An Inquiry into the Measure of the Force of Bodies in Motion: With a Proposal of an Experimentum Crucis, to Decide the Controversy about It. By James Jurin, M. D. Fellow of the Royal College of Physicians, London, and of the Royal Society

1092 Stachys *Canariensis* frutescens, verbasci folio. *Tourn.* 1093 Tamariscus latiore folio. *Park.* Germanicus. *Tourn.* 1094 Tamariscus tenuiore folio. *Park.* Narbonensis. *Tourn.* 1095 Tacamahac foliis serratis. *Pluken. Phyt.* 228. *Fig. 2.* 1096 Tanacetum vulgare luteum. *C. B.* 1097 Tanacetum foliis crispis. Ibid. 1098 Thlaspi *Creticum* purpureum. *Park.* 1099 Trachelium umbelliferum *Ponx.* 1100 Valeriana major hortensis. *Morison.* Phu. Off. XIV. An Inquiry into the Measure of the Force of Bodies in Motion: With a Proposal of an Experimentum Crucis, to decide the Controversy about it. By James Jurin, M.D. Fellow of the Royal College of Physicians, London, and of the Royal Society. Read May 30. 1745. Mechanical Forces may be reduced to two Sorts; one of a Body at Rest, the other of a Body in Motion. The Force of a Body at Rest, is that of a Body lying still upon a Table, or hanging by a Rope, or supported upon a Spring, &c. This is called by the Name of Pressure, Tension, Force, or *Vis mortua*. The Measure of this Force is the Weight with which the Table is pressed, or the Rope is stretched, or the Spring is bent. And that Measure being acknowledged by all Writers, there is no Dispute about this Sort of Force, notwithstanding the Diversity of Appellations by which it is called. The Force of a Body in Motion is on all hands agreed to be a Power residing in that Body, so long as it continues its Motion; by means of which it is able to remove Obstacles lying in its Way; to lessen, destroy, or overcome, the Force of any other moving Body, which meets it in an opposite Direction; or to surmount any dead Pressure or Resistance, as Tension, Gravity, Friction, &c. for some time; but which will be lessen'd or destroy'd by such Obstacles, or by such Resistance, as lessens or destroys the Motion of the Body. This is called moving Force, Vis motrix, and by some late Writers, Vis viva, to distinguish it from the Vis mortua spoken of before: And by these Appellations, however different, the same Thing is understood by all Mathematicians; namely, That Power of displacing Obstacles, withstanding opposite moving Forces, or overcoming any dead Resistance, which resides in a moving Body, and which, in Whole or in Part, continues to accompany it, so long as the Body moves. But about the Measure of this Sort of Force, Mathematicians are divided into two Parties: And, in order to state the Case fairly between them, it will be necessary to shew how far the two Parties agree, and in what Point their Disagreement consists. Both Sides agree, That the Measure of this Force depends partly upon the Mass, or Weight, of the Body, and partly upon the Velocity with which it moves; so that, upon any Increase either of the Weight, or of the Velocity, the moving Force will become greater. They also agree, That if the Velocity continue the same, but the Mass, or Weight of the Body, be increased in any Proportion, the moving Force is increased in the same Proportion: So that, in this Case, the Measure of the moving Force is the same with that of the Weight: Or, when two Bodies move with the same Velocity, if the Weight of the second be double, triple, quadruple, of that of the first, the moving Force of the second will also be double, triple, quadruple, of that of the first. But, when two Bodies are equal, and the Velocities with which they move are different, the two Parties no longer agree about the Measure of the moving Force. One Side maintains, That, when the Velocity of the second Body is double, triple, quadruple, of that of the first, the Measure of the moving Force of the second is also double, triple, quadruple, of that of the moving Force, being the same with that of the Velocity. The other Side pretend, That, in the same Case, the moving Force of the second Body is four times, nine times, sixteen times, as great as that of the first; the Measure of the moving Force being the same with that of the Square of the Velocity. In consequence of the Agreement in the first of these two Cases, and the Disagreement in the second, the one Side pretends, That the Measure of the moving Force is, in all Cases, the Product of the Weight. Weight into the Velocity; and the other, That it is the Product of the Weight into the Square of the Velocity. This Controversy was first started by the famous Mr. Leibnitz, and has been carried on by him and his Followers for near threescore Years; during which Time a great Number of Pieces have been published on both Sides of the Question, and a great Number of Experiments have been made, or proposed to be made, in order to decide it. But tho' both Parties agree in the Event of the Experiments, whether actually made, or only proposed; yet, as the Writers on each Side have found a Way of deducing from those Experiments a Conclusion suitable to their own Opinion, the Disagreement still continues as wide as ever, to the great