It is characteristic of our history that a gap, almost entirely unbridged, exists between the early period and the sixteenth century in the story of the development of methods of precision in determining geographical position. We have already referred to early efforts to estimate the size of the earth, and in this connection have mentioned that simple instrument of unknown origin, the gnomon. Aristarchus improved upon the mere upright rod whose shadow was measured, by setting one upright in a bowl, the length of the rod and the radius of the bowl being equal; by means of this instrument, which was called the scaph, the angle of altitude could be read on a scale of circles inscribed on the inside of the bowl. Among other early instruments were the astrolabe, an invention attributed to Hipparchus, which served mariners and others down to the seventeenth century; the diopter, which appears to have resembled an alidade mounted on a stand, and may be regarded as a prototype of the theodolite; and Ptolemy’s rods, or the triquetum, in which a rod working upon two others, one vertical while the other pointed to the observed object, enabled the angular zenith distance to be read. It is true that some additions to this list of instruments were made by, or for the benefit of, mediæval mariners before the pregnant period about the beginning of the sixteenth century. Thus the less cumbrous quadrant was early brought into use, to the partial displacement of the circle of the astrolabe. The cross-staff, for measuring the angle between the horizon and the sun, is first described, so far as is known, in 1342. [Illustration: Fig. 9.--Scaph.] [Illustration: (Front.) (Back.) Fig. 10.--Astrolabe.] [Illustration: Fig. 11.--Quadrant.]