§ 40. It is possible that this conclusion was really intended to belong to the Fourth Part of the treatise, and was written by way of instalment. See the Prologue, ll. 67-72. It is curious that in all the best MSS. (P. excepted) the last sentence should be incomplete. 13. This sentence is very awkward. It seems to mean--'and then set I the point of F upward in the same sign, because that the latitude was north, upon the latitude of Venus; that is to say, (I set it upward) keeping it in the 6th degree of Capricorn.' _Upward_ means inward, i.e. towards the centre or towards the north; the opposite being expressed by southward, or outward, or toward the border, as in l. 48 below. _Upon the latitude of Venus_ means that the point F of the compass was set above the second degree of latitude, so that the space between the legs of the compass became equal to 2 degrees, as said in l. 16. Lastly, the words _that is to seyn, in the 6 degree_, &c., are an explanation of the vaguer expression _in the same signe_. The repetition of the words _that is to seyn_, &c. (ll. 12 and 14), is intended to draw attention to the necessity of keeping _both_ legs of the compass in the same degree of longitude (A on the zodiac, and F to the north of it). 57. Possibly Chaucer left the sentence incomplete. The words 'thou shalt do well enough' may easily have been added by another hand to bring the sentence to an apparent, though not wholly satisfactory, conclusion. The colophon is written (in a later hand) in MS. A. at the bottom of the page, a part of which, after the words 'howre after howre,' is left blank. 41-43. I have mended the text as well as I could by inserting words, and adopting different readings. Nearly all the emendations rest on authority; see the Critical Notes. The text is not a good one, but I do not see why these sections may not have been written by Chaucer. For a definition of the terms 'Umbra Extensa' and 'Umbra Versa' see sections 5 and 6 of the Practica Chilindri of John Hoveden, published by the Chaucer Society. The _umbra extensa_ or _recta_ is the shadow cast on a plain by any perfectly upright object; but the restriction is commonly introduced, that the altitude of the sun shall exceed 45º. The _umbra versa_ is the shadow cast _perpendicularly_ downwards along a wall by a style which projects from the wall at right angles to it; the restriction is commonly introduced, that the sun's altitude shall be less than 45°. The _umbra versa_ is the one which appeared on the 'chylindre'; hence John de Hoveden explains how to calculate the altitude of an object by it.