[Footnote 373: See Humboldt's _Kosmos_, bd. i. p. 294; Klaproth, _Lettre a M. de Humboldt sur l'invention de la boussole_, pp. 41, 45, 50, 66, 79, 90. But some of Klaproth's conclusions have been doubted: "Pour la boussole, rien ne prouve que les Chinois l'aient employee pour la navigation, tandis que nous la trouvons des le xi^{e} siecle chez les Arabes qui s'en servent non seulement dans leurs traversees maritimes, mais dans les voyages de caravanes au milieu des deserts," etc. Sedillot, _Histoire des Arabes_, tom. ii. p. 130.] [Footnote 374: Is it not a curious instance of human perversity that while customary usage from time immemorial has characterized as "acts of God" such horrible events as famines, pestilences, and earthquakes, on the other hand when some purely beneficent invention has appeared, such as the mariner's compass or the printing press, it has commonly been accredited to the Devil? The case of Dr. Faustus is the most familiar example.] [Footnote 375: This version is cited from Major's _Prince Henry the Navigator_, p. 58.] [Sidenote: Latitude and longitude.] That time was after all not so long in arriving, for by the end of the thirteenth century the compass had come to be quite generally used,[376] and the direction of a ship's course could be watched continuously in foul and fair weather alike. For taking the sun's altitude rude astrolabes and jack-staffs were in use, very crazy affairs as compared with the modern quadrant, but sufficiently accurate to enable a well-trained observer, in calculating his latitude, to get somewhere within two or three degrees of the truth. In calculating longitude the error was apt to be much greater, for in the absence of chronometers there were no accurate means for marking differences in time. It was necessary to depend upon the dead-reckoning, and the custom was first to sail due north or south to the parallel of the place of destination and then to turn at right angles and sail due east or west. Errors of eight or even ten degrees were not uncommon. Thus at the end of a long outward voyage the ship might find itself a hundred miles or more to the north or south, and six or seven hundred miles to the east or west, of the point at which it had been aimed. Under all these difficulties, the approximations made to correct sailing by the most skilful mariners were sometimes wonderful. Doubtless this very poverty of resources served to sharpen their watchful sagacity.[377] To sail the seas was in those days a task requiring high mental equipment; it was no work for your commonplace skipper. Human faculty was taxed to its utmost, and human courage has never been more grandly displayed than by the glorious sailors of the fifteenth and sixteenth centuries.