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From the Theorem of the Broken Chord to the Beginning of the Trigonometry

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From the Theorem of the Broken Chord to the Beginning of the Trigonometry From the Theorem of the Broken Chord to the Beginning of the Trigonometry MARIA DRAKAKI Advisor Professor:Michael Lambrou Department of Mathematics and Applied Mathematics University of Crete May 2020 1/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 1 / 18 Introduction. Τhe Theorem Some proofs of it Archimedes and Trigonometry The trigonometric significance of the Theorem of the Broken Chord The sine of half of the angle Is Archimedes the founder of Trigonometry? ”Who is the founder of Trigonometry?” an open question Conclusion Contents The Broken Chord Theorem 2/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 2 / 18 Archimedes and Trigonometry The trigonometric significance of the Theorem of the Broken Chord The sine of half of the angle Is Archimedes the founder of Trigonometry? ”Who is the founder of Trigonometry?” an open question Conclusion Contents The Broken Chord Theorem Introduction. Τhe Theorem Some proofs of it 2/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 2 / 18 The trigonometric significance of the Theorem of the Broken Chord The sine of half of the angle Is Archimedes the founder of Trigonometry? ”Who is the founder of Trigonometry?” an open question Conclusion Contents The Broken Chord Theorem Introduction. Τhe Theorem Some proofs of it Archimedes and Trigonometry 2/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 2 / 18 Is Archimedes the founder of Trigonometry? ”Who is the founder of Trigonometry?” an open question Conclusion Contents The Broken Chord Theorem Introduction. Τhe Theorem Some proofs of it Archimedes and Trigonometry The trigonometric significance of the Theorem of the Broken Chord The sine of half of the angle 2/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 2 / 18 Conclusion Contents The Broken Chord Theorem Introduction. Τhe Theorem Some proofs of it Archimedes and Trigonometry The trigonometric significance of the Theorem of the Broken Chord The sine of half of the angle Is Archimedes the founder of Trigonometry? ”Who is the founder of Trigonometry?” an open question 2/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 2 / 18 Contents The Broken Chord Theorem Introduction. Τhe Theorem Some proofs of it Archimedes and Trigonometry The trigonometric significance of the Theorem of the Broken Chord The sine of half of the angle Is Archimedes the founder of Trigonometry? ”Who is the founder of Trigonometry?” an open question Conclusion 2/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 2 / 18 Βibliography Becker, Oskar Das mathemathische Denken der Antike, VIII, S. 110 Göttingen, 1966. Boyer, Carl A history of Mathematics, Wiley editions, USA, 1968 Von Braunmühl, Anton Vorlesungen über Geschichte der Trigonometrie, Teubner Verlag, Leipzig, 1900. Van Brummelen, GlenThe Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton, 2009. Heath, Thomas, A History of Greek Mathematics, Vol II, XVII, 1921. Heiberg, J.L., Cl.Ptolemaei Opera quae exstant omnia, Bd I, Leipzig 1898-1903. Miller, G.A, Archimedes and Trigonometry, Science, Vol. 67, June 1928, page 555. 3/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 3 / 18 Bibliography Rome, A., Premiers essais de trigonométrie rectiligne chez les grecs, L’antiquite classique, Tome 2, pp 177-192. classique, 2, 177-192, 1933. Schoy, Carl, Die trigonometrischen Lehren des persischen Astronomen Al Biruni, Hannover 1927. Suter, Heinrich, Das Buch der Auffindung der Sehnen im Kreise von Abul-Kaihän Muh. el-Biruni, Zeitschrift für Geschichte der Mathematischen Wissenschaften, Teubner, Leipzig 1910-1911. Tannery, Paul, Recherches sur l’histoire de lastronomie ancienne, Paris, 1893. Toomer, G.J., The Chord Table of Hipparchus and the Early History of Greek Trigonometry, Centaurus, Vol. 18, Issue 1, p.6-28, 1974. Tropfke, Johannes, Archimedes und die Trigonometrie, Archiv für die Geschichte der Mathematik, der Naturwissenschaften und der Technik, 10, 1928, 432-461. Crux Mathematicorum, Volume 6, June-July 1980. 