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306 account of Tycho's lunar theory is very short, and gives no account of the successive steps which led Tycho to his great discoveries in this branch of astronomy. When he wrote the report on his labours at Hveen for his Mechanica, he was already in possession of the discovery of the third inequality in longitude (variation), and of the periodical change of the inclination and of the motion of the node. During his stay at Wittenberg (if not before) he had from observations of eclipses perceived the necessity of introducing an equation in longitude with a period of a year, but the theory had already required so many circles and epicycles that Longomontanus thought it simplest to allow for this equation by using a different equation of time for the moon; and when Tycho did not appear very delighted with this makeshift, the pupil answered his master somewhat rudely, that he might try to find another method himself, which would agree better with the observations.

It must have been a great satisfaction to Tycho to see his researches on the moon reach at least a temporary conclusion, as his mind had of late years been so full of anxiety for the future that he could doubly enjoy a ray of sunshine. The Emperor appears always to have been most friendly to him, and Tycho wrote in March 1600 to his sister Sophia that Rudolph had not only been very kind to him while the plague was raging during the previous winter, and had offered to send him and his family to Vienna while it lasted, but that the Emperor also took the greatest interest in his