Newton's 'Principia' and Its Legacy || The 'Principia' and Continental Mathematicians

The 'Principia' and Continental MathematiciansAuthor(s): E. A. FellmannSource: Notes and Records of the Royal Society of London, Vol. 42, No. 1, Newton's 'Principia'and Its Legacy (Jan., 1988), pp. 13-34Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/531367 .

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Notes Rec. R. Soc. Lond. 42, 13-34 (I988) Printed in Great Britain

THE PRINCIPIA AND CONTINENTAL MATHEMATICIANS

BY E. A. FELLMANN

Euler-Edition, Arnold-B6cklin-Strasse 37, CH-4oSl Basle, Switzerland

Elephants are always drawn smaller than life -but fleas bigger.

Swift

I

The coincidence alone that I am from Basle might cause the audience to expect that I should be able to tell you something new about the

Bernoulli dynasty in respect to today's topic: unfortunately I have to disappoint you right away. The reason for this is that all the existing material has been largely exhausted by your own specialists. I am thinking mainly of the last three volumes of The Correspondence of Isaac Newton (2)*, which have been so excellently edited, and of the masterly eight-volume edition of The Mathematical Papers of Isaac Newton (3), but also of a whole series of monographs and treatises by various scientists covering our subject. Thus there is not much new that I could offer you, indeed, absolutely nothing that is new to the specialists in the history of exact science among you. I shall therefore try to give merely a synoptic survey of the reception of Newton's Principia by some important continental scientists. A lecture, however, does not allow an in-depth treatment of technical details, which I can touch on only lightly in a few specific cases.

In his very impressive Ninth Gibson Lecture (4) of October 1969, Professor Whiteside states that:

...in Newton's own lifetime only a handful of talented men working without distraction at the frontiers of current research-the Dutch scientist Christiaan Huygens, the German uomo universale Leibniz, the French priest Pierre Varignon, the Huguenot expatriate Abraham de Moivre and Newton's most able editor Roger Cotes- had, each in his own way, achieved a working knowledge of the Principia's technical content.

On the whole I shall follow this 'programme' during the first part of my lecture, yet I will include the 'fox' Johann Bernoulli but not de Moivre. In our context we cannot consider him as a 'continental mathematician'

* Numbers given in this form refer to entries in the references at the end of the text.

[ 3]



14

because at about the time Newton's opus summum was published he went to live and work in England where he remained until the end of his long life (5). Likewise I do not consider Nicolas Fatio de Duillier, even though he was born in Basle, as continental.

II

Let us begin with Christiaan Huygens, the oldest of all serious Principia recipients on the Continent. The essence of the scientific relationship between Huygens and Newton is thoroughly treated in volume 4 of Whiteside's Mathematical Papers, in Rupert Hall's study of 1976 (6), in the third volume of Newton's Correspondence, in the great Huygens's edition (7), and in some excellent articles in the Huygens symposium volume of 1979 (8). Now Professor Hall (6, pp. 45-47) has reminded us:

...that Huygens was thirteen years older than Newton, that Huygens had already a wide reputation when Newton was still a schoolboy and that, when Huygens read the Principia mathematica [in the winter of I688], his scientific style was well established...,

at the latest since the appearance of his masterpiece Horologium oscillatorium (io), to which Newton referred more than once. (Newton's own copy, bearing Huygens's dedication in his own handwriting, is now in the United States; Marie Boas Hall has most felicitously described Newton's first reaction to Huygens's principal work (II).)

It is generally known that by his conservative adherence to a strictly geometrical conception of infinitesimal mathematical problems, Huy- gens was prevented from following up the decisive innovations in the infinitesimal calculus in the sense of Newton or Leibniz, and he similarly failed in the field of dynamics: by clinging to the mechanistic principles underlying his conception of matter and the universe, Huygens was kept from reaching the main results Newton presented in his great book. This notwithstanding, Huygens immediately recognized the colossal significance of the Principia, even though he emphatically objected to Newton's use of an attractive force as a fundamental explanatory principle. Huygens himself demanded a deeper mechanistic explanation of the occurrence of such attractive forces, although in his own early studies the mechanistic point of view had been of importance only as a source of inspiration rather than as an explanatory principle. And in fact the Horologium oscillatorium, too, is completely free of mechanistic philosophy (12). To evaluate properly the important role that the (quasi Cartesian) vortex theory played in Huygens's conceptions-one need only recall the discovery of Saturn's ring or the refractive properties of



15

Iceland spar-I must refer you to Eric Aiton's remarkably detailed book here (I3).

In the summer of 1689 Huygens visited England. This stay has become famous, above all, because it produced two results of the utmost

importance: (i) Since 1678 his Traite de la lumiere had been left unfinished. But in

1690 Huygens finally published it, with an addendum entitled Discours de la cause de la pesanteur. I fully agree with Mrs Hall's statement: 'Without that visit to England it seems probable that he [Huygens] would never have had the energy to complete these works' (II, p. 79);

(ii) The 'first and original neo-Cartesian', as Rupert Hall has called him, only came round to accepting the elliptic orbit for planets in the sense of the first and second Keplerian laws by studying the Principia and

being in direct contact with Newton. At a first glance this might seem

astonishing, whereas on closer inspection one realizes that in the I7th century the reception and acceptance of these two laws (which intrinsically are so closely interrelated) ensued in a slow and hesitant fashion, even in the case of Newton himself, as proven convincingly by Whiteside in his study 'Newton's early thoughts on planetary motion: a fresh look' (I4).

III

After these first comments let us turn to Leibniz now. I shall devote some more time to this most spectacular of all persons for several reasons. First of all, this universal genius is the 'show boy' of continental science of his century and he certainly imparted more to the infinitesimal calculus than merely its currently customary form. Furthermore, through his multifarious efficacy and his abundant generosity in communications, he formally was also instrumental in the break-through to the new analysis via the 'school' ranging from the Bernoullis to Euler, Clairaut, d'Alembert and their followers, thus putting them into a position to forge an adequate and elaborate vessel for the precious contents of Newton's physics. After all, we have a fascinating document at our disposal, namely the copy of the first edition of Newton's Principia bearing marginal notes in Leibniz's own handwriting. It was quite a sensation in 1969 when this copy came to light, but I have a feeling that the book (I5) written about this is known to a few specialists only and has not reached wider circles.

