House / Legend / Astrolabe

41. Umbra Recta.

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 the

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 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, than is 5 12-partyes

of the heyght the space be-tween thee and the toure; with adding

of thyn owne heyght.

42. Umbra Versa.

Another maner of werkinge, by vmbra 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.

43. 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 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.

44. 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 shall 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 directe 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 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 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.

45. 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.

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 withdraw 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 directe 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 Decembere. And if thou woldest wete the

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

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.

46. 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 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

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

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

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.

[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

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.

41b. 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

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.

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

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 shulde

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

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.

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

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.

COMMENTARY ("FOOTNOTES").

1. The Ring. See figs. 1 and 2. The Latin name is Armilla suspensoria; the Arabic name is spelt alhahuacia in MS. Camb. Univ. Ii. 3. 3, but Stöffler says it is Alanthica, Alphantia, or Abalhantica. For the meaning of 'rewle,' see § 13.

2. The Turet. This answers nearly to what we call an eye or a swivel. The metal plate, or loop, to which it is fastened, or in which it turns, is called in Latin Ansa or Armilla Reflexa, in Arabic Alhabos.

3. The Moder. In Latin, Mater or Rotula. This forms the body of the instrument, the back of which is shewn in fig. 1, the front in fig. 2. The 'large hole' is the wide depression sunk in the front of it, into which the various discs are dropped. In the figure, the 'Rete' is shewn fitted into it.

4. See fig. 1; Chaucer describes the 'bak-half' of the instrument first. The centre of the 'large hole amydde' is the centre of the instrument, where a smaller hole is pierced completely through. The Southe lyne (marked Meridies in figs. 1 and 2) is also called Linea Meridiei; the North lyne is also named Linea Mediæ Noctis.

5. The Est lyne is marked with the word Oriens; the West lyne, with Occidens.

6. The rule is the same as in heraldry, the right or dexter side being towards the spectator's left.

7. As the 360 degrees answer to 24 hours of time, 15° answer to an hour, and 5° to twenty minutes, or a Mile-way, as it is the average time for walking a mile. So also 1° answers to 4 minutes of time. See the two outermost circles in fig. 1, and the divisions of the 'border' in fig. 2.

10. See the seventh, eighth, and ninth circles in fig. 1. The names of the months are all Roman. The month formerly called Quinctilis was first called Julius in B.C. 44; that called Sextilis was named Augustus in B.C. 27. It is a mistake to say that Julius and Augustus made the alterations spoken of in the text; what Julius Cæsar really did, was to add 2 days to the months of January, August (Sextilis), and December, and 1 day to April, June, September, and November. February never had more than 28 days till he introduced bissextile years.

11. See the two inmost circles in fig. 1. The names given are adopted from a comparison of the figures in the Cambridge University and Trinity MSS., neither of which are quite correct. The letters of the 'Abc.' are what we now call the Sunday letters. The festivals marked are those of St. Paul (Jan. 25), The Purification (Feb. 2), The Annunciation (Mar. 25), The Invention of the Holy Cross (May 3), St. John the Baptist (June 24), St. James (July 25), St. Lawrence (Aug. 10), The Nativity of the Blessed Virgin (Sept. 8), St. Luke (Oct. 18), St. Martin of Tours (Nov. 11), and St. Thomas (Dec. 21).

12. The 'scale' is in Latin Quadrans, or Scala Altimetra. It is certain that Chaucer has here made a slip, which cannot be fairly laid to the charge of the scribes, as the MSS. agree in transposing versa and recta. The side-parts of the scale are called Umbra versa, the lower part Umbra recta or extensa. This will appear more clearly at the end of Part II. (I here give a corrected text.)

13. See fig. 3, Plate III. Each plate turns on a hinge, just like the 'sights' of a gun. One is drawn flat down, the other partly elevated. Each plate (tabella vel pinnula) has two holes, the smaller one being the lower. This Rewle is named in Arabic Alhidada or Al´idāda; in Latin Verticulum, from its turning easily on the centre; in Greek Dioptra, as carrying the sights. The straight edge, passing through the centre, is called the Linea Fiduciæ. It is pierced by a hole in the centre, of the same size as that in the Mother.

14. See fig. 4, Plate III. The Pin is also called Axis or Clavus, in Latin-Arabic Alchitot; it occupies the position of the Arctic or North Pole, passing through the centre of the plates that are required to turn round it. The Wedge is called cuneus, or equus restringens, in Arabic Alfaras or the horse, because it was sometimes cut into the shape of a horse, as shewn in fig. 7, Plate IV, which is copied from MS. Univ. Camb. Ii. 3. 3.

15. See fig. 2, Plate II. In the figure, the cross-lines are partly hidden by the Rete, which is separate and removable, and revolves within the border.

16. The Border was also called Margilabrum, Margolabrum, or Limbus. It is marked (as explained) with hour-letters and degrees. Each degree contains 4 minutes of time, and each of these minutes contains 60 seconds of time.

