Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturae secundum quam volunt a Deo eandem semper quantitatem motus conservari, qua et in re mechanica abutuntur

CAUSATION 🌍 International Journal of Science Impressum: Causation International Journal Of Science Chief-Editor / V. i. S. d. P.: Ilija Barukˇ ci´ c Hegelitzer Str. 22 DE-26409 Wittmund-Ardorf (Germany) ©Editorial Team: Editorial Team mWebsite: www.causation.eu kE-Mail: [email protected] TPhone: +(49) 4466 - 333 Year: 2026; Volume: 23; Issue: 3; pp. 5–18. BReceived: December 1, 2025 VAccepted: December 1, 2025 ¥Published: December 1, 2025 iDeutsche Nationalbibliothek Frankfurt YISSN: 1863-9542 DOI:10.5281/zenodo.17774513 !Find in: Zenodo Orcid Academia ... Brevis demonstratio erroris memorabilis Cartesii & aliorum circa legem naturae, secundum quam volunt a Deo eandem semper quantitatem motus conservari; qua & in re mechanica abutuntur. Reprint of original article Godefridus Guilelmus Leibnitius; Edited by Ilija Barukˇ ci´ c1 1Internist, Horandstrasse, 26441 Jever, Germany *Correspondence: E-Mail: [email protected]; Tel: +49-4466-333; Fax: +49-4466-333. 6 Abstract Background: The prevailing doctrine among natural philosophers, championed by Monsieur Descartes and his adherents, maintains that the quantity of motion—the product of a body’s mass and its velocity—is conserved by Divine ordinance throughout the universe. This principle, while elegant, leads to manifest contradictions when subjected to rigorous mechanical analysis, particularly in the consideration of falling bodies and simple machines. Material and methods: I employ the established principles of mechanics, as confirmed by the experiments of Galileo, concerning the velocities acquired by bodies falling from different heights. Through a geometrical and demonstrative method, I analyze the case of two bodies of different mass falling through different distances, comparing their acquired vis viva (living force) with their Cartesian quantitas motus (quantity of motion). The argument proceeds by clear suppositions and logical deduction, appealing to no authority other than reason and the observable laws of nature. Results: The analysis reveals a profound and essential distinction. Two bodies—one of mass 1 falling 4 ells, another of mass 4 falling 1 ell—acquire equal vis viva, or capacity to perform work (e.g., to re-ascend to their original height). Yet their quantities of motion are found to be in the ratio of 1:2. Thus, vis viva must be proportional to mass multiplied by the square of the velocity (mv2), not the velocity itself. This living force is the true conserved quantity in elastic interactions and mechanical effects, not the Cartesian mv. Conclusion: The true measure of a body’s motive power is not its quantity of motion, but its vis viva, estimated by the effect it can produce (such as the height to which it can raise a weight). The conservation of this mv2is the fundamental law of dynamics, resolving the errors of Descartes and his followers. This finding restores harmony to mechanical philosophy and provides the correct foundation for estimating forces, thereby advancing our understanding of the divine and immutable order of nature. Keywords: Vis viva; Quantity of motion; Conservation law; Force; Mechanics; Descartes. ©Reviewer 1: None/Author. §Reviewer 2: None. §Reviewer 3: None. Causation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18 7 MENSIS MARTII A. MDCLXXXVI. 161 Godefridus Guilelmus Leibnitius Brevis demonstratio erroris memorabilis Cartesii & aliorum circa legem naturae, secundum quam volunt a Deo eandem semper quantitatem motus conservari; qua & in re mechanica abutuntur. Communicata in litteris d.6. Jan. 1686. datis. Complures Mathematici cum videant in quinque machinis vulgaribus celeritatem & molem inter se compensari, generaliter vim motricem affirmant aequari quantitati motus, sive producto ex multiplicatione corporis in celeritatem suam. Vel ut magis geometrice loquar, vires duorum corporum (ejusdem speciei) in motum concitatorum, ac sua mole pariter ac motu agentium, esse dicunt in ratione composita corporum seu molium, & earum quas habent velocitatum. Itaque cum rationi consentaneum sit, eandem motricis potentiae summam in natura conservari; & neque imminui, quoniam videmus nullam vim ab uno corpore amitti, quin in aliud transferatur; neque augeri, quia vel ideo motus perpetuus mechanicus nuspiam succedit; quod nulla machina, ac proinde ne integer quidem mundus suam vim intendere potest sine novo externo impulsu; inde factum est ut Cartesius, qui vim motricem, & quantitatem motus pro re equivalente habebat, pronunciaverit eandem quantitatem motus a Deo in mundo conservari. Ego vero, ut off` endam quantum inter haec duo intersit, suppono, primo corpus cadens ex certa altitudine acquirere vim suˆ aCausation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18 8 English Translation of Leibniz’s Argument: MONTH OF MARCH, A.D. 1686. 