François du Verdus

Noel Malcolm (ed.), The Clarendon Edition of the Works of Thomas Hobbes, Vol. 6: The Correspondence, Vol. 1: 1622–1659

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pg 231 letter 76[20 february/] 1 march 1656François du Verdus to Hobbes, from Bordeaux

Chatsworth, Hobbes MSS, letter 11 (original).

First enclosure: Chatsworth, Hobbes MSS, letter 82 (original).

Second enclosure: Chatsworth, Hobbes MSS, letter 32 (original).

Monsieur,

Je vous écriuis il peut y auoir six semaines ou du mois une longue lettre1 que j'adressay a Mr Martel le priant de vous la faire tenir. Il l'aura fait sans doute Et vous aurés ueû que sur vos ordres je suspans l'edition de la Traduction que j'ay faite de vostre Liure de Corpore. vous aurés ueû aussi le dessein que j'ay de vous dédier cette version Ce que je vous pric d'agreër. J'ose vous dire qu'elle est exacte Et (si ce n'est pas trop me uanter) que ceux a qui j'en ay leu quelque chose trouuent que j'y ay eu du genie et de la bone fortune. Je ne vous done pour exemple que votre premiere periode de votre Epitre liminaire a Mr le pg 232Comte de Deuonsire (si Comes Deuoniae ne se traduit pas ainsi je vous prie aprenés moy le mot françois[)]2 mais voicy cette période. Monseigneur Je vous presente cette première section des Elemens de la Philosophie, que j'ay fait long temps atendre après l'edition de la Troisieme et qu'enfin j'ay acheuée; et je vous l'offre qu'a l'auenir elle soit un Monument de vos bontés enuers moy et du respect que j'ay pour vous.3 vous uoyés Monsieur que tout y est et qu'auec cela le tour y est francois sans parenthese ni transposition qui est une des délicatesses de nostre langue. Mais je suis encore plus exacte dans le corps de l'oeuure car j'y rends tout mot a mot et si j'y eusse trouué le benignitatis de l'epitre j'eusse traduit benignité.

Ce que je vous disois aussi dans ma lettre que j'adressay a Mr Martel: C'est que je uerrois d'auoir le sentiment de nos Geometres sur vostre Liure. Quand je seray a Paris j'en scauray dauantage Car Mr Roberual4 par exemple et Mr Pascal m'en parleront uolontiers, qui mal-aysément se resoluent a écrire de peur de se couper. Cepandant Mr Mylon m'en a écrit a deux diuerses fois sur la semonçe que je luy en fis. La premiere fois il m'enuoya son premier Memoire. la s[econ]de5 fois le second6 Mais voicy come je répondis au premier.

Au 1er et 2d nombres. Que dans uostre définition. Deux lignes quelles qu'elles soyent (soit droites ou courbes) et mesme deux surfaçes sont paralleles sur lesquelles deux lignes droites tombans quelque part que ce puisse estre et faisans angles égaux auec l'une ou l'autre d'elles seront égales entre elles.7 que dans cette définition dis-je c'est qu'il faut entendre que les lignes tombans sur les paralleles y soient interceptées Ce qui est fort aisé a entendre; et qu'en un besoin on peut enonçer ainsi. Deux lignes quelle que ce soit, droites ou courbes et mesme deux surfaçes sont paralleles entre lesquelles deux lignes droites interceptées quelque part que ce soit qui y font angles égaux sur l'une ou l'autre sont égales entre elles.

Au 4e8 Que ce qu'il ne uous accorde que come une hypothese vous le démontrés en ce qu'il ne peut y auoir de dureté infinie.9 Et que ce qu'il ueut que DesCartes ait dit mieux In Circulum l'a jetté dans l'absurde que nous trouuons dans l'art. 34 de sa seconde partie qu'en ce mouuement en cercle il se fait une diuision infinie de la matière Car affin que cela fut il faudroit que les [cotes deleted > extremités] de ses cercles ne receussent point du tout d'impression de la matiere qui se meut [au ded deleted] entre deux c'est a dire que la resistance de ces cercles là fut infinie.10

