Chaucer's Works (ed. Skeat) Vol. III/Treatise on the Astrolabe

Litell Lowis my sone, I have perceived wel by certeyne evidences thyn abilite to lerne sciencez touchinge noumbres and proporciouns; and as wel considere I thy bisy preyere in special to lerne the Tretis of the Astrolabie. Than, for as mechel as a philosofre seith, 'he wrappeth him in his frend, that condescendeth to the rightful preyers of his frend,' ther-for have I geven thee a suffisaunt Astrolabie as for oure orizonte, compowned after the latitude of Oxenford; up-on which, by mediacion of this litel tretis, I purpose to teche thee a certein nombre of conclusions apertening to the same instrument. I seye a certein of conclusiouns, for three causes. The furste cause is this: truste wel that alle the conclusiouns that han ben founde, or elles possibly mighten be founde in so noble an instrument as an Astrolabie, ben un-knowe perfitly to any mortal man in this regioun, as I suppose. A-nother cause is this; that sothly, in any tretis of the Astrolabie that I have seyn, there ben some conclusions that wole nat in alle thinges performen hir bihestes; and some of hem ben to harde to thy tendre age of ten yeer to conseyve. This tretis, divided in fyve parties, wole I shewe thee under ful lighte rewles and naked wordes in English; for Latin ne canstow yit but smal, my lyte sone. But natheles, suffyse to thee thise trewe conclusiouns in English, as wel as suffyseth to thise noble clerkes Grekes thise same conclusiouns in Greek, and to Arabiens in Arabik, and to Iewes in Ebrew, and to the Latin folk in Latin; whiche Latin folk han hem furst out of othre diverse langages, and writen in hir owne tonge, that is to sein, in Latin. And god wot, that in alle thise langages, and in many mo, han thise conclusiouns ben suffisantly lerned and taught, and yit by diverse rewles, right as diverse pathes leden diverse folk the righte wey to Rome. Now wol I prey meekly every discret persone that redeth or hereth this litel tretis, to have my rewde endyting for excused, and my superfluite of wordes, for two causes. The firste cause is, for that curious endyting and hard sentence is ful hevy atones for swich a child to lerne. And the seconde cause is this, that sothly me semeth betre to wryten un-to a child twyes a good sentence, than he for-gete it ones. And Lowis, yif so be that I shewe thee in my lighte English as trewe conclusiouns touching this matere, and naught only as trewe but as many and as subtil conclusiouns as ben shewed in Latin in any commune tretis of the Astrolabie, con me the more thank; and preye god save the king, that is lord of this langage, and alle that him feyth bereth and obeyeth, everech in his degree, the more and the lasse. But considere wel, that I ne usurpe nat to have founde this werk of my labour or of myn engin. I nam but a lewd compilatour of the labour of olde Astrologiens, and have hit translated in myn English only for thy doctrine; and with this swerd shal I sleen envye.

I. The firste partie of this tretis shal reherse the figures and the membres of thyn Astrolabie, bi-cause that thou shalt han the grettre knowing of thyn owne instrument.

II. The second partie shal teche thee werken the verrey practik of the forseide conclusiouns, as ferforth and as narwe as may be shewed in so smal an instrument portatif aboute. For wel wot every astrologien that smalest fraccions ne wol nat ben shewed in so smal an instrument, as in subtil tables calculed for a cause.

III. The thridde partie shal contienen diverse tables of longitudes and latitudes of sterres fixe for the Astrolabie, and tables of declinacions of the sonne, and tables of longitudes of citeez and of townes; and as wel for the governance of a clokke as for to finde the altitude meridian; and many another notable conclusioun, after the kalendres of the reverent clerkes, frere I. Somer and frere N. Lenne.

IV. The ferthe partie shal ben a theorik to declare the moevinge of the celestial bodies with the causes. The whiche ferthe partie in special shal shewen a table of the verray moeving of the mone from houre to houre, every day and in every signe, after thyn almenak; upon which table ther folwith a canon, suffisant to teche as wel the maner of the wyrking of that same conclusioun, as to knowe in oure orizonte with which degree of the zodiac that the mone ariseth in any latitude; and the arising of any planete after his latitude fro the ecliptik lyne.

V. The fifte partie shal ben an introductorie after the statutz of oure doctours, in which thou maist lerne a gret part of the general rewles of theorik in astrologie. In which fifte partie shaltow finde tables of equacions of houses aftur the latitude of Oxenford; and tables of dignetes of planetes, and other noteful thinges, yif god wol vouche-sauf and his modur the mayde, mo than I be-hete, &c.

