Hipparchus of Nicaea, working in the 2nd century BC, compiled the very first trigonometric table, earning him the title of the father of trigonometry. Before his work, Sumerian astronomers had divided circles into 360 degrees, and Babylonians had studied ratios of similar triangles, but no one had turned these observations into a systematic method for calculating sides and angles. Hipparchus focused on chords, which are line segments connecting two points on a circle, and used them to solve problems in astronomy and geometry. His tables were analogous to modern sine tables, allowing astronomers to predict the positions of celestial bodies with unprecedented accuracy. This innovation laid the groundwork for future mathematicians to develop more complex functions and theories.
The Indian Sine Revolution
In the 5th century AD, the Indian mathematician Aryabhata introduced the concept of sine, marking a significant shift from the Greek focus on chords. Aryabhata's work, documented in the Surya Siddhanta, provided the earliest known tables of values for trigonometric ratios, which were far more advanced than anything seen in the West at the time. These tables were not just lists of numbers but represented a deep understanding of the relationships between angles and side lengths. The Indian approach to trigonometry was more algebraic and less geometric than the Greek method, allowing for more flexible calculations. This knowledge was later translated and expanded by medieval Islamic mathematicians, bridging the gap between ancient Greek and modern European mathematics.The Islamic Golden Age
During the Islamic Golden Age, Persian mathematicians like Habash al-Hasib al-Marwazi and Abū al-Wafā' al-Būzjānī made groundbreaking contributions to trigonometry. In 830 AD, Habash produced the first table of cotangents, while Abū al-Wafā' developed sine tables with 0.25° increments to 8 decimal places of accuracy. Abū al-Wafā' also created accurate tables of tangent values and made important innovations in spherical trigonometry. The Persian polymath Nasir al-Din al-Tusi is credited with creating trigonometry as a mathematical discipline independent from astronomy. He developed spherical trigonometry into its present form and listed the six distinct cases of a right-angled triangle in spherical trigonometry. His work on the law of sines and the law of tangents for spherical triangles provided proofs that were crucial for future developments in the field.The European Renaissance
Trigonometry reached Western Europe through Latin translations of Ptolemy's Greek Almagest and the works of Persian and Arab astronomers. In the 15th century, the German mathematician Regiomontanus wrote De Triangulis, one of the earliest works on trigonometry by a northern European mathematician. He was encouraged to write this work by the Byzantine Greek scholar cardinal Basilios Bessarion, who provided him with a copy of the Almagest. At the same time, another translation of the Almagest from Greek into Latin was completed by the Cretan George of Trebizond. Trigonometry was still so little known in 16th-century northern Europe that Nicolaus Copernicus devoted two chapters of his De revolutionibus orbium coelestium to explain its basic concepts. The term 'trigonometry' was first used by Bartholomaeus Pitiscus in his 1595 publication Trigonometria, marking the formal recognition of the field as a distinct branch of mathematics.