Trig to Ptolemy

The Babylonians set the stage for the development of trigonometry, because they began the recorded study of the sky. They divided the circle into 360 degrees , and further subdivided the degree into 60 minutes and the minute into 60 seconds in order to record the postions of stars. It was apparent from observation that the stars are fixed to a great celestial sphere and that we (the observers) are on a much smaller sphere (the earth) located at the center of the universe. They observed that over the course of a 24 hour day, the sun and stars traced out circular paths from east to west centered roughly about the North star ( Polaris ) in the night sky. Since they themselves felt no movement, the census was that the sphere of the earth is fixed, while the celestial sphere is rotating once a day. The great circle of that revolution is the equator. They observed that over the course of a year, the sun traverses from west to east a great circle (the ecliptic) inclined at an angle of [Maple Math] 51' 12" to the equator. In order to mark the passage, they picked out the 12 constellations of the zodiac in the ecliptic. In addition to the sun, they observed 6 other wanderers of the ecliptic: the moon, and five planets: venus, mars, mercury, jupiter, and saturn. Early on, the astronomers wished to develop a model for the universe which would allow them to predict the movements of these wanderers. This led to the development of trigonometry.

Drawing exercise: Make a sketch of the celestial sphere, with Polaris at the top and the equator the horizontal great circle. Sketch in a small, but not too small, version of earth at the center of the universe. Now sketch in the ecliptic inclined at the appropriate angle intersecting the equator at a point V in front and A in back: these are the vernal and autumnal equinoxes . Mark the high and low points on the ecliptic S and W: these are the s ummer and winter solstices . Finally, draw arrows on the ecliptic and equator showing the direction the sun moves in this sketch.

The classical model of the universe

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Drawing Exercise. Take your sketch and pick a spot on the earth which could represent Lexington (38 degrees N, 84 degrees W). This represents an observer . Draw a short arrow at that spot pointing up at the Zenith (directly overhead). Then draw the great circle on the celestial sphere which is perpendicular to the arrow. This is the horizon (of the observer). Since in the classical model the earth is fixed, the observer and his horizon are fixed also.

Problem. At the vernal equinox, the length of the day in Lexington is 12 hours. (why?) What is the length of the day at the summer solstice? Hint: Find the fraction of the suns path on that day that is above the horizon of an observer at Lexington.

The observer added to the model

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We have spoken of Eudoxus and Apollonius, who developed the geocentric theory of the universe. Apollonius first came up with the idea of epicycles to account for the retrograde motion of mars and other planets. The idea is simple enough: At certain times, Mars appears to stop dead in its tracks and starts back the other way along the ecliptic, then at some later time returns to it normal dircection of motion. Here is a recent example:

"The planet Mars will be going retrograde on Wednesday, February 5, 1997 at 7:24 PM EST at 5 degrees
55 Minutes of Libra. It will be in retrograde motion until Sunday, April 27, 1997 at 3:10 PM EDT when it
goes into direct motion at 16 degrees 44 minutes of Virgo."

Retrograde motion of Mars

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It wasn't until Hipparchus (100 AD) and later Ptolemy (150 AD) defined and tabulated the chord function that the correct parameters to account for the Mars retrograde motion were calculated. Ptolemy went on in the Almagest to establish the theory of plane and spherical trigonometry that could be used to solve many interesting and important problems. Here are a couple of Ptolemys' problems modified to Lexington: (note: these problems can be solved without spherical trig.)

Problem: Find the length of the noon shadow of a 20' pole in Lexington, Kentucky, at the vernal equinox. Note: you could get a pretty good estimate tomorrow at noon. What about at the summer solstice?

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Problem : Find the rising time of the sun in Lexington at the summer solstice.

Hint: This is easy once you know the length of the day.

The chord of an angle alpha was defined by Ptolemy as the length of the chord struck off by a central angle alpha in a circle of some fixed radius R (Usually taken as some power of 60 or of 10). These chords were tabulated. Ptolemy's theorem was used here to express the chord of the difference of two angles in terms of the chords of angles.

Another theorem used by Ptolemy was Menelaus' theorem

Theorem: Given a plane triangle ABC and 3 points X, Y and Z on the sides AB, BC, and CA of the triangle, then X, Y, and Z are collinear if and only if [Maple Math]

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Problem: Prove this theorem: Use vectors. The scribblings below nearly contain a proof.

solution scribbles

This theorem has a 'dual' statement which is called Ceva's theorem .

Theorem: Given a plane triangle ABC and 3 points X, Y and Z on the sides AB, BC, and CA of the triangle, then XC, YA, and ZB are concurrent if and only if [Maple Math]

Problem: Draw a diagram for this theorem and prove the theorem.

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