Scandal of the Learned World. Now, if we examine carefully into the Reason of this, and would see by what means it happens, that two opposite Conclusions are so often drawn from the same Experiment, we shall find it not so much owing to false Reasoning on either Side, (That would be easily detected, and set right), as to another Cause; namely, to their Disagreement in the Principles upon which the Reasoning is founded. For, whereas whatever is laid down on either Side as a Principle, ought to be something all the World agrees in, at least what is admitted by the other Party; without which, all Reasoning upon it is to no Purpose; this Conduct has been so little observed in the present Dispute, that what has been offered on the one Side as an undoubted Principle or Axiom, has commonly been something that the opposite Party does not admit, nay, even absolutely denies. Of this it were easy to produce a Number of Examples; but I shall content myself with two only, one taken from each Side. Those who maintain, That the moving Force is as the Weight into the Velocity, lay down for a Principle, or Axiom, That When two Bodies meet one another in contrary Directions, if their moving Forces be equal, neither Body will prevail over the other: And if their moving Forces be unequal, the stronger will always prevail over the weaker. This the Leibnitian Party deny. They maintain, That one of these Bodies may prevail over the other, though their moving Forces be equal: Nay, that, in many Cases, the weaker will prevail over the stronger. It is therefore to no Purpose to alledge, That the Principle above laid down is founded on common Sense; or that it was always universally received, till this Dispute began: For, since the opposite Party now reject it, all Reasoning upon it can have no Weight with them; you must have recourse to something else. On the other hand, those who adhere to Mr. Leibnitz's Sentiment, lay down for a Principle, That Equal Effects always arise from equal Causes; provided the Causes be entirely consumed in producing those Effects. This their Opponents do not admit, unless in the Case where those equal Effects are produced in equal Times: And therefore, till both Sides shall agree in admitting admitting this Principle, no Argument can be drawn from it by one Party, that will be of any Service to convince the other. But as this Principle is chiefly made use of in reasoning upon Experiments made with Springs, many of which have been produced by both Parties, in Support of their Opinions, it may be worth while more particularly to consider, What Right there is on the one Side to impose this Principle, and what Reasons may be given on the other for rejecting it. When one End of a Spring, wholly unbent, leans against an immovable Support, and the opposite End is struck upon by a Body in Motion, which, bending the Spring to some certain Degree, does thereby lose its whole moving Force; the moving Force of the Body may be considered as the Cause of bending the Spring; and the Bending of the Spring may be looked upon as the Effect of that Cause, which is wholly spent and consumed in producing it. Now if two unequal Bodies, moving with unequal Velocities, strike in this manner upon two equal Springs, and each of them bend the Spring it strikes upon, exactly to the same Degree; and by so doing, the moving Force of each Body be entirely consumed; Here, say the Leibnitian Writers, are two equal Effects produced; for the Springs are equal, and are now equally bent; and the moving Forces, which are the Causes of those Effects, are wholly consumed in producing them; and therefore, by virtue of the Principle above laid down, those Causes must be equal; that is, the moving Forces of the two Bodies must be equal. But But their Antagonists will reply, That this Principle is not admitted by them, except the Times of producing those Effects are equal; and that they are not so in the Case before us: For the greater Body will take up a longer Time in producing its Effect, or in bending its Spring. If therefore the Leibnitian Party pretend, That equal Effects, when produced in unequal Times, do always arise from equal Causes, they must not impose this upon their Opponents by way of Principle or Axiom, but must demonstrate it. Till this be done, there will be Room to doubt, at least, whether the two Bodies have equal moving Forces, though they bend equal Springs to the same Degree. For the larger and slower of these two Bodies will bend the one Spring more slowly; and, consequently, will be resisted for a longer Time, than the smaller and swifter Body will be resisted in bending the other Spring to the same Degree. May not therefore the total Resistance of a Spring be greater, if that Resistance continues for a longer Time? And, if the total Resistance be greater, must not the moving Force, which is destroyed and consumed by that Resistance, be also greater? Is there not Reason then to doubt, whether the moving Forces of these two Bodies be equal, though they bend equal Springs to the same Degree? In like manner, when a Spring, already bent to some certain Degree, does, by unbending, drive before it a Body which gives way to its Pressure, is there not Room to doubt, whether the Pressure of the Spring may not produce a greater Effect, when that Pressure continues for a longer Time? That Pressure may be said to produce three Effects, all of which may, if we please, be considered as different from one another. 