The Arabians used geometry, although they added little to its advancement. Geometry had reached at this period an advanced stage of progress in the problems of Euclid. It was to the honor of the Arabians that they were the first of any of the Western peoples to translate Euclid and use it, for it was not until the sixteenth century that it was freely translated into the modern languages. But in trigonometry the Arabians, by the introduction of the use of the sine, or half-chord, of the double arc in the place of the arc itself, made great advancement, especially in the calculations of surveying and astronomy. In the universities and colleges of Spain under Arabian dominion we find, then, that students had an opportunity of mastering nearly all of the useful elementary mathematics. Great attention was paid to the study of astronomy. Here, as before, they used the Greek knowledge, but they advanced the study of the science greatly by the introduction of instruments, such as those for measuring time by the movement of the pendulum and the measurement of the heavenly bodies by the astrolabe. Likewise they employed the word "azimuth" and many other terms which show a more definite knowledge of the relation of the heavenly bodies. They were enabled, also, to {312} measure approximately a degree of latitude. They knew that the earth was of spheroid form. But we find astrology accompanying all this knowledge of astronomy. While the exact knowledge of the heavenly bodies had been developed to a certain degree, the science of star influence, or astrology, was cultivated to a still greater extent. Thus they sought to show the control of mind forces on earth, and, indeed, of all natural forces by the heavenly bodies. This placed mystical lore in the front rank of their philosophical speculations. _Geography and History_.--In the study of the earth the Arabians showed themselves to be practical and accurate geographers. They applied their mathematical and astronomical knowledge to the study of the earth, and thus gave an impulse to exploration. While their theories of the origin of the earth were crude and untenable, their practical writings on the subject derived from real knowledge, and the practical instruction in schools by the use of globes and maps, were of immense practical value. Their history was made up chiefly of the histories of cities and the lives of prominent men. There was no national history of the rise and development of the Arabian kingdom, for historical writing and study were in an undeveloped state.