• Peter Damerow
  • Gideon Freudenthal
  • Peter Mclaughlin
  • Jürgen Renn
Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)


All sources quoted in the text and documented in this chapter are cited in English translation. Wherever a good translation was already available we have adopted its rendering as far as possible. Almost every adopted translation has undergone some significant changes, which are not indicated individually. The extensive quotations from Galileo’s De motu (Drabkin) and Discorsi (Drake) have, however, been taken over largely unchanged.


Heavy Body Equable Motion Geometric Proportion Determined Motion Natural Descent 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Descartes mistakenly writes1/8 for ag and 7/8 for gb. Google Scholar
  2. 2.
    But the problem could be posed in a different way, such that the attractive force of the Earth is equal to that in the first moment: and while the first remains, a new one is produced. In this case the problem would be resolved by a pyramid. But to lay down the principles of this science [I should say] that an always uniform motion is represented by a line or by a rectangular surface or by a parallelogram or by a parallelopiped; what is augmented by one cause [is represented] by a triangle; by two [causes] by a pyramid, as above; by three [causes] by other figures.Google Scholar
  3. 3.
    Most published versions of this figure contain minor errors and are internally inconsistent. The figure in AT I, 72, which is reprinted in Descartes Correspondance (ed. Adam and Milhaud) vol. 1, p. 85, has 3 more lines overall and also one space more in the right half than in the left. Tannery’s original publication (1891 p. 531) was differently inaccurate; the diagram given by de Waarde in Mersenne’s Correspondance (vol. 2, p. 316) is accurate and internally consistent but the triangle is isosceles. Descartes’ original drawing is preserved in the National Library in Paris (Paris, Bibl. nat., f. fr., nouv. acq. 5160, fol. 48 recto et verso). We should like to thank Gad Freudenthal for securing us a photocopy of the manuscript.Google Scholar
  4. 4.
    The greater part of the translation given here is based on those passages translated by John Cottingham in The Philosophical Writings of Descartes,(transi. by J. Cottingham, R. Stoothoff, and D. Murdoch) Cambridge University Press, 1985 (vol. 1, pp. 240–245); it is used with the permission of Cambridge University Press. We have modified it liberally to standardize the terminology as used in our other translations and also to make some renderings more literal (even at the risk of making them on occasion somewhat clumsy). This translation is based on the Latin text; the (contemporary) French translation was consulted — as were various German and English translations — but its renderings were not accepted when they deviated from the Latin. On difficulties with the French version, see Costabel 1967.Google Scholar
  5. 5.
    The translation given here is by John Cottingham, taken from The Philosophical Writings of Descartes,(transl. by J. Cottingham, R. Stoothoff, and D. Murdoch) Cambridge University Press, 1985 (vol. 1, pp. 156–162); it is used with permission of Cambridge University Press. We have made some minor changes to standardize terminology.Google Scholar
  6. 6.
    Descartes uses the terms “angle of incidence” and “reflection” here to mean the angle made with the surface not with the normal to the surface; “angle of refraction” refers to the refracted ray’s deviation from the original direction not its angle to the normal.4 As for gravity [Pesanteur],I do not imagine anything else than that all the subtle matter which is between here and the moon and which turns very quickly around the Earth, chases toward it all the bodies which cannot move as quickly. Hence it pushes them with more force when they have not yet begun to descend than when they already descend. Finally, when it happens that they [the bodies] descend as quickly as it [the subtle matter] moves, it will not push them at all anymore, and if they [the bodies] descend more quickly, it [the subtle matter] will resist them. From here you may see that there are many things to be considered before anything concerning the speed can be determined, and it is this which has always distracted me; but many things can be also explained by these principles which were not grasped before.Google Scholar
  7. 7.
    VI, 93–101; translation by Cottingham, Descartes 1985, vol. 1, pp. 156–162. For interpretation see 2.5.1.) AT VI, 95; see Fig. 5.16. Figure 5.25 given here was originally published in Clerselier’s edition of Descartes’ correspondence in 1664.Google Scholar
  8. 8.