1. Thyn Astrolabie hath a ring to putten on the thoumbe of thy right hand in taking the heighte of thinges. And tak keep, for from hennes-forthward, I wol clepe the heighte of any thing that is taken by thy rewle, the altitude, with-oute mo wordes.

2. This ring renneth in a maner turet, fast to the moder of thyn Astrolabie, in so rowm a space that hit desturbeth nat the instrument to hangen after his righte centre.

3. The Moder of thyn Astrolabie is the thikkeste plate, perced with a large hole, that resseyveth in hir wombe the thinne plates compowned for diverse clymatz, and thy riet shapen in manere of a net or of a webbe of a loppe; and for the more declaracioun, lo here the figure.

4. This moder is devyded on the bak-half with a lyne, that cometh dessendinge fro the ring down to the nethereste bordure. The whiche lyne, fro the for-seide ring un-to the centre of the large hole amidde, is cleped the south lyne, or elles the lyne meridional. And the remenant of this lyne downe to the bordure is cleped the north lyne, or elles the lyne of midnight. And for the more declaracioun, lo here the figure.

5. Over-thwart this for-seide longe lyne, ther crosseth him another lyne of the same lengthe from est to west. Of the whiche lyne, from a litel croys + in the bordure un-to the centre of the large hole, is cleped the Est lyne, or elles the lyne Orientale; and the remenant of this lyne fro the forseide + un-to the bordure, is cleped the West lyne, or the lyne Occidentale. Now hastow here the foure quarters of thin Astrolabie, devyded after the foure principals plages or quarters of the firmament. And for the more declaracioun, lo here thy figure.

6. The est side of thyn Astrolabie is cleped the right side, and the west side is cleped the left side. Forget nat this, litel Lowis. Put the ring of thyn Astrolabie upon the thoumbe of thy right hand, and thanne wole his right syde be toward thy left syde, and his left syde wol be toward thy right syde; tak this rewle general, as wel on the bak as on the wombe-side. Upon the ende of this est lyne, as I first seide, is marked a litel +, wher-as evere-mo generaly is considered the entring of the first degree in which the sonne aryseth. And for the more declaracioun, lo here the figure.

7. Fro this litel + up to the ende of the lyne meridional, under the ring, shaltow finden the bordure devyded with 90 degrees; and by that same proporcioun is every quarter of thin Astrolabie devyded. Over the whiche degrees ther ben noumbres of augrim, that devyden thilke same degrees fro fyve to fyve, as sheweth by longe strykes by-twene. Of whiche longe strykes the space by-twene contienith a mile-wey. And every degree of the bordure contieneth foure minutes, that is to seyn, minutes of an houre. And for more declaracioun, lo here the figure.

8. Under the compas of thilke degrees ben writen the names of the Twelve Signes, as Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, Pisces; and the nombres of the degrees of tho signes ben writen in augrim above, and with longe devisiouns, fro fyve to fyve; devyded fro tyme that the signe entreth un-to the laste ende. But understond wel, that thise degrees of signes ben everich of hem considered of 60 minutes, and every minute of 60 secondes, and so forth in-to smale fraccions infinit, as seith Alkabucius. And ther-for, know wel, that a degree of the bordure contieneth foure minutes, and a degree of a signe contieneth 60 minutes, and have this in minde. And for the more declaracioun, lo here thy figure.

9. Next this folweth the Cercle of the Dayes, that ben figured in maner of degrees, that contienen in noumbre 365; divyded also with longe strykes fro fyve to fyve, and the nombres in augrim writen under that cercle. And for more declaracioun, lo here thy figure.

10. Next the Cercle of the Dayes, folweth the Cercle of the names of the Monthes; that is to seyen, Ianuare, Februare, Marcius, Aprile, Mayus, Iuin, Iulius, Augustus, Septembre, October, Novembre, Decembre. The names of thise monthes were cleped in Arabiens, somme for hir propretees, and some by statutz of lordes, some by other lordes of Rome. Eek of thise monthes, as lyked to Iulius Cesar and to Cesar Augustus, some were compowned of diverse nombres of dayes, as Iuil and August. Thanne hath Ianuare 31 dayes, Februare 28, March 31, Aprille 30, May 31, Iunius 30, Iulius 31, Augustus 31, September 30, Octobre 31, Novembre 30, December 31. Natheles, al-though that Iulius Cesar took 2 dayes out of Feverer and put hem in his moneth of Iuille, and Augustus Cesar cleped the moneth of August after his name, and ordeyned it of 31 dayes, yit truste wel, that the sonne dwelleth ther-for nevere the more ne lesse in oon signe than in another.