Au 5eme Que ce qu'un côté du parallelograme a esté obmis, c'est une pg 233faute d'impression et que le mot Semissem y etant deux fois l'Imprimeur a pris le premier [> semissem] pour le second et a sauté au mot nam haec &c.11

Au 6éme et 7eme Qu'il se trompe en disant que vous prenés qu' AJ soit accelerate pars acceleratae AH. qu'au contraire vous prenés AJ parcouruë uniformement; que ce que l'a brouïllé c'est qu'il ne distingue pas AB d'elle mesme Et que quand vous dites Dico esse ut AH ad AB ita AB ad AJ vous entendés que AH qui est toute la longueur parcouruë par le mouuement hasté dans le temps AC, est à AB qui est toute la longueur parcourue uniformement dans le mesme temps comme AB prise comme une longueur parcouruë par le mouuement hasté et laquelle est partie de AH, est a AJ prise come longueur parcourue uniformement &c.12

Au 8eme Que si vous estes contraire a Mss Roberual et DesCartes ils sont contraires a la verité13 Ce que vous dites de la refraction etant fort bien demontré.

Au 9éme Que je n'ay point l'esprit de comprendre en quoy vous vous trompés au chap. 18éme. Que si vous uous y estes trompé (ce que j'ay peine a croire) il ne doit pas se fascher qu'on l'ait découuert puis qu'au contraire vous mesmes Monsieur en serés bien aise qui estes un home ingenu Et aymant la verité. Et qu'enfin ie le prie de m'enuoyer cette prétenduë démonstration de Mr Zulichem.

Il me l'a donc enuoyée dans son second Memoire. Je la reçeus par le dernier courrier Et je vous l'enuoye14 sans perdre temps que vous preniés la peine d'y faire réflection.

Au reste Monsieur j'ay a vous dire de Mr Mylon Qu'il parle de vous auec toute sorte d'estime et de bone uolonté voicy quelque chose de sa première lettre. On dit icy que vous aués traduit la Physique de Mr hobbes pour la faire imprimer; vous obligerés toute la france. Je scay bien que l'ecole ne done pas trop son aprobation a cet Ouurage et croy que Joh: Wallesius STD et Geometra in Acad. Oxon. professor Saluïenus l'aura combatu par Aristote;15 mais ni vous ni l'autheur ne vous en souciés pas beaucoup, vous ne prétendés de plaire qu'a ceux qui sont hors des préjugés et conduisent bien leur raison. J'ay leu ce liure de Mr Hobbes auec tres grand plaisir Et y ay tout admiré horsmis les endroits que je vous cotte sur l'entre-papier. receués le come d'une persone tres affectionée a l'autheur et si je me suis méconté faites moy la faueur de me montrer ma faute.

Et dans sa seconde lettre Quand il me dit Que pour décrier le meilleur liure du Monde il ne faut que dire qu'il y a des paralogismes je pg 234vois effectiuement que c'est un trait de sa bone uolonté qu'il souhaiterait que les Chapitres De Dimensione Circuli et des droites égales aux paraboliformes16 ne fussent pas dans le vostre; luy qui n'a pas la mesme opinion de ces Chapitres là Que du reste du Liure qu'il admire.

Ce qu'on m'écrira de plus ou que je scauray autrement sur votre sujet je vous l'écriray de mesme en confidançe. Car je suis asseuré Monsieur Que vous receurés toutes choses de ma part auec [les deleted] autant de bonté que j'ay d'estime et de uénération pour vous. Cepandant soit que vous ueuïlliés me dire en quoy se trompe Mr Zulichem (que je croiray toujours qui se trompe jusqu'a ce que vous m'ayés dit que non); soit que vous me ueuilliés enuoyer vos Cahiers de homine17 que je souhaiterois de reçeuoir a mesure qu'ils s'imprimeront pour y trauailler en mesme temps et auoir vos Trois sections traduites tout a la fois: ou qu'enfin vous ueuilliés de temps en temps m'aprendre de vos Nouuelles que je prie Dieu qui soyent toujours bones Je vous prie Monsieur écriués moy de droiture. Car j'ay esté auerty que le Comis de Paris enuoye les lettres de Bordeaux a Londres et celles de Londres a Bordeaux. Obligés moy donc de mettre ainsi le dessus a vos lettres. / A Monsieur Monsieur de Cheneuas18 Maistre de Bureau de la Poste pour Mr de S.t Hilaire A Bourdeaux. /.

