# Catalogue Entry: THEM00235

Source: A Collection of Papers, Which passed between the late Learned Mr. Leibniz, and Dr. Clarke, In the Years 1715 and 1716, Samuel Clarke (ed.) (London: 1717).

[1] § 3.

[2] § 14.

[3] See Appendix, No 4.

[4] § 15.

[5] See Appendix, No 12.

[6] § 16, 17, 18, 19, and 69.

[7] § 20.

[8] § 16, 17, 69, and 66.

[9] § 16 and 69.

[10] § 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.

[11] See my Sermons at Mr. Boyle's Lecture, Part I. Pag. 106. Edit. 4.

[12] § 11.

[13] See Mr. Leibnitz's Fourth Paper, § 2, 3, 6, 13, & 15.

[14] § 20.

[15] § 26.

[16] § 26.

[17] § 27.

[18] § 28.

[19] § 29.

[20] § 30.

[21] § 30, & 8, & 73.

[22] § 29.

[23] Fourth Paper, § 21.

[24] § 29.

[25] See Appendix, No 10.

[26] § 31.

[27] § 31.

[28] § 32.

[29] Ibid.

[30] Epist. 69, Partis primæ.

[31] § 29, 33, 34, 35, 62, 63.

[32] § 62.

[33] § 30, 32, & 73.

[34] § 35.

[35] Otherwise, What makes the Body of the Earth more difficult to be moved, (even the same way that its Gravity tends,) than the smallest Ball?

[36] § 34.

[37] Ibid.

[38] § 36, 37.

[39] § 38.

[40] § 39.

[41] § 40.

[42] § 41.

[43] § 42.

[44] See above, in my Third Reply. § 3; and Fourth Reply, § 11.

[45] § 43.

[46] § 44.

[47] Acts xvii. 27, 28.

[48] § 45.

[49] See above, the Note on my Fourth Reply, § 10.

[50] § 40.

[51]

* Note: The principal Occasion or Reason of the Confusion and Inconsistencies, which appear in what most Writers have <305> advanced concerning the Nature of Space, seems to be This: that (unless they attend carefully,) men are very apt to neglect That Necessary Distiction, (without which there can be no clear Reasoning,) which ought always to be made between Abstracts and Concretes, such as are Immensitas & Immensum; & also between Ideas and Things, such as are The Notion (which is Within our own Mind) of Immensity, and the real Immensity actually existing Without us.

All the Conceptions (I think) that ever have been or can be framed concerning Space, are these which follow. That it is either absolutely Nothing, or a mere Idea, or only a Relation of one thing to another, or that it is Body, or some other Substance, or else a Property of a Substance.

That it is not absolutely Nothing, is most evident. For of Nothing there is No Quantity, No Dimensions, No Properties. This Principle is the First Foundation of All Science whatsoever; expressing the Only Difference between what does, and what does not, exist.

That it is not a mere Idea, is likewise most manifest. For no Idea of Space, can possibly be framed larger than Finite; and yet Reason demonstrates that 'tis a Contradiction for Space itself not to be actually Infinite.

That it is not a bare Relation of one thing to another, arising from their Situation or Order among themselves, is no less apparent: Because Space is a Quantity, which Relations (such as Situation and Order) are not; As I have largely shown below, on § 54. Also because, if the material Universe is, or can possibly be, Finite; there cannot but be, actual or possible, Extramundane Space: See on § 31, 52, and 73.

That Space is not Body, is also most clear. For then Body would be necessarily infinite; and No Space could be <307> void of Resistance to Motion. Which is contrary to Experience.

That Space is not Any kind of Substance, is no less plain. Because infinite Space is Immensitas, not Immensum; whereas infinite Substance is Immensum, not Immensitas. Just as Duration is not a Substance: because infinite Duration is æternitas, not æternum; but infinite Substance is æternum, not æternitas.

It remains therefore, by Necessary Consequence, that Space is a Property, in like manner as Duration is. Immensitas, is του Immensi; just as Æternitas is is του Æterni.

[53] § 52.

[54] § 54.

[55] § 54.

[56] Fourth Paper, § 16.

[57] § 56.

[58] § 55, 57, 58,—63.

[59] Fourth Paper, § 15.

[60] § 70.

[61] § 73.

[62] Fourth Paper, § 21. and Fifth Paper, § 29.

[63] § 74.

[64] § 75.

[65] § 79, 80.

[66] § 80.

[67] § 81.

[68] § 82.

[69] § 83.

[70] § 87.

[71] See Appendix, No 2.