11. Than folwen the names of the Halidayes in the Kalender, and next hem the lettres of the Abc. on which they fallen. And for the more declaracioun, lo here thy figure.

12. Next the forseide Cercle of the Abc., under the cros-lyne, is marked the scale, in maner of two squyres, or elles in manere of laddres, that serveth by hise 12 poyntes and his devisiouns of ful many a subtil conclusioun. Of this forseide scale, fro the croos-lyne un-to the verre angle, is cleped umbra versa, and the nether partie is cleped the umbra recta, or elles umbra extensa. And for the more declaracioun, lo here the figure.

13. Thanne hastow a brood Rewle, that hath on either ende a square plate perced with a certein holes, some more and some lesse, to resseyven the stremes of the sonne by day, and eek by mediacioun of thyn eye, to knowe the altitude of sterres by nighte. And for the more declaracioun, lo here thy figure.

14. Thanne is ther a large Pyn, in maner of an extree, that goth thorow the hole that halt the tables of the clymates and the riet in the wombe of the Moder, thorw which Pyn ther goth a litel wegge which that is cleped 'the hors,' that streyneth alle thise parties to-hepe; this forseide grete Pyn, in maner of an extree, is imagined to be the Pol Artik in thyn Astrolabie. And for the more declaracioun, lo here the figure.

15. The wombe-side of thyn Astrolabie is also devyded with a longe croys in foure quarters from est to west, fro south to north, fro right syde to left syde, as is the bak-syde. And for the more declaracioun, lo here thy figure.

16. The bordure of which wombe-side is devyded fro the poynt of the est lyne un-to the poynt of the south lyne under the ring, in 90 degres; and by that same proporcioun is every quarter devyded as is the bak-syde, that amonteth 360 degrees. And understond wel, that degrees of this bordure ben answering and consentrik to the degrees of the Equinoxial, that is devyded in the same nombre as every othere cercle is in the heye hevene. This same bordure is devyded also with 23 lettres capitals and a smal croys + above the south lyne, that sheweth the 24 houres equals of the clokke; and, as I have said, 5 of thise degrees maken a mile-wey, and 3 mile-wey maken an houre. And every degree of this bordure conteneth 4 minutes, and every minut 60 secoundes; now have I told thee twye. And for the more declaracioun, lo here the figure.

17. The plate under thy riet is descryved with 3 principal cercles; of which the leste is cleped the cercle of Cancer, by-cause that the heved of Cancer turneth evermor consentrik up-on the same cercle. In this heved of Cancer is the grettest declinacioun northward of the sonne. And ther-for is he cleped the Solsticioun of Somer; whiche declinacioun, aftur Ptholome, is 23 degrees and 50 minutes, as wel in Cancer as in Capricorne. This signe of Cancre is cleped the Tropik of Somer, of tropos, that is to seyn 'agaynward'; for thanne by-ginneth the sonne to passe fro us-ward. And for the more declaracioun, lo here the figure. The middel cercle in wydnesse, of thise 3, is cleped the Cercle Equinoxial; up-on whiche turneth evermo the hedes of Aries and Libra. And understond wel, that evermo this Cercle Equinoxial turneth iustly fro verrey est to verrey west; as I have shewed thee in the spere solide. This same cercle is cleped also the Weyere, equator, of the day; for whan the sonne is in the hevedes of Aries and Libra, than ben the dayes and the nightes ilyke of lengthe in al the world. And ther-fore ben thise two signes called the Equinoxies. And alle that moeveth with-in the hevedes of thise Aries and Libra, his moeving is cleped northward; and alle that moeveth with-oute thise hevedes, his moeving is cleped south-ward as fro the equinoxial. Tak keep of thise latitudes north and sowth, and forget it nat. By this Cercle Equinoxial ben considered the 24 houres of the clokke; for everemo the arysing of 15 degrees of the equinoxial maketh an houre equal of the clokke. This equinoxial is cleped the girdel of the firste moeving, or elles of the angulus primi motus vel primi mobilis. And nota, that firste moeving is cleped 'moeving' of the firste moevable of the 8 spere, whiche moeving is fro est to west, and eft agayn in-to est; also it is clepid 'girdel' of the first moeving, for it departeth the firste moevable, that is to seyn, the spere, in two ilyke parties, evene-distantz fro the poles of this world.