St hilaire est un nom suposé que j'ay doné a Mr Cheneuas parce qu'on me faisoit perdre de mes lettres. sous cette suscription les vostres me seront renduës ponctuellement Et je les reçeuray auec mes sentimens inuiolables d'estime et d'amitié qui suis a la vie et a la mort

  • Monsieur
  • Vôtre tres humble et tres obeissant seruiteur
  • du uerdus.

A Bordeaux le 1er Mars 1656.

[endorsed by James Wheldon:] Mar. 1.st 1656 Monsr du Verdus

[enclosed: Mylon's objections to De corpore in Mylon's hand, with an annotation in du Verdus's hand, '1er Memoyre de Mr. Mylon':]

    Ad physicam Domini hobs.

  1. 1. Chap. 14. Art. 12. La definition des paralleles doit estre supplée

  2. pg 2352. pag. 114. Linea 13. Il ne conclut pas juste par ce qu'il se fonde sur Sa definition des paralleles19

  3. 3 pag. 114. Art, 13 me semble obscur20

  4. 4. Chap. 15. Art. 7. on peut Luy accorder comme vne hypothese que Motus siue Conatos tam In vacuo, quam In pleno propagatur In Infinitum: Il me semble que Monsr DesCartes dit mieux quod propagatur Iste motus In Circulum.21

  5. 5. pag. 128. Linea 6. Il compose vn parallelogramme auec vn seul costé. Il doit estre ainsi. Vel denique per parallelogrammum cujus vnum latus medium proportionale Inter Impetum maximum (siue vltimò acquisitum) et et [sic] Impetus ejusdem maximi semissem, alteram verò latus sit medium proportionale Inter totum tempus et ejusdem totius temporis Semissem, nam Duo haec parallelogramma et Inter se, et [blotted tri]angulo quod fit ex tempore toto et Impetu crescente sunt aequalia.22

  6. 6. Ad cap. 16. Artic. 17. Enunciatur generaliter, demonstrator vero singulariter, vel non Intelligo. Nam In fig. 8.a AB est Longitudo vniformis respondens tempori AC. AH est Longitudo accelerata, respondens eidem tempori AC. Iam sumere AI Longitudo accelerata, pars accelerata AH; debet etiam sumere Longitudinem vniformem vt AO quae sit pars Vniformis AB, atque demonstrare differentiam totius AH supra totam, AB esse ad differentiam partis AI supra partem vt AO; vt ratio Longitudinum vt AH, AB (vel quod Idem est vt ratio partium Longitudinum AJ. AO) ad rationem temporum AC, AM.

  7. 7. Articulo 17. Cap. 16. Non Intelligo demonstrationem.

  8. 8. pag. 114. Linea 4. Il est tout contraire a M. DesCartes et a M. Roberual pour la refraction.23

  9. 9. Cap. 18. Le probleme general qui donne des lignes droites egales aux Paraboles, seroit tres beau s'il estoit vray, mais ce qui me fasche, Monsieur Hugens de Zulichem en a demonstré La fausseté il en doit escrire d'hollande a Monsr. hobs. Je vous l'Enuoyeray quand Il vous plaira si vous auez La Curiosité de voir cette demonstration.

[The following text, written in Mylon's hand on a single folio sheet and endorsed by James Wheldon, 'Monsr. Mylon 1656', is probably to be identified with the geometrical demonstration enclosed with the second commentary by Mylon and forwarded by du Verdus to Hobbes:]

Soit vn triangle rectangle ADC dont AD soit 5. CD soit 12. et AC soit 13. autour duquel Soit vne parabole Conique ABC, son Axe AD, sa base en ordonnée DC, soit diuiseè egalement AD en E, et soit tireè L'ordonneé EB Laquelle soit prolongée de sorte que EF soit egale à pg 236DC, soit prise eg moyenne proportionelle entre EF, et EB. tirant Ag, gc Je Dis que Ag plus gc n'est point egale à La parabole ABC.

