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Carnot’s theory of transversals and its applications by Servois and Brianchon: the awakening of synthetic geometry in France

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Abstract

In this paper we discuss in some depth the main theorems pertaining to Carnot’s theory of transversals, their initial reception by Servois, and the applications that Brianchon made of them to the theory of conic sections. The contributions of these authors brought the long-forgotten theorems of Desargues and Pascal fully to light, renewed the interest in synthetic geometry in France, and prepared the ground from which projective geometry later developed.

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Notes

  1. See (La Hire 1673, 1685), (L’Hospital 1707, Book VI), (Le Poivre 1704).

  2. We have in mind essentially (Poncelet 1822), see below.

  3. In the late eighteenth century geometry appeared divided into two main branches: analytic geometry, and pure, or synthetic, geometry. Roughly speaking the first referred to the Cartesian method of coordinates, used equations, algebraic analysis, and infinitesimal calculus in order to reach the solution of proposed problems; the second, which had its roots in the methods of the ancient Greeks, privileged geometrical reasoning on the figure, and (mainly) used proportions rather than equations. For the “rivalry” between analytic and pure geometry in the late eighteenth century see (Chemla 1998), and for a broader overview of the history of analytic/synthetic distinction we refer to (Lorenat 2015a), (Lorenat 2015b).

  4. Most likely Poncelet was referring to the Menelaus–Ptolemy theorem, see Sect. 3.2.

  5. Poncelet was referring to (Brianchon 1806, 1810b), see Sect.  5.

  6. Poncelet was referring to the several notes and memoirs concerning the Geometry of the ruler, which were published in these journals between 1810 and 1820; in particular, the papers by Servois and Gergonne appeared in the Annales, here commented in Sect. 5 and 6.

  7. For an in-depth analysis of the efforts made by Carnot to import into pure geometry the generality of analysis, we refer to (Chemla 1998).

  8. Since Brianchon wanted to compare his own methods with Newton’s, these will be expounded in an “Appendix” to this paper.

  9. The question of the influence of Brianchon’s works on Poncelet’s early research will be investigated in a forthcoming paper devoted to an in-depth study of the notebooks that Poncelet composed during his captivity in Saratov (Russia).

  10. For more information on this subject see (Langins 1990). We recall that Poncelet also attended the École polytechnique and the École d’Application de l’artillerie et du génie at Metz.

  11. With this term Monge referred to architects, engineers, but also painters and engravers.

  12. Monge did not use these terms, they were introduced by F.-J. Servois (1811b, 337), and J. D. Gergonne (1813a, 297) respectively, see Sects. 4, 5; we adopt them since now for convenience.

  13. That is by considering a conic section as projections of the circle.

  14. Monge took for granted that the curve of contacts was a plane curve, this was proved in (Brianchon 1806, Art. XIV).

  15. We note in passing that this concept, which was already present in Kepler’s treatment of conic sections (Del Centina 2016b), and in Desargues’ Brouillon project, was central in Boscovich’s (1754), who first tried to make it operational through a series of rules (canons), see (Del Centina, Fiocca 2018a, 2018b).

  16. An algebraic identity which holds in the real case extends over complex numbers, see for instance (Bourbaki 1960, 131).

  17. For a thorough discussion of the subject of generality in pure geometry in the early nineteenth century we refer to (Chemla 2016, 56–70).

  18. Problems, and solutions deemed worthy, were published in the journals of the School.

  19. For a complete biography see (Gillespie 1971), (Dhombres J. and N. 1997).

  20. This argument has been discussed in (Chemla 1990, 1998), and (Nabonnand 2011).

  21. We refer to (Chemla 1998) and to (Nabonnand 2008) for some detail.

  22. Carnot’s approach was later criticized by Poncelet, see (Nabonnand 2008,39–42).

  23. “Lettre du cit. Carnot, membre de l’Institut national, ministre de la guerre, au cit. Bossut, membre de l’Institut national examinateur du corps di génie”, dated “30 fructidor an 8 de la république français”, 16th September 1799 (Bossut 1800, vol. 2, 401–421).

  24. See (Chemla 1990, 531–532).

  25. See (Taton 1951, 1955), and (Del Centina 2020).

  26. Actually this theorem was not completely new, see footnote 46 (Sect. 6).

  27. In the Essai of 1806, Carnot perfected the definition calling complete quadrilateral the assemblage of four straight lines joining four points in a plane, ordered at will, and then passing from the first to the second, from the second to the third, from the third to the fourth, and from the fourth to the first. Carnot remarked that quadrilaterals can have three different forms: that “ordinaire”, as ABDC, that he named of first kind; that “à angle rentrant”, as AFDG, that he named of second kind; and that “qui a la forme de deux triangles opposes par le sommet”, as BFCG, that he named of third kind.

