Abstract
The geometer Michel Chasles, one of the nineteenth–century personages who was both a master and a masterly historian of his favorite subject, discussed the contributions of the “illustrious Carnot” after the descriptive geometry of Monge and before the projective geometry of Poncelet in the sequence to which he attributed revival of the methods of Desargues and Pascal and a consequent renaissance of geometry following its eclipse by eighteenth–century analysis (Chasles 1875, p 210). It was a central feature of the careers of those Chasles singled out that each of them had his place in the tradition of engineering mechanics and occupied himself in the science of machines concurrently with the new geometry.
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Notes
- 1.
Hauff’s translation appeared under the title, Betrachtungen über die Theorie der Infinitesimalrechnung von dem Burger Carnot (Carnot 1800b). The relevant papers are in carton 28 in the Carnot family archives at Nolay .
- 2.
Geometrie der Stellung oder über die Anwendung der Analysis auf Geometrie, 2 vols (Carnot 1810, II).
- 3.
Ettore Carruccio , “Bellavitis ”, Dictionary of Scientific Biography (Gillispie 1970–1980, I, pp 590–592). See also Vorlesungen über die Entwicklung der Mathematik im 19 (Klein 1926, I, p 79). Lazare Carnot introduced the concept and word “equipollence” (Carnot 1803b, § 82, pp 83–84) to signify the equivalence, not just of values, but of any mathematical objects such as points or curves that could be substituted for one another.
- 4.
On “inverse quantity” see: Carnot (1806a, p 101, p 102, p 103, p 104, p 111).
- 5.
On “direct quantity” see: Carnot (1806a, p 100, 102, p 111).
- 6.
Digression sur la nature des quantités dites négatives (Carnot 1806a, p 102).
- 7.
It is interesting that the first published hint of the correlation of figures, and also the first original scientific work to which Carnot seems to have turned after the Revolution , appeared in the form of a “Lettre du citoyen Carnot au citoyen Bossut , contenant quelques vues nouvelles sur la trigonométrie”. It is appended to Bossut’s Géométrie et application de l’algèbre à la géométrie (Bossut 1800a, II, pp 401–421). Bossut recounts that just as his book was about to go to press, Carnot , then serving as Minister of War , mentioned to him several ingenious and stimulating ideas on trigonometry. At Bossut’s persuasion, he agreed to write them up for publication in a way suited to an elementary text. He had leisure to give only the results, in any case, leaving the proofs to the students as an exercise. The subject is that which he developed fully in section 4 of Géométrie de position , the relation between the linear quantities of a triangle eliminating all angular quantities from the expressions.
- 8.
In a letter to Lacroix of “4 Nivôse an 10 [25 December 1801]” Lazare Carnot observes that he is spending his leisure developing the ideas in the De la corrélation des figures de géométrie , which had suffered from the haste with which it was composed, for a new edition containing a chapter on the applications of algebra to geometry and another on the theory of curves, neither of them a treatise in itself, but applications of his theory of positive and negative quantity and of transversals (Bibliothèque de l’Institut de France , Mss. 2397).
- 9.
This example is worked in great detail on Carnot (1803b, pp 81–87).
- 10.
Carnot (1803b, Problem XXIV, § 189, p 255).
- 11.
Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (Klein 1926, I, pp 79–80). This proposition appears as “Theorem V ” (Carnot 1803b, § 219, pp 276–278).
Fig. 5.3 
Simplified performance model adapted by Klein (1926)
The triangle GBA being intersected by the transversal CF,
$$ \overline {\mathit{AF}} \cdot \overline {\mathit{BD}} \cdot \overline {\mathit{GC}} \cdot \overline {\mathit{AC}} \cdot \overline {\mathit{BF}} \cdot \overline {\mathit{GD}} $$ - 12.
Carnot (1803b, p 483).
- 13.
Carnot (1803b, p xxxiij).
- 14.
Carnot (1803b, § 298, p 338).
- 15.
Page references are to the most recent edition, that published in Paris in 1921 in the series Les maitres de la pensée scientifique (Carnot 1803b). Paragraph numbers apply equally to other, earlier French editions.
- 16.
See above, Chapter 3, pp 66–67.
- 17.
Carnot (1921, I, § 1, p 2).
- 18.
On that see Théorie des fonctions analytiques contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits, de limites et de fluxions, et réduits à l’analyse algébrique des quantités finies (Lagrange 1797 ).
- 19.
Carnot (1921, I, § 37, pp 39–40).
- 20.
- 21.
In his introductory essay to the Russian translation of the Réflexions sur la métaphysique du calcul infinitésimal translated by Soloviev N.M., with a critical introduction by A. P. Youschkevitch and a biographical sketch of Lazare Carnot by M. E. Podgorny (Carnot 1933, pp 16–18).
- 22.
Carnot (1921, II, § 129, pp 33–34).
- 23.
Carnot (1921), Note Relative au n° 162 de L’Ouvrage précédent, § 20, pp 103–104. Lazare Carnot incorporated a more extensive comparison of analysis to synthesis in the Digression sur la nature des quantités dites négatives (Carnot 1806a).
- 24.
It is very interesting that his first discussion of the method should have occurred in the concluding passages of the theoretical portion of the 1780 prize memoir on the theory of machines (Carnot 1780 in Gillispie 1971, Appendix C, §§101–160, pp 299–343). His hope clearly was to qualify the very abstract reasoning of his memoir for the contest by relating it to the effects of friction and binding. This, the “useful part” (Ivi, § 160, pp 337–340) of the problem could be stated, “The forces applied to a given machine being known and such that neglecting friction and stiffness in cords there would be equilibrium, determine what must be added to the impelling forces to put the machine on the point of moving so that if these forces were increased, motion would begin” (Ibidem). A rigorous solution being extremely difficult, the method that Carnot was proposing would deliberately introduce an erroneous supposition in order to achieve a solution exact enough to satisfy “any reasonable man” (Ibidem). The supposition was that friction on a point is proportional to pressure, and the resulting error , a small one at worst, was then to be reduced “as far as one wishes” (Ibidem) by successive applications of Carnot ’s empirically determined formulas for the relation of friction to pressure. See Lazare Carnot Savant (Gillispie 1971, Appendix C, §§101–160, pp 299–343).
References
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Gillispie, C.C., Pisano, R. (2014). An Engineering Justification of Algebra and the Calculus. In: Lazare and Sadi Carnot. History of Mechanism and Machine Science, vol 19. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8011-7_5
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