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towards the end of the same No. 2, when he is subdividing his celerities and forces; for as in continuing the division to eternity by his method of %c, ic, \c, }c, &c . you can never come to a fraction of velocity that is equal to Oc, or no celerity at all; so, dividing the force in the same manner, you can never come to a fraction of force that will not produce an equal fraction of celerity.
Where then is the mighty vis inertia, and what is its strength, when the greatest assignable mass of matter will give way to, or be moved by, the least assignable force? Suppose two globes equal to the sun and to one another, exactly equipoised in Jove's balance; suppose no friction in the centre of motion, in the beam or elsewhere; if a musqueto then were to light on one of them, would he not give motion to them both, causing one to descend and the other to rise 1 If it is objected, that the force of gravity helps one globe to descend, I answer, the same force opposes the other's rising. Here is an equality that leaves the whole motion to be produced by the musqueto, without whom those globes would not be moved at all. What then does vis inertia do in this case? and what other effect could we expect if there were no such thing? Surely, if it were any thing more than a phantom, there might be enough of it in such vast bodies to annihilate, by its opposition to motion, so trifling a force!
Our author would have reasoned more clearly, I think, if, as he has used the letter a for a certain quantity of matter, and c for a certain quantity of celerity, he had employed one letter more, and put f, perhaps, for a certain quantity of force. This let us suppose to be done; and then, as it is a maxim that the force of bodies in motion is equal to the quantity of matter multiplied by the celerity, (or/=cXa); and as the force
VOL. VI. 12 H*
received by and subsisting in matter, when it is put in motion, can never exceed the force given; so, if f moves a with c, there must needs be required 2/ to move a with 2c; for a moving with 2c would have a force equal to 2/, which it could not receive from 1/; and this, not because there is such a thing as vis inertia, for the case would be the same if that had no existence; but because nothing can give more than it has. And now again, if a thing can give what it has, if If can to la give lc, which is the same thing as giving it 1/, (that is, if force applied to matter at rest, can put it in motion, and give it equal force,) where then is vis inertia? If it existed at all in matter, should we not find the quantity of its resistance subtracted from the force given?
In No. 4, our author goes on and says, "The body a requires a certain force to be impressed on it to be moved with a celerity as c, or such a force is necessary; and therefore it makes a certain resistance, &x.; a body as 2a requires twice that force to be moved with the same celerity, or it makes twice that resistance; and so on." This I think is not true; but that the body 2a, moved by the force If, (though the eye may judge otherwise of it) does really move with the same celerity as it did when impelled by the same force; for 2a is compounded of la + la; and if each of the la's, or each part of the compound, were made to move with lc (as they might be by 2/), then the whole would move with 2c, and not with lc, as our author supposes. But If applied to 2a makes each a move with -J-c; and so the whole moves with lc; exactly the same as la was made to do by If before. What is equal celerity but a measuring the same space by moving bodies in the same time? Now if la, impelled by 1/, measures one hundred yards in a minute; and in 2a, impelled by 1/, each a measures fifty yards in a minute, which added make one hundred; are not the celerities, as the forces, equal? And, since force and celerity in the same quantity of matter are always in proportion to each other, why should we, when the quantity of matter is doubled, allow the force to continue unimpaired, and yet suppose one half of the celerity to be lost ? * I wonder the more at our author's mistake in this point, since in the same number I find him observing; "We may easily conceive that a body, as 3a, 4a, &c., would make three or four bodies equal to once a, each of which would require once the first force to be moved with the celerity c." If then, in 3a, each a requires once the first force /, to be moved with the celerity c, would not each move with the force /, and celerity c 1 and consequently the whole be 3a moving with 3/ and 3c ?/ After so distinct an observation, how could he miss of the consequence, and imagine that lc and 3c were the same? Thus, as our author's abatement of celerity in
* Dr. Franklin's reasoning seems only to prove, that, where bodies of different masses have equal force, they "measure equal space in equal times." For, allowing that 2a moves one hundred yards in a minute (because it moves two separate fifty yards in that time), yet surely that space is not the same with that of the one hundred yards moved by la, in the same time, though it may be equal to it; for the body 2a (that is, a and a), in the first case, describes a broad double space; and the body la, in the second case, describes a long and single space. There is a farther consideration which may show the difference of celerity and force. For when Dr. Franklin says, in his second paragraph, that "there is no mass of matter, how great soever, but may be moved, with any velocity, by any continued force, how small soever," I ask whether the moving body must not have its force rather in the shape of much celerity, than of much matter, for this purpose; since without much celerity it would not move fast enough to apply its force to give the required velocity; even though its quantity of matter, and consequently of force, were infinite. "Equal celerity therefore in moving bodies is their measuring equal space, along a continued line, in equal time." Equal space measured along a number of smaller parallel lines, suits cases of equal motion indeed, but, according to this corrected definition, not of equal celerity. — B. V
the case of 2a moved by 1/ is imaginary, so must be his additional resistance. And here again, I am at a loss to discover any effect of the vis inertia.
In No. 6, he tells us, "that all this is likewise certain when taken the contrary way, viz. from motion to rest; for the body a, moving with a certain velocity, as c, requires a certain degree of force or resistance to stop that motion," &c. &c.; that is, in other words, equal force is necessary to destroy force. It may be so. But how does that discover a vis inertia.? Would not the effect be the same, if there were no such thing? A force If strikes a body la, and moves it with the celerity \c, that is, with the force If; it requires, even according to our author, only an opposing 1/to stop it. But ought it not (if there were a vis inertia) to have not only the force 1/, but an additional force equal to the force of vis inertia, that obstinate power by which a body endeavours with all its might to continue in its present state, whether of motion or rest? I say, ought there not to be an opposing force equal to the sum of these? The truth, however, is, that there is no body, how large soever, moving with any velocity, how great soever, but may be stopped by any opposing force, how small soever, continually applied. At least, all our modern philosophers agree to tell us so.
Let me turn the thing in what light I please, I cannot discover the vis inertia, nor any effect of it. It is allowed by all, that a body la, moving with a velocity lc, and a force If, striking another body la at rest, they will afterwards move on together, each with lc and \f; which, as I said before, is equal in the whole to lc and If. If vis inertia, as in this case, neither abates the force nor the velocity of bodies, what does it, or how does it discover itself?
I imagine I may venture to conclude my observations on this piece, almost in the words of the author; that, if the doctrines of the immateriality of the soul and the existence of God, and of divine providence, are demonstrable from no plainer principles, the deist (that is, theist) has a desperate cause in hand. I oppose my theist to his atheist, because I think they are diametrically opposite; and not near of kin, as Mr. Whitefield seems to suppose, where (in his Journal) he tells us, "M. B. was a deist, I had almost said an atheist;" that is, chalk, I had almost said charcoal.
The din of the Market * increases upon me; and that, with frequent interruptions, has, I find, made me say some things twice over; and, I suppose, forget some others I intended to say. It has, however, one good effect, as it obliges me to come to the relief of your patience with
Your humble servant,
TO CADWALLADER COLDEN.
Baxter's Book on the Vis Inertia of Matter. — Manufacture of Electrical Apparatus. — Colden's Philosophical Treatise.
Philadelphia, 6 August, 1747.
Sir, The observations I sent you on Baxter's book were wrote on a sheet or two of paper in folio.f He builds his whole argument on the vis inertia of matter. I
* Philadelphia Market, near which Dr. Franklin lived. f Probably the same in substance as the preceding letter to Mr. Hoplanson. — Editor.