Présentation Nancy 26 juin 2015

Transcript of Présentation Nancy 26 juin 2015

Gabriel Cramer (1704-1752) Introducing the Introduction Reception of the treatise (1750-1900) Conclusion

Geometry of algebraic curves (1750-1850) :mathematicians and historians readings ofGabriel Cramers Introduction lanalysedes lignes courbes algbriques (1750)

Thierry JOFFREDO

PhD Student, Archives Poincar, Universit de Lorraine, [email protected]

1750-1850 : rupture(s) et continuit(s) en gomtrie26 juin 2015, Nancy

Thierry JOFFREDOMathematicians and historians readings of Gabriel Cramers Analyse des courbes (1750)

Plan

Gabriel Cramer (1704-1752)

Introducing the Introduction

Differential analysis of the reception of the treatise(1750-1900) : encyclopedists, historians, mathematicians

Conclusion and perspectives


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A few biographical landmarks




I Born in 1704 in a wealthy protestant patrician family ofGeneva,

I Professor of Mathematics (1724) and Philosophy (1750)at the Academy of Geneva,

I Member of the royal societies or academies of Montpellier(1743), Bologna (1744), Berlin (1746), London (1749)and Lyon (1750),

I Member of the Council of Two Hundred of the Republicof Geneva (1734) and the Council of Sixty (1749),

I Died (quite young) in 1752 in the south of France(Bagnols-sur-Cze).


Scientific legacy

Scientific legacy

I Published writings :I his masterpiece : Introduction lanalyse des lignes

courbes algbriques (1750),I plus a few dissertations, memoirs or letters on various

topics (sound, celestial mechanics, history ofmathematics, opticks, agronomy, philosophy...),

I Manuscripts : Cours de logique (and probability), Coursde gomtrie.

I Scientific editor :I Johann I and Jacob Bernoullis complete works (1742 &

1744 resp.),I Correspondence Johann I Bernoulli-Leibniz (1745).


Scientific legacy

Scientific legacy








Scientific legacy

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Scientific legacy

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Scientific legacy

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Scientific legacy

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Scientific legacy

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Social activities

Social activities

I Well integrated into the networks of the Republic ofLetters :

I traveled across Europe (1727-1729 and 1747-1748),I in active correspondence, on various scientific and

philosophical topics, with the Bernoullis, Stirling, Buffon,Dortous de Mairan, Clairaut, Bonnet, Euler, dAlembert,Condillac, etc.

I strongly invested in the main scientific research anddiscussions of his time : vivid forces, sound, auroraborealis, hydrodynamics, etc.

I Estimated teacher in the Academy of Geneva (CharlesBonnet, Georges-Louis II Lesage, etc.)


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Quick overview

Introduction lanalyse des lignes courbes algbriques(Gabriel Cramer, Frres Cramer et Cl. Philibert, Genve, 1750)

I Classification of the algebraic curves of the first fiveorders, extending Newtons works on lines of the thirdorder

I Almost 700 pages, 33 boards of drawings, 13 chaptersand 3 short (but famous !) appendices

I Infinite branches, singular points, tangent lines, extrema,curvature, sketching...

I Totally calculus free ! Algebraic methods onlyI For beginners : numerous and various examples


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Genealogy

Attempt of a genealogy

I Heritage (according to Cramers preface) :I Descartes, Gomtrie (1637)I Newton, Enumeratio linearum tertii ordinis (1704)I Stirling, Lineae tertii ordinis Neutonianae (1717)I sGravesande, Matheseos universalis elementa (1727)I Nicole et Bragelongne, Mmoires de lAcadmie Royale

des Sciences (1731)I de Gua de Malves, Usages de lAnalyse de Descartes

(1740)I Exploring footnotes references :

I LHpital, Intelligence des infiniment petits (1696),I Maclaurin, Geometria Organica (1720).


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Genealogy

The case of Eulers IntroductioIntroductio analysin infinitorum (vol.2, Euler, 1748) has thesame object and goals ... reached by different ways and means.

