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of the same tour. Ensample: I sette caas thy rewle falle upon 8; than is 8 two-thrid partyes of 12; so the space is the two-thrid partyes of the tour.

42a. Umbra Versa.

To knowe the heyghth by thy poyntes of umbra versa. Yif thy rewle falle upon 3, whan thou seest the top of the tour, set a prikke there-as thy foot stont; and go ner til thou mayst see the same top at the poynt of 4, and sette ther another lyk prikke. Than mete how many foot ben be-tween the two prikkes, and adde the lengthe up to 5 thyn eye ther-to ; and that shal be the heyght of the tour. And note, that 3 is [the] fourthe party of 12, and 4 is the thridde party of 12. Now passeth 4 the nombre of 3 be the distaunce of I; therfore the same space, with thyn heyght to thyn eye, is the heyght of the tour. And yif it so be that ther be 2 or 3 distaunce in the nombres, so shulde Io the mesures be-tween the prikkes be twyes or thryes the heyghte of the tour.

43a. Ad cognoscendum altitudinem alicuius reiper umbram rectam.

To knowe the heyghte of thinges, yif thou mayst nat come to the bas of a thing. Sette thy rewle upon what thou wilt, so that thou may see the top of the thing thorw the two holes, and make a marke ther thy foot standeth ; and go neer or forther, til thou mayst see thorw another poynt, and marke ther a-nother marke. And loke than what 5 is the differense be-twen the two poyntes in the scale; and right as that difference hath him to 12, right so the space be-tween thee and the two markes hath him to the heyghte of the thing. Ensample: I set caas thou seest it thorw a poynt of 4; after, at the poynt of 3. Now passeth the nombre of 4 the nombre of 3 be the difference of 1 ; io

then EB = , BC = 3 BC. The only difference is that it inverts the equation in the last article.

42a. This is only a particular case of Art. 42. If we can get br=3, and &'c' = 4, the equations become EB = 4BC, E/B = 3BC; whence EE' = BC, a very convenient result. See fig. 17.

43a. The reading versam (as in the MS.) is absurd. We must also read “nat come,’ as, if the base were approachable, no such trouble need be taken; see Art. 41. In fact, the present article is a mere repetition of Art. 43, with different numbers, and with a slight difference in the method of expressing the result. In fig. 18, if &c' = 3, &c = 4, we have EB = , BC, EB = 1, BC; or, subtracting, EE' = ** BC; or BC = 12 EE'. Then add the height of E, viz. Ea, which = AB.

and right as this difference I hath him-self to 12, right so the mesure be-tween the two markes hath him to the heyghte of the thing, putting to the heyghte of thy-self to thyn eye; and thus mayst thou werke fro I to 12.

42b. Per umbram versam.

Furthermore, yif thou wilt knowe in umbra versa, by the craft of umbra recta, I suppose thou take the altitude at the poynt of 4, and makest a marke; and thou goost neer til thou hast it at the poynt of 3, and than makest thou ther a-nother mark. Than muste thou 5 devyde 144 by eche of the poyntes be-fornseyd, as thus: yif thou devyde 144 be 4, and the nombre that cometh ther-of schal be 36, and yif thou devyde 144 be 3, and the nombre that cometh ther-of schal be 48, thanne loke what is the difference be-tween 36 and 48, and ther shalt thou fynde 12; and right as 12 hath him to 12, right so the space 1o be-tween two prikkes hath him to the altitude of the thing.

42b. 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 bF be called mm, and mm, Ec be produced to meet in o, then mo : mE: ; bF : be ; or mo: 12 : : 12 : br; or moa-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. 41a and 41b are mere repetitions of 41; 42a and 426, of 42; and 43a, 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.

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PRologue. l. 26. thise B; bese C; miswritten this A.; see above, ll. 2 I, 22.

32. curious BC; miszwritten 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.

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* As far as I can ascertain.

§ 14, ll. 2, 5. The word halt for holdeth, and the expression to-hepe, together, both occur in Troil. iii. 1764:— “And lost were al, that Love halt now to-hepe.” § 17, l. 1. principal C; tropikal AB; M om. The reading tropikal is absurd, because there are but two such ; besides which, see 1.34 below. 17. the nyht (over an erasure) B; thee nyht (over an erasure) A ; peniştes C; be nystes M. $ 20, l. 4. figure; here (and sometimes elsewhere) miswritten vigur A. Throughout the whole treatise, the scribe has commonly written “vigur'; in many places, it has been corrected to ‘figure.” § 21, l. 15. the (before sterres) supplied from BC. 27. where as C; wher A.B. 56. ouerkeruyd A.; ouerkerued B; ouerkerueth (the latter part of the word over an erasure) C; first time only.

