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The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure.

Preface to Mecanique Analytique.

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# Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion

## Planetary Periods

The Babylonians found good approximations to the sidereal and synodic periods of each planet by carefully observing the patterns of the planet’s motion over a length of time.   For example, they observed that Mars regularly completes one sidereal cycle (around the ecliptic) in a little under 2 years, and one synodic cycle (from one retrograde motion to the next) in a little over 2 years.  Hence, the sidereal period of Mars is a little under two years and its synodic period is a little over 2 years.

By observing the planet over a longer length of time, more accurate estimates were obtained.  For example, in just under 15 years, Mars completes 7 synodic cycles and just under 8 sidereal cycles; thus the sidereal period of Mars is approximately 1.9 years and its synodic period is approximately 2.1 years.  After a little more than 17 years, Mars completes 8 synodic cycles and a little more than 9 sidereal cycles; adding this 17-year cycle to the previous 15-year cycle, we might expect to get a more accurate 32-year cycle:  Mars completes 15 synodic cycles and just about 17 sidereal cycles in just about 32 years.  Direct observation reveals that this is indeed the case, giving the more accurate estimates of 2.13 years and 1.88 years for the synodic period and the sidereal period, respectively.

In a similar manner, adding the 32-year cycle to the 15-year cycle results in a still more accurate 47-year cycle, and the 47-year cycle added to the 32-year cycle gives the even more accurate 79-year cycle:  Mars completes approximately 42 sidereal cycles and 37 synodic cycles in approximately 79 years.  This last 79-year cycle gives a synodic period of 2.135 years and a sidereal period of 1.881 years.

Cyclical data for each planet is given below.  This data was known to the Babylonians and was used by Claudius Ptolemy in the Almagest to initially develop his models [13, pp. 151-152].  In his final models, Ptolemy modified the data somewhat to give better long-term results [20, pp. 423-424].

Mercury:  46 sidereal cycles and 145 synodic cycles in 46 years

Venus:  8 sidereal cycles and 5 synodic cycles in 8 years

Mars:  42 sidereal cycles and 37 synodic cycles in 79 years

Jupiter:  6 sidereal cycles and 65 synodic cycles in 71 years

Saturn:  2 sidereal cycles and 57 synodic cycles in 59 years

By observing the times of the equinoxes and solstices, ancient astronomers were able to approximate the number of days in one year.  The modern value for this is well known:  there are 365.24 days in one year.  This important parameter together with the data above determines the sidereal and synodic rates of motion (in degrees per day) for each planet.  For example, Mars completes 42 sidereal cycles in 79 years.  Therefore in 79 years it completes 42 x 360° in 79 x 365.24 days.  Dividing, this gives a sidereal rate of 0.524°/day.

Table 1 gives the sidereal and synodic periods and rates of motion for each planet determined by the cyclical data listed above.

 Sidereal Period T (in years) Sidereal Rate (°/day) Synodic Period S (in years) Synodic Rate (°/day) Mercury 1 0.986 0.317 3.11 Venus 1 0.986 1.6 0.616 Mars 1.88 0.524 2.14 0.462 Jupiter 11.8 0.0833 1.09 0.902 Saturn 29.5 0.0334 1.04 0.952

Table 1

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Caravella, Sandra M., "Triangles in the Sky: Trigonometry and Early Theories of Planetary Motion," Loci (October 2008), DOI: 10.4169/loci003120