"Cest trop dhonneur que Vous faites mon trait sur lameme matiere, en le mettant en parallel avec le Votre, carmaintenant je viens de reconnoitre, que je nai pas pris toutesles prcautions necessaires pour bien developper la nature desbranches qui setendent linfini, et que suivant ma methodeje me pourrois souvent tromper meme sur lexistence de cesbranches. [...] A cause de ce defaut et encore dautres dontmon ouvrage est defigur, je Vous felicite davoir enrichi lepublic dun ouvrage accompli sur cette matiere, qui etantdelivr de tout defaut, explique la theorie des lignes courbesaussi solidement, que clairement."

Letter from Euler to Cramer, 15 oct 1750Thierry JOFFREDOMathematicians and historians readings of Gabriel Cramers Analyse des courbes (1750)

A closer look

Contents of the treatise

A systematic study of the algebraic curves (with the only helpof algebra) for the classification of the curves of the first fiveorders, based on infinite branches and singular points :I Generalities on algebraic curves and their equations (chap

I-V)I Diameters and centers (chap VI)I Method of series, analytical triangle (chap VII)I Infinite branches (chap VIII)I General classification of the lines of the first five orders

(chap IX)I Multiple and singular points (chap X and XIII)I Tangent lines, inflections, extrema (chap XI)I Curvature (chapter XII)


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A favored tool : the analytical triangle

The analytical triangle is used to identify the overriding termsof the equation and calculate series developments.


A closer look

A favored tool : the analytical triangle

On formera le Triangle avec des points disposs enquinconce, & on en changera en une toile, ou en une petitecroix, chaque point qui tient la place dun des termes delquation.[...] Si lon suppose x ou y infiniment petite, oncherchera, avec la Rgle, quelles sont les Cases pleines par lecentre desquelles peut passer une Droite, sans laisserau-dessous delle aucune Case pleine. Cette Droite, ou cesDroites, car il peut y en avoir plus dune, se nommeront desDterminatrices infrieures, parce quelles dterminent les plusgrands termes de lquation.

Introduction, pp 165-166


A closer look

An example from the Introduction

Lets consider the quartic curve defined by the followingequation :

x4 ax2y axy 2 14aayy = 0 (1)If we represent the different terms of this equation on theanalytical triangle, we obtain a single "dterminatriceinfrieure" (for a study at the origin) :


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This "dterminatrice" provides the following equation

x4 ax2y + 14aayy = 0 (2)which can be resolved by

y = 2xxaTherefore, the first term of the development is 2xxa .To calculate the next term of this development, letssubstitute

(2xxa + u

)to y in the equation (2).

We obtain this new equation :

4x5

a 4x3u axu2 + 14a

2u2 = 0. (3)Thierry JOFFREDOMathematicians and historians readings of Gabriel Cramers Analyse des courbes (1750)

A closer look





(2xxa + u



4x5

a 4x3u axu2 + 14a


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(2xxa + u



4x5

a 4x3u axu2 + 14a


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(2xxa + u



4x5

a 4x3u axu2 + 14a


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Lets put the equation (3) onto the analytical triangle ; we get :

We can see one new "dterminatrice infrieure", whichprovides the following equation :

4x5

a +14a

2u2 = 0 (4)Thierry JOFFREDOMathematicians and historians readings of Gabriel Cramers Analyse des courbes (1750)

A closer look




4x5

a +14a


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4x5

a +14a


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This equation, for x > 0, has two solutions given by :

u = 4x52

a 32= 4xx

axaa .

The development is thus divided in two distinct series from thesecond term :

y = 2xxa 4xxax

aa +8x3a3 + . . . .

which means that there are two different branches at theorigin.


A closer look


This equation, for x > 0, has two solutions given by :

u = 4x52

a 32= 4xx

axaa .

The development is thus divided in two distinct series from thesecond term :

y = 2xxa 4xxax

aa +8x3a3 + . . . .

which means that there are two different branches at theorigin.


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Finally we get this figure, with a beak cusp at the origin(Introduction, plate XXVI, p 600, fig. 201)


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The appendices present two important results, the first beingknown today as Cramers rule for the resolution of a systemof linear equations (Appendix I, p 656-659) :


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... and the second as Bzouts theorem, which affirms thatthe number of intersections of two curves is equal to theproduct of their degrees (Appendix II, p 660-676)


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We must also mention the so-known Cramers paradox.Cramer establishes that a curve of degree n is fully determinedby n(n+3)2 points : for example, a cubic is defined by

3(3+3)2 = 9

given points. But it is also known that two different cubiccurves intersect in 3 3 = 9 points...

