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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 wiste 10 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, and 15 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 the

20 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 mene 25 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 mote 30 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 mene 35 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

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.

with-drawe alle the nethere signes and degrees fro the signes and 40 degrees, minutes, and secoundes of other signes with thy rote; and thy residue that leveth is thy mene mote for that day.

46. For to knowe at what houre of the day, or of the night, shal be flode or ebbe.

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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 that 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 of 10 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 or 15 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 the 20

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

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 evermore 25 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 labelle 30 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.]

41a. 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; and 5 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

I to 12.

416. Umbra Recta.

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, right 5 so the mesure be-tween thee and the tour hath him-self to the heighte

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 12 EB.

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416. 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,

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.

Yif thy

To knowe the heyghth by thy poyntes of umbra versa. 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 10 the mesures be-tween the prikkes be twyes or thryes the heyghte of the tour.

43a. 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 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; 10

then EB =

=

BC 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'd 4, the equations become EB = 4BC, E'B = 3BC; whence EE' BC, a very convenient result. See fig. 17.

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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'd = 3, bc = 4, we have E'B = BC, EB = BC; or, subtracting, EE'

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BC; or BC = 12 EE'. Then add the height of E, viz. Ea,

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 1 to 12.

426. 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 10 be-tween two prikkes hath him to the altitude of the thing.

426. 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. those in articles 41, 42, and 43, corresponding to figs. 16, 17, and 18 respectively. Arts. 41a and 416 are mere repetitions of 41; 42a and 426, of 42; and 43a, of 43.

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