Abstract. To get accurate knowledge about the longitudinal motion of the Moon we must use the oldest preserved interpretable depictions and texts from the ancient cultures. 33 ancient total or almost total solar eclipses, back to 3653 BC, have been used to determine the lunar sidereal secular acceleration in longitude, -30.128 ±0.0035 arcseconds/(century) 2 ("/cy 2 ). The lunar secular acceleration, from the Lunar Laser Range (LLR) measurements, -25.856 ±0.003 "/cy 2 , must be corrected for the relativistic effect, -3.604 "/cy 2 , in the Earth-Moon inertial system, to get the lunar sidereal secular acceleration, -29.460 "/cy 2 , corrected for General Relativity. The difference between this value and the new calibration, -0.668 ±0.0046 "/cy 2 , may be interpreted as the cosmological acceleration by Dvali et al. in a Modified Theory of gravity. This value is equivalent to a Mass of the Graviton = 1.306 ±0.009 x 10 -56 grams and there is no need for the hypothetical Dark Energy.