Footnote 67
Supp. Prop., §41.  Sections 41-43 and 41a-42b are from the MS. in St. John's College, Cambridge.  For the scale of umbra recta, see fig. 1, Plate I.  Observe that the umbra recta is used where the angle of elevation of an object is greater than 45°; the umbra versa, where it is less.  See also fig. 16, Plate VI; where, if AC be the height of the tower, BC the same height minus the height of the observer's eye (supposed to be placed at E), and EB the distance of the observer from the tower, then bc : Eb : : EB : BC.  But Eb is reckoned as 12, and if bc be 4, we find that BC is 3 EB, i.e. 60 feet, when EB is 20.  Hence AC is 60 feet, plus the height of the observer's eye.  The last sentance is to be read thus -- `And if thy "rewle" fall upon 5, then are 5-12ths of the height equivalent to the space between thee and the tower (with addition of thine own height).'  The MS. reads `5 12-partyes þe heygt of þe space,' &c.; but the word of must be transposed, in order to make sense.  It is clear that, if bc = 5, then 5 : 12 : : EB : BC, which is the same as saying that EB = 5/12 BC.  Conversely, BC is 12/5 EB = 48, if EB = 20.