17. We may place under the Rete any plates we please. If only the Mother be under it, without any plate, we may suppose the Mother marked as in fig. 2. The plate or disc (tympanum) which was usually dropped in under the Rete is that shewn in fig. 5, Plate III, and which Chaucer now describes. Any number of these, marked differently for different latitudes, could be provided for the Astrolabe. The greatest declination of the sun measures the obliquity of the ecliptic, the true value of which is slightly variable, but was about 23° 31′ in Chaucer's time, and about 23° 40′ in the time of Ptolemy, who certainly assigns to it too large a value. The value of it must be known before the three circles can be drawn. The method of finding their relative magnitudes is very simple. Let ABCD (fig. 8, Pl. IV) be the tropic of Capricorn, BO the South line, OC the West line. Make the angle EOB equal to the obliquity (say 23½°), and join EA, meeting BO in F. Then OF is the radius of the Equatorial circle, and if GH be drawn parallel to EF, OH is the radius of the Tropic of Cancer. In the phrase angulus primi motus, angulus must be taken to mean angular motion. The 'first moving' (primus motus) has its name of 'moving' (motus) from its denoting motion due to the primum mobile or 'first moveable.' This primum mobile (usually considered as the ninth sphere) causes the rotation of the eighth sphere, or sphæra stellarum fixarum. See the fig. in MS. Camb. Univ. Ii. 3. 3 (copied in fig. 10, Pl. V). Some authors make 12 heavens, viz. those of the 7 planets, the firmamentum (stellarum fixarum), the nonum cœlum, decimum cœlum, primum mobile, and cœlum empyræum.

18. See fig. 5, Pl. III. This is made upon the alt-azimuth system, and the plates are marked according to the latitude. The circles, called in Latin circuli progressionum, in Arabic Almucantarāt, are circles of altitude, the largest imperfect one representing the horizon (horizon obliquus), and the central dot being the zenith, or pole of the horizon. In my figure, they are 'compounded by' 5 and 5, but Chaucer's shewed every second degree, i.e. it possessed 45 such circles. For the method of drawing them, see Stöffler, leaf 5, back.

19. Some Astrolabes shew 18 of these azimuthal circles, as in my figure (fig. 5, Pl. III). See Stöffler, leaf 13, where will be found also the rules for drawing them.

20. If accurately drawn, these embelife or oblique lines should divide the portions of the three circles below the horizon obliquus into twelve equal parts. Thus each arc is determined by having to pass through three known points. They are called arcus horarum inequalium, as they shew the 'houres inequales.'

21. In fig. 2, Pl. II, the Rete is shewn as it appears when dropped into the depression in the front of the instrument. The shape of it varied much, and another drawing of one (copied from Camb. Univ. MS. Ii. 3. 3, fol. 66 b) is given in fig. 9, Pl. IV. The positions of the stars are marked by the extreme points of the metal tongues. Fig. 2 is taken from the figures in the Cambridge MSS., but the positions of the stars have been corrected by the list of latitudes and longitudes given by Stöffler, whom I have followed, not because he is correct, but because he probably represents their positions as they were supposed to be in Chaucer's time very nearly indeed. There was not room to inscribe the names of all the stars on the Rete, and to have written them on the plate below would have conveyed a false impression. A list of the stars marked in fig. 2 is given in the note to § 21, l. 4. The Ecliptic is the circle which crosses the Equinoctial at its East and West points (fig. 2). In Chaucer's description of the zodiac, carefully note the distinction between the Zodiac of the Astrolabe and the Zodiac of Heaven. The former is only six degrees broad, and shews only the northern half of the heavenly zodiac, the breadth of which is imagined to be 12 degrees. Chaucer's zodiac only shewed every other degree in the divisions round its border. This border is divided by help of a table of right ascensions of the various degrees of the ecliptic, which is by no means easily done. See Note on l. 4 of this section. I may add that the Rete is also called Aranea or Volvellum; in Arabic, Al´ancabūt (the spider).

22. The Label. See fig. 6, Pl. III. The label is more usually used on the front of the instrument, where the Rete and other plates revolve. The rule is used on the back, for taking altitudes by help of the scale.

23. The Almury; called also denticulus, ostensor, or 'calculer.' In fig. 2, it may be seen that the edge of the Rete is cut away near the head of Capricorn, leaving only a small pointed projecting tongue, which is the almury or denticle, or (as we should now say) pointer. As the Rete revolves, it points to the different degrees of the border. See also fig. 9, where the almury is plainly marked.

Part II, § 1. [The Latin headings to the propositions are taken from the MS. in St. John's College, Cambridge.] See fig. 1. Any straight edge laid across from the centre will shew this at once. Chaucer, reckoning by the old style, differs from us by about eight days. The first degree of Aries, which in his time answered to the 12th of March, now vibrates between the 20th and 21st of that month. This difference of eight days must be carefully borne in mind in calculating Chaucer's dates.

2. Here 'thy left side' means the left side of thine own body, and therefore the right or Eastern edge of the Astrolabe. In taking the altitude of the sun, the rays are allowed to shine through the holes; but the stars are observed by looking through them. See figs. 1 and 3.