  28. We note in passing that Brianchon applied this latter in order to prove the theorem on the circumscribed hexagons to a conic section, which bears his name, see Sect. 5.

  29. This theorem was known to Apollonius (Conics, III, props.17–21), see also Sect. 7.4. For future discussion we also recall that Newton used the chords theorem in order to get the important results on conic sections he presented in the fifth section of the first book of the Principia, and in particular to determine the conic sections subject to five conditions (see the “Appendix”). We also note that L’Hospital and J. Stirling recognized its fundamental value for the theory of conic section. For a history of this theorem, and a discussion of the above questions, see (Del Centina, Fiocca 2020).

  30. Carnot referred to Newton’s Enumeratio Linearum Tertii Ordinis, appended to (Newton 1704), see also (Guicciardini 2009, 117–121).

  31. Likely by “etc.” he meant some of the authors quoted in the Dissertation preliminaire.

  32. This theorem re-emerged, also in the general form for conic sections, in the following years. It had, with its dual, a not secondary role in the history of polarity; see later on.

  33. See, Nouveaux Mémoires de l’Académie Royale de sciences et belles lettres, 1776, pp. 265–283, and pp. 284–311.

  34. Annibale Giordamo, was born in S. Giuseppe d’Ottajano, a small town very close to Naples. He was seventeen when he presented his paper. Carnot believed “di Ottajano” to be a noble patronymic, and, as we will see, this led Brianchon, as well as other French mathematicians, to make the same mistake.

  35. See, Memorie di matematica e fisica della Società Italiana, IV, 1788, pp. 4–17, and pp. 201–205.

  36. He was referring to (Simson 1776), published posthumously and containing a work on Pappus’ porisms.

  37. The works which Carnot referred to are (Stewart 1763, 1746).

  38. Namely the Pappus three lines problem.

  39. To denote these points Carnot actually used a quite complicated notation, but we have not followed him.

  40. The hexagon ABEDCF in Carnot’s figure 162 corresponds to the hexagon AECDFB in figure 82 in (Simson 1735).

  41. Although in the long preface to his book Carnot mentioned Maclaurin, and quoted Simson in the text, it seems that he was unaware of (Maclaurin 1748) and (Simson 1735), in which a synthetic proof of Pascal’s was given, see (Del Centina 2020).

  42. We will return to this in the next section.

  43. For the second part, and a discussion on the teaching in the Artillery schools, we refer to (Aebischer, Longuereau 2010).

  44. For Servois’ contribution to the theory of differential operator see (Bradley 2002).

  45. Servois used to label relevant equations by capital letters.

  46. Servois quoted Art. CLV (Proposition I) of the fourth volume of Bernoulli’s complete works (1742). It is worth noting that Bernouilli considered only the case in which the chosen point is inside the triangle, and proved the theorem by using only the similarity of triangles.

  47. “Sont deux diagonales homologues de deux quadrilatères équiangles”, Servois’ words.

  48. Servois adopted the definition given by Carnot.

  49. It is this proof that was inserted into (Carnot 1806, 92–93).

  50. For the history of this theorem see (Mackay, 1892).

  51. This was the third of three propositions announced in 1804 by M. Livet (instructor at the École) in the note Du contact des surfaces coniques avec les surfaces du second degré, whose (analytical) proofs were later published in the Journal. de l’École polytechnique in 1806. Brianchon’s paper appearing in the same issue was inserted immediately after that of Livet.

  52. See Sect. 3.3.

  53. Curiously enough, Brianchon denoted the projections of points L, M, N, l, m, n, by the same upside down letters, which cannot be reproduced here, thus we have used the same letters but in bold font.

  54. We wonder if Brianchon realized that he had proved Desargues’ theorem on homological triangles, and if he was aware of what was already proved by Servois.

  55. The only difference between Carnot’s original statement and Brianchon’s, is the change in meaning of points C and E; compare Fig. 17a with Fig. 8a.

  56. Oddly enough Brianchon did not remark that the triangles DEF and ABC are in perspective with respect to P, and so the points H, I, k are necessarily collinear, that is k belongs to the straight line HIK; similarly, one can see that also i, h lie on HIK. Thus we presume that he did not know Desargues’ theorem on homological triangles.

  57. Since the point P through which pass all contact chords is the “pole” of the straight line HI, we observe that the “pole-polar” property is implicitly stated. We will return to this later on.

  58. Brianchon will develop this topic in (1807), see below.

  59. In this regard it is interesting to read the letter that Brianchon sent to Hachette from Toledo (Spain) on April 1810, see below.

