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128 than the Alfonsine, hardly claimed, and certainly did not possess, minute accuracy. Coppernicus had once told Rheticus that he would be extravagantly pleased if he could make his theory agree with observation to within 10'; but as a matter of fact discrepancies of a much more serious character were noticed from time to time. The comparatively small number of observations available and their roughness made it extremely difficult, either to find the most satisfactory numerical data necessary for the detailed development of any theory, or to test the theory properly by comparison of calculated with observed places of the celestial bodies. Accordingly it became evident to more than one astronomer that one of the most pressing needs of the science was that observations should be taken on as large a scale as possible and with the utmost attainable accuracy. To meet this need two schools of observational astronomy, of very unequal excellence, developed during the latter half of the 16th century, and provided a mass of material for the use of the astronomers of the next generation. Fortunately too the same period was marked by rapid progress in algebra and allied branches of mathematics. Of the three great inventions which have so enormously diminished the labour of numerical calculations, one, the so-called Arabic notation (chapter, § 64), was already familiar, the other two (decimal fractions and logarithms) were suggested in the 16th century and were in working order early in the 17th century.

97. The first important set of observations taken after the death of Regiomontanus and Walther (chapter, § 68) were due to the energy of the Landgrave William IV. of Hesse (1532-1592). He was remarkable as a boy for his love of study, and is reported to have had his interest in astronomy created or stimulated when he was little more than 20 by a copy of Apian's beautiful Astronomicum Caesareum, the cardboard models in which he caused to be imitated and developed in metal-work. He went on with the subject seriously, and in 1561 had an observatory built at Cassel, which was remarkable as being the first which had a revolving roof, a device now almost universal. In this he made extensive observations (chiefly of fixed stars) during the next six years. The death of his father then compelled