At all times the historians of exact science have known that Leibniz did read the Principia, but there was no documentation whatsoever as to when, where and how he did so. The letter that Leibniz wrote to Huygens in October 1690 (7, t. IX, 521; I6, Bd.6, I89) contains one of



the few precise hints: 'Having given intensive thought to Mr. Newton's book which I saw for the first time in Rome, I admired the number of beautiful subjects which he raises therein...'. Later Leibniz repeatedly protested that he had not seen the Principia itself before the publication of his Tentamen (17) (I shall come back to this point shortly), but solely the excerpt of twelve pages that had appeared as an anonymous review of the Principia in the June I688 issue of the Acta Eruditorum. (Most likely the anonymous author was not the editor, Mencke, himself, as presumed by I. B. Cohen in his Introduction (I8, p. 155), but Mencke's son-in-law, Christoph Pfautz (1645-1711), asJ. E. Hofmann (d. 1973) once told me during a personal conversation.)

Leibniz's Tentamen de motuum coelestium causis appeared in February 1689, almost two years after the Principia. By means of his circulatio harmonica and the sollicitatio paracentrica, roughly corresponding to Newton's vis centripeta, Leibniz reaches similar conclusions in this paper regarding planetary motion as the great Briton, whom, however, he does not mention with one single word! Today we know the reasons: of course, it has absolutely nothing to do with the quarrel about the invention of the calculus; as a Fellow of the Royal Society Leibniz was entitled to a copy of the Principia. Still, he did not receive it at once as he was travelling from autumn 1687 until June 1690. During his 'grand tour' he settled down in Rome for a while, establishing his headquarters there from April to November 1689 (24, 25). It was only there that he obtained a copy of Newton's Principia.

Back to the Tentamen now, the substance of which has been thoroughly analyzed and presented in a modern way by Aiton in a number of works (I9-23). As in the case of Leibniz's first outline of the differential calculus, i.e. the Nova methodus pro maximis et minimis published in 1684, the Tentamen was disfigured by printing errors and other mistakes. Owing to the severe criticism that Huygens expressed in a letter with respect to the Tentamen as well as to some corrections made by Varignon, Leibniz felt compelled to insert a public emendation in the October 1706 issue of the Acta Eruditorum. Much later Newton provided a direct commentary on the 1689 edition of the Tentamen, probably in connection with his reply to John Keill's letter of 2 May I714. These Notae are printed in the fifth volume of the Correspondence (2, no. Io69a) where good guidance is offered in elucidatory comments, which inter alia correct Newton's harsh verdict that 'Leibniz did not yet (in I689) understand the differential calculus in so far as second differences come under consideration' (no. I5).

Alas, I am digressing too much. Let us return to Rome in late summer of I689 and take a closer look at how Leibniz actually read the Principia for the first time. To avoid raising your hopes for spectacular findings,



17

PHILOSOPHIE NATURALIS

PRINCIPIA

MATHE MATICA

Autore S. NEYWTON, Trin. Coll. Cantab. Soc. Mathefeos Profeffore Lucafiano, & Societatis Regalis Sodali.

IMP RIMA TU R S. P E P Y S, Reg. Soc. PR ISES.

Julii 5. x686.

LO NDIN I, Juffu Societatis Regie ac Typis yofepbi Str,eaer. Proftant'Vcna-

les apud Sam. Smith ad infiynia Principis IWllUi in Ceimitcrio D. Panli, aliofq; nonnullos Bibliopolas. Amno MDCLXXXVII.

FIGURE I. Title page of Leibniz's Principia.

I can tell you right away that grosso modo his annotations prove to be disappointingly insignificant for the specialists in the history of exact science and did not measure up to the exalted expectations that one had hoped to see fulfilled by a genius of Leibniz's calibre. All in all, he made marginal notes on only 25 pages of the Principia, and marked only 17

:l~~~~~~~~~"



i8

C 4 1]

Prop. IV. Theor. IV.

Corporut qude diverfos cirtnlos equabili motu de/cribunt, Vires cen- tripetas ad centra eorlundem circulorulm tendere, ene intcr fc ut arcuw fi ,nlmd defcriptortrn quadrata applicata ad circuloruNm radios. Corpora B, b in circumferentiis circulorum B D, bd pyran-

tia, Imlul defcribant arcus B D, bd. Qo niaom tola vi iniita dc- fcribercnt tangentcs BC, b c his arcubus qu.csf 1fiftu1m eft quod vircs centripr t lunt que . , ,. t-

perpetuo rerrahunt corpora de 't,' ,, c K tangenribus ad circumterentniaf,' , \ . d

circulorum, atq; adco hx funt:;.",^ , t ad inviccm in rarione prima fpa-'"`'v/f : j ~ \ tiorum nafcentium C D, c d: ten- / \ dunt vcro ad centra circulo- I rum per Theor. II, propterea quod aree radiis defcript.x- p/ / nunrur tcmporibus proportiona- \/ Ics.. Fiat ligura .kb. figurae D s ,. CB ilmii.'s, &-per Lerrna'V " -''^" ¥'": -^ . n liun ola C D erit ad ];neolani ut _ ^ , A t / 'ariis^B D ad arcuni b t: ncec non, per Lemma x , lineota nanc/ns a$E:^ tl; ad lincolam naiccntem dcc ut bt qiad. ad bd quad. & ex z- .~

B

quo lineola nafccns DC ad lineolam nafcentem dc ut B D xbt ,aO~%

ad b d q;tad. fcu quod perinde eft, ut BDxt ad '-ItI a-d Sb Sb

oh, , . Zbt BD B D quad. b d rv.. dcoq, ( ob xquales rationes - &-. ) , t B qa d b '"

C. E. D. , . c Corol. i. Hinc vires centripetae fun t ut velocitatum cudrta,

applicata ad radios circulorum. r," .v ,,^ ec ' ". · ,-, Corol. 2. Et keciplroc ut IuIarata r'my um,velo~oulr n a r-

v, :.,s' ,~a.,-.t-.u . ,,' ,( . r e .c .f,-;', -*. 4 {e-.' ,,, t , ,: '(~ - . "-"-:r ;,; _* b1i-

FIGURE 2. Leibniz's annotations on page 41.

