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LI. TO THOMAS HOPKINSON 2 - Benjamin Franklin, The Works of Benjamin Franklin, Vol. II Letters and Misc. Writings 1735-1753 [1904]

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The Works of Benjamin Franklin, including the Private as well as the Official and Scientific Correspondence, together with the Unmutilated and Correct Version of the Autobiography, compiled and edited by John Bigelow (New York: G.P. Putnam’s Sons, 1904). The Federal Edition in 12 volumes. Vol. II (Letters and Misc. Writings 1735-1753).

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TO THOMAS HOPKINSON2

Philadelphia, 1747.

According to my promise, I send you in writing my observations on your book1 ; 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 further. 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 in 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 2c, 3c, &c., or ½c, ⅓c, &c., respectively. Also the body doubled, tripled, or halved, thirded, will be 2a, 3a, or ½a, ⅓a, 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 2c, 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, and 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 ½c, ⅓c, ¼c, ⅕c, &c., you can never come to a fraction of velocity that is equal to 0c, 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 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? 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 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 × 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 2f to move a with 2c; for a moving with 2c would have a force equal to 2f, which it could not receive from 1f; 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. And now again, if a thing can give what it has, if 1f can to 1a give 1c, which is the same thing as giving it 1f (that is, 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 it makes a certain resistance, &c.; 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 1f (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 1a + 1a; and if each of the 1a’s, or each part of the compound, were made to move with 1c (as they might be by 2f), then the whole would move with 2c, and not with 1c, as our author supposes. But 1f applied to 2a makes each a move with ½c; and so the whole moves with 1c; exactly the same as 1a was made to do by 1f before. What is equal celerity but a measuring the same space by moving bodies in the same time? Now if 1a, impelled by 1f, measures one hundred yards in a minute; and in 2a, impelled by 1f, 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?1 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 f to be moved with the celerity c, would not each move with the force f and celerity c? and consequently the whole be 3a moving with 3f and 3c? After so distinct an observation, how could he miss of the consequence, and imagine that 1c and 3c were the same? Thus, as our author’s abatement of celerity in the case of 2a moved by 1f 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 1f strikes a body 1a, and moves it with the celerity 1c—that is, with the force 1f; it requires, even according to our author, only an opposing 1f to stop it. But ought it not (if there were a vis inertiæ) to have not only the force 1f, 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 1a, moving with a velocity 1c and a force 1f, striking another body 1a at rest, they will afterwards move on together, each with ½c and ½f; which, as I said before, is equal in the whole to 1c and 1f. 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 (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 Market1 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.

[2 ]Thomas Hopkinson was born in London, in April, 1709, had been a student at Oxford, came to America while young, married and settled in Philadelphia, where he died in 1751. He was an intimate friend of Franklin, and associated with him in his electrical and philosophical experiments. Mr. Hopkinson was chosen the first president of the American Philosophical Society, instituted in the year 1744, and also took an active part in founding the City Library and the College of Philadelphia. He left several children, among whom was Francis Hopkinson, one of the signers of the Declaration of Independence, well known as a writer, and for his valuable public services during and after the revolution.—Editor.

[1 ]It was a book by Andrew Baxter, entitled An Inquiry into the Nature of the Human Soul, wherein its Immateriality is evinced, &c. One of the chief objects of this book was to prove, that a resistance to any change is essential to matter, consequently inconsistent with active powers in it; and that, if matter wants active powers, an immaterial being is necessary for all those effects, &c., ascribed to its own natural powers. After stating the several proofs, questioned by Dr. Franklin, of a Vis inertiæ, or force of inertness, in matter, the author adds. “If the immateriality of the soul, the existence of God, and the necessity of a most particular, incessant providence in the world, are demonstrable from such plain and easy principles, the atheist has a desperate cause in hand.” (See the third edition, pp. 1-8.) In fact, Mr. Baxter’s doctrine seems to establish, rather than disprove, an activity in matter, and consequently to defeat his own conclusion, were not that conclusion to be found from other premises Primâ facie, it seems better for Mr. Baxter’s system to suppose matter incapable of force or effort, even in the case, as he calls it, of resisting change, which case appears to me no other than the simple one of matter not altering its state without a cause, and a cause exactly proportioned to the effect.—B. V.

[1 ]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 1a, 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 1a, 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, “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.

[1 ]Philadelphia Market, near which Dr. Franklin lived.