The wydeste of thise three principal cercles is cleped the Cercle of Capricorne, by-cause that the heved of Capricorne turneth evermo consentrik up-on the same cercle. In the heved of this for-seide Capricorne is the grettest declinacioun southward of the sonne, and ther-for is it cleped the Solsticioun of Winter. This signe of Capricorne is also cleped the Tropik of Winter, for thanne byginneth the sonne to come agayn to us-ward. And for the more declaracioun, lo here thy figure.

18. Upon this forseide plate ben compassed certein cercles that highten Almicanteras, of which som of hem semen perfit cercles, and somme semen inperfit. The centre that standith a-middes the narwest cercle is cleped the Senith; and the netherest cercle, or the firste cercle, is clepid the Orisonte, that is to seyn, the cercle that devydeth the two emisperies, that is, the partie of the hevene a-bove the erthe and the partie be-nethe. Thise Almicanteras ben compowned by two and two, al-be-it so that on divers Astrolabies some Almicanteras ben devyded by oon, and some by two, and somme by three, after the quantite of the Astrolabie. This forseide senith is imagened to ben the verrey point over the crowne of thyn heved; and also this senith is the verrey pool of the orisonte in every regioun. And for the more declaracioun, lo here thy figure.

19. From this senith, as it semeth, ther come a maner crokede strykes lyke to the clawes of a loppe, or elles like to the werk of a womanes calle, in kerving overthwart the Almikanteras. And thise same strykes or divisiouns ben cleped Azimuthz. And they devyden the orisonte of thyn Astrolabie in four and twenty devisiouns. And thise Azimutz serven to knowe the costes of the firmament, and to othre conclusiouns, as for to knowe the cenith of the sonne and of every sterre. And for more declaracioun, lo here thy figure.

20. Next thise azimutz, under the Cercle of Cancer, ben ther twelve devisiouns embelif, moche like to the shap of the azimutes, that shewen the spaces of the houres of planetes; and for more declaracioun, lo here thy figure.

21. The Riet of thyn Astrolabie with thy zodiak, shapen in maner of a net or of a loppe-webbe after the olde descripcioun, which thow mayst tornen up and doun as thy-self lyketh, conteneth certein nombre of sterres fixes, with hir longitudes and latitudes determinat; yif so be that the makere have nat erred. The names of the sterres ben writen in the margin of the riet ther as they sitte; of whiche sterres the smale poynt is cleped the Centre. And understond also that alle sterres sittinge with-in the zodiak of thyn Astrolabie ben cleped 'sterres of the north,' for they arysen by northe the est lyne. And alle the remenant fixed, out of the zodiak, ben cleped 'sterres of the south;' but I sey nat that they arysen alle by southe the est lyne; witnesse on Aldeberan and Algomeysa. Generally understond this rewle, that thilke sterres that ben cleped sterres of the north arysen rather than the degree of hir longitude, and alle the sterres of the south arysen after the degree of hir longitude; this is to seyn, sterres fixed in thyn Astrolabie. The mesure of this longitude of sterres is taken in the lyne ecliptik of hevene, under which lyne, whan that the sonne and the mone ben lyne-right or elles in the superfice of this lyne, than is the eclips of the sonne or of the mone; as I shal declare, and eek the cause why. But sothly the Ecliptik Lyne of thy zodiak is the outtereste bordure of thy zodiak, ther the degrees ben marked.