more pages by underlining or otherwise. The outcome is sparser still in the second book of the Principia where merely a few minor marginal notes can be found on a total of six pages only, with the exception, however, of page 254 (which I will show you), and pencil marks in three

places only. As pars pro toto I want to show you at least some pictures of



I9

41 A 1 vires centripetae mobilium ( et )

unicum circulum aequabiliter descri- bentium, cd

2 et Kt sunt in duplicata ratione velocitatum bd, bt seu be, bK

3 Vires centripetae Kt, CD duorum mobilium ] et 9 duos circulos eodem tempore

4 absolventium, seu velocitate radiis

proportionali, sunt radiis propor [-] 5 tionales sb: sB 6 Centrip. vis mobilis 0

ad ) ::v2:(v)2 7 vis ) ad 9 ::sb:sB 8 Ergo vis C ad 9:: v2sb: (v)sB 9 et (v) ((v))sb:sB

10 vis 0 ad 9 v2sb:sb2((v))2sB2 11 seu vis ( ad 9 :: v: b:((v))2:

41 B 1 suspectum hoc Lemma generale. in

circulo tamen res 2 vera speciali ratione, quia 3 abscissae in 4 circulo sunt 5 ut quadrata 6 chordarum

41 C (cancelled)

1 Vires centripetae ad radios

circulorum diversorum aunt ut quadrata temporum. Vires centripetae ad radios

2 diversorum circulorum ut temporum

periodicorum quadrata recipro[ce]

41 D 1 vires centripetae ut vel[ocitatum]

qu[adrata]: rad[ios.] circumf[erentiae] seu rad[ii] ut temp[ora]

2 period.[ica] et vel[ocitates]. Ergo Temp. per. ut radius: vel. et vel. ut circumf. rad. :temp. per.

3 circumf. ut rad. Ergo vires centrip. ut rad. quadr.: temp. period. quadr. rad.

4 seu vires centrip. ut rad.: temp. per. quadr.

5 Si vir. ceutrip. ut ([ d qull : rad. qu. erunt temp. period. quadr. ut cub. rad. Ergo rad. qu:vel. qu. ut cub. rad. Ergo radii ut

6 1: quadrata velocitatum. Itaque perinde moventur planetae, si in mediis distantiis circulari

ponerentur, ac si projecti essent 7 Iviribuol velocitatibus in subduplicata

reciproca ratione distantiarum a sole, et interim jessent geeae ] in duplicata impellerentur ad centrum a gravitate

FIGURE 3. Transcription of Leibniz's annotations on page 41.

the 'holy original' with a few words of comment. (Also this may be an appropriate moment to interpolate a personal remark: I would like to thank Professor Whiteside once more for his spontaneous and active cooperation in the adventure of the transcription and evaluation of Leibniz's marginal notes during the winter of 1969/70. His good advice was indeed very helpful and greatly appreciated.)

It can be seen from the title page (figure I) that Leibniz's copy was a 'three line issue'; the page-numbers visible at the bottom were written in ink by Leibniz himself. In some aspects one of the nicest marginal notes is the one that contains the fourth proposition of the first book where Newton refers to Huygens's well-known formula for circular motion (a = v2/r).*

* For the transcription, see figure 3 and reference 15 (with commentary).



20

b c K

B I

S FIGURE 4.

The handwritten passages (figure 2) are marked A, B, C and D. B contains an invalid critical remark against Newton's Lemma XI; A, C and D are mere transpositions of the Principia's text (IS, pp. 73-75). They show that Leibniz did not have the slightest difficulty here in mastering Newton's arguments in detail.

The only annotation of real interest on this page as marked by Leibniz is the sign 'delta' in the upper right hand corner. One could be tempted to interpret this in the meaning of 'deleatur', whereas for Leibniz it always stands for 'destilletur' (reminiscences' of the realm of alchemy ?) with the implication: 'This still needs to be thought over thoroughly'. You see, the 'delta' refers to that part of the text underlined by Leibniz: '[BC, bc]...his arcubus aequales'. When the mass-points execute circular motions (see figure 4) and by the action of force are to be diverted out of their tangential movements BC, bK towards the centre of the circle S, then, according to Newton's concept of centripetal force, the differential parts of the paths of attraction cd, Kt of course are not straight lines but infinitesimal parts of circle involutes that in the points c and K are perpendicular to bcK. That is to say that the equality of the arcs bd = bc, respectively bt = bK, alleged in the Principia, applies in the strictest mathematical sense and not only in approximation. In this respect Newton's printed figure on page 41 indeed is an enormous simplification, and without doubt he was very well aware of this because in his pre- Principia manuscript, De motu corporum, he actually has approximated



21

E 327 3 em eundem CB generatur, minus refiftitur quam folidum prius; fi mode utrumque fecundum plagam axis fui AB progrediatur, & utriufque terminus B precedat. QO.am quidem propofitio- nem in conftruendis Navi- bus non inurilem futuram - effe cenfeo.

Quod fi figura DNFB ejufinodi fit ur, fi ab ejus A M i R

pun&o quovis N ad axem A B demittatur perpendi- , /

culum NM, & a pun&o E dato G ducatur re&a G R quae parallela fit re&z figuram tangenti in N, & axem produ&um fecet in R, fuerit MN ad GR ut G R cub. ad 4BR xGBq: So- lidum quod figurae hujus revolutione circa axem A B fa&a defcri- bitur, in Medio raro & Elaftico ab A verfus B velociffime mo- vendo, minus refifletur quam aliud quodvis eadem longitudine & latitudinedefcnprpumSolidum circulare. r'"t

Prop. XXXVI Prob. VIII.

Invenire refiftentiam corwris Spberici in Fluido rara & Elaftico velocifJne progredientis. ( Vide Fig. Pag. 32 5.)

Defignet A1BKIcorpus Sphbaricum centro C femidiametro CA defcriptum. Producatur CA primoad S deinde ad R, ut fit AS parstertia ipfius CA, & CR fit ad CS ut denfitascorporis Sphae- rici ad denfitatem Medii. Ad C R erigantur perpendicula PC, R X centroque R & Afymptotis C R, R X defcribarur Hyper- bola quaevis P VT. In C R capiatur CT longitudinis cujufvis, & erigatur perpendiculum TV abfcindens aream Hyperbolicam P CTV, & fit C Z latus hujus area applicata ad re&am PC. Di- co quod motus quem globus, defcribendo fpatiun C Z, ex refi- ftentia Medii amittet, erit ad ejus motum totum fub initio ut lon-

gitudo CT ad longitudinem C quamproxime. Nam



22

r ' 54 Tangentes, & fimilia peragendi, qua in terminis furdis exque ac in rationalibus procederet, & literis tranfpofitis hanc fententiam involventibus [ Data axquatione quotcunq; fluentes quantitates involvente, fluxiones invenire, & vice verfa ] candem celarem re- k:ripfit Vir Clarifl;.nus fe quoq; in ejufmodi mnethodum incidiffe, & methodum fuam communicavit a minea vix abludentem preter- quam in verborum & notaruni formulis. Utriufq; fundamentum continetur in hoc Lemmate.