1. The Pressure carries the Body thro' a certain Space; by which Space the Length of the bent Spring is increased, in returning to its natural Situation. 2. The Pressure gives to the Body a certain Quantity of Motion. 3. It gives the Body a certain moving Force. Now, the first of these Effects is greater, when the Pressure acts for a longer Time. For, if the Pressure of a bent Spring, by acting for one Second upon the Body 1, carry that Body 1 thro' the Space 1; the Pressure of the same, or of an equal Spring equally bent, by acting for two Seconds upon the Body 4, will carry that Body 4 thro' the same Space 1. Likewise the second Effect is greater, when the Pressure continues for a longer Time. For, in the Case just now mentioned, the Body 4 will have twice the Quantity of Motion that the Body 1 has; though these two Quantities of Motion arise from the Pressure of the same, or, which is all one, of equal Springs equally bent. Why therefore are we take it for granted, or to have it imposed upon us by way of Principle or Axiom, That the third Effect is not greater, when the Time, in which it is produced by the Pressure of the same, or equal Springs, is longer, nay, infinitely longer? But we are told, that all the Force, which resided in the Spring, while bent, is now, upon the Unbending of the Spring, communicated to the Body moved. I ask therefore, What was that Force, or what kind of Force was that, which resided in the Spring, while bent, and without Motion? Was it a bare Pressure, or a moving Force? A Vis mortua, or a Vis viva? You must acknowledge, it was a Vis mortua, a bare Pressure, and nothing more. But the Force communicated to the Body, and which now resides in the Body in Motion, is a Vis viva, a moving Force. This therefore is not the same Force, nor a Force of the same kind, as that which resided in the bent Spring. It will be said, however, That the Force of the bent Spring is intirely exhausted in giving the Body its moving Force. I ask therefore again, What is it I am to understand by these Words, The Force of the Spring is intirely exhausted? If the Meaning be, that the Spring could not possibly give that same Body any greater moving Force, than what it has already given, I allow it: But this does not prove, that the same Spring, bent afresh to the same Degree, or an equal Spring equally bent, cannot give a greater Force to a greater Body. But if the Meaning of these Words be, That the Spring cannot give a greater moving Force to any Body whatsoever, I must answer, That this is taking for granted the very Point which is in Dispute. For the opposite Party pretend, That a Body of four Times the Bulk, will receive twice the moving Force in twice the Time, from the Pressure of the same Spring in unbending itself, or, if you please, in exhausting all its Force. It is plain, therefore, that the Followers of Mr. Leibnitz have no Right to say, A Body has such or such a Force, because such or such a Spring has put it in Motion by unbending itself, or can be bent by it. This is not a Position to be taken for granted, but stands in need of a Demonstration, which nobody has as yet attempted to give, at least from any unconverted Principle; and, till this be done, the laying down any such Position can have no other Effect, than to perplex the Controversy more and more, without Hopes of ever coming to an End of it. For which Reason I propose to take a quite different Method in what follows, and to lay down nothing, by way of Principle or Axiom, but what is allowed of by all the World, or, at least, has never yet been contradicted \textit{a priori}. \textbf{Axiom I.} When a bent Spring does, by unbending itself, push a Body before it, the greater the Body is, the more slowly will the Spring unbend itself. \textbf{Axiom II.} The more any Spring is bent, the greater is its Pressure. \textbf{Axiom III.} A greater Pressure produces a greater moving Force, if the Time be given. Proposition I. Moving Forces are not proportional to the Masses of the Bodies, and the Squares of their Velocities. Demonstration. Let there be two Springs, equal, and equally bent, $A$ and $B$, which, by unbending themselves, push before them two unequal Bodies; the Spring $A$ pushing before it the greater Body. Now, by Axiom I. the Spring $A$ will unbend more slowly than the other: from which it follows, that, at every Instant of the Time which the Spring $B$ takes up in unbending itself, the Spring $A$ will have unbent itself less than $B$, or will be more bent than $B$. Therefore, by Axiom II. the Pressure of the Spring $A$ will, at any Instant of that Time, be greater than the Pressure of the Spring $B$ at that same Instant. Hence, by Axiom III. the nascent, or infinitely small moving Force, which is produced by the Pressure of the Spring $A$ in every infinitely small Part of that