Introduction pp 78-79Thierry JOFFREDOMathematicians and historians readings of Gabriel Cramers Analyse des courbes (1750)

Reading and reception : a dynamical analysis


I through different periods of time (from 1750 to 1900),I through different geographical spaces (France, Germany,

England...),I through different types of readers : encyclopedists,

historians, mathematiciansI through different mathematical fields or practices

(teachers, geometers, algebrists...)



How can this analysis contribute to the question ofruptures and continuities in the history of geometry

between 1750 and 1850 ?

At the turn of the XIXth century :I success of the differential methods in the study of curves

(Lacroix, Cauchy),I return of the synthetic methods in geometry (Poncelet,

Steiner)Is Cramers treatise quickly turning obsolete ? Or is it stillread ? By who, and what for ? What kind of posterity forCramers Analyse des courbes ?


Cramers work seen through the prism of encyclopaedias and dictionaries

Cramers work seen through the prism ofencyclopaedias and dictionaries :

I Encyclopdie ou Dictionnaire raisonn des sciences, desarts et des mtiers (Paris, 1751)

I Encyclopdie ou dictionnaire universel raisonn desconnaissances humaines (Yverdon, 1772)

I Encyclopdie Mthodique, Mathmatiques (Paris, 1784)I Mathematical and philosophical dictionary (London,

1795)I Dictionnaire des sciences mathmatiques pures et

appliques (Paris, 1835)I Encyclopaedia Metropolitana (London, 1845)



Encyclopdie ou Dictionnaire raisonn des sciences, desarts et des mtiers (dAlembert, Paris, 1751)

DAlembert reads carefully the Cramers treatise and discussesseveral points in three letters written between september 1750and january 1751. He finally says :

"A legard de votre ouvrage sur les courbes je persistetoujours dans lide que jen ay, et je viens dajouter larticlecourbe de lEncyclopedie le jugement que jen porte."




Article Courbe :"Les meilleurs ouvrages dans lesquels on puisse sinstruire

de la thorie des courbes, sont 1o lenumeratio linearum tertiiordinis de M. Newton, do une partie de cet article COURBEest tir ; 2o louvrage de M Stirling sur le mme sujet, &Geometria organica de M. Maclaurin, dont nous avons parl ;3o les usages de lanalyse de Descartes par M. labb de Gua,dj cits ; [...]"




"[...] ouvrage original & plein dexcellentes choses, maisquil faut lire avec prcaution ; 4o lintroduction lAnalysedes Lignes Courbes, par M. Cramer, ouvragetrs-complet, trs-clair et trs-instructif, & dans lequelon trouve dailleurs plusieurs mthodes nouvelles ; 5louvrage de M. Euler, qui a pour titre, introductio in analys.infinitorum, Lausan. 1748."




Article Branche :"On trouvera une thorie trs-complette des branches

infinies des courbes dans le huitieme chapitre de lIntroduction lanalyse des lignes courbes par M. Cramer. Il y donne lamthode de dterminer les diffrentes branches dune courbe,& leurs asymptotes droites ou courbes. Comme cette thorienous conduiroit trop loin, nous renvoyons l-dessus sonouvrage."




Article Asymptote :"Voyez [...] lIntroduction lanalyse des Lignes courbes,

par M. Cramer, p. 344. art. 147. Ce dernier ouvrage contientune excellente thorie des asymptotes des courbesgomtriques & de leurs branches, chap. viii."

Article Centre :"M. Cramer, dans son Introduction lanalyse des lignes

courbes, donne une mthode trs-exacte pour dterminer lescentres gnraux."




Article Paralllogramme :"On peut voir dans les usages de lanalyse de Descartes

de M. labb de Gua, & dans lintroduction lanalyse deslignes courbes de M. Crammer, la dmonstration, les diffrensusages, & les applications de cette regle, suivant les cas quipeuvent se prsenter ; il suffit ici den donner lesprit. Il estbon dobserver que MM. de Gua & Crammer transforment leparalllogramme en un triangle quils appellent analytique, cequi ne change rien au fond."