Prop. VIII. Theor. VI.

Si corpus in Medio ,niformi, Gravitate uniformiter agonte, re&da af- ceridat vel defcendat, & fpatiurn totrnm defcriptnrn diftinguatur in partes ,equales, inq, principiis fingilarum partilir ( addendo refiftentiamu Medii ad vimi grsivitatis, quando corpus afcendit, velfiibducendo ipfam quando corpuo defcendit ) colligantur 'vires abfohftx;3 dico quod vires iUx abfolute funt in progrefione Geo- metrica.

·i a Exporratur eiitti vis gravitatis per datam lineam A; refiften-

tia per lineam indefinitain AK; vi.abfoluta in dcfccnf~ corporIs, : A; . j,nru,,n

x Kt:" per differeentiai n K C_ vclcit,as corporis per lineam Al ( quz.e-:-

fit media proportionali inter AK & AC, idcoq, in dinidiara ra-r . tione refiftentix) incremeDtum refiftentiz data temporis particu-,' la faEtum per lincolaip K L, & contemporaneum velociratis incre-, menturnmper lineolamP OQ.; & ccntro C Afymptotis re6ianeu1 is - CA, C H defcribatur Hyperbola quzvis B. .'., ere&tis perpendicu-r li A B, KN, LO, KP Q S occurrens in B, N,, R, S. Quo-u Po nium A KIcft ut A,4' q., erit hujus momentum K L ut illius mo- e . nientumn 2 4 f Q__ id eft ut Al in K C. Na, velocittatis incre-1t'.> mentuni Po, per motus Leg.,i. pr9portionale eft vi generanti.`,': K C. Componatur ratio iius J mcu,ratione ipfius K N, & Ect re&anguumr'"I x K N ut A. x K Qx g N; hoc eft, ob da- tm rc&angulum K C x K N,ut A P. Atqlui arcs Hyperbolic:x / ,,/V da.6,1- -ov . ~__ r' . ,, Jv i

L i..-- -p: a-A KNrFr

q-(4r t ,

'I J a Ae ,f a ' gAr e7 ' _,

FIGURE 6. Leibniz's reworking of Proposition 8, Book II.



23

254 A (left hand margin) NB vereor ne subsit error, nam, cum velocitates sunt in dimidiata ratione resistentiae supponendum est temporis incrementa non minus quam gravitatis impressiones esse aequales. Adde sign. NB, p. 257

254 B (right hand margin) Si fuisset v = gr fieret dv = g dr = = g-r. Ergo dr g: g-r = dt. Et fiet t = dr g:g-r jam v = ds. [Ergo fiot 1 = gr et r =

g-gdr Ergo fiet V = gg- g2dr Ergo ds gg l-dr et | ds =t-r, nam g = dt = 1. aliter: dr =dv: g et dv = dds Ergo ds =

frl -dds. Ergo s t-ds.

254 C (foot of page) 1 KN =aa:g-r 2 constans g. resistentia r. conatus

absolutus -r = dv vv = gr. 2vdv = gdr

3 2v = gdr: g-r = ds 4 Ergo 2vdt = gdr: g-r. jam vdt = s.

Ergo s = gdr: g-r seu s existentibus 5 logarithmis, g-r seu incrementa

velocitatis sunt numeri. quoniam autem dt

6 est constans, hine, ob 2v = ds gdr: g-r

7 et v = g, fit 2dt = gdr: g-r gr seu 2t= gdr: g-r ygr

FIGURE 7. Transcription of Leibniz's annotations on page 254.

the arcs cd, Kt by the chords cd, Kt, which are inclined towards bS by one third of their centric angles. It is possible that Leibniz did not feel like 'fathoming all depths' at the first time of his reading the Principia or perhaps he did not devote too much time to it right away-the indication of doubt manifested by this 'delta' certainly speaks in his favour.

I would like to give you one more example of how Leibniz read the Principia. As you may know, in proposition 35 of the second book Newton gave the differential equation of the meridian curve of the solid of revolution of least resistance, however, without derivation or proof.

1 2 3 4 5 6 7 8 9

10

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23



24

Figure 5 shows page 327 of the first edition of the Principia with a marginal note written by Leibniz. Let us now in the usual way imagine in C a system of Cartesian coordinates and let us call the 'curve of least resistance'f(x), where the point Nhas the coordinates N(x/y), and so on. Now let us write for the first derivative y' = f'(x) = p and let us put this into Newton's equation, then there results the differential equation of the fourth degree.

p4 +4yp3 +2p2I = O,

for the solution of which Euler has given the representation in parameters (26) as follows:

x = k 4 + +lnp +k2, y = k1 p3 _4p p~ P

In his comments in Mathematical Papers volume 6, Whiteside has stated that 'the immediate reaction of Newton's contemporaries to this scholium on its publication in the 1687 Principia was one of nearly total incomprehension'. (Newton's own derivation appeared in print for the first time in the third volume of the Correspondence (2, pp. 375-377).) Whereas Huygens was the only one who was able to reconstruct more or less Newton's suppressed analysis (3, vol. 6, 466n), Leibniz allowed himself to be provoked to the following short, obscurely irresolute note: 'investigandum ex isolabis facillime progrediens', or as Whiteside preferred to read it: 'investigandum est isoclinis facillime progrediens'; translated: 'It can easily be found by means of the isolaba' (or isoclines, D.T.W.).

Indeed, one is in doubt as to what Leibniz might have been thinking at that moment; first he wrote isoperimetris, then crossed it out and wrote isolabis (my reading) over it, a made-up word that he invented himself. Up to now I personally have come across no document showing that he possibly could have tried an analytic solution of'Newton's problem' as it is called in today's literature about the calculus of variations. Nevertheless, Leibniz's rather isolated and vague marginal note can be taken as an indication that, alerted by his inborn instinct, he might at least have sensed in Newton's scholium the historical break-through-in dubio pro reo.

Now, let us quickly look at page 254 of the Principia that Leibniz annotated so impressively (figure 6). I admit it does not offer anything new regarding Newton's Proposition 8 of Book II, which treats the up- and-down movement of a body in a resisting medium; however, Leibniz's marginal notes show that he did not find it particularly difficult to convert the substance of Newton's text into his own mathematical form.