Soit descrit le Cercle AJC dont le Centre soit dans l'axe AD prolongé. L'arc AJC sera hors la parabole ce qui est aisé a demonstrer comme a fait M.r hugenius dans la prop. 17. du Liure de Circuli magnitudine. Soient tirées les droites AK, CK qui touchent le cercle AJC en A et C. Soit tireé Kh diuisant egalement AC en h. Le point h sera l'Intersection de EF, AC, a cause des paralleles ef, DC, et de la Ligne AD my partie en E.

A Cause de la similitude des triangles DCA, hAK, comme DC à CA, ainsi hA à AK ou ainsi 2hA, à 2AK, c'est a dire ainsi CA; à AK plus KC. donc AK + KC sera Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux. donc Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAC + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAK + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxCK sera Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux qui sera plus grand que l'arc de Cercle AJC par La 9.eme prop. de hugenius de Circuli magnitudine.

Mais EB est √72. car AD est à AE comme CD.q à EB.q

Donc Eg.q est √10368 lequel est vn peu plus grand que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Mais AE.q est Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Donc AE.q + eg.q c'est a dire Ag q sera plus grand que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux

Et la droite Ag sera plus grande que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Or la Racine quarreé prochainement moindre du nombre Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux est Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Soit pris Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux ou Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux pour valeur trop forte de la droite eg, car le quarré quarré de Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux est plus grand que 10406. qui est plus grand que 10368 qui est la valeur de eg.q q.

Donc fg sera plus grande que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux. Mais [fc quarré]24 est Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Donc Fg.q + FC.q c'est a dire cg q sera plus grand que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux

Donc cg sera plus grande que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux

Donc Ag + gc sera plus grande que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux qui est plus grand que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux

Car Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux vaut Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux Et Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux vaut Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux

Donc Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux est plus grand que la valeur de Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAC + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAK + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxCK

Et partant si la parabole [>ABC] est egale à Ag + gc, elle sera aussi [>beaucoup deleted] plus grande que Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAC + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAK + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxCK.

Et par consequent [ag + gc sera beaucoup pl deleted] la mesme parabole ABC est beaucoup plus grande que l'arc AJC. ce qui est absurd. Donc la Construction precedente n'est pas vraye ce qu'il falloit demonstrer.

pg 237 Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux

Translation of Letter 76

Sir,

I wrote a long letter to you, perhaps six weeks or a month ago,1 which I sent to M. de Martel, asking him to convey it to you. No doubt he has done so, and you have seen that, obeying your request, I am suspending publication of my translation of your book De corpore. You will also have seen that I plan to dedicate this translation to you—something which I beg you to accept. I am so bold as to tell you that it is exact, and (if this is not boasting too much) that, according to those who have heard me read some extracts, it is a talented and felicitous translation, I shall give you just one example, the first sentence of your dedicatory epistle to the Earl ['M. le Comte'] of Devonshire (if that is not the correct translation of 'Comes Devoniae', please tell me what the French term is)(2). Here is the sentence: 'My Lord, I present to you this first section of the Elements of Philosophy, which I have long deferred after the publication of the third, and which I have at last finished; and I present it to you so that it may be in future a monument of your bounty to me and my respect for you.'3 You see, Sir, everything is there, and at the same time the turn of phrase is French, with no parenthetical or transposed clauses—which is one of the finer points of our language. But my translation is even more exact in the body of the work, since there I give a word-for-word rendering of everything; if I had found the dedicatory epistle's word 'benignitatis' ['of bounty'] there, I would have translated it as 'benignity' ['benignité'].

What I also told you in the letter which I sent to M. de Martel was that I would try to get the judgements of our geometers on your book. I shall find out more when I am in Paris; for M. Roberval,4 for example, and M. Pascal, who are afraid to write for fear of contradiction, will be happy to talk to me about it. However, M. Mylon wrote to me about it on two separate occasions, after I had admonished him about it. He pg 238sent me his first commentary the first time, and his second,(5) the second.6 But here is how I replied to the first one:

To his first and second points: these concern your definition, 'any two lines whatsoever, strait or crooked, as also any two superficies, are parallel; when two equal strait lines, wheresoever they fall upon them, make always equal angles with each of them'.7 In this definition, I say, one must understand that the lines falling on the parallel lines are intersected there. This is very easy to understand; and if need be one can put it as follows. 'Any two lines whatsoever, straight or curved, and likewise any two surfaces, are parallel, when any two straight lines between them, intersected anywhere, make equal angles and are of equal length.'

