In this paper Euler presents a method for determining solar noon, the time at which the Sun crosses the meridian. The method requires the times of two observations of the Sun, one in the morning and one in the afternoon, at equal altitudes above the horizon. Solar noon is approximately the midpoint between two such observations, but, since the declination of the Sun will have changed during the day, a correction term, called the equation of noon, is required. Euler explains that this term is too large to ignore and discusses the table of values constructed by de la Hire; this table applies only at the latitude of Paris and relies on laborious calculations. For his own method, Euler describes the apparent motion of the Sun using spherical trigonometry and then uses differentials to complete the calculation with sufficient accuracy for his purposes. He provides examples and claims that his method makes it practical to construct a table at whatever latitude is required.
Headley, Patrick T.
"A Method for Calculating the Equation of Noon (an English translation of Methodus Computandi Aequationem Meridiei),"
Euleriana: 2(2), pp.67-78.
Available at: https://scholarlycommons.pacific.edu/euleriana/vol2/iss2/3