Thy Zodiak of thyn Astrolabie is shapen as a compas which that conteneth a large brede, as after the quantite of thyn Astrolabie; in ensample that the zodiak in hevene is imagened to ben a superfice contening a latitude of twelve degrees, wheras al the remenant of cercles in the hevene ben imagined verrey lynes with-oute eny latitude. Amiddes this celestial zodiak ys imagined a lyne, which that is cleped the Ecliptik Lyne, under which lyne is evermo the wey of the sonne. Thus ben ther six degrees of the zodiak on that on side of the lyne, and six degrees on that other. This zodiak is devided in twelve principal devisiouns, that departen the twelve signes. And, for the streitnes of thin Astrolabie, than is every smal devisioun in a signe departid by two degrees and two; I mene degrees contening sixty minutes. And this forseide hevenissh zodiak is cleped the Cercle of the Signes, or the Cercle of the Bestes; for zodia in langage of Greek sowneth 'bestes' in Latin tonge; and in the zodiak ben the twelve signes that ban names of bestes; or elles, for whan the sonne entreth in any of the signes, he taketh the propretee of swich bestes; or elles, for that the sterres that ben there fixed ben disposed in signes of bestes, or shape like bestes; or elles, whan the planetes ben under thilke signes, they causen us by hir influence operaciouns and effectes lyk to the operaciouns of bestes. And understonde also, that whan an hot planete cometh in-to an hot signe, than encresseth his hete; and yif a planete be cold, thanne amenuseth his coldnesse, by-cause of the hote signe. And by this conclusioun maystow take ensample in alle the signes, be they moist or drye, or moeble or fix; rekening the qualitee of the planete as I first seide. And everich of thise twelve signes hath respecte to a certein parcelle of the body of a man and hath it in governance; as Aries hath thyn heved, and Taurus thy nekke and thy throte, Gemini thyn armholes and thyn armes, and so forth; as shal be shewed more pleyn in the fifte partie of this tretis. This zodiak, which that is part of the eighte spere, over-kerveth the equinoxial; and he over-kerveth him again in evene parties; and that on half declineth southward, and that other northward, as pleynly declareth the tretis of the spere. And for more declaracioun, lo here thy figure.

22. Thanne hastow a label, that is schapen lyk a rewle, save that it is streit and hath no plates on either ende with holes; but, with the smale point of the forseide label, shallow calcule thyne equaciouns in the bordure of thin Astrolabie, as by thyn almury. And for the more declaracioun, lo here thy figure.

23. Thyn Almury is cleped the Denticle of Capricorne, or elles the Calculer. This same Almury sit fix in the bed of Capricorne, and it serveth of many a necessarie conclusioun in equaciouns of thinges, as shal be shewed; and for the more declaracioun, lo here thy figure.

1. To fynde the degree in which the sonne is day by day, after hir cours a-boute.

[Hic incipiunt Conclusiones Astrolabii; et prima est ad inveniendum gradus solis in quibus singulis diebus secundum cursum sol est existens.]

Rekene and knowe which is the day of thy monthe; and ley thy rewle up that same day; and thanne wol the verray point of thy rewle sitten in the bordure, up-on the degree of thy sonne. Ensample as thus; the yeer of oure lord 1391, the 12 day of March at midday, I wolde knowe the degree of the sonne. I soughte in the bak-half of myn Astrolabie, and fond the cercle of the dayes, the which I knowe by the names of the monthes writen under the same cercle. Tho leide I my rewle over this forseide day, and fond the point of my rewle in the bordure up-on the firste degree of Aries, a litel with-in the degree; and thus knowe I this conclusioun. Another day, I wolde knowe the degree of my sonne, and this was at midday in the 13 day of Decembre; I fond the day of the monthe in maner as I seide; tho leide I my rewle up-on this forseide 13 day, and fond the point of my rewle in the bordure up-on the first degree of Capricorne, a lite with-in the degree; and than hadde I of this conclusioun the ful experience. And for the more declaracioun, lo here thy figure.

2. To knowe the altitude of the sonne, or of othre celestial bodies.

[De altitudine solis et aliorum corporum supra celestium.]

Put the ring of thyn Astrolabie up-on thy right thoumbe, and turne thy lift syde agayn the light of the sonne. And remeve thy rewle up and doun, til that the stremes of the sonne shyne thorgh bothe holes of thy rewle. Loke thanne how many degrees thy rewle is areised fro the litel crois up-on thyn est line, and tak ther the altitude of thy sonne. And in this same wyse maistow knowe by nighte the altitude of the mone, or of brighte sterres. This chapitre is so general ever in oon, that ther nedith no more declaracion; but forget it nat. And for the more declaracioun, lo here the figure.

3. To knowe every tyme of the day by light of the sonne, and every tyme of the night by the sterres fixe, and eke to knowe by night or by day the degree of any signe that assendeth on the Est Orisonte, which that is cleped communly the Assendent, or elles Oruscupum.

[Ad cognoscendum quodlibet tempus diei per solis indicacionem, et quodlibet tempus noctis per quasdam stellas in celo fixas; ac eciam ad inveniendum et cognoscendum signum super orizontem qui communiter vocatur ascendens.]