To his fourth8 point: that which he grants you only as a hypothesis, is in fact demonstrated by you, in that there cannot be such a thing as infinite hardness,9 And that which he claims Descartes expressed better in his section on the circle has made him fall into the absurdity which we find in part 2, section 34: namely that in this circular motion matter is infinitely divided. For in order for that to happen, the extremities of his circles would have to receive no impression whatsoever from the matter which moves between them—in other words, the resistance of those circles would need to be infinitely great.10

To his fifth point: the fact that one side of the parallelogram was omitted is a printing error. And because the word 'semissem' ['half'] occurred there twice, the printer mistook the first 'semissem' for the second, and jumped to the words 'nam haec … ' etc.11

To his sixth and seventh points: he is mistaken in saying that you take AJ as the accelerated part of the accelerated motion AH. On the contrary, you take AJ as traversed in uniform motion. What has confused him is his failure to distinguish between AB and itself. And when you write 'I say, that as AH is to AB, so will AB be to AJ,' you mean that AH (the entire length traversed by accelerated motion in the time AC) is to AB (the entire length traversed by uniform motion in the same time) as AB (taken as a length which is traversed by accelerated motion, and which is part of AH) is to AJ (taken as a length traversed by uniform motion), and so on.12

To his eighth point: if your theories contradict those of MM. Roberval and Descartes, then their theories contradict the truth.13 What you say about refraction is very well demonstrated.

To his ninth point: I do not have the wit to understand what your error may be in ch. XVIII. If you have made an error there (which I find pg 239hard to believe), then he should not worry about the fact that someone has exposed it: on the contrary, you yourself, Sir, will be very happy to see it exposed, since you are an intellectually honest man and a lover of truth. And finally I ask him to send me that so-called demonstration by M. Huygens.

So he did send it to me, enclosed in his second commentary. I received it by the last carrier, and without further delay I send it to you,14 so that you may take the trouble to think about it.

Otherwise, Sir, I must tell you that M. Mylon speaks about you with very high esteem and goodwill. Here is an extract from his first letter: 'It is said here that you have translated Mr Hobbes's "Physics" for publication; you will do a service thereby to the whole of France. I know that the universities do not approve too highly of this work, and I believe that the geometer Dr John Wallis, Savilian Professor in the University of Oxford, has used Aristotle to attack it;15 but neither you nor the author can be very worried by that. You are only claiming to please those who are free of prejudice and reason correctly. I have read this book by Mr Hobbes with very great pleasure, admiring everything in it (with the exception of the points which I list for you on the enclosed paper). Receive this as from someone who is very well disposed towards the author; and if I have made any mistakes, please do me the favour of demonstrating my errors.'

And in his second letter, when he says that in order to denigrate the best book in the world, one need only say that it contains some paralogisms, I see that it is indeed a sip of his goodwill that he would prefer it if your book did not contain the chapters 'Of the Dimension of a Circle' and 'Of the Equation of Strait Lines with the Crooked Lines of Parabolas and other Figures made in Imitation of Parabolas':16 he does not have the same opinion of those chapters as he does of the rest of the book, which he admires.

If there are any other judgements concerning you which I receive in letters, or learn about by other means, I shall also write to you about them in confidence. For I am sure, Sir, that you will accept everything I send you with as much goodwill towards me as I feel esteem and veneration towards you. Anyway, should you wish to tell me what error M. Zuylichem has committed (and I shall always assume that he is in error unless you tell me otherwise); should you wish to send me your sheets of De homine,17 which I should like to receive as they are printed, in order to work on them at the same time and have your three sections translated simultaneously; or, finally, should you wish from time to pg 240time to tell me your news—which I pray to God may always be good—please write to me, Sir, directly. For I have been told that the Paris post office sends letters from Bordeaux to London, and letters from London to Bordeaux. So please address your letters as follows: 'To M. de Chenevas,18 Director of the Post Office, for M. de St Hilaire, at Bordeaux'.