  60. For this construction see also Sect. 5.3.

  61. This name indicated the surface described by all straight lines (generators) that lean on three skew given straight lines (directive lines), which is the double ruled surface also known as one-sheet hyperboloid.

  62. In a footnote it was indicated “Cl. Ptolemei. Venetiis, 1515, in fol.”. Brianchon’s interest in the history of geometry fully emerged in (1817), see later on.

  63. For a comparison between Ptolemy’s statements of the theorem and Carnot’s see (Chemla 1990, 530–532).

  64. In fact, one can suppose that the given curve is a circle, thus, if the given straight line is l, it is about finding an oblique cone with base the given circle such that a sub-contrary section is parallel to the plane passing through the vertex of the cone and the line l.

  65. See (Poncelet 1862, 387).

  66. He did not say this but obviously he had the following construction in mind: issue from F any two secants as FHG and FLK, intersect GL and KH, GK and HL, then draw the straight line through the two points of intersection; this straight line (XY) intersects the conic section at the contact points of the tangent through F. Clearly XY is the polar of F, see later on.

  67. This means that XY is the polar of F.

  68. Clearly referring to (Carnot 1803, Th. XLV).

  69. That is, the tangent lines to the conic section at the extremities of each chord through P meet on HK.

  70. Brianchon referred to Art. XIII, most likely a misprint.

  71. Brianchon’s “projective view” is indisputable.

  72. Brianchon denoted the circles and their centres by the same letters.

  73. In fact, OO′ is equal to the radius of C′, and the perpendicular to CC′ issued from m is the polar of C′ with respect to C, but Brianchon omitted to say this.

  74. See (Poncelet 1862, 408).

  75. D. Encontre was a professor at the Faculty of Sciences of the Academy of Montpellier.

  76. Obviously the poles of the given straight lines, but Encontre did not say it.

  77. Likely Servois confused Mdccxxxiii for Mdcccxxviii, so the reference is to (Braikenridge 1733).

  78. It is not clear to us to which proofs Servois was referring to.

  79. It is worth noticing that Poncelet referred to the geometry “cultivée par les modernes” as that which proceeds “par la voie d’intuition” (Poncelet 1817, 143).

  80. (Brianchon 1807, 1810b), see and Sect. 5.2.

  81. J. D. Gergonne was born in Nancy, and he attended the local College, but he was essentially self-taught. In 1792 he enrolled in the Army, and, examined by Lacroix, he was admitted to the Artillery school at Châlon. As Lieutenant he participated in the war of Spain. From 1795, under the influence of Monge, he concentrated on mathematics. Gergonne left the army and became professor of transcendental mathematics at Nimes, then, in 1816, he was appointed professor of astronomy at the University of Montpellier, of which he became rector in 1832. In 1810 he founded the first journal entirely devoted to mathematics, the Annales de mathématiques pures et appliquées, which played a leading role in the development of geometry in France, see (Gerini 2002).

  82. We point out that in this case the three intersection points of the opposite sides lie on the line at infinity.

  83. To do that, Gergonne tacitly admitted that the straight line joining the two intersection points of the two pairs of opposite sides does not meet the circle, see below.

  84. See Annales des mathématiques pures et appliquées, vol. 4 (1813–1814), Questions proposées, p. 193.

  85. This note appeared as “par M. B.***, abonné”. We do not know the reason why this note was published semi-anonymously.

  86. On the contest between Poncelet and Gergonne see (Lorenat 2015c, 551–553).

  87. That is numbered at will.

  88. This is a direct consequence of the law of sines.

  89. In a footnote Poncelet stated exactly Brianchon’s proposition quoting (Brianchon 1817).

  90. See Sect. 3.3. Brianchon took the opportunity to recall that this principle was known to the ancients, who had extended it to the sphere, making it the basis of all their trigonometry; and he referred to the Almagest, and to (Maurolico 1575, 281).

  91. In a footnote Brianchon pointed out that, in 1639, Desargues had printed a treatise on conic sections that no longer existed (1817, 15). A manuscript copy of Desargues’ Brouillon project due to La Hire was found by Chasles in 1845, and an original copy was found in the Biblithèque nationale de France by R. Taton in 1951.

  92. Brianchon’s notation.

  93. It is worth noticing that this means that Brianchon determined C, K as the fixed points of the involution defined on the transversal AB by the opposite sides of the (degenerate) quadrilateral UUXX formed by tangents at U and X (as opposite sides), and UX counted twice (as the other two opposite sides).

