Hermann Weyl succeeded in presenting a consistent overarching analysis that accounts for symmetry in (1) material artifacts, (2) natural phenomena, and (3) physical theories. Weyl showed that group theory is the underlying mathematical structure for symmetry in all three domains. But in this study Weyl did not include appeals to symmetry arguments which, for example, Einstein expressed as “for reasons of symmetry” (wegen der Symmetrie; aus Symmetriegründen). An argument typically takes the form of a set of premises and rules of inference that lead to a conclusion. Symmetry may enter an argument both in the premises and the rules of inference, and the resulting conclusion may also exhibit symmetrical properties. Taking our cue from Pierre Curie, we distinguish two categories of symmetry arguments, axiomatic and heuristic; they will be defined and then illustrated by historical cases.

Abraham ibn Ezra the Spaniard (d. 1167) was one of the foremost transmitters of Arabic science to the West. His astrological and astronomical works, written in Hebrew and later translated into Latin, were considered authoritative by many medieval Jewish and Christian scholars. Some of the works he translated from Arabic are no longer extant in their original form, and on occasion his treatises provide information about earlier sources that is otherwise poorly preserved, if at all. Ibn Ezra seems to be the earliest scholar to record one of the seven methods for setting up the astrological houses, and this method was subsequently used by Levi ben Gerson (d. 1344) in southern France.

At the end of the sixteenth century astronomers and others felt compelled to choose among different cosmologies. For Tycho Brahe, who played a central role in these debates, the intersection of the spheres of Mars and the Sun was an outstanding problem that had to be resolved before he made his choice. His ultimate solution was to eliminate celestial spheres in favour of fluid heavens, a crucial step in the abandonment of the Ptolemaic system and the demise of Aristotelian celestial physics. These debates involved issues that had not previously been part of astronomy, and had the effect of undermining the traditional hierarchy of the sciences. While this complicated story involves many scientific personalities of the sixteenth and seventeenth centuries, in the present paper we will concentrate on one figure who has been assigned an unnecessarily minor role in most histories of science: Christoph Rothmann. In the present paper we show that ‘the dissolution of the celestial spheres’ depends on arguments about the substance of the heavens, following a mistaken argument of Gemma Frisius, elaborated by Joannes Pena and appropriated by Rothmann. We next consider the status and origin of the doctrine, as presented by Brahe, that the heavenly spheres are solid, and the impact on Brahe of Rothmann's treatise on the comet of 1585. Rothmann provided several key ideas that enabled Brahe to develop his system, and we suggest in passing that Rothmann may also have precipitated Brahe's re-evaluation of his attempt to detect the parallax of Mars during the opposition of 1582–83. We offer a new account of this central piece of evidence for the Tychonic system.