Essay on the History of Trigonometry

McGill, Sept 2007 Essay on the history of trigonometry Trigonometry is an entity born out of astrological phenomena. It does not span from any one man or civilization. Much of the work done by the Greeks at around the 2nd century C.E. involved the geometric analysis of the celestial bodies. It analyzed their movement patterns to aid navigation, climate prediction etc. This led the Greeks, viz. Ptolemy (100-178 AD), to conduct extensive work in the field of plane and spherical geometry. The Greeks were already theorizing the shape and celestial makeup of the earth. They believed it was spherical, stationary in the middle of a celestial sphere. Logic led them to this conclusion, and through observation of the celestial bodies they noticed patterns in climate and the occurrence of celestial events (e.g. solar eclipse). So study of astrology has potentially vast economic benefit. Initially, Ptolemy was intrigued by the nature of the observed movements; circular paths over spherical spaces. For example, the ‘great circle’ path of the sun, west-to-east through the stars. It was called the ecliptic, and many other paths were recorded by Ptolemy. His work marries well with the work of Eudoxus, who invented the ‘two sphere’ model, turning astronomy into a mathematical science, using spheres as computational devices. But before Ptolemy, the Greeks were already experimenting with different ideas. Appollonius (250-175 BCE) did much to advance astronomical calculation in the way of eccenters and epicylces. They parameterized the problem of varying lengths of season by centering the orbit of the sun away from earth. These geometric models led to the invention of trigonometry as we know it. In particular, the epicycle was an advanced concept for the time, involving the movement of a body (e.g. the sun) in a small circle, the centre of which orbits another body (the earth). Complicated planetary movements could be explained by this notion and since the relative position of any planet could be found by appropriate triangles, trigonometry naturally followed. It was Hipparchus (190-120 BCE) that did much to deal quantitatively with the positions of the bodies. He initiated the division of the circle circumference into 360 equal parts. But instead of using the ecliptic coordinate system (there is evidence of its early use by Babylonians), he used a system of angle ascension and declination based on the celestial equator. This led him to attempt the tabulation of lengths which would enable plane triangles to be solved. One basic element of his analysis was a chord subtending a given arc in a circle of fixed radius. He produced a table of chords and even came up with a result equivalent to sin^2 + cos^2 = 1, and the standard half angle formula. These are the early roots of (Greek) trigonometry. 1 McGill, Sept 2007 However, it was Ptolemy who amalgamated important Greek works on astronomy into the 13 books ‘Mathematical Collection’. His first calculation established the chord of 36 degrees (length of a side of a decagon inscribed in a circle). It is from this analysis that he found the relation crd(72) = root(r-sq + crd(36)-sq) and by working through Euclidean geometric propositions, he was able to produce extensive chord tables. His work into the properties of chords led him to formulate ‘Ptolemy’s Theorem’ i.e. for a quadrilateral inscribed in a circle, the product of the diagonals equals the sum of the products of the opposite sides. Ptolemy’s chord tables were remarkable because of how they were formulated. He evidently used formulas equivalent to the sine difference formula, the cosine sum formula and the half angle formula (originally the work of Hipparchus). This greatly improved the accuracy of his tables, which were, in modern terms, accurate to 5 decimal places. To improve the ease of calculation and therefore to improve the efficiency of his tables, Ptolemy used a circle of radius 60 to derive his chord lengths. The stage was now set for the analysis of plane triangles. Ptolemy adjusted the tabulated chord length values where necessary (i.e. where a circle of radius 60 didn’t suffice). Using his tables, he was able to solve many angular problems involving triangles. One famous example was calculating the length of the noon shadow cast by a pole of length 60 in Rhodes, Greece. Without going into detail regarding the proofs and technicalities, his calculations led him to solve for the ‘leg’ of the right triangle formed. Another example shows how Ptolemy calculated the parameters for the eccentric model of the sun. A further example is Ptolemy’s solution for an oblique triangle. Again, this had much to do with the eccentric model for the sun. His application of early Greek geometric principles to astronomical phenomena evolved trigonometric concepts. His various calculations even led him to use the equivalent of the law of sines. Hipparchus’ and Ptolemy’s contributions to trigonometric measurement were, to a large extent, specialized to the movement of celestial bodies. The same can be said of Chinese and Indian contributions to trigonometry. In the eighth century CE Chinese astronomers did use genuine trigonometric methods involving tables of tangents calculated for various angles. The motivation behind this was to predict various celestial events such as eclipses. Early Indian trigonometry involved spherical trigonometric calculations derived from astronomical observations. They found it easier to use a table of half-chords rather than using chords, as Ptolemy did (in order to solve triangle problems, Ptolemy had to use half the chord with double the angle). The concept of half-chord translates to the modern notation of sine. Therefore, sometime during the 5th century, Indian mathematicians had constructed a table of sines and by the 6th century, cosines. The motivation was again astronomical, and aside from the obvious economic benefits of climate prediction, mathematicians of ancient times were to a greater extent motivated by the solace of the heavens. As civilized societies formed, mathematical concepts emerged. Driven by the curiosity of man, many of these concepts flourished into practical economic tools. From the place value notation to the decimal fraction system, concepts have been scrutinised, enhanced and fused. Brought together by vast empires and wise men, these concepts of counting, notation and arithmetic have shaped the world in which we live. 2 .

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