A Letter from the Rev. Dr. Samuel Clarke to Mr. Benjamin Hoadly, F. R. S. Occasion'd by the Present Controversy among Mathematicians, concerning the Proportion of Velocity and Force in Bodies in Motion

the finest Injection; opening the Stomach and Intestines they contained a mucous Matter, the greatest Part of which was Blood, and the fine villous Coat which is so visible in these Animals was entirely destroyed. About an Hour before he was bit he had a plentiful Meal of coarse Beef, of which there was not the least Appearance. Opening the Thorax, the Pleura and other Membranes looked as if injected; the Heart was turgid with Blood, as were also its Vessels. The Vessels of the Membranes of the Brain made a most beautiful Figure from the Quantity of Blood contained in them, as did likewise the Blood-Vessels of the Nerves; there was a small Quantity of Water between the two Hemispheres. The Blood contained in the Heart and its Vessels was an even Mass about the Consistence of Cream. The Cat had upon opening nearly the same Appearances, and lived about five Hours. II. A Letter from the Rev. Dr. Samuel Clarke to Mr. Benjamin Hoadly, F. R. S. occasion'd by the present Controversy among Mathematicians, concerning the Proportion of Velocity and Force in Bodies in Motion. SIR, It has often been observed in general, that Learning does not give Men Understanding; and that the abstrusest Things in the World have been asserted and maintained, by Persons whose Education and Studies dies should seem to have furnish'd them with the greatest Extent of Science. That Knowledge in many Languages and Terms of Art, and in the History of Opinions and Romantick Hypotheses of Philosophers, should sometimes be of no Effect in correcting Mens Judgment, is not so much to be wonder'd at. But that in Mathematicks themselves, which are a real Science, and founded in the necessary Nature of Things; men of very great Abilities in abstract Computations, when they come to apply those Computations to the Nature of Things, should persist in maintaining the most palpable Absurdities, and in refusing to see some of the most evident and obvious Truths; is very strange. An extraordinary Instance of this, we have had of late Years in very eminent Mathematicians, Mr. Leibnitz, Mr. Herman, Mr. 'sGravesande, and Mr. Bernoulli; who (in order to raise a Dust of Opposition against Sir Isaac Newton's Philosophy, the Glory of which is the Application of abstract Mathematicks to the real Phænomena of Nature,) have for some Years insisted with great Eagerness, upon a Principle which subverts all Science, and which may easily be made appear (even to an ordinary Capacity) to be contrary to the necessary and essential Nature of Things. What they contend for, is, That the Force of any Body in Motion, is proportional, not to its Velocity, but to the Square of its Velocity. The Absurdity of which Notion, I shall first make appear, and then shew what it is that has led these Gentlemen into Error. In the Nature of Things, 'tis evident, every Effect must necessarily be proportionate to the Cause of that Effect; Effect; that is, to the Action of the Cause, or the Power exerted at the Time when the Effect is produced. To suppose any Effect proportional to the Square or Cube of its Cause, is to suppose that an Effect arises partly from its Cause, and partly from* Nothing. In a Body in Motion, there may be considered distinctly, the Quantity of the Matter, and the Velocity of the Motion. The Force arising from the Quantity of the Matter as its Cause, must necessarily be proportional to the Quantity of the Matter: And the Force arising from the Velocity of the Motion as its Cause, must necessarily be proportional to the Velocity of the Motion. The whole Force therefore arising from these two Causes, must necessarily be proportional to these two Causes taken together. And therefore in Bodies of equal Bigness and Density, or in one and the same Body, the Quantity of Matter continuing always the same, the Force must necessarily be always proportional to the Velocity of the Motion. If the Force were as the Square of the Velocity, all that Part of the Force, which was above the Proportion of the Velocity, would arise either out of Nothing, or (according to Mr. Leibnitz's Philosophy) out of some * Which is just like the Supposition made by those Mathematicians, who have taken it for granted, that $\frac{1}{\infty}$ is equal to Infinite; that is, that as $0$ to $1$, so $1$ is to Infinite; that is, that Infinite multiplied by $0$, is equal to $1$, or an infinite Number of Nothings equal to Something; which is palpably false. The true Proportion is, not as $0$ to $1$, so $1$ to Infinite; but as an Infinitesimal is to $1$, so is $1$ to Infinite. And as the Infinitesimal of an Infinitesimal (that is, a second Fluxion, or the second Power of Infinitesimal) is to $1$, so is $1$ to Infinity of Infinites, or the second Power of Infinite; that is, (for Instance) 'tis as a finite [physical] Line to an Infinite Surface, or as a finite [physical] Surface to an infinite Solid. And as $0$ (which is beyond all Proportion lower than the Infinite'th Power of an Infinitesimal) is to $1$, so is $1$ to that which is beyond all Proportion higher than the Infinite'th Power of Infinite. Which clearly removes the Foundation of all the ridiculous Consequences, which have been drawn from the Supposition of the fore-mention'd false Proportion. living Soul essentially belonging to every Particle of Matter. Whenever any Effect whatsoever, is in a duplicate Proportion, or as the Square of any Cause; 'tis always either because there are two Causes acting at the same Time, or that one and the same Cause continues to act for a double Quantity of Time. The Resistance made to a Body moving in any fluid Medium, is in a duplicate Proportion to the Velocity of its Motion; because, in Proportion to its Velocity, it is resisted by a greater Number of Particles in the same Time; and again, in Proportion to its Velocity, 'tis resisted by the same Particles singly with a greater Force, as being to be moved out of their Places with greater Velocity. Light decreases in a duplicate Proportion of its Distance from the Sun; because the Rays divaricate according to two Dimensions; according to the Dimension upwards or downwards, and according to the Dimension side-ways. But according to the third Dimension forwards from the Sun, a Ray of Light undergoes no Alteration; because the Particles, of which it consists, being emitted all of them with an equal Velocity, continue everywhere at an equal Distance from each other. One and the same Cause, acting in a double Quantity of Time, produces the same Effect, as two equal Causes acting in a single Quantity of Time. One and the same Force, in two Parts of Time, will cause a Body in Motion to describe the same Space, as double the Force would do in one Part of Time. The Space described therefore by a Body in Motion, is not as the Force; but as the Force and the Time taken together. A Body, with any the least assignable Force, will move through infinite Space, if it meets with no Resistance, in an infinite Time. And in Spaces where there is an uniform Resistance to Motion, the Space described before the Motion ceases, must needs be as the Force and as the Time together: Because a double Force will carry a Body twice as far in the same Time, and will also cause the Motion to be twice as long Time in destroying by an uniform Resistance. The Space described therefore before the Motion ceases, is in this Case demonstrably as the Square of the Force. A Body thrown upwards with double Force, will be carried four Times as high, before its Motion be stopp'd by the uniform Resistance of Gravity; because the double Force will carry it twice as high in the same Time, and moreover require twice the Time for the uniform Resistance to destroy the Motion. The Case is the same in accelerated Motion; in Bodies accelerated by a Succession of clastic Impressions, or falling with a Motion accelerated by the uniform Power of Gravity, or by any other uniform Power whatsoever. The Space described must needs be as the Force, and as the Time wherein the Force operates. What I have thus demonstrated concerning any Force, considered as the Cause producing an Effect; and concerning the Time, during which the Force operates; is on all Hands acknowledged to be true concerning Velocity. And therefore Velocity and Force, in this Case, are one and the same Thing. So that to affirm Force to be as the Square of the Velocity, is to affirm that the Force is equal to the Square of it self. Now from hence appears very clearly the Ground of the Error these Gentlemen have fallen into, and of their Misapplication of the Experiments they build upon. The Effect of a Force impress'd on a moveable Body, is the Motion of that Body from one Place to another. Now forasmuch as the Effect cannot but be proportional to its Cause, hence Mr. Leibnitz (whom the other Gentlemen have follow'd) contends that the Space describ'd by a Body in falling, is proportional to the