[72] § 91.

[73] See Appendix, No 11.

[74] § 83, 87, 89, 90.

[75] See Appendix, No 5.

[76] § 84.

[77] § 84.

[78] § 85.

[79] § 86, 87, 88, 82.

[80] § 92, 95, and 116.

[81] See Appendix, No 13.

[82] See Appendix, No 12.

[83] See more, on § 99.

[84]

There appears a great Confusion and Inconsistency in Mr Leibnitz's whole Notion of this Matter. For the Word, <329> Force, and Active Force, signifies in the present Question, the Impetus or relative Impulsive Force of Bodies in Motion: See my Third Reply, § 13. Mr. Leibnitz constantly uses the Word in this Sense: As when he speaks [§ 93, 94, 99, and 107, of this last Answer,] of Bodies not changing their Force after Reflexion, because they return with the same Swiftness: Of a Body's receiving a new Force from another Body, which loses as much of its own : Of the Impossibility, that one Body should acquire any new Force, without the Loss of as much in others: Of the new Force which the whole material Universe would receive, if the Soul of Man communicated any Force to the Body: And of Active Forces continuing always the same in the Universe, because the Force which un-elastick Bodies lose in their Whole, is communicated to and dispersed among their small Parts. Now this Impetus, or relative Impulsive Active Force of Bodies in Motion, is evidently both in Reason and Experience, always proportional to the Quantity of Motion. Therefore, according to Mr. Leibnitz's Principles, this impulsive active Force being always the same in Quantity, the Quantity of Motion also must of necessity be always the same in the Universe. Yet elsewhere, he inconsistently acknowledges, [§ 99,] that the Quantity of Motion is Not always the same: And in the Acta Eruditorum, ad Ann. 1686, pag. 161, he endeavours to Prove that the Quantity of Motion in the Universe is Not always the same, from that very Argument, and from that single Argument only, (of the Quantity of Impulsive Force being always the same,) which, if it was true, would necessarily infer on the contrary, that the Quantity of Motion could not but be always the same. The Reason of his Inconsistency in this Matter was his computing, by a wonderfully unphilosophical Error, the Quantity of Impulsive Force in an Ascending Body, from the Quantity of its Matter and of the Space described by it in Ascending, without considering the Time of its ascending. [85] "Suppono, says he, tantâ vi opus esse ad elevandum corpus A unius libræ usque ad altitudinem quatuor ulnarum, quantâ opus est ad <331> elevandum corpus B quatuor librarum usque ad altitudinem Unius Ulnæ. Omnia hæc à Cartesianis pariter ac cæteris Philosophis & Mathematicis nostri temporis conceduntur. Hinc sequitur, corpus A delapsum ex altitudine quatuor ulnarum, præcisè tantum acquisivisse virium, quantum B lapsum ex altitudine Unius Ulnæ". That is: "[I suppose the same Force is requisite to raise a Body A of one Pound Weight, to the Height of four Yards; which will raise the Body B of four Pounds Weight, to the Height of One Yard. This is Granted both by the Cartesians, and other Philosophers and Mathematicians of our Times. And from hence it follows, that the Body A, by falling from the Height of four Yards, acquires exactly the same Force, as the Body B by falling from the Height of One Yard".] But in this Supposition, Mr. Leibnitz is greatly mistaken. Neither the Cartesians, nor any other Philosophers or Mathematicians ever grant this, but in such Cases only, where the Times of Ascent or Descent are equal. If a Pendulum oscillates in a Cycloid; the Arch of the Cycloid described in ascending, will be as the Force with which the pendulous Body begins to ascend from the lowest Point; because the Times of ascending are equal. And if equal Bodies liberate upon the Arm of a Balance, at various Distances from the Axis of the Balance; the Forces of the Bodies will be in Proportion as the Arches described by them in librating, because they librate in the same Time. And if two equal Globes lying upon an Horizontal Plain, be impelled by unequal Forces, they will in equal Times describe Spaces proportional to the Forces impelling them. Or if unequal Globes be impelled with equal Forces, they will in equal Times describe Spaces reciprocally proportional to their Masses. And in all these Cases, if equal Bodies be impelled by Unequal Forces, the Forces impressed the Velocities generated, and the Spaces described in equal Times, will be proportional to one another. And if the Bodies be unequal, the Velocity of the bigger Bodies will be so much less, as the Bodies are bigger; And therefore the Motion (arising from the Mass and Velocity together) will he <333> in all these Cases, and in all Other Cases consequently, proportional to the Force imprest. [From whence, by the way, it plainly follows, that if there be always the same impulsive Force in the World, as Mr. Leibnitz affirms; there must be always the same Motion in the World, contrary to what he affirms.]

