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 shalt 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 the poynt of 4; than mete I the space between 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 the 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, thanne is 5 12-partyes of the heyght the space be-tween thee and the toure; with adding of thyn owne heyght. (F67)
 
 

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 ther 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; and 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; 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 sic de ceteris. (F68)
 
 

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 shalt 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, adde the stature of thyn heyghte to thyn eye. (F69)
 
 

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 the 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 loke 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 the 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 direct of 20, and if hit be more, as 6 or 18, than take that thou findest in directe 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; and 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 whiche thou hast purposed; and if hit so be that thou wolt weten thy mene mote for any day, or for any fraccion 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 from the 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 in 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. (F70)
 
 

Whan thou wolt make the mene mote of any planete to be by Arsechieles tables, tak 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. 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 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 direct under the same planete that I wroghte for, and 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 December. And if thou woldest wete the mene mote of any planete in March, April, 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 December, the yere of oure lord 1400; and so with monethes and dayes entere in-to thy table ther thou findest thy mene 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 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 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 and degrees, minutes, and secoundes of other signes with thy rote; and thy residue that leveth is thy mene mote for that day. (F71)
 
 

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 thanne 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 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 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 coniunccion, 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 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 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 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. (F72)
 

The following sections are spurious; they are numbered so as to show what propositions they repeat. (E78)
 

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 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. (F73)
 
 

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 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-thrid partyes of 12; so the space is the two-thrid partyes of the tour. (F74)
 
 

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 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 eb 2 or 3 distaunce in the nombres, so shulde the mesures be-tween the prikkes be twyes or thryes the heyghte of the tour. (F75)
 
 

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 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; 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. (F76)
 
 

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 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 be-tween two prikkes hath him to the altitude of the thing. (F77)