Front Page Titles (by Subject) SUPPLEMENTARY PROPOSITIONS. - The Complete Works of Geoffrey Chaucer, vol. 3 (House of Fame, Legend of Good Women, Treatise on Astrolabe, Sources of Canterbury Tales)
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SUPPLEMENTARY PROPOSITIONS. - Geoffrey Chaucer, The Complete Works of Geoffrey Chaucer, vol. 3 (House of Fame, Legend of Good Women, Treatise on Astrolabe, Sources of Canterbury Tales) 
The Complete Works of Geoffrey Chaucer, edited from numerous manuscripts by the Rev. Walter W. Skeat (2nd ed.) (Oxford: Clarendon Press, 1899). 7 vols.
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Yif it so be that thou wilt werke by umbra recta, and thou may come to the bas of the toure, in this maner thou schalt werke. Tak the altitude of the tour by bothe holes, so that thy rewle ligge even in a poynt. Ensample as thus: I see him thorw at the5 poynt of 4; than mete I the space be-tween me and the tour, and I finde it 20 feet; than be-holde I how 4 is to 12, right so is the space betwixe thee and the tour to the altitude of the tour. For 4 is the thridde part of 12, so is the space be-tween thee and the tour the thridde part of the altitude of the tour; than thryes 20 feet is the10 heyghte of the tour, with adding of thyn owne persone to thyn eye. And this rewle is so general in umbra recta, fro the poynt of oon to 12. And yif thy rewle falle upon 5, than is 5 12-partyes of the heyght the space be-tween thee and the toure; with adding of thyn owne heyght.
Another maner of werkinge, by umbra versa. Yif so be that thou may nat come to the bas of the tour, I see him thorw the nombre of 1; I sette ther a prikke at my fote; than go I neer to the tour, and I see him thorw at the poynt of 2, and there I sette a-nother prikke; and I beholde how 1 hath him to 12, and ther5 finde I that it hath him twelfe sythes; than beholde I how 2 hath him to 12, and thou shalt finde it sexe sythes; than thou shalt finde that as 12 above 6 is the numbre of 6, right so is the space between thy two prikkes the space of 6 tymes thyn altitude. And note, that at the ferste altitude of 1, thou settest a prikke; and10 afterward, whan thou seest him at 2, ther thou settest an-other prikke; than thou findest between two prikkys 60 feet; than thou shalt finde that 10 is the 6-party of 60. And then is 10 feet the altitude of the tour. For other poyntis, yif it fille in umbra versa, as thus: I sette caas it fill upon 2 , and at the secunde upon 3;15 than schalt thou finde that 2 is 6 partyes of 12; and 3 is 4 partyes of 12; than passeth 6 4, by nombre of 2; so is the space between two prikkes twyes the heyghte of the tour. And yif the differens were thryes, than shulde it be three tymes; and thus mayst thou werke fro 2 to 12; and yif it be 4, 4 tymes; or 5, 5 tymes; et sic20de ceteris.
Umbra Recta[ ] .
An-other maner of wyrking be umbra recta. Yif it so be that thou mayst nat come to the baas of the tour, in this maner thou schalt werke. Sette thy rewle upon 1 till thou see the altitude, and sette at thy foot a prikke. Than sette thy rewle upon 2, and beholde what is the differense be-tween 1 and 2, and thou shalt5 finde that it is 1. Than mete the space be-tween two prikkes, and that is the 12 partie of the altitude of the tour. And yif ther were 2, it were the 6 partye; and yif ther were 3, the 4 partye; et sic deinceps. And note, yif it were 5, it were the 5 party of 12; and 7, 7 party of 12; and note, at the altitude of thy conclusioun,10 adde the stature of thyn heyghte to thyn eye.
* * * * * * *
Another maner conclusion, to knowe the mene mote and the argumentis of any planete. To know the mene mote and the argumentis of every planete fro yere to yere, from day to day, from houre to houre, and from smale fraccionis infinite.[ ][ ]
[Ad cognoscendum medios motus et argumenta de hora in horam cuiuslibet planete, de anno in annum, de die in diem.]