But Mr. Leibnitz confounds these Cases where the Times are equal, with the Cases where the Times are unequal: And chiefly That of Bodies rising and falling at the Ends of the unequal Arms of a Balance [Acta Erudit. ad Ann. 1686, Pag. 162; & ad Ann. 1690, Pag. 234; & ad Ann. 1691, Pag. 439; & ad Ann. 1695, Pag. 155;] is by him confounded with That of Bodies falling downwards and thrown upwards, without allowing for the Inequality of the Time. For a Body with one and the same Force, and one and the same Velocity, will in a longer Time describe a greater Space; and therefore the Time is to be considered; and the Forces are not to be reckoned proportional to the Spaces, unless where the Times are equal. Where the Times are unequal, the Forces of equal Bodies are as the Spaces applied to the Times. And in This, the Cartesians and other Philosophers and Mathematicians agree; all of them making the impulsive Forces of Bodies proportional to their Motions, and measuring their Motions by their Masses and Velocities together, and their Velocities by the Spaces which they describe, applied to the Times in which they describe them. If a Body thrown upwards does, by doubling its Velocity, ascend four Times higher in twice the Time; its impulsive Force will be increased, not in the proportion of the Space described by its Ascent, but in the Proportion of that Space applied to the Time; that is, in the Proportion of $\frac{4}{2}$ to $\frac{1}{1}$ or 2 to 1. For if, in this Case, the Force should be increased in the Proportion of 4 to 1; and, in oscillating in a Cycloid, the same Body, with the same Velocity doubled, describes only a doubled Arch, and its Force is therefore only doubled; this Body, with one and the same Degree of Velocity, would have twice as <235> much Force when thrown upwards, as when thrown horizontally: Which is a plain Contradiction. And there is the same Contradiction in affirming, that although a Body at the end of the unequal Arms of a Balance, by doubling its Velocity, acquires only a double impulsive Force, yet, by being thrown upwards with the same doubled Velocity, it acquires a quadruple impulsive Force; in this Assertion, I say, there is the same Contradiction: For equal Bodies with equal Velocities, cannot have unequal impulsive Forces.

Upon the Supposition of Gravity being Uniform, Galilæo demonstrated the Motion of Projectiles in Mediums void of Resistance; and his Propositions are allowed by all Mathematicians, not excepting Mr. Leibnitz himself. Now, supposing the Time of a falling Body to be divided into equal Parts; since Gravity is uniform, and, by being so, acts equally in equal Parts of Time, it must by its Action impress and communicate to the falling Body, equal impulsive Forces, Velocities, and Motions, in equal Times. And therefore the impulsive Force, the Velocity, and the Motion of the falling Body, will increase in Proportion to the Time of falling. But the Space described by the falling Body, arises partly from the Velocity of the Body, and partly from the Time of its falling; and so is in a compound ratio of them Both, or as the Square of either of them; and consequently as the Square of the impulsive Force. And by the same way of arguing, it may be proved, that when a Body is thrown upwards with any impulsive Force, the Height to which it will ascend, will be as the Square of that Force: And that the Force requisite to make the Body B, of four Pound Weight, rise up one Yard, will make the Body A, of One Pound Weight, rise up (not four Yards, as Mr. Leibnitz represents but) sixteen Yards, in quadruple the Time. For the Gravity of <337> four Pounds Weight in One part of Time, acts as much as the Gravity of one Pound Weight in Four Parts of Time.

But Mr. Herman, in his Phoronomia, Pag. 113, (arguing for Mr. Leibnitz against Those who hold that the Forces acquired by falling Bodies are proportional to the Times of falling, or to the Velocities acquired,) represents that this is founded upon a False Supposition, that Bodies thrown upwards receive from the Gravity which resists them, an equal Number of Impulses in equal Times. Which is as much as to say, that Gravity is not uniform; and, by Consequence, to overthrow the Theory of Galilæo concerning Projectiles, allowed by all Geometers. I suppose, he means that the swifter the Motion of Bodies is upwards, the more numerous are the Impulses; because the Bodies meet the [imaginary] gravitating Particles. And thus the Weight of Bodies will be greater when they move upwards, and less when they move downwards. And yet Mr. Leibnitz and Mr. Herman themselves allow, that Gravity in equal Times generates equal Velocities in descending Bodies, and takes away equal Velocities in ascending Bodies; and therefore is Uniform. In its action upon Bodies for generating Velocity, they allow it to be uniform; in its action upon them for generating impulsive Force, they deny it to be uniform: And so are inconsistent with themselves.

