The serious mechanical difficulties connected with the assumption of a rapid motion of the earth which is quite imperceptible to its inhabitants could only be met by further progress in mechanics, and specially in knowledge of the laws according to which the motion of bodies is produced, kept up, changed, or destroyed; in this direction no considerable progress was made before the time of Galilei, whose work falls chiefly into the early 17th century (cf. chapter VI., §§ 116, 130, 133). The objection to the Coppernican scheme that the stars shewed no such apparent annual motions as the motion of the earth should produce (chapter IV., § 92) would also be either answered or strengthened according as improved methods of observation did or did not reveal the required motion. Moreover the _Prussian Tables_, although more accurate 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 III., § 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.