It should be recalled, however, that this supposition of tangible spheres for the various planetary and stellar orbits was a mediaeval interpretation of Ptolemy's theory rather than an interpretation of Ptolemy himself, there being nothing to show that the Alexandrian astronomer regarded his cycles and epicycles as other than theoretical.
An interesting practical discovery made by Tycho was his method of determining the latitude of a place by means of two observations made at an interval of twelve hours.

Hitherto it had been necessary to observe the sun's angle on the equinoctial days, a period of six months being therefore required.

Tycho measured the angle of elevation of some star situated near the pole, when on the meridian, and then, twelve hours later, measured the angle of elevation of the same star when it again came to the meridian at the opposite point of its apparent circle about the polestar.

Half the sum of these angles gives the latitude of the place of observation.
As illustrating the accuracy of Tycho's observations, it may be noted that he rediscovered a third inequality of the moon's motion at its variation, he, in common with other European astronomers, being then quite unaware that this inequality had been observed by an Arabian astronomer.

Tycho proved also that the angle of inclination of the moon's orbit to the ecliptic is subject to slight variation.
The very brilliant new star which shone forth suddenly in the constellation of Cassiopeia in the year 1572, was made the object of special studies by Tycho, who proved that the star had no sensible parallax and consequently was far beyond the planetary regions.