3. Drop the disc (fig. 5) within the border of the mother, and the Rete over it. Take the sun's altitude by § 2, and let it be 25½°. As the altitude was taken by the back of the Astrolabe, turn it over, and then let the Rete revolve westward till the 1st point of Aries is just within the altitude-circle marked 25, allowing for the ½ degree by guess. This will bring the denticle near the letter C, and the first point of Aries near X, which means 9 A.M. At the same time, the 20th degree of Gemini will be on the horizon obliquus. See fig. 11, Pl. V. This result can be approximately verified by a common globe thus; elevate the pole nearly 52°; turn the small brass hour-circle so that the figure XII lies on the equinoctial colure; then turn the globe till IX lies under the brass meridian. In the next example, by the Astrolabe, let the height of Alhabor (Sirius) be about 18°. Turn the denticle Eastward till it touches the 58th degree near the letter O, and it will be found that Alhabor is about 18° high among the almicanteras, whilst the first point of Aries points to 32° near the letter H, i.e. to 8 minutes past 8 P.M.; whilst at the same time, the 23rd degree of Libra is almost on the Horizon obliquus on the Eastern side. By the globe, at about 8 minutes past 8 P.M., the altitude of Sirius is very nearly 18°, and the 23rd of Libra is very near the Eastern horizon. See fig. 12, Pl. V.

4. The ascendent at any given moment is that degree of the zodiac which is then seen upon the Eastern horizon. Chaucer says that astrologers reckoned in also 5 degrees of the zodiac above, and 25 below; the object being to extend the planet's influence over a whole 'house,' which is a space of the same length as a sign, viz. 30°. See § 36 below.

6. This depends upon the refraction of light by the atmosphere, owing to which light from the sun reaches us whilst he is still 18° below the horizon. The nadir of the sun being 18° high on the W. side, the sun itself is 18° below the Eastern horizon, giving the time of dawn; and if the nadir be 18° high on the E. side, we get the time of the end of the evening twilight. Thus, at the vernal equinox, the sun is 18° high soon after 8 A.M. (roughly speaking), and hence the evening twilight ends soon after 8 P.M., 12 hours later, sunset being at 6 P.M.

7. Ex. The sun being in the first point of Cancer on the longest day, its rising will be shewn by the point in fig. 5 where the horizon obliquus and Tropicus Cancri intersect; this corresponds to a point between P and Q in fig. 2, or to about a quarter to 4 A.M. So too the sunset is at about a quarter past 8, and the length of the day 16½ hours; hence also, the length of the night is about 7½ hours, neglecting twilight.

8. On the same day, the number of degrees in the whole day is about 247½, that being the number through which the Rete is turned in the example to § 7. Divide by 15, and we have 16½ equal hours.

9. The 'day vulgar' is the length of the 'artificial day,' with the length of the twilight, both at morn and at eve, added to it.

10. If, as in § 7, the day be 16½ hours long, the length of each 'hour inequal' is 1 h. 22½ m.; and the length of each 'hour inequal' of the night is the 12th part of 7½ hours, or 37½ m.; and 1 h. 22½ m., added to 37½ m., will of course make up 2 hours, or 30°.

11. This merely repeats that 15° of the border answer to an hour of the clock. The '4 partie of this tretis' was never written.

12. This 'hour of the planet' is a mere astrological supposition, involving no point of astronomy. Each hour is an 'hour inequal,' or the 12th part of the artificial day or night. The assumptions are so made that first hour of every day may resemble the name of the day; the first hour of Sunday is the hour of the Sun, and so on. These hours may be easily found by the following method. Let 1 represent both Sunday and the Sun; 2, Monday and the Moon; 3, Tuesday and Mars; 4, Wednesday and Mercury; 5, Thursday and Jupiter; 6, Friday and Venus; 7, Saturday and Saturn. Next, write down the following succession of figures, which will shew the hours at once.

Ex. To find the planet of the 10th hour of Tuesday. Tuesday is the third day of the week; begin with 3, to the left of the upright line, and reckon 10 onwards; the 10th figure (counting 3 as the first) is 6, i.e. Venus. So also, the planet of the 24th hour of Friday is the Moon, and Saturday begins with Saturn. It may be observed that this table can be carried in the memory, by simply observing that the numbers are written, beginning with 1, in the reverse order of the spheres, i.e. Sun, Venus, Mercury, Moon; and then (beginning again at the outmost sphere) Saturn, Jupiter, Mars. This is why Chaucer takes a Saturday; that he may begin with the remotest planet, Saturn, and follow the reverse order of the spheres. See fig. 10, Pl. V. Here, too, we have the obvious reason for the succession of the names of the days of the week, viz. that the planets being reckoned in this order, we find the Moon in the 25th place or hour from the Sun, and so on.

13. The reason of this is obvious from what has gone before. The sun's meridional altitude is at once seen by placing the sun's degree on the South line.

14. This is the exact converse of the preceding. It furnishes a method of testing the accuracy of the drawing of the almikanteras.

SUPPLEMENTARY PROPOSITIONS.

COMMENTARY ("FOOTNOTES").