In this maner shalt thou worche: consider thy rote first, the whiche is made the beginning of the tables fro the yere of oure lord 1397, and entere hit in-to thy slate for the laste meridie of December; and than consider the yere of oure lord, what is the5 date, and be-hold whether thy date be more or lasse than the yere 1397. And yf hit so be that hit be more, loke how many yeres hit passeth, and with so many entere into thy tables in the first lyne ther-as is writen anni collecti et expansi. And loke where the same planet is writen in the hede of thy table, and than loke10 what thou findest in directe of the same yere of oure lord whiche is passid, be hit 8, or 9, or 10, or what nombre that evere it be, til the tyme that thou come to 20, or 40, or 60. And that thou findest in directe wryte in thy slate under thy rote, and adde hit to-geder , and that is thy mene mote, for the laste meridian of the15 December, for the same yere whiche that thou hast purposed. And if hit so be that hit passe 20, consider wel that fro 1 to 20 ben anni expansi, and fro 20 to 3000 ben anni collecti; and if thy nombere passe 20, than take that thou findest in directe of 20, and if hit be more, as 6 or 18, than take that thou findest in directe20 there-of, that is to sayen, signes, degrees, minutes, and secoundes, and adde to-gedere un-to thy rote; and thus to make rotes; and note, that if hit so be that the yere of oure lord be lasse than the rote, whiche is the yere of oure lord 1397, than shalt thou wryte in the same wyse furst thy rote in thy slate, and after entere in-to thy table in the same yere that be lasse, as I taught be-fore; and25 than consider how many signes, degrees, minutes, and secoundes thyn entringe conteyneth. And so be that ther be 2 entrees, than adde hem togeder, and after with-drawe hem from the rote, the yere of oure lord 1397; and the residue that leveth is thy mene mote fro the laste meridie of December, the whiche30 thou hast purposed; and if hit so be that thou wolt weten thy mene mote for any day, or for any fraccioun of day, in this maner thou shalt worche. Make thy rote fro the laste day of Decembere in the maner as I have taught , and afterward behold how many monethis, dayes, and houres ben passid from35the meridie of Decembere, and with that entere with the laste moneth that is ful passed, and take that thou findest in directe of him, and wryte hit in thy slate; and entere with as mony dayes as be more, and wryte that thou findest in directe of the same planete that thou worchest for; and in the same wyse in40 the table of houres, for houres that ben passed, and adde alle these to thy rote; and the residue is the mene mote for the same day and the same houre.
Another manere to knowe the mene mote.[ ]
Whan thou wolt make the mene mote of eny planete to be by Arsechieles tables, take thy rote, the whiche is for the yere of oure lord 1397; and if so be that thy yere be passid the date, wryte that date, and than wryte the nombere of the yeres. Than withdrawe the yeres out of the yeres that ben passed that rote.5 Ensampul as thus: the yere of oure lord 1400, I wolde witen, precise, my rote; than wroot I furst 1400. And under that nombere I wrote a 1397 ; than withdrow I the laste nombere out of that, and than fond I the residue was 3 yere; I wiste10 that 3 yere was passed fro the rote, the whiche was writen in my tables. Than after-ward soghte I in my tables the annis collectis et expansis, and amonge myn expanse yeres fond I 3 yeer. Than tok I alle the signes, degrees, and minutes, that I fond directe under the same planete that I wroghte for, and15 wroot so many signes, degrees, and minutes in my slate, and afterward added I to signes, degrees, minutes, and secoundes, the whiche I fond in my rote the yere of oure lord 1397; and kepte the residue; and than had I the mene mote for the laste day of Decembere. And if thou woldest wete the20 mene mote of any planete in March, Aprile, or May, other in any other tyme or moneth of the yere, loke how many monethes and dayes ben passed from the laste day of Decembere, the yere of oure lord 1400; and so with monethes and dayes entere in-to thy table ther thou findest thy mene25 mote y-writen in monethes and dayes, and take alle the signes, degrees, minutes, and secoundes that thou findest y-write in directe of thy monethes, and adde to signes, degrees, minutes, and secoundes that thou findest with thy rote the yere of oure lord 1400, and the residue that leveth is the mene mote30 for that same day. And note, if hit so be that thou woldest wete the mene mote in ony yere that is lasse than thy rote, withdrawe the nombere of so many yeres as hit is lasse than the yere of oure lord a 1397, and kepe the residue; and so many yeres, monethes, and dayes entere in-to thy tabelis of thy mene35 mote. And take alle the signes, degrees, and minutes, and secoundes, that thou findest in directe of alle the yeris, monethes, and dayes, and wryte hem in thy slate; and above thilke nombere wryte the signes, degrees, minutes, and secoundes, the whiche thou findest with thy rote the yere of oure lord a 1397; and with-drawe alle the nethere signes and degrees fro the signes and40 degrees, minutes, and secoundes of other signes with thy rote; and thy residue that leveth is thy mene mote for that day.
