Carnot's Theorem is a theorem in plane trigonometry. It's a generalization of Pythagoras' Theorem to any triangle; it relates the lengths of the sides of a triangle to the cosine of one of its angles. This theorem dates back to Euclid's Elements; we can find it in the propositions II.12 and II.13 in a non-trigonometric form. Analyzing some commentaries to the Elements, written in different historical periods by eminent mathematicians, we notice different approaches and applications of the theorem. Some authors give a "classical" proof of the theorem; others show simply generalizable examples; Deidier gives a proof based on an alternative geometric construction. During the 18th century, Simpson and Playflair generalize proposition II.13 to any triangle (not only acute triangles). Carnot's Theorem, in his non-trigonometric form, found applications to research as well: for instance, the Dutch scientist Christian Huygens applied it to solve analytically the so-called inclination's problem or square's problem. At present, the trigonometric form of the theorem is the most used. For this reason, Chapter 2 of the thesis opens with a short history of trigonometry. Ptolemy's trigonometric tables represent the most influential work in Ancient Times. After them, important developments came from Indian (i.e. Aryabhata and Bhaskara II) and Arabian (i.e. al-Burini, Ulug-Beg, al-Kashi) mathematician-astronomers. During Europe's Middle Ages and Renaissance, mathematicians as Copernico, Regiomontanus and Viète gave the greatest contributions. Both Regiomontanus and Viète also dealt with spherical trigonometry; Viète found the analogue of Carnot's theorem for spherical trigonometry (Law of Cosines). Chapter 3 is dedicated to Lazare Carnot's work. He was a famous French politician and mathematician, lived at the turn of the 18th and 19th centuries. He gave many important contributions to the development of plane and spherical geometry. In his greatest work, Géométrie de position, he proposed a theory on the use of negative numbers (not only in geometry) and about the so-called correlative systems. He applied these theories reaching important results such as a generalization of the theorem bearing his name to any polygon (both plane and skew) and to the faces of any polyhedron. Later on, we present a detailed study about Stewart's Theorem, that can be considered as a Carnot's Theorem generalization. We'll see also how, in the 20th century, the mathematician Alfredo Bassi uses it to solve triangles. Chapter 4 starts with a short excursus on the history of non-Euclidean geometries. Then we introduce the Taurinus' logarithmic-spherical geometry, today known as elliptic geometry. He shows that starting from the Cosines' Law in this geometry and considering an infinitely small radius, we obtain Carnot's Theorem for plane geometry. In the last century, the Italian mathematician Tullio Levi-Civita proved that every formula valid for geodetic triangles on a sphere is also valid for small enough curved triangles on any surface. Following his indications, we show how to adapt Carnot's Theorem, valid for rectilinear plane triangles, to the case of any small curved triangle on the plane. In closing, we search for reasons and origins of the eponym Carnot's Theorem by which this old result is known, at least in Italy.
Il Teorema di Carnot è un teorema trigonometrico della geometria piana di facile e diffusissima applicazione. Si tratta di una generalizzazione del Teorema di Pitagora a triangoli qualsiasi; esso permette di trovare la lunghezza del lato di un triangolo qualsiasi dati gli altri due lati e l’angolo tra essi compreso. Le sue origini risalgono agli Elementi di Euclide che lo enunciò nelle proposizioni II.12 e II.13 in forma non trigonometrica. Analizzando diversi commentari agli Elementi scritti in diverse lingue e diverse epoche, anche da matematici illustri, si possono notare approcci e applicazioni differenti del teorema. Alcuni autori esibiscono una dimostrazione "classica", altri si limitano a fornire esempi generalizzabili; Deidier dà una dimostrazione basata su una costruzione geometrica alternativa rispetto alle altre. Durante il XVIII secolo, Simpson e Playfair generalizzano la proposizione II.13 a triangoli qualsiasi (non solo acutangoli). Il Teorema di Carnot in forma geometrica ha trovato applicazioni anche in ambito di ricerca: il matematico olandese Christian Hugyns ne farà uso nella ricerca di una soluzione analitica al cosiddetto problema delle inclinazioni o del quadrato. La forma oggi più utilizzata del teorema è quella trigonometrica e prevede il calcolo del coseno dell’angolo noto. Per questo, il capitolo 2 si apre con una breve storia sulla trigonometria. L’opera più influente dell’antichità è costituita dalle tavole trigonometriche di Tolomeo. Sviluppi importanti arrivarono ad opera di astronomi-matematici indiani (ad esempio Aryabhata e Bhaskara II) e arabi della scuola di Samarcanda (Al-Biruni, Ulug-Beg, al-Kashi). Nell’Europa medioevale e rinascimentale, i maggiori contributi giunsero da Regiomontano, Niccolò Copernico e François Viète. Sia Regiomontano che Viète si occuparono anche di trigonometria sferica e a quest’ultimo fu attribuita la paternità dell’analogo del Teorema di Carnot per la trigonometria sferica, conosciuto come legge del coseno o Formula di Eulero. Il capitolo 3 è dedicato