St Hilaire is a fictitious name which I have given to M. Chenevas because people were intercepting my mail. Addressed like this, your letters will be delivered on time; and I shall receive them with the feelings of unalterable esteem and friendship of one who is, in life and death,

  • Sir,
  • Your most humble and obedient servant,
  • du Verdus

Bordeaux, 1 March 1656

[enclosed: Mylon's objections to De corpore in Mylon's hand, with an annotation in du Verdus's hand, 'M. Mylon's first commentary';]

    On Mr Hobbes's Physics

  1. 1. XIV. 12: the definition of parallels must be supplemented.

  2. 2. Page 114, line 13: he reaches a false conclusion, because his argument is based on Ms definition of parallels.19

  3. 3. Page 114, article 13: this seems obscure to me.20

  4. 4. XV. 7: one can grant him, as a hypothesis, that 'motion or endeavour, whether in space which is empty, or in space which is filled, is propagated to an infinite distance'; but I think M. Descartes argues better when he says that 'that motion is propagated in a circle'.21

  5. 5. Page 128, line 6: here he constructs a parallelogram with only one side. It should go like this: 'or lastly, by a parallelogram, one of whose sides is a mean proportional between the greatest impetus (or the last impetus to have been acquired) and half that greatest impetus, and whose other side is the mean proportional between the whole time and half that whole time. For both these parallelograms are equal to one another, and severally equal to the triangle which is made of the whole line of time and the increasing impetus'.22

  6. 6. XVI. 17: the proposition is made in general terms, but the demonstration is true only in a particular case; or else I do not pg 241understand it. For in fig. 8, AB is a uniform length corresponding to the time AC. AH is the accelerated length corresponding to the same time AC. The accelerated length AI is taken to be part of the accelerated length AH; so one should also take the uniform length AO as part of the uniform length AB, and demonstrate that the ratio between the difference between the whole of AH and the whole of AB, and the difference between the part AI and the part AO, is the same as that between the ratio between the lengths AH and AB (or the ratio of the parts of those lengths, AI and AO) and the ratio between the times AC and AM.

  7. 7. XVI. 17: I do not understand this demonstration.

  8. 8. Page 114, line 4: his account of refraction is the opposite of that of M. Descartes and M. Roberval.23

  9. 9. XVIII: the general problem which he uses to equate straight lines with parabolas would be fine if it were true; but the trouble is, M. Huygens of Zuylichem has demonstrated that it is false. He is due to write about that from Holland, to Mr Hobbes. If you are curious to see this demonstration of his, I shall send it to you whenever you like.

Editor’s Note1.0[Second enclosure from Mylon]

Let there be a right-angled triangle ADC, of which AD is 5, CD is 12, and AC is 13. Round it, let there be a conical parabola ABC, with AD as its axis and DC as its base or ordinate. Let AD be bisected at E, and let the ordinate EB be drawn and continued so that EF is equal to DC. Let EG be the mean proportional between EF and EB. Drawing AG and GC, I say that AG plus GC is not at all equal to the parabola ABC.

Let the circle AJC be drawn, the centre of which lies on the continuation of the axis AD. The arc AJC will lie outside the parabola: that is easily demonstrated, as Mr Huygens has done in proposition 17 of his book De circuli magnitudine. Let the straight lines AK and CK be drawn, touching the circle AJC at A and C. Let KH be drawn, bisecting AC at H. The point H will be the intersection of EF and AC, because EF and DC are parallel lines, and because E is the mid-point of the line AD.

Because of the similarity of the triangles DCA and HAK, as DC is to CA, so HA is to AK, or 2HA is to 2AK; in other words so is CA to AK plus KC. Therefore AK + KC will be Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Therefore Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAC + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAK + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxCK will be Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux, which is greater than the arc of the circle AJC, according to the ninth proposition of Huygens's De circuli magnitudine.

pg 242But EB is √72; for AD is to AE as CD2 is to EB2.