For to knowe at what houre of the day, or of the night, shal be flode or ebbe.
First wite thou certeinly, how that haven stondeth, that thou list to werke for; that is to say in whiche place of the firmament the mone being, maketh fulle see. Than awayte thou redily in what degree of the zodiak that the mone at that tyme is inne. Bringe furth than the labelle, and set the point therof in that5 same cost that the mone maketh flode, and set thou there the degree of the mone according with the egge of the label. Than afterward awayte where is than the degree of the sonne, at that tyme. Remeve thou than the label fro the mone, and bringe and sette it iustly upon the degree of the sonne. And the point of10 the label shal than declare to thee, at what houre of the day or of the night shal be flode. And there also maist thou wite by the same point of the label, whether it be, at that same tyme, flode or ebbe, or half flode, or quarter flode, or ebbe, or half or quarter ebbe; or ellis at what houre it was last, or shal be next by night or15 by day, thou than shalt esely knowe, &c. Furthermore, if it so be that thou happe to worke for this matere aboute the tyme of the coniunccioun, bringe furthe the degree of the mone with the labelle to that coste as it is before seyd. But than thou shalt understonde that thou may not bringe furthe the label fro the20 degree of the mone as thou dide before; for-why the sonne is than in the same degree with the mone. And so thou may at that tyme by the point of the labelle unremeved knowe the houre of the flode or of the ebbe, as it is before seyd, &c. And evermore25 as thou findest the mone passe fro the sonne, so remeve thou the labelle than fro the degree of the mone, and bringe it to the degree of the sonne. And worke thou than as thou dide before, &c. Or elles knowe thou what houre it is that thou art inne, by thyn instrument. Than bringe thou furth fro thennes the labelle30 and ley it upon the degree of the mone, and therby may thou wite also whan it was flode, or whan it wol be next, be it night or day; &c.
[The following sections are spurious; they are numbered so as to shew what propositions they repeat.]
Umbra Recta.[ ]
Yif thy rewle falle upon the 8 poynt on right schadwe, than make thy figure of 8; than loke how moche space of feet is be-tween thee and the tour, and multiplye that be 12, and whan thou hast multiplied it, than divyde it be the same nombre of 8, and kepe the residue; and5 adde therto up to thyn eye to the residue, and that shal be the verry heyght of the tour. And thus mayst thou werke on the same wyse, fro 1 to 12.
An-other maner of werking upon the same syde. Loke upon which poynt thy rewle falleth whan thou seest the top of the tour thorow two litil holes; and mete than the space fro thy foot to the baas of the tour; and right as the nombre of thy poynt hath him-self to 12, right5 so the mesure be-tween thee and the tour hath him-self to the heighte of the same tour. Ensample: I sette caas thy rewle falle upon 8; than is 8 two-third partyes of 12; so the space is the two-third partyes of the tour.
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 to5 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 1; 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 shulde10 the mesures be-tween the prikkes be twyes or thryes the heyghte of the tour.
Ad cognoscendum altitudinem alicuius rei per 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 what5 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;10 and right as this difference 1 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 1 to 12.
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 thou5 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 space10 be-tween two prikkes hath him to the altitude of the thing.
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 sentence 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 heyȜt 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= BC. Conversely, BC is EB=48, if EB=20.
42. See fig. 1, Plate I. See also fig. 17, Plate VI. Let Eb=12, bc =1; also E′b′=12, b′c′=2; then EB=12 BC, E′B=6 BC; therefore EE′=6 BC. If EE′=60 feet, then BC=⅙ EE′=10 feet. To get the whole height, add the height of the eye. The last part of the article, beginning ‘For other poyntis,’ is altogether corrupt in the MS.
43. Here versa (in M.) is certainly miswritten for recta, as in L. See fig. 18, Plate VI. Here Eb=E′b′=12; b′c′=1, bc=2. Hence E′B= BC, EB = BC, whence EE′ = BC. Or again, if bc become = 3, 4, 5, &c., successively, whilst b′c′ remains = 1, then EE′ is successively = or ⅙, or ¼, , &c. Afterwards, add in the height of E.