25

At the beginning I warned you of the rather disappointing outcome of Leibniz's first confrontation with the Principia, although the commentator's tendency to 'fault-picking', as Whiteside once called it, might to a certain extent account for that. To be fair, one should not fail to imagine the historical background and the conditions under which Leibniz first read the Principia; in Italy, and especially in the Vatican City, Leibniz was occupied with countless important activities of a political, scientific or theological nature and he was at all times intent on making the acquaintance of all kinds of interesting personalities. Also some traces suggest that our philosopher may have read the British scholar's masterpiece in bed mainly, perhaps before falling asleep! (I give the reasons for this hypothesis in my book (15, p. 121).)

However this may be, it is a manifestation of human inadequacy always to expect only the greatest of thoughts from the greatest of minds. Yet in Leibniz's case another factor come into play. I have a feeling that it took Leibniz quite some time to approach this 'book of books' with a different attitude, mainly aroused in the course of his later correspondence with Johann Bernoulli and his reading of the latter's 'Principia critique' of I7Io/I I, which stimulated him towards fathoming some of the depths in more detail and inspired in him an ever-growing respect for Newton as well as for the quarrelsome man from Basle.

IV

From Leibniz it is but a stone's throw to the Dutch mathematician, physician and philosopher, Bernard Nieuwentijt (I654-I718). I do not want to leave this truly interesting personality out of my summary, even if this can only be at the expense of my own compatriots. However, they have been immortalized in such prolific scientific writing and, after all, their influence has been well acknowledged in those great British editions mentioned earlier on (2, 3), whereas this burgomaster of Purmerend has remained less known by far, although at the threshold of the i8th century he made history not only as one of the first scientific popularizers but equally as a mathematician in his own right (28). Hans Freudenthal, the well-known mathematician from Utrecht, confirms that Nieuwentijt with his wide spectrum of knowledge was up to date with all natural sciences of his time. From the standpoint of moder historiography Nieuwentijt has not yet been sufficiently examined in his capacity as a mathematician, apart from the (as yet unpublished) studies of Beatrice Bosshart in Basle.

In I695 Nieuwentijt sent two of his works to Leibniz, i.e. his Considerationes... (29) and the Analysis infinitorum... (30) in which he

rejected Leibniz's approach to analysis. Briefly put, Nieuwentijt did not



26

admit infinitesimals of a higher order. His method consists, in modern terms, of adjoining to the real field an element e with e2 = o. Leibniz immediately replied with a not altogether convincing treatise (3 ) in the Acta Eruditorum, which Nieuwentijt at once parried in 1696 with his Considerationes secundae... (32). Finally, Jakob Hermann was instigated to take the field and fight for the Leibniz-Bernoulli calculus (33), an undertaking that brought him membership of the Berlin Academy as a reward.

However, in our context interest above all centres on Nieuwentijt's relationship to Newton's Principia. No doubt the former Spinoza adherent and 'ex-Cartesian' did read it, yet the depth of his penetration cannot be determined from his popular book of 1714 (34), which was translated into English by E. J. Chamberlayne in 1718 and later also into German by J. A. Segner. Nieuwentijt only used some physical data out of the Principia's third book to support his physico-theological argument for the existence of God. (It is indeed an interesting aperfu that Voltaire, too, thoroughly studied Nieuwentijt's book, marking it with many marginal notes.)

On the other hand there are several concrete mathematical references to the Principia in Nieuwentijt's Considerationes... (29). For instance, he begins his Sectio secunda with Newton's Lemma I and emphatic praise of its 'illustrious author'. Nieuwentijt's Analysis infinitorum... (30) can be called the first comprehensive book about the calculus as it appeared one year before de l'Hospital's Analyse des infiniment petits (35). To begin with Nieuwentijt, like Newton, gives a handful of lemmas of which the 2Ist (out of 52) is identical with Newton's Lemma VI about the angle of contact. In the second chapter, specifically mentioning Newton by name, he extracts the root of a binomial, strictly following Newton's example. For our consideration, however, the most interesting place might be in chapter 8. Here Nieuwentijt deduces the rules of differentiation for the sum and difference, the product, the quotient and the power of two variables-symbolis Leibnitianis in his terminology. But this is followed immediately (in paragraph 5) by a sentence that would be of considerably greater importance if then, in I695, the quarrel about the invention of the calculus had already been in progress. I translate his Latin:

If, however, we want to call the variables fluents, and their infinitesimals fluxions, moments, time increments, the demonstration of the calculations the famous Newton made in his profoundly erudite work Philosophia naturalis principia mathematica comes out of the very same source.

Summing up, it should be understood that Nieuwentijt, who did not think too highly of the so-called 'pure mathematics', rather was in favour of Newton's physical conception than against it.



27

V

We now turn to the French priest, Pierre Varignon (I654-I722), in our chronological sequence. Our interest in this most influential scientist at the Paris Academy is not so much for his role as conciliatory mediator in the priority dispute between the Continentals, mainly Johann Bernoulli, and Newton, but in his capacity as an early recipient of the Principia and as 'mechanician'. Varignon, for whose biography we must mainly depend upon Fontenelle's Eloge, was the first Frenchman to interpret the mechanics of Newton's Principia to his countrymen (21) and one of the first continental mathematicians to interpret Newton's work in terms of the differential calculus proper, having received his first instructions in the calculus from Johann Bernoulli during the latter's visit to Paris in 1692 as the guest of Guillaume de l'Hospital, who, as is well known, had to pay Bernoulli rather exorbitantly for his private lessons.

In a series of papers presented to the Academie Royale des Sciences in Paris (see reference 36) between 1700 and 1706 Varignon, with explicit reference to Newton and Leibniz, used the differential calculus to develop a general theory of central forces. Taking into account the respective achievements ofJakob Hermann and Johann Bernoulli, Aiton has in several important works analyzed Varignon's substantial contributions mainly to the inverse problem of central forces, in a masterly way. There is a direct path leading from Varignon's transposition (from 1700 onwards) of the most important propositions of the Principia mathematica into symbolism conformable to Leibniz's own, to Roger Cotes's second edition of the Principia in 1713 and to Jakob Hermann's Phoronomia (37), which was part of the mathematical 'mother's milk' upon which Leonhard Euler was nourished in Basle until 1727, when he left his native town never to return to it again.