all’opera di Lazare Carnot. Egli fu un importante uomo politico francese e matematico. Diede diversi contributi allo sviluppo della trigonometria piana e sferica. Nella sua opera maggiore, Géométrie de position, esporrà la sua teoria sul significato dei numeri negativi (non solo in geometria) e sui cosiddetti sistemi correlativi. Applicherà poi questa teoria conseguendo alcuni importanti risultati: una generalizzazione del teorema a poligoni qualsiasi piani o sghembi e una generalizzazione alle facce di poliedri qualsiasi. Segue un approfondimento sul teorema di Stewart, che può essere considerato come una generalizzazione del teorema di Carnot. Vedremo che il matematico italiano Alfredo Bassi all’inizio del XX secolo lo utilizzerà per la risoluzione dei triangoli. Il capitolo IV si apre con un breve excursus sulla storia delle geometrie non euclidee per poi introdurre il lavoro di Taurinus sulla geometria logaritmico-sferica, oggi conosciuta come geometria ellittica. Egli, tra le altre cose, mostrò che a partire dal teorema del coseno per questa geometria, considerando un raggio infinitamente grande, ritroviamo il Teorema di Carnot per la geometria piana. Nel secolo scorso, il matematico italiano Levi Cività dimostrò che le formule valide per i triangoli geodetici sulla sfera sono valide anche per triangoli curvilinei sufficientemente piccoli su superfici qualsiasi. Seguendo le sue indicazioni, si mostra come adattare il Teorema di Carnot, valido per triangoli rettilinei piani, al caso di triangoli curvilinei qualsiasi sul piano. Nel capitolo conclusivo si cercano le motivazioni e le origini della denominazione Teorema di Carnot con cui è conosciuto questo risultato, almeno in Italia.
Il teorema di Carnot. Anzi, di Euclide.
SBERNA, ROBERTA
2020/2021
Abstract
Carnot's Theorem is a theorem in plane trigonometry. It's a generalization of Pythagoras' Theorem to any triangle; it relates the lengths of the sides of a triangle to the cosine of one of its angles. This theorem dates back to Euclid's Elements; we can find it in the propositions II.12 and II.13 in a non-trigonometric form. Analyzing some commentaries to the Elements, written in different historical periods by eminent mathematicians, we notice different approaches and applications of the theorem. Some authors give a "classical" proof of the theorem; others show simply generalizable examples; Deidier gives a proof based on an alternative geometric construction. During the 18th century, Simpson and Playflair generalize proposition II.13 to any triangle (not only acute triangles). Carnot's Theorem, in his non-trigonometric form, found applications to research as well: for instance, the Dutch scientist Christian Huygens applied it to solve analytically the so-called inclination's problem or square's problem. At present, the trigonometric form of the theorem is the most used. For this reason, Chapter 2 of the thesis opens with a short history of trigonometry. Ptolemy's trigonometric tables represent the most influential work in Ancient Times. After them, important developments came from Indian (i.e. Aryabhata and Bhaskara II) and Arabian (i.e. al-Burini, Ulug-Beg, al-Kashi) mathematician-astronomers. During Europe's Middle Ages and Renaissance, mathematicians as Copernico, Regiomontanus and Viète gave the greatest contributions. Both Regiomontanus and Viète also dealt with spherical trigonometry; Viète found the analogue of Carnot's theorem for spherical trigonometry (Law of Cosines). Chapter 3 is dedicated to Lazare Carnot's work. He was a famous French politician and mathematician, lived at the turn of the 18th and 19th centuries. He gave many important contributions to the development of plane and spherical geometry. In his greatest work, Géométrie de position, he proposed a theory on the use of negative numbers (not only in geometry) and about the so-called correlative systems. He applied these theories reaching important results such as a generalization of the theorem bearing his name to any polygon (both plane and skew) and to the faces of any polyhedron. Later on, we present a detailed study about Stewart's Theorem, that can be considered as a Carnot's Theorem generalization. We'll see also how, in the 20th century, the mathematician Alfredo Bassi uses it to solve triangles. Chapter 4 starts with a short excursus on the history of non-Euclidean geometries. Then we introduce the Taurinus' logarithmic-spherical geometry, today known as elliptic geometry. He shows that starting from the Cosines' Law in this geometry and considering an infinitely small radius, we obtain Carnot's Theorem for plane geometry. In the last century, the Italian mathematician Tullio Levi-Civita proved that every formula valid for geodetic triangles on a sphere is also valid for small enough curved triangles on any surface. Following his indications, we show how to adapt Carnot's Theorem, valid for rectilinear plane triangles, to the case of any small curved triangle on the plane. In closing, we search for reasons and origins of the eponym Carnot's Theorem by which this old result is known, at least in Italy.È consentito all'utente scaricare e condividere i documenti disponibili a testo pieno in UNITESI UNIPV nel rispetto della licenza Creative Commons del tipo CC BY NC ND.
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https://hdl.handle.net/20.500.14239/13344