44. Sections 44 and 45 are from MS. Digby 72. This long explanation of the method of finding a planet’s place depends upon the tables which were constructed for that purpose from observation. The general idea is this. The figures shewing a planet’s position for the last day of December, 1397, give what is called the root, and afford us, in fact, a starting-point from which to measure. An ‘argument’ is the angle upon which the tabulated quantity depends; for example, a very important ‘argument’ is the planet’s longitude, upon which its declination may be made to depend, so as to admit of tabulation. The planet’s longitude for the given above-mentioned date being taken as the root, the planet’s longitude at a second date can be found from the tables. If this second date be less than 20 years afterwards, the increase of motion is set down separately for each year, viz. so much in 1 year, so much in 2 years, and so on. These separate years are called anni expansi. But when the increase during a large round number of years (such as 20, 40, or 60 years at once) is allowed for, such years are called anni collecti. For example, a period of 27 years includes 20 years taken together, and 7 separate or expanse years. The mean motion during smaller periods of time, such as months, days, and hours, is added in afterwards.
45. Here the author enters a little more into particulars. If the mean motion be required for the year 1400, 3 years later than the starting-point, look for 3 in the table of expanse years, and add the result to the number already corresponding to the ‘root,’ which is calculated for the last day of December, 1397. Allow for months and days afterwards. For a date earlier than 1397 the process is just reversed, involving subtraction instead of addition.
46. This article is probably not Chaucer’s. It is found in MS. Bodley 619, and in MS. Addit. 29250. The text is from the former of these, collated with the latter. What it asserts comes to this. Suppose it be noted, that at a given place, there is a full flood when the moon is in a certain quarter; say, e. g. when the moon is due east. And suppose that, at the time of observation, the moon’s actual longitude is such that it is in the first point of Cancer. Make the label point due east; then bring the first point of Cancer to the east by turning the Rete a quarter of the way round. Let the sun at the time be in the first point of Leo, and bring the label over this point by the motion of the label only, keeping the Rete fixed. The label then points nearly to the 32nd degree near the letter Q, or about S.E. by E.; shewing that the sun is S.E. by E. (and the moon consequently due E.) at about 4 a.m. In fact, the article merely asserts that the moon’s place in the sky is known from the sun’s place, if the difference of their longitudes be known. At the time of conjunction, the moon and sun are together, and the difference of their longitudes is zero, which much simplifies the problem. If there is a flood tide when the moon is in the E., there is another when it comes to the W., so that there is high water twice a day. It may be doubted whether this proposition is of much practical utility.
41a. This comes to precisely the same as Art. 41, but is expressed with a slight difference. See fig. 16, where, if bc = 8, then BC = EB.
41b. Merely another repetition of Art. 41. It is hard to see why it should be thus repeated in almost the same words. If bc = 8 in fig. 16, then EB = BC = ⅔ 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 bc=3, and b′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 b′c′ = 3, bc = 4, we have E′B = BC, EB = BC; or, subtracting, EE′ = BC; or BC = 12 EE′. Then add the height of E, viz. Ea, which = AB.
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 bE be called nm, and mn, Ec be produced to meet in o, then mo : mE : : bE : 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. thouse 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 42b, of 42; and 43a, of 43.
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.]
[8.]M omits as, above, and is þe; L has 12 passethe 6 the.
[11.]seest] so in LR; miswritten settest M.
[12.]60] so in LNR; sexe M.
[13.]M omits from 10 is to 10 feet, which is supplied from NLPR.
[14.]For] so in LNR; fro M.
[15.]For 2, M has 6; so also R. For 3, M has 4.
[16.]For 2, M has 6; for 6, M has 2; and the words and 3 is 4 partyes of 12 are omitted, though L has—& 4 is the thrid partye of 12.
[17.]betwen R] by-twene L; bitwixe P; miswritten be M; cf. sect. 41, 7.
[19.]thre R] 3 LP; miswritten þe M.
[§ 43.]Rubric in M, Umbra Versa; obviously a mistake for Recta. The error is repeated in l. 1. LPR rightly read Recta.
[3.]M omits 1, which is supplied from LPR; see l. 5.
[11.]After heythe (as in M), LNR add to thyn eye. In place of lines 9-11, P has—& so of alle oþer, &c.
[§ 44.]From MS. Digby 72 (N). Also in LMOR.
[2.]fro] so in LO; for M.
[3.]into] so in L; in M. for] so in O; fro M.
[6.]Ȝeris M; LNO omit.
[7.]tabelis NO; table M; tables L.
[8.]where L; qwere O; wheþer N.
[9.]loke LM; N omits.
