Trigonometry follows a similar path as algebra: it
was developed in the ancient Middle East and through trade and
immigration moved to Greece, India, medieval Arabia and finally Europe
(where consequently, colonialism made it the version most people are
taught today). The timeline of trigonometric discovery is complicated by
the fact that India and Arabia continued to excel in the study for
centuries after the passing of knowledge across cultural borders. For
example, Madhava’s 1400 discovery of the infinite series of sine was
unknown to Europe up through Isaac Newton’s independent discovery in
1670. Due to these complications, we’ll focus exclusively on the
discovery and passage of sine, cosine, and tangent.
Beginning in the Middle East, seventh-century B.C. scholars of Neo-Babylonia determined a technique for computing the rise times of fixed stars on
the zodiac. It takes approximately 10 days for a different fixed star
to rise just before dawn, and there are three fixed stars in each of the
12 zodiacal signs; 10 × 12 × 3 = 360. The number 360 is close enough to
the 365.24 days in a year but far more convenient to work with. Nearly
identical divisions are found in the texts of other ancient
civilizations, such as Egypt and the Indus Valley. According to Uta Merzbach in “A History of Mathematics”
(Wiley, 2011), the adaptation of this Babylonian technique by Greek
scholar Hypsicles of Alexandria around 150 B.C. was likely the
inspiration for Hipparchus of Nicea (190 to 120 B.C.) to begin the trend
of cutting the circle into 360 degrees. Using geometry, Hipparchus
determined trigonometric values (for a function no longer used) for
increments of 7.5 degrees (a 48th of a circle). Ptolemy of Alexandria (A.D. 90 to 168), in his A.D. 148 “Almagest”, furthered the work of Hipparchus by determining trigonometric values for increments of 0.5 degrees (a 720th of a circle) from 0 to 180 degrees.
The oldest record of the sine function comes from fifth-century India
in the work of Aryabhata (476 to 550). Verse 1.12 of the “Aryabhatiya” (499), instead of representing angles in degrees, contains a list of sequential differences of sines of twenty-fourths of a right angle (increments of 3.75 degrees). This was the launching point for much of trigonometry for centuries to come.
The next group of great scholars to inherit trigonometry were from the
Golden Age of Islam. Al-Ma'mun (813 to 833), the seventh caliph of the
Abbasid Caliphate and creator of the House of Wisdom in Baghdad,
sponsored the translation of Ptolemy’s “Almagest” and Aryabhata’s
“Aryabhatiya” into Arabic. Soon after, Al-Khwārizmī (780
to 850) produced accurate sine and cosine tables in “Zīj al-Sindhind”
(820). It is through this work that that knowledge of trigonometry first
came to Europe. According to Gerald Toomer in the “Dictionary of Scientific Biography 7,” while the original Arabic version has been lost, it was edited around 1000 by al-Majriti of Al-Andalus (modern Spain), who likely added tables of tangents before Adelard of Bath (in South England) translated it into Latin in 1126.
For people looking for History of Mathematics resources!
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