    AT VI, 97; see Fig. 5.17.Google Scholar
  9. 9.
    But in order clearly to destroy the proposition we must examine two sorts of compound motions made on two straight lines. Let us consider [Fig. 5.27] the two [lines] DA and AO which make the angle DAO, of whatever size you will; and let us imagine a heavy body [un grave] at point A, which descends in the line ACD at the same time that the line advances along AN such that it always makes the same angle with AO, and that the point A of the same line ACD is always in the line AN. If the two motions, that of the line ACD along AO and that of the same heavy body in the line ACD are uniform, as we may suppose them to be, it is certain that the compound motion will always conduct the heavy body in a straight line such as AB; and if you take a point such as B from which you draw the lines BN and BC parallel to lines DA and AO; then the heavy body will be at point B, in the same time that it would have been at point C if there had only been the motion on ACD and that it would have been at point N if there had been only the other motion alone; and the proportion of the force that conducts it on AD to the force that conducts it along AO is as AC to AN that is to say, as BN to BC. It is this sort of compound motions that was used by Archimedes and the other ancients in the composition of their helices; their principal property is that the two motive forces do not impede each other at all, remaining always the same. But since this motion cannot be applied very well, it is necessary to consider another version and to engage in a particular speculation.Google Scholar
  10. 10.
    Let us suppose in the same figure a heavy body at point A, which is impelled at the same time by two forces, one of which pushes it along AO and the other along AD, with the result that the line of direction of the first motion is AO and that of the second is AD. If there were only the first force alone, the body would always remain on AO, and on AD if there were only the second [force]. But since both forces mutually impede and resist one another, let us suppose (and it should be remembered that we also suppose these motions to be uniform, for otherwise the compound motions would not be carried out in straight lines) that in one minute of an hour, for example, the second force makes the body depart from its direction AO according to the length NB which it must describe parallel to AD; for the body that is transported on AD by the second force, finding itself hindered by the first, will continue on and advance from A toward D by parallels to AD. Let us suppose as well that in the same minute of an hour, the first force makes the body depart from its direction AD according to the length CB, parallel (for the reasons given above) to line AO. It is completely certain that in one minute of an hour the body will be found at point B, which is the intersection of the two lines BN and BC. The compound motion will occur on the line AB and we can say that the body traverses the line AB in one minute.Google Scholar
  11. 11.
    Let us suppose now the angle DAO to be changed [Fig. 5.28] and, for example, to be greater. In the next figure, the same things being posited, I say that in one minute of an hour, as before, the body departs from the direction AO according to the line BN — equal to that to which we have given the same name in the preceding figure. For, since the forces are the same, the second will equally diminish the determination of the first, and will in equal times remove the body from its direction as much as before, because there is always the same resistance. We may conclude the same thing for line BC.Google Scholar
  12. The compound motion will thus occur here on the line AB, and the line AB will be traversed as before in one minute of an hour. But because in the two avoid prolixity, that the sine of angle DAB is to the sine of angle OAB in the first figure as the sine of angle DAF to the sine of angle OAF in the second figure.Google Scholar
  13. 13.
    This thus assumed and demonstrated, let us consider the figure on page 20 of the Dioptrics,9 in which the author supposes that the ball, having first been impelled from A to B, being at point B, is impelled in such a manner by the racket CBE, which (doubtless in the sense of the author) pushes along BG. Therefore, from the two motions of which the one pushes along BD and the other along BG a third is made which conducts the ball in the line BI.Google Scholar
  14. 14.
    Let us now imagine a second figure similar to this one, in which the force of the ball and that of the racket are the same, and that only the angle DBG is greater in this second figure. It is certain by the demonstrations we have just given that there will be the same proportion of sine of the angle GBI to sine of angle IBD in this second figure [Fig. 5.29], which we imagine to be drawn and which we omit to avoid lengthiness. Whereas if the propositions of the author were true, there would be the same proportion of the sine of angle GBD to the sine of angle GBI in the figure of the author that the sine of the angle GBD to the sine of the angle GBI in this second figure which we have imagined. But since this proportion is different from the other, it follows that it cannot hold.Google Scholar
  15. 15.
