Christopher M. Graney writes a new paper:
Tycho Brahe, the most prominent and accomplished astronomer of his era, made measurements of the apparent sizes of the Sun, Moon, stars, and planets. From these he showed that within a geocentric cosmos these bodies were of comparable sizes, with the Sun being the largest body and the Moon the smallest. He further showed that within a heliocentric cosmos, the stars had to be absurdly large - with the smallest star dwarfing even the Sun. (The results of Tycho's calculations are illustrated in this paper.) Various Copernicans responded to this issue of observation and geometry by appealing to the power of God: They argued that giant stars were not absurd because even such giant objects were nothing compared to an infinite God, and that in fact the Copernican stars pointed out the power of God to humankind. Tycho rejected this argument.He quotes an 1836 encyclopedia:
The stars, to the naked eye, present diameters varying from a quarter of a minute of space, or less, to as much as two minutes. The telescope was not then invented which show that this is an optical delusion, and that they are points of immeasurably small diameter. It was certain to Tycho Brahé, that if the earth did move, the whole motion of the earth in its orbit did not alter the place of the stars by two minutes, and that consequently they must be so distant, that to have two minutes of apparent diameter, they must be spheres as great a radius at least as the distance from the sun to the earth. This latter distance Tycho Brahé supposed to be 1150 times the semi-diameter of the earth, and the sun about 180 times as great as the earth. Both suppositions are grossly incorrect; but they were common ground, being nearly those of Ptolemy and Copernicus. It followed then, for any thing a real Copernican could show to the contrary, that some of the fixed stars must be 1520 millions of times as great as the earth, or nine millions of times as great as they supposed the sun to be.. Delambre, who comments with brief contempt upon the several arguments of Tycho Brahé, has here only to say, `We should now answer that no star has an apparent diameter of a second.' Undoubtedly, but what would you have answered then, is the reply. The stars were spheres of visible magnitude, and are so still; nobody can deny it who looks at the heavens without a telescope; did Tycho reason wrong because he did not know a fact which could only be known by an instrument invented after his death?The modern view is that motion is relative, and the motion of the Earth depends on your frame of reference. So Copernicus and Tycho were more or less equally correct about the motion of the Earth. Much more important was the collection and assimilation of the data, and the reasoning about the data. Tycho failed to accurately measure the diameter and parallax of the stars, but his reasoning was brilliant based on what he had.