PART II. § 2, 1.8. euer M ; euere C; euery (wrongly) AB. § 3, 11. 31, 32. A has 12 degres, corrected to 18 degres; B. has 12 degrees; C has 18. The numbers in the MSS. in these propositions are somewhat uncertain; it seems probable that some alteration was made by Chaucer himself. The readings in MS. B give one set of calculations, which are no doubt the original ones; for in MS. A the same set is again found, but altered throughout, by the scribe who drew the diagrams. The sets of readings are these :— Ll. 31, 32. 12 degrees B; so in A, but altered to 18; C has 18. 37. passed 9 of the clokke the space of Io degrees B; so in A, with 9 altered to 8, and 10 altered to 2 ; C has ij for 9, but agrees with A in the reading 2. 39. fond ther Io degrees of taurus B; so in A originally, but 1o has been corrected to 23, and libra is written over an erasure. C agrees with neither, Aaving 20 for Io, but agreeing with A as to libra. The later MSS. sometimes vary from all these. 42. an supplied from C; AB omit. $ 4, 1. 5. largest C; largesse AB. 6. upon C; vn (!) AB. 8. forseide degree of his longitude] forseyde same degre of hys longitude C; forseid same gre of his longitude P; forseyde latitude his longitude (sic!) AB. 9. planete ys C; miswritten planetes AB, but is is added in margin of A. 16. For ‘25 degrees,' all the MSS. have “15 degrees.” The mistake is probably Chaucer's own; the correction was made by Mr. Brae, who remarks that it is a mere translation from the Latin version of Ptolemy's Tetrabiblos, which has—“Signum ascendentis, quod est a quinque gradibus qui super horizontem ante ipsum ascenderant usque ad viginti quinque qui ad ascendentem remanserint’; Lib. iii. c. Io. In fact, it is clear that 25 must be added to 5 to make up the extent of a ‘house,' which was 30 degrees. 16. ys like C ; is lik P; miswritten illyk AB. 17. in is supplied from GM; ABC omit it. 23. second the supplied from CP; AB omit. 32. wel supplied from CPM ; AB omit. 36. than] }an CM; benne P; AB omit. 40. The number 1 o is supplied from C; AB omit. 42. some folk supplied from CPG ; AB omit. 44, yit is] AB wrongly have yit it is; but CPGM omit it.

$ 5, 1.3. by 2 and 2 ACG ; by 3 and 3 P.; left blank in B. Either reading makes sense, but it is clear that divisions representing three degrees each must have been very awkward. 10. of supplied from CPGM : AB omit. § 6, l. 5. est C; west A (which is absurd); west (corrected to est) B. 9. signe CGP; signes ABM. § 10, l. 3. than B; pan C; A has & by nyht, which is absurd. 4, 5. A omits day with the howr inequal of the, which is supplied from BCP; the number 30 is also supplied from BCM, as A has a blank space here; see l. Io. § 11, l. 12. The number 4 is from CP; AB omit; old edd. fourthe. 13. ther supplied from PM ; pere C ; AB omit. § 12, l. 1. the supplied from BC; A omits. 8. The figure 2 is from BCP; G has secunde; A omits. § 14, 1. 9, 10. The last clause supplied from B. $ 15, 1.6. pointe] point P; pointes A; pointz B; poyntes C; but grammar requires the singular. 9. the supplied from CP; AB omit. § 16, l. 5. AB wrongly insert the before Cancer; CP omit it. 8. y-lyke] Ilyke G; ilik P; y-like C; ilke AB; see 1.7. $ 17. Latin rubric; for latitudinem (as in M) read longitudinem. l. 18. heued B; hed ACP; see sect. 16, l. 3. The word “the ' (rightly placed in BCMP) is, in A, wrongly placed defore “Aries' instead of before ‘ende.’ 23. second the] pe C; AB omit. § 19. Latin Rubric; for orizon (as in M) read statio. $ 20. Latin Rubric ; the MS. (M) transposes the words in and a, having a zodiaco in circulo, which contradicts the sense. § 22. Latin Rubric; for centri (as in M) read regionis. § 23, 1. 21. The figure “8” is omitted in AB. 23. than] A omits ; thanne inserted afterwards in B. § 25, l. 3. first the] supplied from B; AC omit. 15. CP om. and Io minutes. 16. CP om. and minutes out. For 51 degrees and 50 minutes, C has 52, pan is 52 degrees; and P has 52. penne is .52. grees. 19. CP om. as I mighte prove. 20. the supplied from CP; AB om. 27. the firste degree] Io degrees C; Io gree P. 28. 58 degrees and Io minutes] almost 56 C (meaning 56 degrees); almost .56. grees P. 29. almost 20) almost 18 C. 31. thee] C om. and odde Minutes] CP om. It thus appears that there is a second set of readings, involving a different calculation. The second set supposes the Sun to be in the loth degree of Leo, his altitude to be 56°, and his declination 18°; the difference, viz. 38°, is the complement of the latitude. Either set of readings suits the sense, but the one in the text agrees best with the former latitude, viz. 51°. 50'. 37. After there, C inserts 38 grees, pat is; and omits the words of the pole, 51 degrees and 50 Minutes. But this is a mere repetition of the “height of the Equinoctial,' and is obviously wrong. After pole, in l. 38, A inserts an that, which is unmeaning, and omitted in B.

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