Encyclopdie ou dictionnaire universel raisonn desconnaissances humaines (de Felice, Yverdon, 1772)

Encyclopdie Mthodique, Mathmatiques (Pancoucke,Paris, 1784)

These encyclopaedias repeat verbatim the text of the articleCurve of the Encyclopdie. However :I the Encyclopdie dYverdon adds two pages on the

method of the analytical triangle,I in the Encyclopdie mthodique, articleApproximation : Newtons parallelogram method is"explique de la manire la plus claire & la plus dtailledans un excellent Ouvrage de M. Cramer, qui a pourtitre : Introduction lAnalyse des Lignes courbesAlgbriques, Gen. 1750."



Mathematical and philosophical dictionary (Hutton,London, 1795)

Article Curve :"The theory of curves forms a considerable branch of the

mathematical sciences. Those who are curious of advancingbeyond the knowledge of the circle and the conic sections,and to consider geometrical curves of a higher nature, and ina general view, will do well to study Cramers Introduction lAnalyse des Lignes Courbes Algebraiques, which the learnedand ingenious author composed for the use of beginners."



Mathematical and philosophical dictionary (Hutton,London, 1795)

Article Asymptote :"See [...] Cramers Introduction lanalyse des lignes

courbes, art 147 & seq., for an excellent theory of asymptotesof geometrical curves and their branches."

Article [Newton or Analytical] parallelogram :"And especially Cramers Analyses des Lignes Courbes, p

148 This author observes, that this invention, which is thetrue foundation of the method of series, was but imperfectlyunderstood, and not valued as it deserved, for a long time."



Dictionnaire des sciences mathmatiques pures etappliques (Par une socit danciens lves de lcole

Polytechnique, dir. Montferrier, Paris, 1835)

Cramer and his Introduction are still cited in several articles :I as main reference in the articles Asymptote, Branchede courbe, Centre, Paralllogramme deNewton, Points singuliers.

I but no mention in the article Courbe, unlike Newton,Maclaurin, Euler.

There is also an article entitled CRAMER (Gabriel) :"Louvrage qui a consacr la clbrit de Cramer est celui

quil intitula modestement : Introduction lanalyse des lignescourbes algbriques [...]. Ce livre est connu de tous lesmathmaticiens."



Encyclopaedia Metropolitana (Pure sciences, vol I,Londres, 1845)

Cramers treatise remains a reference book cited at the end ofthe article Analytical geometry, in company withMaclaurin and Euler :

"For farther information on the subject of Curves and ofSurfaces, the reader is referred to the following works : [...]Cramers Introduction lAnalyse des Lignes CourbesAlgbriques ; Eulers Introductio in Analysin Infinitorum, tom.ii [...]."


Historian readings of Cramers Analyse des courbes

Historian readings of Cramers Analyse des courbes :

I Histoire des mathmatiques (Montucla, Paris, 1802)I Histoire gnrale des mathmatiques (Bossut, Paris,

1802)I Trait des proprits projectives des figures (Poncelet,

Paris, 1823)I Aperu historique sur lorigine et le dveloppement des

mthodes en gomtrie (Chasles, Paris, 1837)I On the Singular Points of Curves, and on Newtons

Method of Coordinated Exponents (De Morgan, Londres,1856)



Histoire des mathmatiques (Montucla et Lalande,Paris, 1802)

"Il manquoit cependant encore jusquen 1750 un livre surce sujet, qui runt la profondeur de la doctrine, lesdveloppements ncessaires pour le rendre accessible tousles gomtres. Cest ce que M Cramer a excut avec le plusgrand succs, par son ouvrage trop modestement intitulIntroduction lAnalyse des Lignes Courbes Algbriques quiparut Genve en 1750, in 4o , ouvrage dailleurs original enplusieurs points, et dans lequel au mrite du fond sajoutecelui de la forme, je veux dire une clart et une mthodetout--fait satisfaisantes. On ne sauroit par toutes ces raisonstrop le conseiller tous ceux qui dsirent approfondir cettethorie."



Histoire des mathmatiques (Montucla et Lalande,Paris, 1802)

Beyond this laudatory words, the authors strongly rely on theanalytical methods implemented by Cramer to explain how tostudy algebraic curves (pp 73-85).

Histoire gnrale des mathmatiques (Bossut, Paris,1802)

"Je ne dois pas oublier de citer avec distinction Cramerparmi les bienfaiteurs de la nouvelle Gomtrie. SonIntroduction lAnalise des lignes courbes algbriques, est letrait le plus complet qui existe sur cette matire. Lauteur nelaisse rien dsirer sur la thorie des branches infinies descourbes, sur leurs points multiples, et en gnral sur tous lessymptmes qui servent les caractriser".