Tak the altitude of the sonne whan thee list, as I have said; and set the degree of the sonne, in cas that it be by-forn the middel of the day, among thyn almikanteras on the est side of thyn Astrolabie; and yif it be after the middel of the day, set the degree of thy sonne up-on the west side; tak this manere of setting for a general rewle, ones for evere. And whan thou hast set the degree of thy sonne up as many almikanteras of heyghte as was the altitude of the sonne taken by thy rewle, ley over thy label, up-on the degree of the sonne; and thanne wol the point of thy label sitten in the bordure, up-on the verrey tyd of the day. Ensample as thus: the yeer of oure lord 1391, the 12 day of March, I wold knowe the tyd of the day. I took the altitude of my sonne, and fond that it was 25 degrees and 30 of minutes of heyghte in the bordure on the bak-syde. Tho turnede I myn Astrolabie, and by-cause that it was by-forn midday, I turnede my riet, and sette the degree of the sonne, that is to seyn, the 1 degree of Aries, on the right syde of myn Astrolabie, up-on that 25 degrees and 30 of minutes of heyghte among myn almikanteras; tho leide I my label up-on the degree of my sonne, and fond the poynte of my label in the bordure, up-on a capital lettre that is cleped an X; tho rekened I alle the capitalles lettres fro the lyne of midnight un-to this forseide lettre X, and fond that it was 9 of the clokke of the day. Tho loked I down up-on the est orisonte, and fond there the 20 degree of Geminis assending; which that I tok for myn assendent. And in this wyse hadde I the experience for ever-mo in which maner I sholde knowe the tyd of the day, and eek myn assendent. Tho wolde I wite the same night folwing the hour of the night, and wroughte in this wyse. Among an heep of sterris fixe, it lyked me for to take the altitude of the feire white sterre that is cleped Alhabor; and fond hir sitting on the west side of the lyne of midday, 18 degres of heighte taken by my rewle on the bak-syde. Tho sette I the centre of this Alhabor up-on 18 degrees among myn almikanteras, up-on the west syde; by-cause that she was founden on the west syde. Tho leide I my label over the degree of the sonne that was descended under the weste orisonte, and rikened alle the lettres capitals fro the lyne of midday un-to the point of my label in the bordure; and fond that it was passed 8 of the clokke the space of 2 degrees. Tho loked I doun up-on myn est orisonte, and fond ther 23 degrees of Libra assending, whom I tok for myn assendent; and thus lerned I to knowe ones for ever in which manere I shuld come to the houre of the night and to myn assendent; as verreyly as may be taken by so smal an instrument. But natheles, in general, wolde I warne thee for evere, ne mak thee nevere bold to have take a iust ascendent by thyn Astrolabie, or elles to have set iustly a clokke, whan any celestial body by which that thow wenest governe thilke thinges ben ney the south lyne; for trust wel, whan that the sonne is ney the meridional lyne, the degree of the sonne renneth so longe consentrik up-on the almikanteras, that sothly thou shalt erre fro the iust assendent. The same conclusioun sey I by the centre of any sterre fix by night; and more-over, by experience, I wot wel that in oure orisonte, from 11 of the clokke un-to oon of the clokke, in taking of a iust assendent in a portatif Astrolabie, hit is to hard to knowe. I mene, from 11 of the clokke biforn the houre of noon til oon of the clok next folwing. And for the more declaracion, lo here thy figure.

4. Special declaracion of the assendent.

[Specialis declaracio de ascendente.]