Time, will be greater than that produced by the Pressure of the Spring $B$ in the same infinitely small Part of the Time. Therefore, the Sum of the infinitely small moving Forces; that is to say, the whole moving Force, which is produced by the Spring $A$, during that Time, will be greater than the moving Force produced by the Spring $B$ in that same Time: Or the moving Force of the greater Body will be greater than that of the lesser, at the Instant that the Spring $B$, being now wholly wholly unbent, ceases to act any longer upon the Body it has pushed before it: And as, after that Instant, the Spring $A$, not being yet wholly unbent, continues to act upon the greater Body, the moving Force of the greater Body will still continue to increase, and consequently will more and more exceed the moving Force of the smaller Body. But every one knows, that the Products of the Masses and Squares of the Velocities are equal in the two Bodies. Therefore the moving Forces, which we have proved to be unequal, are not proportional to the Products of the Masses and Squares of the Velocities. Which was to be demonstrated. To consider this in a particular Example, let us suppose the Masses of the two Bodies exposed to the Pressure of the Springs $A$ and $B$, to be 4 and 1 respectively; and let the Spring $B$ unbend itself, and thereby give the Body 1 its whole moving Force in one Second of Time. Then, at the End of that Second, the moving Force of the Body 4 will already exceed that of the Body 1, and will still grow greater during another Second of Time. For the Times are as the square Roots of the Masses. Also, if the Masses be 100 and 1, the moving Force of the Body 100, will, at the End of the first Second of Time, be greater than that of the Body 1, and will continue to increase during the Space of nine other Seconds. Corollary. When a bent Spring does, by unbending itself, drive a Body before it, the larger that Body is, the greater will be the moving Force which it receives from the Spring. Having Having now clearly proved, that the moving Forces are not proportional to the Squares of the Velocities, I proceed next to demonstrate, that they are proportional to the Velocities themselves: And, in order thereto, I shall, as I have hitherto done, make use of no other Principles or Axioms, than such as are admitted on both Sides, or, at least, have never yet been controverted \textit{a priori} by either Party. \textbf{Axiom IV.} Springs of unequal Lengths, when bent alike, have equal Pressures. We speak here of Springs equal in all Respects, except the Length only; and, by being bent alike, we mean, that they are so compressed, as that the Lengths they are now reduced to, are exactly proportional to their natural Lengths, or to the Lengths they are of when no way compressed. In this Condition, if one be directly opposed to the other, they will mutually sustain each other's Pressure, so as to maintain a perfect \textit{Equilibrium}: Or, if each be placed separately in a vertical Situation, they will sustain equal Weights. And in one or the other of these Cases, it is evident, that they must exercise equal Pressures. \textbf{Axiom V.} Equal Pressures in equal Times produce equal moving Forces. Proposition II. Moving Forces are proportional to the Masses and Velocities jointly. Demonstration. Let there be two Springs, of the Lengths 1 and 2, but equal in all other Respects, and bent alike: And, in unbending themselves, let the Spring 1 drive before it a Body whose Mass is 2; and the Spring 2 another Body of the Mass 1. Now, by Coroll. 11. of my general Theorem concerning the Action of Springs, these two Springs will unbend themselves exactly in the same Time; and, consequently, the Spring 2 will unbend itself with a Velocity double of that of the Spring 1: And, by Coroll. 12. of the same Theorem, it will give to the Body 1 a Velocity double of that, which the Body 2 will receive from the Spring 1. Also, as the two Springs were supposed to be bent alike at first, and the Spring 2 unbends itself with a Velocity double to that of the Spring 1, it is manifest, that, during the whole Time of their Expansion, they will be always bent alike, one to the other. Therefore, by Axiom IV. their Pressures will be constantly equal one to the other: And hence, by Axiom V. the infinitely small moving Forces produced by each of these Springs, in every infinitely small Part of Time, will be equal one to the other. Consequently, the Sums of those infinitely small moving Forces, that is, the whole moving Forces, produced by the two Springs, will be equal one to the other. And the Masses of the two Bodies being 2 and 1, and and their Velocities being 1 and 2 respectively, it is plain, that the moving Forces are proportional to the Masses and Velocities jointly. Which was to be demonstrated. For the greater Facility of examining this Demonstration, we have suited it to a single Case only, and that the most simple that can be supposed: But everybody will see, how easy it is to form a general one upon the same Principles. As we do not think, that any Flaw can be found in either of the Demonstrations above laid