Varignon's first major work, Projet de nouvelle me'chanique (38), appeared 300 years ago, almost simultaneously with Newton's Principia with which, of course, it must not be compared. Still, French scientists are of the opinon that in France more interest was aroused on behalf of the Projet exactly because of this simultaneous appearance of the two books than actually would have been the case without the famous English book (36, p. 584). However this may be, at that time Varignon was mainly interested in the application of his principle of the composition of forces in statics. In his treatises of the years 1692 and 1693 (39) there are some expressions, such as 'premieres et deuxiemes vitesses', 'premieres forces', that imply the influence of his acquaintance with the Principia. Yet, already here it is obvious that Varignon was unreceptive



28

to one of the most vital points in Newton's basic conception: the law of inertia. I was told by Pierre Costabel from Paris that this unreceptivity is just as evident in the correspondence between Johann Bernoulli and Varignon (which has not yet appeared in print; the first volume, however, can be expected this year), and it continues right up to Varignon's last paper of I720 about the generalized movement of the fall of heavy bodies.

In a letter of 24 May 1696, addressed to Bernoulli, Varignon's first explicit quotation from Newton's Principia can be found. He refers to section X of the first book but, according to Costabel, Varignon's reference is without any particular interest. On the other hand, in volume II of this correspondence, which will contain the letters exchanged between 1702 and 1713, Newton's Principia is quoted some 75 times-thus there is something to look forward to. But, for the time being, these few remarks must suffice.

VI

It is high time now to return to Basle. However, concerning Jakob Bernoulli, the first (and perhaps profoundest) of the Bernoulli dynasty of famous mathematicians, nothing of substantial significance can be added today to the global picture drawn exactly 50 years ago by Eduard Fueter (40). The influence of Newton's opus summum on Jakob Bernoulli was negligible. That can be explained by a number of reasons: first of all as a mathematician this 'centaur' from Basle was a Leibnizian, but as a physicist grosso modo a Cartesian, as shown by his works in natural philosophy (4I)-you only have to think inter alia of his abortive cometary theory at the beginning of his Opera omnia. Furthermore, at the time of the Principia's publication, Jakob Bernoulli was passionately absorbed in the Leibnizian analysis, which he reached by having fought his way there all on his own from 1684 onwards, that is ever since the appearance of the rudimentary and rather badly edited Nova methodus by Leibniz. Thus he became a true pioneer of the calculus. In natural philosophy, however, this eminent mathematician has to be considered pre-Newtonian, as after 1687 he did not write any major treatises of purely physical or astronomical content. Yet in the field of what today is called pure mathematics he often refers to Newton, for whom, after initial hesitation, he expressed the greatest admiration in his letters. No doubt Bernoulli had to accept that Newton, for instance, anticipated substantially the theorema aureum for the general determination of the



29

radius of curvature, to which Bernoulli has given the elegant analytical form

r = (I + y'2)3/2/y".

It was, by the way, thanks to this formula that in Paris early in the I69os

Jakob's 'little brother' Johann could reap some of his most spectacular successes.

Now a few words about Johann. First of all there is one striking similarity between the two: as mathematicians they were Leibnizians, as natural philosophers Cartesians. Whereas, as we have just heard, it was quite irrelevant in Jakob's case, this fact is enormously significant as far asJohann is concerned. It is useful though to distinguish two main phases in the latter's 'Cartesian activities': namely the period before and the one after the outbreak of the priority dispute, which is the subject of the book written by A. R. Hall a few years ago (42). It is not necessary to rehash these things once more; the rather double-edged role played by Johann Bernoulli-'lion by night, jackal by day'-has been aptly summarized, for example, in Gjertsen's Newton Handbook (43). But let us look at the first period now.

The earliest proof of Johann's first reading of the Principia is to be found in de l'Hospital's letter to Bernoulli of 2 January 1693, which treats the proposition on conics in corollary 3 of the lemma 25 in the first book (45). Hence Bernoulli must have read the Principia by 1692 at the latest. At that time-and even more so later on the occasion of Newton's immediate masterly solution to the problem posed by Bernoulli of the brachistochrone, the curve of quickest descent-Bernoulli still expressed his utmost admiration for Newton as a mathematician (46) whose authorship of the anonymously submitted 'cycloid solution' he immediately recognized tamquam ex ungue leonem. This unstinted praise might seem amazing considering Newton's fierce opposition to Descartes's physics and cosmology with which Bernoulli still fully concurred. It can easily be understood that Newton's adoption of concepts that were by no means 'clear and distinct' (clara et distincte), of gravitational force as an action at a distance and of the vacuum, necessarily appeared to Johann Bernoulli in his first period as a relapse into scholasticism in the sense of peripatetic dictatorship. This can be verified by many references collected by among others, Pierre Brunet (47) and Eduard Fueter (40).

The beginning of the second period can be put around I7I0 with Bernoulli's famous critique of the tenth proposition of the Principia's second book, a critique that through de Moivre's mediation in 1712



30

led to the well-known personal meeting in London between Johann's nephew, Nikolaus I. Bernoulli, and Sir Isaac, and that brought both men from Basle into the Fellowship of the Royal Society (48).

In the context of Johann Bernoulli's breaking his Cartesian lances on the strong British shields in his innumerable treatises and prize-essays with truly ingenious and exquisite mathematical means, the following circumstances should not be overlooked:

(I) Bernoulli's doubtlessly genuine Cartesian convictions were certainly mixed with a generous sprinkling of opportunism: he highly valued praise from Paris, and awards from the Academie des Sciences, and all the more so as his opposition to the English with respect to the priority dispute became more pronounced-and nearly all the French Academicians were Cartesians.

(2) It must have been hard on the ageing father to witness first his son Daniel's desertion to the 'English camp', and then also the defection of Leonhard Euler, his finest pupil whose genius he had discovered and decisively formed by his privatissima held every Saturday, Euler who, in his first public speech on the occasion of his Master's Examination in Basle compared the systems of Descartes and Newton. In this context it is interesting to note that Bernoulli, in his extensive correspondence with Euler, did not even once touch upon this tender subject. Whenever one of these two mentions Newton by name, their discussion is strictly confined to pure mathematical and technical details.

VII

Permit me to offer you the last part of my survey like a bunch of flowers. In Goethe's Elective Affinities the following aperfu of profound wisdom can be found: 'Only the inadequate is productive'. With this in mind I would now like to sketch some inadequacies or gaps in Newton's Principia.