161 A Brief Demonstration of the Remarkable Error of Descartes and Others Concerning the Law of Nature by Which They Think That God Always Conserves the Same Quantity of Motion; in Which Matter They Also Err in Mechanics. Communicated in a letter dated January 6, 1686. Godefridus Guilelmus Leibnitius Since many mathematicians see that in the five simple machines speed and mass compensate each other, they generally assert that motive force equals the quantity of motion, or the product from the multiplication of the body by its speed. Or, to speak more geometrically, they say that the forces of two bodies (of the same kind) put into motion, and acting equally by their mass as well as their motion, are in the ratio composed of the bodies or masses and the velocities which they have. Therefore, since it is consistent with reason that the same sum of motive power is conserved in nature; and neither diminished (since we see no force lost from one body without being transferred to another) nor increased (because for this reason mechanical perpetual motion never occurs; since no machine, and therefore not even the whole world, can increase its force without a new external impulse); hence it happened that Descartes, who considered motive force and quantity of motion as equivalent things, declared that the same quantity of motion is conserved by God in the world. But I, to show how much difference there is between these two, suppose first that a body falling from a certain height acquires a force Causation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18 9 162 ACTA ERUDITORUM que rursus assurgendi, si directio ejus ita ferat, nec quicquam externorum impediat: exempli causa, pendulum ad altitudinem ex qua dimissum est praecise rediturum esse, nisi aeris resistentia similiaque impedimenta exigua alia nonnihil de vi ejus absorberent, a quibus nos quidem nunc animum abstrahimus. Suppono item secundo, tanta vi opus esse ad elevandum corpus A unius librae usque ad altitudinem CD quatuor ulnarum, quanta opus est ad elevandum corpus B quatuor librarum, usque ad altitudinem EF unius ulnae. Omnia hae a Cartesianis pariter ac ceteris Philosophis & Mathematicis nostri temporis conceduntur. Hinc sequitur corpus A delapsum ex altitudine CD, praecise tantum acquisivisse vim, quantum corpus B lapsum ex altitudine EF. Nam corpus (A) postquam lapsu ex C pervenit ad D, ibi habet vim reassurgendi usque ad C, per suppos. 1, hoc est vim elevandi corpus unius librae (corpus scilicet proprium) ad altitudinem quatuor ulnarum. Et similiter corpus (B) postquam lapsu ex E pervenit ad F, ibi habet vim reassurgendi usque ad E, per suppos. 1, hoc est vim elevandi corpus quatuor librarum (corpus scilicet proprium) ad altitudinem unius ulnae. Ergo per suppos. 2, vis corporis (A) existentis in D, & vis corporis (B) existentis in F, sunt aequales. Videamus jam an & quantitas motus utrobique eadem sit. Verum ibi praeter spem discrimen maximum reperietur. Quod ita ostendo. Demonstratum est a Galileo, celeritatem acquisitam lapsu CD, esse duplum celeritatis acquisitae lapsu EF. Multiplicemus ergo corpus A quod est ut 1, per celeritatem suam quae est ut 2, productum seu quantitas motus erit ut 2, rursus multiplicemus corpus B quod est ut 4, per suam celeritatem quae est ut 1, productum seu quantitas motus erit ut 4. Ergo quantitas motus quae est corporis (A) existentis in D, est dimidia quantitatis motus quae est corporis (B) existentis in F, & tamen paulo ante vires utrobique inventae sunt aequales. Itaque magnum est discrimen inter vim motricem & quantitatem motus, ita ut unum per alterum aestimari non possit, quod ostendendum susceperamus. Ex his apparet, quomodo vis aestimanda sit a quantitate effectus, quem producere potest, exempli gratia ab altitudine ad quam ipsa corpus grave data magnitudinis & speciei potest elevare, non vero a celeritate quam corpori potest imprimere. Non