down; and the Axioms, upon which they are founded, have never yet been disputed, as far as we know; we presume, that the Leibnitian Opinion about the Measure of moving Forces, is incontestably overthrown by the first Proposition, and the opposite Sentiment is as evidently established by the second. But, if any Reader shall be of a different Opinion, we must beg Leave to propose to his Consideration the following Experiment, which we hope may justly deserve the Name of an Experimentum Crucis; and, as such, may put a final Period to this Controversy. It is not new indeed, having been proposed before by myself and others; but, as the Manner, in which it was formerly offered, has given Occasion to some Objections, which, tho' not affecting the Substance of the Argument drawn from it, may yet have amused and embarrassed the less attentive Readers, I shall now propose it in such a Manner, as may obviate all those Difficulties, and, I think, will render it absolutely decisive. To me, I am sure, it will be so, since I shall immediately embrace the K k k Leibnitian Leibnitian Doctrine, if my Argument drawn from it can receive a clear and satisfactory Answer. Experiment. Upon an horizontal Plane, at Rest, but moveable with the least Force, suppose upon a Boat in a stagnant Water, let there be placed, between two equal Bodies, a bent Spring, by the unbending of which the two Bodies may be pushed contrary Ways. In this Case it is evident, that the Velocities, which the two Bodies receive from the Spring, will be exactly equal, and their moving Forces will also be exactly equal; and that the Plane they move upon, and also the Boat upon which it lies, will have no Motion given them either Way. Let us call the Velocity of each Body $1$, and the moving Force also $1$. Now, let us suppose the Spring to be bent afresh to the same Degree as before, and to be again placed between the two Bodies lying at Rest; then let the Plane, upon which the Spring and the Bodies lie, be carried uniformly forwards, in the Direction of the Length of the Spring, with this same Velocity $1$. In this Case it is manifest, that each of the Bodies will have the Velocity $1$, and the moving Force $1$, both in the Direction of the Axis of the Spring. During this Motion, let the Spring again unbend, and push the two Bodies contrary Ways, as before, the one forwards, the other backwards: Then the Spring will give to each of these Bodies the Velocity $1$, as before, when the Plane was at Rest. By this means the hindmost Body, or that which is pushed backwards, will have its Velocity $1$, which it had before by the Motion of the Plane, now intirely destroy'd, and will be absolutely at Rest. But the Body, which is pushed forwards, will now have the Velocity $2$, namely $1$ from the Motion of the Plane, and $1$ from the Action of the Spring. Thus far every body agrees in what will be the Event of this Experiment. But the Question is, What will be the moving Force of the foremost Body, or of that which is pushed forwards, and which has the Velocity $2$; viz. $1$ from the Motion of the Plane, and $1$ from the Action of the Spring. By the Leibnitian Doctrine, its moving Force must be $4$: And, if so, it must have received the moving Force from the Action of the Spring; for it had only the moving Force $1$ from the Motion of the Plane. Let us examine, whether this be possible, or reconcileable to their own Doctrine. Their Doctrine is, That equal Springs, equally bent, will, by unbending themselves, give equal moving Forces to the Bodies they act upon, whatever those Bodies are. We agree to this, not generally indeed; but in the Case before us, where the Bodies are of equal Masses or Weights, we agree to it. Let us therefore imagine the bent Spring, which is placed between the two Bodies, to be divided transversly into two equal Parts. In this Case it is plain, that the two Halves of the Spring, may be considered as two entire Springs, equal, and equally bent, each of which rests at one End in Aequilibrio against the other Spring, and at the opposite End, presses against the Body it is to move. Consequently, by the Leibnitian Doctrine, to which, in this particular Case, where the Bodies are equal, we also agree, the two Springs will give equal moving Forces to the two Bodies. But the moving Force received by the hindmost Body from the hinder Spring, was undoubtedly the moving Force 1: For by that Force given it in the Direction backwards, the moving Force 1, which it had before from the Motion of the Plane in the Direction forwards, is exactly balanced and destroyed, the Body remaining, as was observed before, in absolute Rest. Therefore the moving Force received by the foremost Body from the foremost Spring, was also the moving Force 1. And this, added to the other moving Force 1, which it had before from the Motion of the Plane, makes the moving Force 2, and not the moving Force 4, as the Leibnitian Philosophers pretend. Consequently, that Body, which had before the Velocity 1, and the moving Force 1, and now has the Velocity 2, has also the moving Force 2: That is, the moving Forces are proportional to the Velocities.