neither more nor less than the reduction of that scienc

the principle of virtual movements is tantam

sent exists, dynamics is founded on statics, whereas it is desirable that in a science which pretends to deductiv

, and in the instruction of individuals, the easy should precede the difficult, the simple the complex, the special the general, yet the mind, when once it has reached a higher point of view, demands the contrary course, in which all statics shall appear simply as a special c

he principle made on me when I first took it up as a student and when I subsequently resumed it after having made historical researches. It first appeared to me insipid, chiefly on account of the pulleys and the cords which did not fit in

plishes almost all, which displeased Lagrange, but which he still had to employ, at least tacitly, in

But the principle cannot properly be based upon mechanical perceptions. For long before the development of mechanics the conviction of its truth existed