MATHS 460 - BALL STATE UNIVERSITY - SUMMER 2000

Introduction

Trig is short for trigonometry, which is the study of relationships between angles and sides of right triangles¹.  The very first theories of trigonometry were developed by Claudius Ptolemy (ca 100-170 AD), a famous astronomer and mathematician.  In his manuscript Almagest, Ptolemy describes his model of our solar system:  the sun, moon, stars, and planets rotating around a fixed Earth.  However, he didn't stop there.  Ptolemy went further in his discussion, attempting to describe the locations of the stars in relevance to other stars.  He discovered most stars are fixed in space, while other "stars" wander around, which is true today.  In fact, these wondering stars would later be know to us as planets.  However at the time, Ptolemy's solar system only contained five planets.  It was these planets that got Ptolemy's attention.  He predicted he could describe their locations with respect to other fixed objects - such as stars and constellations like the Ursa Major (Big Dipper) and Ursa Minor (Little Dipper).  In order to describe the movements of these planets and predict their locations, Ptolemy developed his own mathematics - what we would later call trigonometry.

Ptolemy's Trigonometry

Every branch of mathematics begins somewhere, and for Ptolemy it began with a circle that had a radius of 60 parts, or units.  Ptolemy's mathematics is based on a sexagesimal system or a base 60 system (as opposed to our base 10 system).  Our numeration system is a place-value system.  We have a units place, tens place, hundreds (10²) place, thousands (10³) place, etc. in our numbers, and a value 0-9 fills in each place in the number. Commas separate the places into groups of three - such as 13,456.  The most important property about our base 10 system is this:  each place differs by a multiple of 10.  For example, 10 units equals one ten, ten hundreds equals one thousand.  

Ptolemy's sexagesimal system works in the same way.  However, instead of each place differing from a multiple of 10, they differ by a multiple of 60.  There is still a ones (or units) place, but instead of the next place being a tens place, it is a 60s place.  The next place is a 3600s place (60²), etc.  In order to distinguish the different values, a semi-colon between the 3600s place and the 60s place, and a comma between the 60s place and the units place.  Ptolemy would write 1;12,30 to represent a group of 3600, 12 groups of 60, and 30 units.  Ptolemy's actual numbers written below and follow the same pattern.  Thinking in this way, using a sexagesimal system, Ptolemy's 60 unit circle is the same as our unit circle today - a circle with radius of 1 unit - the foundation of our trigonometric functions we see in today's mathematics textbooks.

Next, he developed a function, called the Crd function (similar to our well-known sine and cosine trigonometric functions).  Using a semi-circle similar to the one in the diagram on the left, he was able to compute some of the common Crd functions.  In order to do this,  Ptolemy first defined some of the distances in his diagram.  AG is the diameter of the circle and is 120 units, and BD, AD, and GD are all radii 60 units long, with BD perpendicular to AG, and BE = EZ.   

From this diagram, Ptolemy discovered the following:

The last point in the list is derived from what we now know as the Pythagorean Theorem which is stated as Proposition 47 in Book I of Euclid's Elements.  Looking at the diagram, we have two right triangles - DBE and DBZ, with BE and BZ the hypotenuse of each triangle, respectively.  Recalling the famous theorem:  The sum of the squares of the legs of any right triangle equals the square of the hypotenuse.  Hence by deduction, the last point in the list is reached, and then we find EZ = √4500.  This is approximately 37;4,55 units.²  This is important because it later gives Ptolemy the ability to calculate Crd 90º and Crd 120º, which are 84;51,10 units and 103;55,23 units, respectively.  According to the figures at the left, R = 60 units.  In figure A, we use the Pythagorean Theorem to determine Crd 90.  Since the chord is a hypotenuse of an isosceles right triangle, the Crd 90º = √60² + 60² or 60√2  which approximates to 84;51,10 sexagesimal units.  In figure B,  Crd 120º = √120² - 60² = 60√3 or approximately 103;55,23 units³

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1.  This definition is provided by the 1982 edition of Webster's New World Dictionary.
2.  These are Ptolemy's actual sexagesimal calculations.
3.  Taken from Classics of Mathematics by Richard Calinger, © 1995, pp. 166-167.

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