38. This problem is discussed in arts. 144 and 145 of Hymes's Astronomy, 2nd ed. 1840, p. 84. The words 'for warping' mean 'to prevent the errors which may arise from the plate becoming warped.' The 'broader' of course means 'the larger.' See fig. 15, Plate VI. If the shadow of the sun be observed at a time _before_ midday when its extremity just enters within the circle, and again at a time _after_ midday when it is just passing beyond the circle, the altitude of the sun at these two observations must be the same, and the south line must lie half-way between the two shadows. In the figure, S and S' are the 2 positions of the sun, OT the rod, Ot and Ot' the shadows, and OR the direction of the south line. Ott' is the metal disc. 39. This begins with an explanation of the terms 'meridian' and 'longitude.' 'They chaungen her Almikanteras' means that they differ in latitude. But, when Chaucer speaks of the longitude and latitude of a 'climate,' he means the length and breadth of it. A 'climate' (_clima_) is a belt of the earth included between two fixed parallels of latitude. The ancients reckoned _seven_ climates; in the sixteenth century there were _nine_. The 'latitude of the climate' is the breadth of this belt; the 'longitude' of it he seems to consider as measured along lines lying equidistant between the parallels of latitude of the places from which the climates are named. See Stöffler, fol. 20 _b_; and Petri Apiani Cosmographia, per Gemmam Phrysium restituta, ed. 1574, fol. 7 _b_. The seven climates were as follows:-- 1. That whose central line passes through Meroë (lat. 17°); from nearly 13° to nearly 20°. 2. Central line, through Syene (lat. 24°); from 20° to 27°, nearly. 3. Central line through Alexandria (lat. 31°); from 27° to 34°, nearly. 4. Central line through Rhodes (lat. 36°); from 34° to 39°, nearly. 5. Central line through Rome (lat. 41°); from 39° to 43°, nearly. 6. Central line through Borysthenes (lat. 45°); from 43° to 47°. 7. Through the Riphæan mountains (lat. 48°); from 47° to 50°. But Chaucer must have included an _eighth_ climate (called _ultra Mæotides paludes_) from 50° to 56°; and a _ninth_, from 56° to the pole. The part of the earth to the north of the 7th climate was considered by the ancients to be uninhabitable. A rough drawing of these climates is given in MS. Camb. Univ. Lib. Ii. 3. 3, fol. 33 _b_.