Consider the following passage:
Historians of science frequently dismiss Tycho Brahe’s 1588 geo-heliocentric cosmological model as an ad hoc theological compromise designed to preserve Aristotelian geocentrism against the mathematical elegance of Copernican heliocentrism. However, this appraisal overlooks the compelling empirical obstacle that the Copernican system encountered prior to the invention of the telescope. If the Earth orbited the Sun, the apparent positions of fixed stars should shift relative to one another throughout the year—a phenomenon known as annual stellar parallax. Because no such parallax was detectable with sixteenth-century instruments, Copernicus had asserted that the stars lay at an almost unfathomable distance from Saturn's orbit. Tycho recognized the profound physical corollary of this assertion: given that naked-eye observers routinely measured the apparent angular diameters of third-magnitude stars at approximately two arcminutes, placing those stars at the minimum distance required to eliminate observable parallax would necessitate that individual stars were physical behemoths vastly larger than the Sun itself. To Tycho, an astronomer committed to physical plausibility, positing millions of disproportionately colossal stars simply to salvage terrestrial motion violated the principle of parsimony. Thus, Tycho’s citation of the two-arcminute stellar disk was not an obstinate retreat into dogma, but a mathematically rigorous application of contemporary observational data against an unverified hypothesis.
Statement: In the context of the passage, the author's reference to the 'apparent angular diameters of third-magnitude stars at approximately two arcminutes' functions primarily to identify an observational measurement that, within the geometric framework of heliocentrism, produced a physical consequence that Tycho considered unacceptable.
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