The fall equinox

The equinox is the period of the year when the suns pass by the ecuador, so the day has got a duration equals to the night duration, from a polar circle to the other one.

"The rotation axis has always the same inclination, while the Earth takes, during the year, all the positions regarding to the Sun (...) the lighting cicle (of the globe), namely the big circle which separates the lighted hemisphere from the shadowed one, passes by the poles on the equinoxes (march 21 and september 23). At this dates, all the points on the globe's surface are brought to the light by the rotation movement, and stay there as long as in the shadow. the day is equal to the night on all the Earth. The name of equinox comes from this fact."

E. DE MARTONNE, Traité de géographie physique, t. I, p. 39.

Etymologically, the word "equinox" comes from latin «æquinoctium», from æquus «equal», and nox, noctis «night».

The moments of the spring equinox and the fall equinox change every year, and are not necessarily march 21 for the spring or september 21 (or 23) for the fall. The first day of the Republican Calendar is the day of the fall equinox (in the northen hemisphere). There is no direct matching between the gregorian calendar dates and the republican calendar dates. Vendemiaire 1 may be september 20, 21, 22, 23 or 24, according to the year.

For more information about how to calculate equinoxes and solstices, the book by Jean de Meeus «Astronomical Algorithms» is highly recomended. Some interesting facts too on the web page by Alain Gauthier, and on Astronomy and numerical software source codes.