enim Causation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18 10 English Translation of Leibniz’s Argument (Continued) 162 ACTA ERUDITORUM which will rise again if its direction carries it so, and nothing external hinders it: for example, a pendulum will return precisely to the height from which it was released, unless the resistance of the air and similar small impediments absorb some of its force, which we indeed now disregard. I suppose secondly, that as much force is needed to raise body A of one pound to the height CD of four ells, as is needed to raise body B of four pounds to the height EF of one ell. All these points are granted equally by the Cartesians as well as by other philosophers and mathematicians of our time. Hence it follows that body A, having fallen from height CD, has acquired precisely as much force as body B having fallen from height EF. For body (A), after having fallen from C, arrives at D, and there it has the force to rise again up to C, by supposition 1, that is, the force to raise a body of one pound (namely its own body) to a height of four ells. And similarly body (B), after having fallen from E, arrives at F, and there it has the force to rise again up to E, by supposition 1, that is, the force to raise a body of four pounds (namely its own body) to a height of one ell. Therefore, by supposition 2, the force of body (A) existing in D, and the force of body (B) existing in F, are equal. Let us now see whether the quantity of motion is also the same in both cases. But here, contrary to expectation, a very great difference will be found. I show it thus. It has been demonstrated by Galileo that the velocity acquired in the fall CD is double the velocity acquired in the fall EF. Let us therefore multiply body A, which is as 1, by its velocity, which is as 2; the product, or quantity of motion, will be as 2. Again, let us multiply body B, which is as 4, by its velocity, which is as 1; the product, or quantity of motion, will be as 4. Therefore the quantity of motion which belongs to body (A) existing in D, is half of the quantity of motion which belongs to body (B) existing in F, and yet a little earlier the forces in both cases were found to be equal. Hence there is a great difference between motive force and quantity of motion, so that one cannot be estimated by the other, which we had undertaken to show. From this it appears how force should be estimated by the quantity of the effect which it can produce, for example by the height to which it can raise a heavy body of given magnitude and kind, and not by the velocity which it can impart to a body. For not Causation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18 11 MENSIS MARTII A. MDCLXXXVI. 163 enim dupla sed majore vi opus est ad duplam eidem corpori dandam celeritatem. Nemo vero miretur in vulgaribus machinis, vecte, axe in peritrochio, trochlea, cuneo, cochlea & similibus aequilibrium esse, cum magnitudo unius corporis celeritate alterius, quae ex dispositione machinae oritura esset, compensatur; seu cum magnitudines (posita eadem corporum specie) sint reciproce ut celeritates; seu cum eadem alterutro modo prodiret quantitas motus. Ibi enim evenit etiam eandem utrobique futuram esse quantitatem effectus, seu altitudinem descensus aut ascensus, in quodcunque equilibrii latus motum fieri velis. Itaque per accidens ibi contingit, ut vis a motus quantitate possit aestimari. Alii vero casus dantur, qualis is est quem supra attulimus, ubi non coincidunt. Cæterum cum nihil sit probatione nostra simplicius, mirum est vel Cartesio vel Cartesianis, viris doctissimis, in mentem non venisse. Sed illum quidem nimia fiducia sui ingenii intransversum egit, hos alieni. Nam Cartesius, solito magnis viris vitio, postremo factus est paulo praesidentior. Cartesiani autem non pauci vereor ne paulatim Peripateticos complures imitari incipiant, quos irrident, hoc est ne pro recta ratione & natura rerum, consuetudinem magistri libris assuescant. Dicendum est ergo vires esse in composita ratione corporum (ejusdem gravitatis specificae seu soliditatis) & altitudinum celeritatis productricium, ex quibus scilicet labendo tales celeritates acquiri potuissent; vel generalius (quia interdum nulla adhuc celeritas producta est) altitudinum productarum: non vero generaliter ipsarum celeritatum, utcunque id plausibile prima specie videatur, & plerisque sit visum; ex quo complures errores nati sunt, qui in scriptis mathematico-mechanicis R.R. PP. Honorati Fabry &Claudii des Chales, itemque Job. Alph. Borelli & aliorum virorum, cæteroqui in his studiis praestantium, deprehenduntur. Quin & hinc factum puto, quod nuper Regula Hugeniana circa centrum oscillationis pendulorum, quae verissima est, a nonnullis viris doctis in dubium fuit vocata. Causation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18 12 English Translation of Leibniz’s Argument (Continued) MONTH OF MARCH, A.D. 1686. 