Trait des proprits projectives des figures (Poncelet,Vol 2, Paris, 1866)

In a text written in 1823, Poncelet gives an interesting andoriginal historian point of view on Cramers treatise :

"Lintroduction lAnalyse des lignes courbes par Cramer,quon doit considrer comme renfermant peu prs tout cequi tait alors connu sur le sujet nest en effet que ledveloppement des principes poss par Newton et Stirlingpour la classification des lignes courbes [...]. On y enseigne, ilest vrai, dterminer, pour chaque point de la courbe, latangente et la normale, le cercle et le rayon de courbure, ainsique les grandeurs ou les lieux qui en dpendent, tels que lesdveloppes, etc."



Trait des proprits projectives des figures (Poncelet,Vol 2, Paris, 1866)

"[...] mais ces recherches, ces solutions de problmesinfiniment utiles aux arts qui se fondent sur le dessin linaire,ne sont l prsentes ou rsolues qu laide des quations descourbes et par le calcul des abscisses et ordonnes ; on noprepresque jamais sur les lignes, sauf dans quelques castrs-simples ; en un mot, cest de lAnalyse algbrique,admirable, il est vrai, par son universalit, mais non de laGomtrie telle que le rclament les divers besoins des arts.Cest encore ainsi que, de nos jours (1823), on traitevolontiers, dans les ouvrages consacrs linstruction, lathorie des lignes courbes."



Aperu historique sur lorigine et le dveloppement desmthodes en gomtrie (Chasles, Bruxelles, 1837)

Chasles gives, in turn, his opinion in the part of his Aperuentitled "Histoire de la gomtrie", after having mentionedMaclaurin, De Gua and Euler :

"Cramer donna, sous le titre : Introduction lanalysedes lignes courbes algbriques (in-4o , 1750), un trait spcial,le plus complet, et encore aujourdhui le plus estim, sur cettevaste et importante branche de la Gomtrie."

But he eventually adds :"Ce sont l, je crois, les derniers perfectionnemens

notables que la science des courbes dut la Gomtrie desanciens et lanalyse de Descartes."



On the Singular Points of Curves, and on NewtonsMethod of Coordinated Exponents (De Morgan,

Transactions of the Cambridge Philosophical Society,London, 1856)

In this article De Morgan questions the methods for studyingthe singular points of curves :

"Two methods have been adopted of treating the inquiryinto the singular points of curves. The first, of which thefullest development ever given is in the well-known and highlyvalued, but (as I shall show) little read work of CramerAnalyse des Lignes Courbes Algbriques, Geneva, 1750 [...]"



On the Singular Points of Curves, and on NewtonsMethod of Coordinated Exponents (De Morgan,

Transactions of the Cambridge Philosophical Society,London, 1856)

A few pages later, he seems to exhume the Newtonsparallelogram method from Cramers works to rehabilitate it :

"But, remarkable as it may be, it is still more remarkablethat Newtons parallelogram, or method of co-ordinatedexponents, as I shall call it, has still more completely falleninto oblivion. [...] Lastly, Cramer has made it the leadingmethod of his well-known work on curve lines."


Mathematician readings of Cramers Analyse des courbes

Mathematician readings of Cramers Analyse descourbes :

Was the Introduction read by mathematicians (geometers)between 1750 and 1850 ? For what purposes ? What placedoes it occupy in the curriculum of students in mathematics ?Lets take a look :I in France, from Bezout to Puiseux,I in Germany, in company of Mbius, Plcker and Clebsch,I and finally, in the United Kingdom (and on the other side

of the Atlantic ocean !)



Cours de mathmatiques lusage des gardes dupavillon et de la Marine (4th part, Bezout, Paris, 1770)

Two references to Cramers Introduction in the fourth part ofthe famous Cours de mathmatiques, "contenant les principesgnraux de la Mchanique, prcds des Principes de Calculqui servent dintroduction aux SciencesPhysico-mathmatiques" :I in a paragraph dealing with singular points (cusps) of

curves,I and at the end of a part concerning approximations for

integration.The treatise is also mentioned twice in the third part of theCours, containing t

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