[11, 2.]NM omit from or what to or; supplied from O, which has—or qwat nombre þat euere it be, tyl þe tyme þat þou come to 20, or 40, or 60. I have merely turned qwat into what, as in L, which also has this insertion.
[13.]wreten N; the alteration to wryte is my own; see l. 23.
[14.]to-geder] too-geder M; miswritten to 2 degreis N; to the 2 degrees L.
[15.]hast M; miswritten laste N; last L.
[16.]that (1); supplied from M; LN omit. For 1 (as in M) LN have 10.
[21.]to-gedere M; to the degreis N; 2 grees O; to degrees L.
[22.]that (2); supplied from M; LNO omit.
[25.]that] so in L; þat MO; if hit N.
[27.]entringe] entre M; entre L. ther] so in M; miswritten the Ȝere N; the Ȝeer L.
[30.]merydie LM; merdie N.
[32.]for LM; fro N (twice).
[34.]thaȜthe N; have tauȜt M; have tawȜt O; haue tauht L.
[36.]the (1); supplied from M; LNO omit.
[40.]in (2)] in-to N; yn M.
[§ 45.]From MS. Digby 72 (N); also in LOR; but not in M.
[4.]that N; the L; þe O (after wryte in l. 3).
[6.]wrytoun O; Iwyton N. But L has I wold wyttyn; read—I wolde witen precise my rote; cf. ll. 19, 30.
[8.]1397] miswritten 1391 LN; O has 1391, corrected to 1397; see l. 3.
[11.]soȜth N; sowte O; sowthe L; read soghte.
[14.]vnder N; vndyr-nethe O; vndre-nethe L.
[20, 1.]oþer in any oþer tyme or monyth N; or any oder tymys or monthys O; or in eny other moneth L.
[27.]adde] supplied from L; NO omit. There is no doubt about it, for see l. 16.
[31.]wete the] so in O; wete thi L; miswritten with thy N; see l. 19.
[35.]and (3)] supplied from LO; N omits.
[§ 46, 5, 6.]þat same E; þe same S.
[10.]it S; E omits.
[13.]þat same (om. tyme) E; þe same tyme S.
[16.]þou þan esely E; than shallt thou easly S.
[17.]tyme of E; tyme of the S.
[20.]S meve (for bringe furþe).
[§ 41a.]This and the remaining sections are certainly spurious. They occur in LMNR, the first being also found in O. The text of 41a-42b is from M.
[3.]hast] supplied from LR; M omits.
[§ 42a, 1.]heyth by þy N; heyth by the L; heythe bi þi R; M om.
[4.]lyk] lykk M; L. omits. mete] mette M; mett L.
[9.]is L; miswritten hys M.
[§ 43a, 1.]nat] not R; nott L; M omits; see the footnote. In the rubric, M has versam; but L has the rubric—Vmbra Recta.
[§ 42b, 5.]as] so in LR; miswritten & M.
[6.]4 is supplied from LR; M omits.
[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 45°. 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.
[44.]This article and the next may possibly be Chaucer’s. It is well known that he speaks of ‘collect’ and ‘expans yeres’ and ‘rotes’ in the Frankeleines Tale; Cant. Ta., F 1275, 6, the note upon which in the glossary to Urry’s Chaucer may be found also in Tyrwhitt’s Glossary, s. v. Expans; but it is worth while to repeat it here. ‘In this and the following verses, the Poet describes the Alphonsine Astronomical Tables by the several parts of them, wherein some technical terms occur, which were used by the old astronomers, and continued by the compilers of those tables. Collect years are certain sums of years, with the motions of the heavenly bodies corresponding to them, as of 20, 40, 60, &c., disposed into tables; and Expans years are the single years, with the motions of the heavenly bodies answering to them, beginning at 1, and continued on to the smallest Collect sum, as 20. A Root, or Radix, is any certain time taken at pleasure, from which, as an era, the celestial motions are to be computed. By ‘proporcionels convenientes’ [C. T., F 1278] are meant the Tables of Proportional parts.’ To which Moxon adds, from Chamber’s Encyclopædia, with reference to C. T., F 1277, that ‘Argument in astronomy is an arc whereby we seek another unknown arc proportional to [or rather, dependent upon] the first.’
[41a-42b.]The fact that these articles are mere repetitions of sections 41-43 is almost conclusive against their genuineness. I do not suppose that sect 46 (at p. 229) is Chaucer’s either, but it is added for the sake of completeness.