Therefore EG2 is √10,368, which is slightly greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

But AE2 is Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Therefore AE2 + EG2, that is, AG2, will be greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

And the straight line AG will be greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Now, the closest square root lower than the number Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux is Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Let Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux or Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux be taken as slightly more than the value of the straight line EG, since the fourth power of Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux is greater than 10,406, which is greater than 10,368, which is the value of EG4.

Therefore FG will be more than 19/10. But [FC2]24 is Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Therefore FG2 + FC2, that is, CG2, will be greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Therefore CG will be greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Therefore AG + GC will be greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux, which is greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

For Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux equals Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux. And Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux equals Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux.

Therefore Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux is greater than the value of Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAC + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAK + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxCK.

And by the same token, if the parabola ABC is equal to AG + GC, it will also be greater than Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAC + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxAK + Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from BordeauxCK.

And consequently the same parabola ABC is much greater than the arc AJC: which is absurd. Therefore the foregoing construction is false: QED.

Letter 76 [20 February/] 1 March 1656 François du Verdus to Hobbes, from Bordeaux pg 243pg 244

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Notes

Editor’s Note
1 This letter does not apparently survive. The odd construction 'du mois' here may perhaps be a slip for 'deux mois' ('two months').
Editor’s Note
2 ⁓) omitted in MS.
Editor’s Note
3 The sentence in the 1656 English translation, which keeps more closely to the 'parenthetical and transposed clauses' of the original, is as follows: 'This first Section of the Elements of Philosophy, the Monument of my Service, & your Lordships bounty, though (after the third Section published) long deferred, yet at last finished, I now present (my most excellent Lord) and dedicate to your Lordship. '
Editor’s Note
4 Gilles Personne de Roberval (1602–75) was a mathematician of humble origins, who seems to have received little formal education before he reached Paris in 1628 and joined Mersenne's circle. He became Professor of Philosophy at the Collège de maître Gervais, Paris, in 1632, and in 1634 won the competition for the Ramus chair at the Collège royale. He remained at the Collège royale for the rest of his life, succeeding Gassendi in the chair of mathematics there in 1655, and in 1666 he was a founder member of the Académie royale des sciences. He published only two scientific books, the Traité de méchanique (1636) and Aristarchi Samii de mundi systemate (1644); Ms mathematical researches were concentrated on the geometry of infinitesimals, but he was secretive with his discoveries (some of which circulated in MS: see the entry for du Verdus in the Biographical Register) and was often involved in quarrels with other mathematicians over questions of precedence and plagiarism.
Editor’s Note
5 Contraction expanded.
Editor’s Note
6 The first commentary is printed as the first enclosure to this letter; the second contained the geometrical demonstration which is printed as the second enclosure.
Editor’s Note
7 De corpore, XIV. 12.
Editor’s Note
8 Du Verdus seems to have inadvertently omitted his reply to the third point.
Editor’s Note
9 Descartes, Principia, part 2, sect. 34 (A&T viii, pp. 59–60). Cf. the claim made in De corpore, XXII. 2 that the terms 'soft' and 'hard' 'are used onely comparatively; and are not different kinds, but different degrees of Quality' ('dicuntur tantum comparative, nec sunt diversa genera, sed diversi gradus qualitatis').
Editor’s Note
