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The ancients thought of the stars as luminous points immovably attached to a spherical shell which covered in the flat earth much as a telescope-dome covers in the telescope, so that when one star differed from another in glory, it was not because the two stars were at different distances from us, but because one was intrinsically more luminous than the other.
Hipparchus introduced the conception of “magnitude” as measuring the brightnesses of the stars, and Ptolemy, in his Almagest, divided the stars into six groups of six different magnitudes. The 20 brightest stars formed the first magnitude stars, while stars which were only just visible to the eye were the sixth magnitude stars. Thus Ptolemy regarded the differences of visible glory as being represented by five steps, each step down being represented as an increase of one magnitude.
According to the well-known physiological law of Fechner, the effect which any cause produces on our senses is proportional to the logarithm of the cause. If we can just, and only just, appreciate the difference between 10 and 11, we shall not notice any difference at all between 20 and 21, but shall just be able to detect the difference between 20 and 22, or between 5 and 5½. Our senses do not supply us with a direct estimate of the intensity of the phenomenon which is affecting them, but of its logarithm.
368. The original aim of cosmogony was to discover the origin of the solar system, but the whole history of cosmogony illustrates how nothing fails so surely in science as the direct frontal attack. The plan of action in the present book has been to study the various transformations which astronomical matter must undergo through the action of physical forces, identifying the formations predicted by theory with those observed in the sky when possible. In this way it has proved possible to trace out the origin and evolution of many astronomical objects, including elliptical and spiral nebulae, star clusters of various forms, binary and multiple stars and (conjecturally at least) Cepheid and long-period variables. But nowhere have we come upon anything bearing the least resemblance to the solar system.
If the sun had been unattended by planets, its origin and evolution would have presented no difficulty. It would have been a quite ordinary star, born out of a nebula in the ordinary way, but endowed with insufficient rotation to carry it on to the later stages of fission into a binary or multiple system; it could in fact be supposed to have had precisely the same evolutionary career as half of the stars in the sky. In support of the conjecture that the sun had stopped short of fission on its evolutionary career we should only have had to note the slowness of its present rotation.