4/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 4 / 18 The History of the Theorem Al Biruni(973-1048) Book on the Derivation of Chords in a Circle Suter (1848-1922) Carl Schoy (1877-1925) Die trigonometrischen Lehren des persischen Astronomen Al Biruni 5/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 5 / 18 The Theorem If AB and BC make up a broken chord in a circle, where BC > AB, and if M is the midpoint of arc ABC, then the foot D of the perpendicular from M on BC is the midpoint of the broken chord ABC. That is AB + BD = DC. 6/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 6 / 18 Proofs of the Theorem of the Broken Chord The 3 Proofs of Archimedes 2. and 3. 1. 7/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 7 / 18 Other proofs Patruno (1980) As-Sidjzi 8/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 8 / 18 The trigonometric significance of the Theorem of the Broken Chord We put arc ΜC=2x and arc BΜ=2y, then arc AB=2x-2y. MC=2sinx, BM=2siny, AB=2sin(x-y) DC=2sinxcosy and DB=2sinycosx From AB + BD = DC i.e AB = DC − BD arises the type: sin(x − y) = sinxcosy − cosxsiny 9/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 9 / 18 Other trigonometric identities (BC = BD + DC) sin(x + y) = sinxcosy + cosxsiny (replacing x with 90-x) cos(x + y) = cosxcosy − sinxsiny and cos(x − y) = cosxcosy + sinxsiny 10/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 10 / 18 The sine of half the angle r 1 − cosa sin a = 2 2 Archimedes Ptolemy 11/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 11 / 18 Archimedes’ work on the heptagon Tabit ibn Qurra (Arabic) Schoy(German) Proposition 14: Let ACB be a semicircle of center Ζ, ΑΒ=d its diameter, AC a chord and D the midpoint of the arc BC. We draw the DB, DC and make AH=AC. We claim that ZB · BH = BD2 12/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 12 / 18 Is Archimedes the founder of Trigonometry? Archimedes has the theoretical background for calculating the x y chords,when the chords x, y are given. knows to calculate the half-angle chord given the whole one. could possess a table of chords. 13/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 13 / 18 1900, Anton von Braunmühl: “..only with Hipparchus, Trigonometry could be considered a true science.” 1910, Suter translated the work of Al-Biruni Book on the Derivation of Chords in a Circle. 1921, Heath: “..The first person to make systematic use of Trigonometry is, as far as we know, Hipparchus.” 1928, Johannes Tropfke: “What Hipparchus achieved in Trigonometry, refers back to Archimedes’ genius.” “Who is the founder of Trigonometry?” an open question 1893, Paul Tannery: “ Hipparchus found prepared ground due to Archimedes and Apollonius”. 14/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 14 / 18 1910, Suter translated the work of Al-Biruni Book on the Derivation of Chords in a Circle. 1921, Heath: “..The first person to make systematic use of Trigonometry is, as far as we know, Hipparchus.” 1928, Johannes Tropfke: “What Hipparchus achieved in Trigonometry, refers back to Archimedes’ genius.” “Who is the founder of Trigonometry?” an open question 1893, Paul Tannery: “ Hipparchus found prepared ground due to Archimedes and Apollonius”. 1900, Anton von Braunmühl: “..only with Hipparchus, Trigonometry could be considered a true science.” 14/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 14 / 18 1921, Heath: “..The first person to make systematic use of Trigonometry is, as far as we know, Hipparchus.” 1928, Johannes Tropfke: “What Hipparchus achieved in Trigonometry, refers back to Archimedes’ genius.” “Who is the founder of Trigonometry?” an open question 1893, Paul Tannery: “ Hipparchus found prepared ground due to Archimedes and Apollonius”. 1900, Anton von Braunmühl: “..only with Hipparchus, Trigonometry could be considered a true science.” 1910, Suter translated the work of Al-Biruni Book on the Derivation of Chords in a Circle. 14/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 14 / 18 1928, Johannes Tropfke: “What Hipparchus achieved in Trigonometry, refers back to Archimedes’ genius.” “Who is the founder of Trigonometry?” an open question 1893, Paul Tannery: “ Hipparchus found prepared ground due to Archimedes and Apollonius”. 1900, Anton von Braunmühl: “..only with Hipparchus, Trigonometry could be considered a true science.” 1910, Suter translated the work of Al-Biruni Book on the Derivation of Chords in a Circle. 1921, Heath: “..The first person to make systematic use of Trigonometry is, as far as we know, Hipparchus.” 14/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 14 / 18 “Who is the founder of Trigonometry?” an open question 1893, Paul Tannery: “ Hipparchus found prepared ground due to Archimedes and Apollonius”.
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