163 For not double but greater force is needed to give double velocity to the same body. But let no one wonder that in common machines–the lever, axle in the pulley, wheel, wedge, screw, and similar devices–there is equilibrium, when the magnitude of one body is compensated by the velocity of another, which would arise from the arrangement of the machine; or when the magnitudes (assuming the same kind of bodies) are reciprocally as the velocities; or when the same quantity of motion results in either way. For there it also happens that the same quantity of effect, or height of descent or ascent, will be on both sides, into whichever side of the equilibrium you wish motion to occur. Therefore, it happens accidentally there that force can be estimated from the quantity of motion. But other cases are given, such as the one we presented above, where they do not coincide. Moreover, since nothing is simpler than our proof, it is astonishing that it did not occur either to Descartes or to the Cartesians, most learned men. But indeed, excessive confidence in his own genius led the former astray, the latter by following another. For Descartes, with the usual fault of great men, finally became a little too presumptuous. And I fear that not a few Cartesians are beginning little by little to imitate many Peripatetics, whom they mock; that is, they are becoming accustomed to consulting the books of their master instead of right reason and the nature of things. It must therefore be said that forces are in the compound ratio of bodies (of the same specific gravity or solidity) and of the heights productive of velocity, from which of course such velocities could have been acquired by falling; or more generally (because sometimes no velocity is yet produced) of the heights produced: not indeed generally of the velocities themselves, however plausible this may seem at first glance, and has seemed to most; from which several errors have arisen, which are found in the mathematical-mechanical writings of Reverend Father Honor´ e Fabri and Claude des Chales, likewise Giovanni Alfonso Borelli and other men, otherwise outstanding in these studies. Indeed, I think it is from this that the recently proposed Huygensian Rule concerning the center of oscillation of pendulums, which is most true, has been called into doubt by some learned men. Causation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18 13 Historical and Philological Annotations Significance of Classical Abbreviations •R.R. PP. =Reverendissimi Patres (Most Reverend Fathers) – An honorific title for clergy •Job. =Johannis (Johann/John) – Latin genitive form •Alph. =Alphonsi (Alphons) – Latin genitive form Register of Mentioned Scientists •Honorati Fabry (Honor´ e Fabri, 1607–1688) – French Jesuit, mathematician, and physicist •Claudii des Chales (Claude Franc¸ois Milliet Dechales, 1621–1678) – French Jesuit mathematician •Job. Alph. Borelli (Giovanni Alfonso Borelli, 1608–1679) – Italian physiologist, physicist, and mathematician Key Concepts in Leibniz’s Argument •Peripatetics: Followers of Aristotle (from ”Peripatos,” the walkway at Aristotle’s school) •Huygensian Rule: Christiana Huygens’ rule for calculating the center of oscillation of compound pendulums •Compound ratio: Mathematical ratio composed of multiple factors (bodies ×heights) •Specific gravity: Density relative to water, an important concept in 17th-century physics Contextual Notes This passage continues Leibniz’s critique of Cartesian physics, specifically: 1. The distinction between force (vis viva = mv²) and quantity of motion (momentum = mv) 2. The accidental nature of equilibrium in simple machines 3. The methodological critique of authorities (Descartes, Peripatetics) over direct reasoning from nature 4. Reference to contemporary scientific debates about pendulum motion Causation ISSN: 1863-9542 https://www.doi.org/10.5281/zenodo.17774513 Volume 23, Issue 3, 5–18