    Moreover, the principal ground of the author’s demonstration is based on the fact that he believes that the motion compounded on BI is always equally swift, even if the angle GBD made by the lines of direction of the two motive forces happens to change; this is false as we have already plainly demonstrated.Google Scholar
  16. 16.
    I do not want to maintain, that in the application of the figure to refraction, which he makes on page 20,10 one should take my proportion and not his; for 1 am not sure whether this compound motion should serve as the rule for refraction; on this matter I shall tell you my sentiments another time at more length.Google Scholar
  17. 17.
    And he continues his paralogism up to the end, where he concludes that the compound motion on BI (that is to say the motion whose speed is compounded) is not always equally swift when the angle GBD made by the lines of direction of the two forces (that is to say by the lines that mark how the determinations of the two forces are compounded) is changed; drawing this conclusion from that which he had already proved concerning the motion whose determination — and not whose speed — is compounded that the speed changes when the angle changes. But you can see the faults better than I and if some difficulty in all this should remain which I have not explained enough please oblige me by pointing it out to me.Google Scholar
  18. 18.
    The impetus in the single points of the parabola bec is therefore determined by the square [potential of the moment acquired in the descent along ab,which will always be the same and which determines the horizontal impetus, and by the square of the other moment acquired in the descent along the vertical. Thus, for instance in e the impetus will be determined by the square root of the sum of the squares of [linea potente] ab and the mean proportional between db, bf,which is bg. Google Scholar
  19. 19.
    The impetus of a body falling from h to a will be 141 in a; deflected along the parabola ae,however, the impetus will be doubled in e,that is, 282. It is therefore true that the impetus of the body coming along the parabola ce to e will be larger than that of the body coming along the parabola ae. And if the projectile [shot] from e,along the elevation eh,has an impetus as 282, it will traverse the parabola ea; but along the elevation ea the projectile traverses the parabola ec,if it has an impetus as 300. Hence it will be thrown farther by the same force along the elevation eh of half a right angle than along the elevation ea,which is smaller than half a right angle.20 The original ms (and Favaro’s transcription) reads simply “da.” Google Scholar
  20. 21.
    The original ms (and Favaro’s transcription) reads “ec, dc”; a number of significant lines, in particular line eh and parabola ea,are missing from the transcribed diagram on EN VIII, 430. See Plate VII.Google Scholar
  21. 22.
    The translation is taken from Galileo Galilei, Two New Sciences,(transl. by Stillman Drake) University of Wisconsin Press, 1974, pp. 165–167; it is used with permission of Stillman Drake, Wisconsin University Press, and Wall and Emerson, Inc.Google Scholar
  22. 23.
    The original has the misprint “AB” instead of “AG”.Google Scholar
  23. 24.
    EN VII, 248/221–222Google Scholar
  24. 25.
    EN VII, 219/193. Here Galileo does indeed explain that the the motion of the projector impresses an impetus upon the projectile to move, when they separate, along a straight line tangent to curve characterizing the motion of the projector at the point of separation, but he does not claim that this motion is continuous.Google Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Peter Damerow
    • 1
  • Gideon Freudenthal
    • 2
  • Peter Mclaughlin
    • 3
  • Jürgen Renn
    • 4
  1. 1.Max-Planck-Institut für BildungsforschungBerlin 33Germany
  2. 2.Cohn Institute for the History and Philosophy of Science and IdeasTel Aviv UniversityRamat-Aviv, Tel AvivIsrael
  3. 3.Fachgruppe PhilosophieUniversität KonstanzKonstanz 1Germany
  4. 4.The Collected Papers of Albert EinsteinBoston UniversityBostonUSA

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