with those near the horizon, so Kepler “went off at a tangent” and tried a totally new set of ideas, which all reduced to the absurdity of a refraction which vanished at the horizon.  These were followed by another set, involving either a constant amount of refraction or one becoming infinite.  He then came to the conclusion that these geometrical methods must fail because the refracted image is not real, and determined to try by analogy only, comparing the equally unreal image formed by a mirror with that formed by refraction in water.  He noticed how the bottom of a vessel containing water appears to rise more and more away from the vertical, and at once jumped to the analogy of a concave mirror, which magnifies the image, while a convex mirror was likened to a rarer medium.  This line of attack also failed him, as did various attempts to find relations between his measurements of refraction and conic sections, and he broke off suddenly with a diatribe against Tycho’s critics, whom he likened to blind men disputing about colours.  Not many years later Snell discovered the true law of refraction, but Kepler’s contribution to the subject, though he failed to discover the actual law, includes several of the adopted “by-laws”.  He noted that atmospheric refraction would alter with the height of the atmosphere and with temperature, and also recognised the fact that rainbow colours depend on the angle of refraction, whether seen in the rainbow itself, or in dew, glass, water, or any similar medium.  He thus came near to anticipating Newton.  Before leaving the subject of Kepler’s optics it will be well to recall that a few years later after hearing of Galileo’s telescope, Kepler suggested that for astronomical purposes two convex lenses should be used, so that there should be a real image where measuring wires could be placed for reference.  He did not carry out the idea himself, and it was left to the Englishman Gascoigne to produce the first instrument on this “Keplerian” principle, universally known as the Astronomical Telescope.

In 1606 came a second treatise on the new star, discussing various theories to account for its appearance, and refusing to accept the notion that it was a “fortuitous concourse of atoms”.  This was followed in 1607 by a treatise on comets, suggested by the comet appearing that year, known as Halley’s comet after its next return.  He regarded comets as “planets” moving in straight lines, never having examined sufficient observations of any comet to convince himself that their paths are curved.  If he had not assumed that they were external to the system and so could not be expected to return, he might have anticipated Halley’s discovery.  Another suggestive remark of his was to the effect that the planets must be self-luminous, as otherwise Mercury and Venus, at any rate, ought to show phases.  This was put to the test not long afterwards by means of Galileo’s telescope.