  94. To see this, Brianchon observed that the triangles EVX, FUW satisfy the conditions of theorem Λ with respect to C, K respectively; that is he used Desargues’ theorem on homological triangles.

  95. See (Maclaurin 1748, Art. 623).

  96. In a footnote Brianchon pointed out, “Depuis la publication du mémoire cite, je suis parvenu à une proposition plus générale; elle est consignée dans le quatrième volume des Annales de Mathématiques, page 196 et 379”. He was referring to the more general result published in (1814b), see Sect. 6.2.

  97. For a historical account on Poncelet’s porism see (Del Centina 2016a).

  98. Brianchon did not give any reference for the theorem he was referring to, and, as far as we know, it does not appear either in (Carnot 1803) or in (Servois 1804). Gauss knew it already in 1810, and provided an analytical proof of it (Gauss 1873, 391), but it is quite improbable that Brianchon was aware of this. However, the theorem in question seems to have been proposed for the first time by J. T. Connor in the Ladies’ Diary for the year 1795, and two solutions were offered shortly afterwards, see (Leybourn 1817, 291).

  99. We do not know which authors Brianchon was referring to.

  100. As it will be seen in the “Appendix”, in solving the same problems Newton used the chords theorem, which, similarly to Pascal’s hexagon theorem, gives a condition for six point to belong to a conic section, but while in the latter case the six points can be taken at will, in the former four of the six points have to lie on parallel straight lines. Thus, although the chords theorem is sufficient to reach the solution, its use affects the simplicity of the proofs.

  101. Brianchon quote it dated 1662, the year of Pascal’s death.

  102. (Poncelet 1822, xiv). For a discussion and justification of this principle in Poncelet we refer to (Nabonnand 2011, 2015), and (Lorenat 2015a).

  103. See for instance (Lorenat 2015a, b, c).

  104. See the fifth section of the first book of the Principia (Newton 1687, 1729).

  105. Newton referred to it simply writing “ex Conicis”, see (Heath 1896, 95).

  106. On this author see (Raynaud 2020).

  107. English translation extracted from (Newton 1729).

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I owe heartfelt thanks to the referees for their thoughtful, inspiring, and helpful suggestions that allowed me to improve this paper.

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Appendices

Appendix

Newton’s methods for determining the conic sections subject to five conditions

To understand the progress made by Brianchon in finding the conic sections subject to five conditions, we compare them with those used by Newton,Footnote 104 which had their roots in the chords theorem that we mentioned previously in Sect. 3.3.

n = 5. Without going into detail, we can say that Newton gave two solutions, one through his Lemma 20, and the other via his organic construction; both depending on the chords theorem.

In the other cases Newton resorted to a special case of the chords theorem, which, see Fig 44. 

Fig. 44
figure 44

a Diagram for the special case of the chords theorem. b This is an enhanced version of the figure provided in (Newton 1687, 84)

Full size image

44a, can be statedFootnote 105: (#) Let two tangents to a conic section be given, PO, QO, touching the curve at P, Q and intersecting each other at O, together with two chords parallel to them, Rr, R′r′, meeting at J. Then

$$\frac{P{O}^{2}}{Q{O}^{2}}=\frac{RJ\times Jr}{R{^{\prime}}J\times r{^{\prime}}J}$$

n = 4. Newton proceeded as follows. He let B, C, D, P be the given points, and IH be the given straight line, which he supposed cut at points I and H by the straight lines CP and BD respectively (Fig. 44b). By applying proposition #, Newton proved that the point of tangency A on IH satisfies the equation

$$\frac{H{A}^{2}}{A{I}^{2}}=\frac{XH\times HY}{BH\times HD}\left(=\frac{CG\times GP}{DG\times GB}\right)\cdot \frac{BH\times HD}{PI\times IC}=\frac{XH\times HY}{PI\times IC}$$

and remarked that A can fall either between H and I, or outside this interval, that is to say that the problem has two solutions.

n = 3. Newton named B, C, D the given points, G the point where the two given straight lines GP and GA meet, and P, A, the points of tangency. He drew the straight line BD, which intersects GP at H and GA at K (Fig. 45a), and called R the point where BD meets the chord of contacts PA. Then he drew the straight line CD, which cuts GP at I, and GA at L; and called S the point where CD meets the chord of contacts PA. By applying proposition (#), he showed that R satisfies the equation

$$H{R}^{2}\times BK\times KR=K{R}^{2}\times BH\times HD$$

and S satisfies the equation

$$I{S}^{2}\times CL\times LD=L{S}^{2}\times CI\times ID$$
Fig. 45
figure 45

a Enhanced version of Newton’s figure for the problem n = 3, which includes one of the sought conic sections. b Diagram for the discussion of the case n = 0

Full size image

He also proved that R, S, P, A are collinear. So if R, S are determined on BD and CD by the above equations, the contact chord, coinciding with SR, is known, and hence also the points of contact P and A are known. Then Newton observed that R may lie between H and K, or outside; and S may lie between I and L, or outside. By observing this Newton implicitly said that there are four conic sections satisfying the required conditions, which can be drawn by resorting to the case n = 5.

