Kepler's First Law

To put it briefly, The Law of Ellipses states that the orbit of a planet traces an elliptical path. All ellipses have two focii, and in this case, one of them is the Sun.

image from UStudy Initiative

     He came to this conclusion after studying Tycho Brahe's observations on Mars. Conventional thought was that all planetary orbits were circular. Kepler's circular model caused the satellite to move too fast at aphelion and perihelion and then move too slowly at the sides.

     As a way to fix this problem, Kepler chose to alter the shape of the orbit. He was able to find the right information from Brahe's observations to support his mathematical calculations and prove that the orbit was elliptical. He extended this theory to the orbits of the other planets and gave evidence to support his assertions after doings thousands of pages of arithmetic.

     Kepler did not yet have calculus to prove this, so he worked through it another way.
He did this by first locating the equant point, the place in the middle of the two focii from which all of the planets appear to have the same angular velocity (rate of change of the angle of displacement of an object in its orbit). It is expressed in this manner:

Sir Isaac Newton later would return to Kepler's laws to apply them to calculus, applying inverse square law to help explain his work.

Mathematically formulated: Intensity ∝ 1/(distance ^2)

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Reference https://www.physicsforums.com/threads/what-does-this-symbol-mean.225342/

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Reference https://www.physicsforums.com/threads/what-does-this-symbol-mean.225342/

--and it can actually be used in many applications of physics, including gravity.

image from Physics Classroom

"The Apple, the Moon, and the Inverse Square Law." The Physics Classroom. N.p., n.d. Web. 22 Apr. 2015. <http://www.physicsclassroom.com/class/circles/Lesson-3/The-Apple,-the-Moon,-and-the-Inverse-Square-Law>.

Hwang, Jenny. "Kepler and the First Law of Planetary Motion." Kepler and the First Law of Planetary Motion. Berkeley University, n.d. Web. 22 Apr. 2015. <https://math.berkeley.edu/~robin/Kepler/>.

Knisley, Jeff. "The Inverse Square Law." Vectors. Dept. of Math., East Tennessee State University, 2001. Web. 22 Apr. 2015. <http://math.etsu.edu/multicalc/prealpha/Capstone/intro.htm>.

Nave, Carl R. "Kepler's Laws." Kepler's Laws. Hyperphysics, Georgia State University, 2000. Web. 22 Apr. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html>