Ptolemy’s two chief works, the _Geography_ in eight books, and ἡ μαθηματικὴ σύνταξις, or _Almagest_ (al-μεγίστη) as the Arabs called it, in thirteen books, have been so often described in histories of mathematics, astronomy, geography, and discovery that such outline of their contents need not be repeated here. The erroneous Ptolemaic theories of a geocentric universe and of an earth’s surface on which dry land preponderated are equally well known. What is more to the point at present is to note that one of these theories was so well fitted to actual scientific observations and the other was thought to be so similarly based, that they stood the test of theory, criticism, and practice for over a thousand years.[494] It should, however, be said that the _Geography_ does not seem to have been translated into Latin until the opening of the fifteenth century,[495] when Jacobus Angelus made a translation for Pope Alexander V, (1409-1410), which is extant in many manuscripts[496] as well as in print.[497] It therefore did not have the influence and fame in the Latin middle ages that the _Almagest_ did or the briefer astrological writings, genuine and spurious, current under Ptolemy’s name. [Sidenote: His mathematical method.] We may briefly state one or two of Ptolemy’s greatest contributions to mathematical and natural science and his probable position in the history of experimental method. Perhaps of greater consequence in the history of science than any one specific thing he did was his continual reliance upon mathematical method both in his astronomy and his geography. In particular may be noted his important contribution to trigonometry in his table of chords, which modern scholars have found correct to five decimal places, and his contribution to the science of cartography by his successful projection of spherical surfaces upon flat maps. [Sidenote: Attitude towards authority and observation.]