We point out that in the cases n = 4, 3, Newton tacitly supposed that the straight line XY to intersect the sought-for conic section. It is worth noticing that in the third edition of the Principia (1726, 87), Newton claimed that the equations to which he was led for the determination of the points of tangency still hold even if XY does not intersect the curve. With this Newton implicitly applied the principle of continuity in the form of permanence of algebraic relations.

n = 2, 1. To treat these cases Newton introduced certain transformations in the plane of the figure which allowed him to solve the problem in a simpler situation, and then to claim the solution of the general problem by applying the inverse transformation. This procedure has been thoroughly discussed in (Milne 1927) and (Whiteside 1961), so we limit ourselves to pointing out a few things. In case n = 2, if B, C are the given points, A is the point where two among the given straight lines meet, and O is the point where the third straight line meets the straight line BC, Newton transformed the given figure in such a way that the first two straight lines become parallel, and the third becomes parallel to the transformation of BC. In case n = 1, Newton transformed the quadrilateral formed by the given straight lines into a parallelogram. Afterwards, in both cases he applied theorem (#) to get the points of tangency.

We observe that in case n = 2 the transformation applied by Newton maps the straight line AO in the line at infinity, while in case n = 1 the transformation applied maps the straight line joining the intersection point of two given lines, say A, with the intersection point of the other two, say O, in the line at infinity. Hence, in both cases the transformations applied by Newton are equivalent to making a perspective of the given figures into a plane parallel to the plane through the point of projection and the straight line AO.

In agreement with Whiteside (1961, 288), we believe that Newton may have drawn inspiration from (La Hire 1673), where similar transformations are introduced, and applied in the study of conic sections. We also point out that La Hire was led to study questions of tangency (La Hire 1672) by Abraham Bosse, an engraver and friend to Desargues, who was interested in solving certain masonry problems connected with the construction of rampant arches. These problems, in fact, require an arc of conic section to be drawn that passes through given points and is tangent to certain straight lines. In this regards, it is interesting to note that Brianchon (1817, 52) instead of (La Hire 1672) quoted. (Blondel 1673),Footnote 106 in which the same problems were treated.

Moreover, it is worth noting that although Newton did not directly resort to the projection method he showed he was aware of it. In fact, after having proved Lemma 22 (1687, 87), Newton wrote “For as often as two conic sections occur…any of them may be transformed…into a circle. So also a right line and a conic section may be transformed into a right line and a circle”.Footnote 107 Enunciations very close to one of the projection principles that we have seen Brianchon put forward.

n = 0. Newton’s method for this case is very long. He needed three more lemmas, the second of which was already known to Apollonius (Conics, III, prop. 42), and some corollaries. We omit all these preliminaries, referring the reader directly to the Principia, or to (Milne 1927). Then, to find the conic section tangent to the five given straight lines, Newton first determined the centre of the curve. Let ABC, BCF, GCD, FDE, EA be the five tangents, and consider the quadrilateral ABFE circumscribed to the sought curve (Fig. 45b). By means of the third lemma, he proved that the straight line joined the midpoints M, N of the diagonals AF, BE is a diameter of the sought curve, and that the same holds true for the straight line joining the midpoints P, Q of the diagonals BD, GF of the circumscribed quadrilateral BGDF. Then the point O intersection of these two diameters is the centre of the sought-for curve. Afterwards he drew LK parallel to BC at such a distance that O is in the middle between LK and BC, K, L being the points where it intersects the tangents FDE and GCD, and denoted R the point where FL meets CK. By a corollary of the second lemma, he proved that the straight line RO meets the tangents CF, KL at the contact points. Thus, determined the five points of contact, the problem was reduced to case n = 5.

As we have seen in Sect. 7.4, Brianchon quoted Newton only when treating the cases n = 0, 1, and to underline that his method was different from Newton’s, but he certainly knew how Newton handled the other cases, and we are left to wonder whether he recognized any similarity with his arguments.

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Del Centina, A. Carnot’s theory of transversals and its applications by Servois and Brianchon: the awakening of synthetic geometry in France. Arch. Hist. Exact Sci. 76, 45–128 (2022). https://doi.org/10.1007/s00407-021-00276-1

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