On the Vis Inertiæ of Matter.

Philadelphia, 1748.

Sir,

According to my promise, I send you in writing my observations on your book[18]: you will be the better able to consider them; which I desire you to do at your leisure, and to set me right where I am wrong.

I stumble at the threshold of the building, and therefore have not read farther. The author's vis inertiæ essential to matter, upon which the whole work is founded, I have not been able to comprehend. And I do not think he demonstrates at all clearly (at least to me he does not) that there is really such a property in matter.

He says, No. 2. "Let a given body or mass of matter be called a, and let any given celerity be called c. That celerity doubled, tripled, &c. or halved, thirded, &c. will be 2 c, 3 c, &c. or ½ c, ⅓ c, &c. respectively: also the body doubled, tripled, or halved, thirded, will be 2 a, 3 a, or ½ a, ⅓ a, respectively." Thus far is clear.—But he adds, "Now to move the body a with the celerity c, requires a certain force to be impressed upon it; and to move it with a celerity as 2 c, requires twice that force to be impressed upon it, &c." Here I suspect some mistake creeps in by the author's not distinguishing between a great force applied at once, or a small one continually applied, to a mass of matter, in order to move it. I think it is generally allowed by the philosophers, and, for aught we know, is certainly true, that there is no mass of matter, how great soever, but may be moved by any force how small soever (taking friction out of the question) and this small force continued, will in time bring the mass to move with any velocity whatsoever.—Our author himself seems to allow this 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 ½ c, ⅓ c, ¼ c, ⅕ c, &c. you can never come to a fraction of velocity that is equal to 0 c, 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 inertiæ, 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 musketo 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? 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 musketo, without whom those globes would not be moved at all.—What then does vis inertiæ 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 f = c X a); and as the force 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 f to move a with 2 c; for a moving with 2 c would have a force equal to 2 f, which it could not receive from 1 f; and this, not because there is such a thing as vis inertiæ, for the case would be the same if that had no existence; but because nothing can give more than it has, if 1 f can to 1 a give 1 c, which is the same thing as giving it 1 f; (i. e. if force applied to matter at rest, can put it in motion, and give it equal force) where then is vis inertiæ? 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 makes a certain resistance, &c. A body as 2 a 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 2 a moved by the force 1 f (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 2 a is compounded of 1 a+1 a: and if each of the 1 a's or each part of the compound were made to move with 1 c (as they might be by 2 f) then the whole would move with 2 c, and not with 1 c, as our author supposes. But 1 f applied to 2 a, makes each a move with ½ c; and so the whole moves with 1 c; exactly the same as 1 a was made to do by 1 f before. What is equal celerity but a measuring the same space by moving bodies in the same time?—Now if 1 a impelled by 1 f measures 100 yards in a minute; and in 2 a impelled by 1 f, each a measures 50 yards in a minute, which added make 100; 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 3 a, 4 a, &c. would make 3 or 4 bodies equal to once a, each of which would require once the first force to be moved with the celerity c." If then in 3 a, each a requires once the first force f to be moved with the celerity c, would not each move with the force f and celerity c; and consequently the whole be 3 a moving with 3 f and 3 c? After so distinct an observation, how could he miss of the consequence, and imagine that 1 c and 3 c were the same? Thus as our author's abatement of celerity in the case of 2 a moved by 1 f is imaginary, so must be his additional resistance.—And here again, I am at a loss to discover any effect of the vis inertiæ.

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 inertiæ? would not the effect be the same if there were no such thing? A force 1 f strikes a body 1 a, and moves it with the celerity 1 c, i. e. with the force 1 f: It requires, even according to our author, only an opposing 1 f to stop it. But ought it not (if there were a vis inertiæ) to have not only the force 1 f, but an additional force equal to the force of vis inertiæ, 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 inertiæ, nor any effect of it. It is allowed by all, that a body 1 a moving with a velocity 1 c, and a force 1 f striking another body 1 a at rest, they will afterwards move on together, each with ½ c and ½ f; which, as I said before, is equal in the whole to 1 c and 1 f. If vis inertiæ, 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 [i.e. 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. Whitfield 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[19] 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,

B. FRANKLIN.