The length of the semi-major axis a of an ellipse is related to the semi-minor axis's length b through the eccentricity e and the semi-latus rectum \ ell, as follows:
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In an ellipse, the semi-latus rectum ( the distance from a focus to the ellipse along a line parallel to the minor axis ) is the harmonic mean of the maximum and minimum distances of the ellipse from a focus.
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The length of the semilatus rectum ( half of the latus rectum ) is the difference between the-coordinates of this point, which is considered as P in the above derivation of the coordinates of the focus, and of the focus itself.
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Also, in terms of r _ { a } and r _ { p }, the semi-major axis a is their arithmetic mean, the semi-minor axis b is their geometric mean, and the semi-latus rectum l is their harmonic mean.
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He also identifies a geometrical criterion for distinguishing between the elliptical case and the others, based on the calculated size of the latus rectum, as a proportion to the distance the orbiting body at closest approach to the center . ( Proposition 17 in the'Principia'.)
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Menaechmus knew that in a parabola, the equation y 2 = " l " x holds, where " l " is a constant called the latus rectum, although he was not aware of the fact that any equation in two unknowns determines a curve.
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