As is well known, Newton solved (prop. 41, book i) the two-body problem, or the inverse problem in the case of a single central force, in the following sense: he obtained the equivalent of the polar equation of the orbit in differential form, leaving only the integrals to be evaluated by substitution of the law of force, and he evaluated the integrals in the particular case of the inverse-cube law of force. In I710 'Jakob Hermann and Johann Bernoulli independently proved that the conic sections were the only possible orbits in the vital case of an inverse-square law of force, thus filling a gap left by Newton' (21, p. 99). In direct continuation, most likely inspired by Varignon's Nouvelle mecanique (1725), Leonhard Euler occupied himself with this complex of problems, publishing his



3

results in his two-volume Analytical mechanics (49) and the succeeding treatises about Celestial mechanics that, in direct succession to Newton, were the foundation for this classical discipline (o5), which after further development by Gauss, Lagrange, Laplace, Jacobi, Leverrier and others culminated in the evaluation of the orbit and finally in the discovery of the planet Neptune (I846), representing its pinnacle of triumph at that time.

I can now do nothing more but hint at a few other 'productive gaps' or shortcomings in the Principia; 'shortcomings' by no means with negative implications because without exception they concern problems that simply could not yet be fully solved considering the state of development of analysis at the time. First of all I am thinking of the problem of mathematical ballistics, the modern beginnings of which, via Huygens, Newton, Johann Bernoulli, Robins, Euler and D'Arqy, have led to the solution by Lambert, who finally succeeded in fixing the explicit formula for the trajectory of a projectile (52; 53, pp. 207-224).

Then there is the problem of the tides for which Aiton (54) has given us an excellent historical summary. Furthermore I am thinking of the theory of lunar motion in which Clairaut and Euler saw the criterion of truth of Newton's theory of attraction. (In this connection it is indispensable to draw attention to Todhunter's colossal work (55), a

gold-mine for the history of exact science in the i8th century, above all.) Then the whole complex of problems regarding the figure of the Earth, which is most closely associated with the names of Maupertuis, Clairaut, Euler, d'Alembert, Daniel Bernoulli, Legendre, Lagrange and Laplace, just to name a few of the greatest after Newton.

Finally, mention must be made of fluid mechanics, the history of which after Newton has been treated several times, in particular by Truesdell (56), I. Szab6 (53) and Mikhailov (57). In all these works a central position is given to the further development of Newton's approaches by the Bernoullis, father and son, up to the classical

perfection of hydrodynamics by Euler (I755). Analogously the same could be said about the rational mechanics of flexible or elastic bodies. Truesdell, for instance, appropriately finishes the first part of his

'introductory volume' (58) with the following words:

For today it is instantly plain that the language of our subject is partial differential equations.... Lacking was a formal calculus of partial derivatives.... What was needed was a man who could express and master the Newtonian view of mechanics in Leibnizians partial differentials. This man was Euler...'. (58, p. I41) (Truesdell's emphasis.)

A last point. There are 'gaps' in the Principia that show up the

necessary limits of the Newtonian system, as reflected by Hypothesis I of



32

the third book: 'That the centre of the world is immovable'. This boundary was only to be overcome by the conceptual combination of inertia, metrics and gravitation so that space and time no longer could be imagined without contents, that is, by the theory of relativity. And all this began with the impact of Newton's Principia, 300 years ago. I cannot end in a better way than with the impressive words that in 1823 the 'continental' Laplace uttered in homage to Newton's Principia mathematica:

This admirable masterpiece contains the germs of all the great discoveries that have since been made concerning the system of the world: the history of their

development by the successors of this great geometer would at the same time be the most useful commentary upon his Work and the best guide to the attainment of new discoveries. (59)

REFERENCES

(I) I. Newton, Philosophiae naturalis principia mathematica (London, 1687; revised edn

Cambridge, 1713). (2) H. W. Turnbull, J. F. Scott, A. R. Hall & L. Tilling (ed.), The Correspondence of

Isaac Newton (7 vols, Cambridge, Cambridge University Press, 1959-77). (3) D. T. Whiteside (ed.), The Mathematical Papers of Isaac Newton (8 vols,

Cambridge, Cambridge University Press, 1967-81). (4) D. T. Whiteside, The Mathematical Principles underlying Newton's Principia

mathematica (University of Glasgow, 1970). (5) I. Schneider, 'Der Mathematiker Abraham de Moivre (I667-I754)', Archivefor

History of Exact Sciences 5 (1968), 177-317.

(6) A. R. Hall, 'Huygens and Newton', The Anglo-Dutch contribution to the civilisation

of early modern society (O.U.P. for the British Academy, 1976), pp. 45-59. (7) C. Huygens, Oeuvres completes (22 vols, The Hague, I888-I950). (8) H.J.M. Bos, M.J. S. Rudwick, H. A. M. Snelders & R. P. W. Visser (ed.),

Studies on Christiaan Huygens (Lisse, Swets & Zeitlinger B.V., 1980).

(9) R. S. Westman, 'Huygens and the Problem of Cartesianism', Studies on Christiaan

Huygens (Lisse, Swets & Zeitlinger B.V., 1980), pp. 83-103. (Io) C. Huygens, Horologium oscillatorium... (Paris, I673). (ii) M. B. Hall, 'Huygens' Scientific Contacts with England', Studies on Christiaan

Huygens (Lisse, Swets & Zeitlinger B.V., 1980), pp. 66-82.

(12) H. J. M. Bos, 'Christiaan Huygens', Dictionary of Scientific Biography 6 (1972).

(I3) E.J. Aiton, The Vortex Theory of Planetary Motion (London, New York, 1972).

(14) D. T. Whiteside, 'Newton's early thoughts on planetary motion: a fresh look', British Journalfor the History of Science 2 (1964), II7-I37.

(IS) E. A. Fellmann, G. W. Leibniz-Marginalia in Newtoni Principia Mathematica

(1687) (Paris, Vrin, 1973). (Collection des travaux de l'Academie Internationale d'Histoire des Sciences, no. I8.)

(I6) G. W. Leibniz, Mathematische Schriften (ed. C. J. Gerhardt) (7 vols, Berlin, Halle, 849-63).

(17) G. W. Leibniz, 'Tentamen de motuum coelestium causis', Acta Eruditorum 2

(I689), 82-96.



33

(i8) I. B. Cohen, Introduction to Newton's Principia (Cambridge, Cambridge University Press, I97I).

(I9) E.J. Aiton, 'The celestial mechanics of Leibniz', Annals of Science 16 (1960), 65-82.

