Eclipses. A brief history of celestial mechanics, astrometry and astrophysics

Solar and lunar eclipses are indeed the first astronomical phenomena which have been recorded since very early antiquity. Their periodicities gave birth to the first luni-solar calendars based on the Methonic cycle since the sixth century before Christ. The Saros cycle of 18.03 years is due to the Chaldean astronomical observations. Their eclipses' observations reported by Ptolemy in the Almagest (Alexandria of Egypt, about 150 a.C.) enabled modern astronomers to recognize the irregular rotation rate of the Earth. The Earth's rotation is some hours in delay after the last three~millenia if we use the present rotation to simulate the 721 b.C. total eclipse in Babylon. This is one of the most important issues in modern celestial mechanics, along with the Earth's axis nutation of 18 yr (discovered in 1737), precession of 25.7 Kyr (discovered by Ipparchus around 150 b.C.) and obliquity of 42 Kyr motions (discovered by Arabic astronomers and assessed from the Middle Ages to the modern era, IX to XVIII centuries). Newtonian and Einstenian gravitational theories explain fully these tiny motions, along with the Lense--Thirring gravitodynamic effect, which required great experimental accuracy. The most accurate lunar and solar theories, or their motion in analytical or numerical form, allow us to predict---along with the lunar limb profile recovered by a Japanese lunar orbiter---the appearance of total, annular solar eclipses or lunar occultations for a given place on Earth. The observation of these events, with precise timing, may permit us to verify the sphericity of the solar profile and its variability. The variation of the solar diameter on a global scale was claimed firstly by Angelo Secchi in the 1860s and more recently by Jack Eddy in 1978. In~both cases, long and accurate observational campaigns started in Rome (1877--1937) and Greenwich Observatories, as well as at Yale University and the NASA and US Naval Observatory (1979--2011) with eclipses and balloon-borne heliometric observations. The IOTA/ES and US sections as well as the ICRA continued the eclipse campaigns. The global variations of the solar diameter over a decadal timescale, and at the millarcsecond level, may reflect some variation in solar energy output, which may explain some past climatic variations (such as the Aller\o d and Dryas periods in Pleistocene), involving the outer layers of the Sun. ``An eclipse never comes alone''; in the eclipse season, lasting about one month, we can have also lunar eclipses. Including the penumbral lunar eclipses, the probability of occurrence is equi-distributed amongst lunar and solar eclipses, but while the lunar eclipses are visible for a whole hemisphere at once, the solar eclipses are not. The color of the umbral shadow on the Moon was known since antiquity, and Galileo (1632, Dialogo sopra i Massimi Sistemi del Mondo) shows clearly these phenomena from copper color to a totally dark, eclipsed full Moon. Three~centuries later, André Danjon was able to correlate that umbral color with the 11-year cycle of solar activity. The forthcoming American total solar eclipse of 8 April 2024 will be probably the eclipse with the largest mediatic impact of the history; we wish that also the scientific impulse toward solar physics and astronomy will be relevant, and the measure of the solar diameter with Baily's beads is indeed one of the topics significantly related to the Sun--Earth connections.