Force by which it is impell'd during its Fall; and that the Force acquir'd by a Body in falling, is proportional to the Space it has described in its Fall. Which Space being agreed to be as the Square of the Velocity (as being proportional to the Velocity and to the Time taken together) hence they infer that the Force likewise is as the Square of the Velocity. But from what has been said, 'tis plain, that the Space described in these and all other the like Cases, is not as the Force only, but as the Force and as the Time wherein the Force acts; that is to say, as the Square of the Force. For the Cause of the Quantity of the Space described, is not barely the Quantity of the Force, but also the Continuance of the Time wherein the Force acts. The Force therefore and the Time taken together, being necessarily as the Space described; as the Velocity and the Time taken together, are on all Hands acknowledg'd to be; it follows that the Velocity and the Force are equal, and not the Force as the Square of the Velocity. When two unequal Bodies fastened at the Ends of the Arms of a Balance of unequal Length, counterpoise each other, and vibrate in equal Times; as they must necessarily do, being fastened to the Arms of the same Balance: which is an Observation Mr. Leibnitz lays great Stress upon: In That Case indeed the Forces will will be as the Spaces described. But not therefore as the Squares of the Velocities. For in That Case, the Velocities themselves are as the Spaces described, because the Times are equal. When a Body projected with a double Velocity, enters deeper into Snow or soft Clay, or into a Heap of springy or elastic Parts, than in Proportion to its Velocity; 'tis not because the Force is more than proportional to the Velocity; but because the Depth it penetrates into a soft Medium, arises partly from the Degree of the Force or Velocity, and partly from the Time wherein the Force operates before it be spent. In the Collision of hard Bodies, 'tis (I think) agreed on all Hands, that 'tis demonstrated by Reason, and confirmed by Experience; that when a perfectly hard Ball, moved with whatever Degree of Velocity, strikes full upon another hard Ball, equal in Bigness and Weight, and without any Motion in it; if the Balls be unelastic, they will both go on together the same Way, dividing the Motion equally between them, with half the Velocity the first Ball had originally: But if they be perfectly elastic, the moving Ball will communicate its whole Motion and Velocity to the quiescent Ball, and itself lie still in the others Place. Were it true now, that the Force of the moving Ball was as the Square of its Velocity; these Experiments would then shew (which is infinitely absurd) that the Force or vis inertiae in the quiescent Ball, the dead Force, was always proportional to the Square of the Velocity (which these Gentlemen affect fantastically to call the living Force) of the moving Ball, whatever its Velocity were. Or the Force in Both might just as reasonably be supposed to be as the Cube, or the quadrato-quadrate, or any other Power of the Velocity of the moving Ball. Which is turning the Nature of Things into Ridicule. Mr. Leibnitz, in some Letters which he wrote into England, intimated that he had a Prospect of a perpetual Motion, founded on the Notion of a Vital Principle, or active Power in Matter. But from the Experiments now mentioned, 'tis evident that if the Force of Bodies in Motion could be exalted even to the infinit'th Power of their Velocity; yet since, to answer the Phenomena of Nature with Regard to Action and Re-action, the same Force must necessarily be allowed to all quiescent Bodies likewise; it could be of No Effect. III. Astronomical Observations made at Vera Cruz, by Mr. Joseph Harris. Revised and communicated by Edm. Halley, L.L.D. Astron. Reg. & R. S. S. The Latitude of this Place I found (by several distant Observations made by a Quadrant of four Feet Radius) to be $19^\circ 12'$ N. On March 11, 1727. O.S. there happen'd here a considerable Eclipse of the Sun, the greatest Obscuration being about $10\frac{1}{2}$ Digits; and having that Morning carefully adjusted the Pendulum Clock, and fixed a Telescope to the Index of the foresaid Quadrant, I observed it to begin in or about the S.E. by S. Part of the Sun's Disk at $49\frac{3}{4}$ Minutes after Noon apparent Time; the Altitude of the Sun's Center then was $67^\circ 53'$. We could not determine exactly the Middle of this Eclipse, but as near as we could judge, it happen'd about $2^h 30^m$ P.M.