If the Force acquired by a Body in falling, be as the Space described; let the Time be divided into equal parts; and if in the first part of Time it gain One part of Force, in the two first parts of Time it will gain four parts of Force, in the three first parts of Time it will gain nine parts of Force, and so on. And by consequence, in the second part of Time it will gain three parts of Force, in the third part of Time it will gain five parts of Force, in the fourth part of Time it will gain seven parts of Force, and so on. And therefore if the Action of Gravity for generating these Forces, be supposed, in the middle of the first part of Time, to be of One degree; it will, in the middle of the second, third, and fourth parts of Time, be of three, five, and seven degrees, and so on: That is, it will be proportional to <339> the Time and to the Velocity acquired: And, by consequence, in the Beginning of the Time it will be none at all; and so the Body, for want of Gravity, will not fall down. And by the same way of arguing, when a Body is thrown upwards, its gravity will decrease as its velocity decreases, and cease when the Body ceases to ascend; and then, for want of gravity, it will rest in the Air, and fall down no more. So full of Absurdities is the Notion of this Learned Author in this Particular.

To decide this question demonstratively; let two pendulous globes of hardned Steel, be suspended by equal radij or Threads of equal Length: So that when they hang down and touch each other, the Radij or Threads may be parallel. Let One of the Globes be constantly the same, and be drawn aside from the Other to one and the same distance in All the subsequent Trials. Let the Other be of Any Bigness, and be drawn aside the contrary way to a Distance reciprocally proportional to its Weight. Let Both of them then be let go at one and the same Moment of Time, so that they may meet each other at the lowest place of their Descent, where they hung before they were drawn aside: And the first Globe will always rebound alike, from the Other. Wherefore the Force of the Other is always the same, when its Velocity is reciprocally proportional to its Weight. And by consequence, if its Weight remains the same, its Force will be proportional to its Velocity. Q. E. D.

[85] Acta Erudit. ad Ann. 1686, pag. 162.

[86] See above, the Note on my Third Reply. § 13.

[87] See above, the Note on § 93,—95.

[88] § 99.

[89] The Vis inertiæ of Matter, is That Passive Force, by which it always continues of itself in the State 'tis in; and never changes That State, but in proportion to a contrary Power acting upon it. 'Tis That Passive Force, not by which (as Mr. Leibnitz from Kepler understands it, See Appendix, No 7,) Matter resists Motion; but by which it equally resists Any Change from the State 'tis in, either of Rest or Motion: So that the very same Force, which is requisite to give any certain Velocity to any certain Quantity of Matter at Rest, is always exactly requisite to reduce the same Quantity of Matter from the same degree of Velocity to a state of Rest again. This Vis inertiæ is always proportional to the Quantity of Matter; and therefore continues invariably the same, in all possible States of Matter, whether at Rest or in Motion; and is never transferred from One Body to another. Without this Vis, the Least Force would give Any Velocity to the Greatest Quantity of Matter at Rest; and the Greatest Quantity of Matter in Any Velocity of Motion, would be stopped by the Least Force, without any the least shock at all. So that, properly and indeed, All Force in Matter either at Rest or in Motion, all its Action and Reaction, all Impulse and all Resistance, is nothing but this Vis inertiæ in different Circumstances.

[90] That is; proportional to the Quantity of Matter and the Velocity; not (as Mr. Leibnitz affirms, Acta Erudit. ad Ann. 1695, pag. 156,) to the Quantity of Matter and the Square of the Velocity. See above, the Note on § 93,——95.

[91] See above, the Note on § 93——95; & Third Reply, § 13.

[92] Fourth Paper, § 40, 20, 21, 22; and Fifth Paper, § 29.

[93] Fourth Paper, § 40, 20, 21, 22; and Fifth Paper, § 29.

[94] See above, Mr. Leibnitz's Postscript to his Fourth Paper.

[95] Fourth Paper, § 40, 20, 21, 22; and Fifth Paper, § 29.

[96] § 104.

[97] § 105.

[98] § 106.

[99] § 108.

[100] § 110.

[101] § 110:

[102] § 111.

[103] § 112.

[104] § 113.

