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

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.

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

√ 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

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

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

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”. 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.” 14/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 14 / 18 1933, Rome took into account the works of Tannery, Suter, a Schoy, Tropfke and refers to the calculation of the sin 2 from Archimedes and to the theorem of Broken Chord, which allowed to calculate the x y chords, when the chords x, y are given.

1966, Becker “.. Archimedes knew and proved the theorem about the sin(x + y), while his predecessor Aristarchos did not seem to possess such a trigonometrical table..”

“Who is the founder of Trigonometry?” an open question

1928, G.A. Miller: “..now we have more substantial reasons for regarding Archimedes as the founder of trigonometry..”

15/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 15 / 18 1966, Becker “.. Archimedes knew and proved the theorem about the sin(x + y), while his predecessor Aristarchos did not seem to possess such a trigonometrical table..”

“Who is the founder of Trigonometry?” an open question

1928, G.A. Miller: “..now we have more substantial reasons for regarding Archimedes as the founder of trigonometry..”

1933, Rome took into account the works of Tannery, Suter, a Schoy, Tropfke and refers to the calculation of the sin 2 from Archimedes and to the theorem of Broken Chord, which allowed to calculate the x y chords, when the chords x, y are given.

15/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 15 / 18 “Who is the founder of Trigonometry?” an open question

1928, G.A. Miller: “..now we have more substantial reasons for regarding Archimedes as the founder of trigonometry..”

1966, Becker “.. Archimedes knew and proved the theorem about the sin(x + y), while his predecessor Aristarchos did not seem to possess such a trigonometrical table..”

15/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 15 / 18 1973, Toomer: “... though Archimedes’ work was important as a basis for later developments, we cannot attribute Trigonometry to him. The credit for the creation of Greek Trigonometry must go to Hipparchus...” 2009, Glen Van Brummelen: “..The extent to which Archimedes was able to practice Trigonometry itself is a matter for interpretation and debate..”

“Who is the founder of Trigonometry?” an open question

1968, Boyer claims that Hipparchus earned the right to be known as “the father of trigonometry”, since he apparently compiled the first trigonometric table.

16/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 16 / 18 2009, Glen Van Brummelen: “..The extent to which Archimedes was able to practice Trigonometry itself is a matter for interpretation and debate..”

“Who is the founder of Trigonometry?” an open question

16/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 16 / 18 “Who is the founder of Trigonometry?” an open question

1968, Boyer claims that Hipparchus earned the right to be known as “the father of trigonometry”, since he apparently compiled the first trigonometric table. 1973, Toomer: “... though Archimedes’ work was important as a basis for later developments, we cannot attribute Trigonometry to him. The credit for the creation of Greek Trigonometry must go to Hipparchus...” 2009, Glen Van Brummelen: “..The extent to which Archimedes was able to practice Trigonometry itself is a matter for interpretation and debate..” 16/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 16 / 18 Conclusion

The question about the founder of Trigonometry is still open.

Hipparchus or Archimedes?

Research will continue!

17/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 17 / 18 Thank you for your attention!

18/18 MARIA DRAKAKI From the Theorem of the Broken Chord to the Beginning of the TrigonometryMay 2020 18 / 18