42_b._: Here, 'by the craft of _Umbra Recta_' signifies, by a method similar to that in the last article, for which purpose the numbers must be adapted for computation by the _umbra recta_. Moreover, it is clear, from fig. 17, that the numbers 4 and 3 (in lines 2 and 4) must be transposed. If the side parallel to _b_E be called _nm_, and _mn_, E_c_ be produced to meet in _o_, then _mo_ : _m_E :: _b_E : _bc_; or _mo_ : 12 :: 12 : _bc_; or _mo_ = 144, divided by _bc_ (= 3) = 48. Similarly, _m'o'_ = 144, divided by _b'c'_ (= 4) = 36. And, as in the last article, the difference of these is to 12, as the space EE' is to the altitude. This is nothing but Art. 42 in a rather clumsier shape. Hence it appears that there are here but 3 independent propositions, viz. those in articles 41, 42, and 43, corresponding to figs. 16, 17, and 18 respectively. Arts. 41_a_ and 41_b_ are mere repetitions of 41; 42_a_ and 42_b_, of 42; and 43_a_, of 43. CRITICAL NOTES. As, in the preceding pages which contain the text, the lower portion of each page is occupied with a running commentary, such Critical Notes upon the text as seem to be most necessary are here subjoined. TITLE. Tractatus, &c.; adopted from the colophon. MS. F has 'tractatus astrolabii.' A second title, 'Bred and mylk for childeren,' is in MSS. B. and E. [The MSS. are as follows:--A. Cambridge Univ. Lib. Dd. 3. 53.--B. Bodley, E Museo 54.--C. Rawlinson 1370.--D. Ashmole 391.--E. Bodley 619.--F. Corpus 424.--G. Trin. Coll. Cam. R. 15. 18.--H. Sloane 314.--I. Sloane 261.--K. Rawlinson Misc. 3.--L. Addit. 23002. (B. M.)--M. St. John's Coll. Cam.--N. Digby 72.--O. Ashmole 360.--P. Camb. Univ. Lib. Dd. 12. 51.--Q. Ashmole 393.--R. Egerton 2622 (B. M.).--S. Addit. 29250 (B. M.) See the descriptions of them in the Introduction.] PROLOGUE. l. 26. thise B; þese C; _miswritten_ this A; see above, ll. 21, 22. 32. curious BC; _miswritten_ curios A. Many similar very slight alterations of spelling have been silently made in the text, and are not worth specifying here. A complete list of them is given in my edition of this treatise for the Early English Text Society. I give, however, the real variations of reading. Thus, in l. 58, A. has _som_ for _sonne_; and in l. 64 omits the second _the_. PART I. § 1, l. 3. wol B; wolde AC. § 2, l. 2. _Rowm_ is here an adjective, meaning _large_, _ample_. It is the right reading; we find Rowm AB rowme C; rvm M. § 3, l. 1. AB _omit_ the. § 9, l. 3. nombre AB; nou_m_bre C; _but_ nombres _in old editions_. § 12, l. 5. The MSS. all[60] read--'_vmbra recta_ or elles _vmbra extensa_, & the nether partie is cleped the _vmbra versa_.' This is certainly wrong. § 13, l. 2. a certein] _so in_ AB; CM _omit_ a. But Chaucer certainly uses the phrase 'a certain'; cf. 'of unces a certain,' C. T., G 776; and see G 1024. § 14, ll. 2, 5. The word _halt_ for _holdeth_, and the expression _to-hepe_, together, both occur in Troil. iii. 1764:--