[105] Quâ causâ efficiente hæ Attractiones peragantur, in id verò hic non inquiro. Quam ego Attractionem appello, fieri sanè potest ut ea efficiatur Impulsu, vel alio aliquo modo nobis ignoto. Hanc vocem Attractionis ita hic accipi velim, ut in universum solummodo vim aliquam significare intelligatur, quâ corpora ad se mutuo tendant; cuicunque demùm causæ attribuenda sit illa vis. Nam ex phænomenis Naturæ illud nos prius edoctos oportet, quænam corpora se invicem Attrahant, & quænam sint Leges & Proprietates istius Attractionis; quàm in id inquirere par sit, quânam Efficiente Causâ peragatur Attractio. Newtoni Optice, Qu. 23, pag. 322. Atque hæc quidem Principia considero, non ut occultas Qualitates, quæ ex Specificis rerum Formis oriri singantur; sed ut universales Naturæ Leges, quibus res ipsæ sunt formatæ. Nam Principia quidem talia revera existere, ostendunt Phænomena Naturæ; licet ipsorum causæ quæ sint, nondum fuerit explicatum. Affirmare singulas rerum species, specificis præditas esse qualitatibus occultis, per quas eæ Vim certam in Agendo habeant; hoc utique est Nihil dicere. At ex phænomenis Naturæ, duo vel tria derivare generalia Motis Principia; & deinde explicare quemadmodum proprietates & actiones rerum corporearum omnium ex Principiis istis consequantur; id verò magnus esset factus in Philosophiâ progressus, etiamsi Principiorum istorum Causæ nondum essent cognitæ: Id. ibid. Pag. 344. Phænomena Cœlorum & maris nostri per Vim Gravitatis exposui, sed causam Gravitatis nondum assignavi. Oritur utique hæc <357> Vis à causa aliqua, quæ penetrat ad usque centra Solis & Planetarum, sine virtutis diminutione; quæque agit non pro quantitate Superficierum particularum in quas agit, (ut solent causæ mechanicæ,) sed pro quantitate materiæ solidæ; & cujus actio in immensas distantias undique extenditur, decrescendo semper in duplicatâ ratione distantiarum. —— Rationem verò harum Gravitatis proprietaum ex Phænomenis nondum potui deducere, & Hypotheses non fingo: Principia Philos. Schol. generale sub finem. i. e. What the efficient Cause of these Attractions is, I do not here inquire. What I call Attraction, may possibly be caused by some Impulse, or some other way unknown to us. I use the Word Attraction, only in general, to signify the Force by which Bodies tend towards each other; whatever be the Cause of that Force. For we must first learn from the Phænomena of Nature, what Bodies attract each other, and what are the Laws and Properties of that Attraction, before 'tis proper to inquire what the efficient Cause of Attraction is. Again: I consider these Principles, not as occult Qualities, imagined to arise from the specifick Forms of Things; but as Universal Laws of Nature, according to which the Things themselves were formed. For, that such Principles do really exist, appears from the Phænomena of Nature; though, what the Causes of them are, be not yet explained. To affirm that every distinct Species of Things, is indued with specifick occult Qualities, by means whereof the Things have certain Active Forces; this indeed is saying Nothing. But to deduce from the Phænomena of Nature, two or three general Principles of Motion; and then to explain how the Properties and Actions of all corporeal Things follow from those Principles; This would be a great Progress in Philosophy, though the Causes of those Principles were not yet discovered. Again: I have explained the Phænomena of the Heavens and the Sea, by the Force of Gravity; but the Cause of Gravity I have not yet assigned. It is a Force arising from some Cause, which reaches to the very Centers of the Sun and Planets, without any diminution of its Force: And it acts, not proportionally to the Surfaces of the Particles it acts upon, as Mechanical Causes use to do; but proportionally to the Quantity of Solid Matter: And its Action reaches every way to immense Distances, decreasing always in a duplicate ratio of the Distances. But the Cause of these Properties of Gravity, I have not yet found deducible from Phænomena: And Hypotheses I make not.

[106] § 109 and 92, and 87, 89, 90.

[107] See Appendix, No 5.

[108] § 92.

[109] See Appendix, No 13.

[110] See Appendix, No 12.

[111] See Sir Isaac Newton's Opticks, Latin Edition, Pag. 224. English Edition, Book 2, Page 65.

[112] § 115, 116.

[113] See Appendix, No 13.

[114] See above, Mr. Leibnitz's Third Paper, § 17.

[115] § 118.

[116] See above, the Note on § 113.

[117] § 112.

[118] § 123.

[119] § 123.

[120] § 124.

[121] § 128.

[122] § 125, &c.

[123] § 125.

[124] See above, on § 1-20, and 21-25.

[125] § 26 & 125, &c.