(20) E.J. Aiton, 'The celestial mechanics of Leibniz in the light of Newtonian criticism', Annals of Science 18 (1962), 3I-41.

(21) E.J. Aiton, 'The inverse problem of central forces', Annals of Science 20 (1964), 8I-99.

(22) E. J. Aiton, 'Leibniz on motion in a resisting medium', Archive for History of Exact Sciences 9 (I972), 257-274.

(23) E.J. Aiton, 'The mathematical basis of Leibniz's theory of planetary motion', Studia Leibnitiana, Sonderheft 13 (1984), 209-225.

(24) E.J. Aiton, Leibniz-a Biography (Bristol and Boston, Adam Hilger Ltd, I985).

(25) K. Miiller & G. Kr6nert, Leben und Werk von Gottfried Wilhelm Leibniz-Eine Chronik (Frankfurt a.M., Klostermann, 1969).

(26) L. Euler, Methodus inveniendi lineas curvas... (Lausanne/Geneva, 1744; Opera omnia I, 24).

(27) E. A. Fellmann, 'Uber eine Bemerkung von G. W. Leibniz zu einem Theorem in Newton's "Principia mathematica "', Verhandl. Naturf. Gesellschaft Basel, 87/88 (1978), 21-28.

(28) H. Freudenthal, 'Bernard Nieuwentijt', Dictionary of Scientific Biography 10

(1974), I20-I2I.

(29) B. Nieuwentijt, Considerationes circa analyseos ad quantitates infinite parvas applicatae principia, et calculi differentialis usum in resolvendis problematibus geometricis (Amsterdam, 1694).

(30) B. Nieuwentijt, Analysis infinitorum seu curvilineorum proprietates ex polygonorum natura deductae (Amsterdam, I695).

(31) G. W. Leibniz, [G.G.L.], 'Responsio ad nonnullas difficultates, a Dn. Bernardo

Nieuwentijt circa methodum differentialem seu infinitesimalem motas', Acta Eruditorum (1695), pp. 3I0-316.

(32) B. Nieuwentijt, Considerationes secundae circa calculi differentialis principia; et

responsio ad virum nobilissimum G. G. Leibnitium (Amsterdam, 1696). (33) J. Hermann, Responsio ad Clar. viri Bernh. Nieuwentijt considerationes secundas circa

calculi differentialis principia editas (Basle, 1700). (34) B. Nieuwentijt, Het regt gebruik der wereltbeschouwingen ter overtuiginge van

ongodisten en ongelovigen, aangetoont door... (Amsterdam, I714).

(35) G. de l'Hospital, Analyse des infiniment petits (Paris, 1696). (36) P. Costabel, 'Pierre Varignon', Dictionary of Scientific Biography 13 (1976),

584-587. (37) J. Hermann, Phoronomia sive de Viribus et Motibus Corporum solidorum etfluidorum

libri duo (Amsterdam, 1716).

(38) P. Varignon, Projet de nouvelle me'chanique (Paris, 1687). (39) P. Varignon, 'Regles du mouvement en general...', Memoires de mathematiques et

de physiques tires des registres de l'Academie... (Paris, 1692), pp. 190-I95. (40) E. Fueter, 'Isaak Newton und die schweizerischen Naturforscher seiner Zeit',

Beiblatt zur Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, no. 28 (I937), Jahrg. 82.

(41) J. Bernoulli, Werke (ed. J. O. Fleckenstein), Bd. i (Basle, Birkhauser, 1969).



34

(42) A. R. Hall, Philosophers at War-The Quarrel between Newton and Leibniz (Cambridge University Press, 1980).

(43) D. Gjertsen, The Newton Handbook (London and New York, Routledge & Kegan Paul, 1986).

(44) J. Hendry, 'Inconsequential Newtonia', New Scientist, 5 February 1987.

(45) J. Bernoulli, Briefwechsel (ed. O. Spiess), Bd. I. (Basle, Birkhauser, I955). (46) J. Beroulli, Opera, vol. i (Lausanne and Geneva, 1742), p. 196. (47) P. Brunet, L'Introduction des theories de Newton en France au I8me siecle, I,

Avant 1738 (Paris, I93I). (48) K. Wollenschlaeger, 'Der mathematische Briefwechsel zwischen Johann I

Bernoulli und Abraham de Moivre', Verh. naturf. Ges. Basel 34 (I931/32), 151-317.

(49) L. Euler, Mechanica sive motus scientia analytice exposita, 2 vols (Petersburg, 1736); Opera omnia, II, I, 2.

(50) L. Euler, Opera omnia, I, 25 (ed. M. Schiirer) (Zurich, 1960). (51) O. Volk, 'Eulers Beitrage zur Theorie der Bewegungen der Himmelsk6rper', in

Leonhard Euler 1707-1783-Beitrdge zu Leben und Werk (ed. J. J. Burckhardt, E. A. Fellmann & W. Habicht) (Basle, Birkhauser, 1983), pp. 345-362.

(52) A. R. Hall, Ballistics in the Seventeenth Century (Cambridge University Press, 1952).

(53) I. Szabo, Geschichte der mechanischen Prinzipien und ihrer wichtigsten Anwendungen, 3, Aufl. (ed. P. Zimmermann & E. A. Fellmann) (Basle, Birkhauser, 1987).

(54) E. J. Aiton, 'The contributions of Newton, Bernoulli and Euler to the theory of the tides', Annals of Science, II (1955), 206-223.

(55) I. Todhunter, A History of the Mathematical Theories of Attraction and the Figure of the Earth, 2 vols, 1873 (New York, Dover Publications Inc., 1962).

(56) C. Truesdell, Essays in the History of Mechanics (Berlin, Heidelberg, New York; Springer-Verlag, 1968).

(57) G. K. Mikhailov, 'Leonhard Euler und die Entwicklung der theoretischen Hydraulik im zweiten Viertel des I8. Jahrhunderts', in Leonhard Euler 1707-1783-Beitrdige zu Leben und Werk (ed. J. J. Burckhardt, E. A. Fellmann & W. Habicht) (Basle, Birkhauser, 1983), pp. 229-241.

(58) C. Truesdell, The rational mechanics of flexible or elastic bodies 1638-1788. Introduction to Leonhardi Euleri Opera omnia vol. X et XI seriei secundae, Opera omnia II, 11/2 (Zurich, I960).

(59) Connaissance des tems pour l'an 1823.



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