10 In Principia, part 2, sect. 33, Descartes argued that in a plenistic universe, every individual motion is part of a chain of motions which is ultimately circular: when one part moves from space A to space B, it displaces another part, and so on—with, at the end of the chain, another part moving into space A. Descartes allowed for the idea that the spaces might be of different sizes by arguing that this was compensated by the different speeds at which matter would move into them. (An analogy to this would be water moving at different speeds through pipes of different diameters.) In sect. 34 Descartes argued that matter must therefore be capable of travelling through an infinite variety of sizes of spaces, and that this implied that matter was infinitely divisible. Du Verdus's objection is obscure; he seems to be conflating the 'circles', i.e. circular motions, of Descartes's account, with the physical limits on the motion of the quantities of matter (in the water analogy, the sides of the pipe).
Editor’s Note
12 See De corpore, XVI, 17, and Mylon's sixth and seventh objections in the first enclosure to this letter.
Editor’s Note
13 For Descartes's theory of refraction, see the second discourse of his 'Dioptrique'. Roberval's views on refraction were not the same as those of Descartes (with whom, in any case, he tended to disagree whenever possible). In the posthumously published version of du Verdus's compilation of Roberval's teachings, Roberval criticized Descartes's theory of refraction and argued that the motion of light is slower in a denser medium (Roberval, Ouvrages de mathématique, pp. 13–14: for the history of this work, see the Biographical Register, 'du Verdus'). In his notes for a treatise on refraction, however, Roberval avoided giving any physical explanation for the phenomenon (BN MS f.fr. n.a. 5175, fos. 14–21), And in the treatise which he planned to publish as a companion to Mersenne's two treatises on optics, L'Optique et la catoptriqtte, he explicitly refused to offer a physical theory of the transmission of light (BN MS f.fr. 12279, 'Liure troisieme de la dioptrique', fos. 3, 8r).
Editor’s Note
14 See the second enclosure to this letter.
Editor’s Note
15 The reasons for this claim are obscure.
Editor’s Note
16 Chs. XX and XVIII respectively.
Editor’s Note
18 François Chenevas is mentioned in a decree of Mar. 1665 as one of several joint holders of the four offices of maître des courriers at Bordeaux (see Vaillé, Histoire générale des postes françaises, iii, p. 275).
Editor’s Note
19 The reference is to XIV. 12, coroll. 5, first sentence. Hobbes cited these comments in Six Lessons: 'The same [sc. criticism] was observed also upon this place by one of the prime Geometricians of Paris, and noted in a Letter to his friend in these words Chap, 14 Art. 12 the Definition of Parallels wanteth somewhat to be supplyed. And of the Consectary, he says it concludeth not, because it is grounded on the Definition of Parallels' (p. 25: EW vii, p. 255).
Editor’s Note
20 XIV. 13.
Editor’s Note
21 See n. 10 above. The words attributed to Hobbes here are a summary of the argument of XV. 7, rather than a direct quotation.
Editor’s Note
22 This is adapting XVI. 1, coroll.; the translation here adapts Of Body, p. 161.
Editor’s Note
23 Mylon appears to have copied out the wrong reference for this objection. Hobbes's account of refraction is in XXIV, pp. 215–22.
Editor’s Note
Commentary on the second enclosure. Hobbes's assertion appears to be that the length of the arc ABC of the parabola equals AG + GC, where G is defined by EG 2 = EB . EF , and E is the bisector of the axis AD. This is refuted here by Mylon. Two propositions of Huygens are used: (1) that the circle through A and C with centre on the axis AD lies outside the parabola, up to the point where the circle and parabola meet at C, and (2) that the length of the arc AC of this circle is less than AK 3 + CK 3 + 2 AC 3 . Combining these propositions one has:
length of arc of parabola ABC < length of circular arc ABC < AK 3 + CK 3 + 2 AC 3 .
Using Pythagoras' theorem, Mylon then shows that AG + GC > AK 3 + CK 3 + 2 AC 3 for the particular lengths AD = 5 , CD = 12 , and hence that Hobbes's assertion is false.
Simple proofs of Huygens's propositions are as follows: Choose AD as the X-axis and AK as the y-axis, and let the equation of the parabola be y 2 = 4 ax . Suppose D is the point ( x c ) 2 + y 2 = c 2 . The coordinates of C are therefore {b,√(4ab)}. Suppose the centre of the circle, which lies on the x-axis, is (c,o). Then its equation must be of the form ( x c ) 2 + y 2 = c 2 , and, since it passes through C, we have c = 2 a + b 2 . The circle is therefore given by y 2 = ( 4 a + b ) x x 2 , and it will lie outside (to the left of) the parabola if ( 4 a + b ) x x 2 4 ax , i.e. if o x b . This proves (1).
To prove (2), suppose the angle subtended at the centre of the circle by its arc AC is 2θ. Simple trigonometry shows that the tangents AK and CK have length r tan θ, and the chord AC has length 2 sin θ, Since the length of the arc is 2rθ, the proposition amounts to tan θ + 2 sin θ 3 θ . Equality holds at θ = o , and it is easy to check that the derivative of tan θ + 2 sin θ 3 θ is positive in the range o < θ < π 2 . G.M.
Editor’s Note
24 fc MS.
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