The assendent sothly, as wel in alle nativitez as in questiouns and elecciouns of tymes, is a thing which that thise astrologiens gretly observen; wher-fore me semeth convenient, sin that I speke of the assendent, to make of it special declaracioun. The assendent sothly, to take it at the largeste, is thilke degree that assendeth at any of thise forseide tymes upon the est orisonte; and there-for, yif that any planet assende at that same tyme in thilke for-seide degree of his longitude, men seyn that thilke planete is in horoscopo. But sothly, the hous of the assendent, that is to seyn, the firste hous or the est angle, is a thing more brood and large. For after the statutz of astrologiens, what celestial body that is 5 degres above thilk degree that assendeth, or with-in that noumbre, that is to seyn, nere the degree that assendeth, yit rikne they thilke planet in the assendent. And what planete that is under thilke degree that assendith the space of 25 degrees, yit seyn they that thilke planete is lyk to him that is in the hous of the assendent; but sothly, yif he passe the bondes of thise forseide spaces, above or bynethe, they seyn that the planete is failling fro the assendent. Yit sein thise astrologiens, that the assendent, and eke the lord of the assendent, may be shapen for to be fortunat or infortunat, as thus: a fortunat assendent clepen they whan that no wykkid planete, as Saturne or Mars, or elles the Tail of the Dragoun, is in the hous of the assendent, ne that no wikked planete have non aspecte of enemite up-on the assendent; but they wol caste that they have a fortunat planete in hir assendent and yit in his felicitee, and than sey they that it is wel. Forther-over, they seyn that the infortuning of an assendent is the contrarie of thise forseide thinges. The lord of the assendent, sey they, that he is fortunat, whan he is in good place fro the assendent as in angle; or in a succedent, where-as he is in his dignitee and conforted with frendly aspectes of planetes and wel resceived, and eek that he may seen the assendent, and that he be nat retrograd ne combust, ne ioigned with no shrewe in the same signe; ne that he be nat in his descencioun, ne ioigned with no planete in his discencioun, ne have up-on him non aspecte infortunat; and than sey they that he is wel. Natheles, thise ben observauncez of iudicial matiere and rytes of payens, in which my spirit ne hath no feith, ne no knowing of hir horoscopum; for they seyn that every signe is departed in 3 evene parties by 10 degrees, and thilke porcioun they clepe a Face. And al-thogh that a planete have a latitude fro the ecliptik, yit sey some folk, so that the planete aryse in that same signe with any degree of the forseide face in which his longitude is rekned, that yit is the planete in horoscopo, be it in nativite or in eleccioun, &c. And for the more declaracioun, lo here the figure.

5. To knowe the verrey equacioun of the degree of the sonne, yif so be that it falle by-twixe thyn Almikanteras.

[Ad cognoscendum veram equacionem de gradu solis, si contigerit fore in duas Almicanteras.]

For as moche as the almikanteras in thyn Astrolabie been compouned by two and two, where-as some almikanteras in sondry Astrolabies ben compouned by on and on, or elles by two and two, it is necessarie to thy lerning to teche thee first to knowe and worke with thyn owne instrument. Wher-for, whan that the degree of thy sonne falleth by-twixe two almikanteras, or elles yif thyn almikanteras ben graven with over gret a point of a compas, (for bothe thise thinges may causen errour as wel in knowing of the tyd of the day as of the verrey assendent), thou most werken in this wyse. Set the degree of thy sonne up-on the heyer almikanteras of bothe, and waite wel wher as thin almury toucheth the bordure, and set ther a prikke of inke. Set doun agayn the degree of thy sonne up-on the nethere almikanteras of bothe, and set ther another prikke. Remewe thanne thyn almury in the bordure evene amiddes bothe prikkes, and this wol lede iustly the degree of thy sonne to sitte by-twixe bothe almikanteras in his right place. Ley thanne thy label over the degree of thy sonne; and find in the bordure the verrey tyde of the day or of the night. And as verreyly shaltow finde up-on thyn est orisonte thyn assendent. And for more declaracioun, lo here thy figure.

6. To knowe the spring of the dawing and the ende of the evening, the which ben called the two crepusculis:

[Ad cognoscendum ortum solis et eius occasum, que vocatur vulgariter  crepusculum.]

Set the nadir of thy sonne up-on 18 degrees of heighte among thyn almikanteras on the west syde, and ley thy label on the degree of thy sonne, and thanne shal the poynt of thy label schewe the spring of day. Also set the nadir of thy sonne up-on 18 degrees of heighte a-mong thyn almikanteras on the est side, and ley over thy label up-on the degree of the sonne, and with the point of thy label find in the bordure the ende of the evening, that is, verrey night. The nadir of the sonne is thilke degree that is opposit to the degree of the sonne, in the seventhe signe, as thus: every degree of Aries by ordre is nadir to every degree of Libra by ordre; and Taurus to Scorpion; Gemini to Sagittare; Cancer to Capricorne; Leo to Aquarie; Virgo to Pisces; and yif any degree in thy zodiak be dirk, his nadir shal declare him. And for the more declaracioun, lo here thy figure.

7. To knowe the arch of the day, that some folk callen the day artificial, from the sonne arysing til hit go to reste.

[Ad cognoscendum archum diei, quem vulgus vocat diem artificialem, in hoc, ab ortu solis usque ad occasum.]

Set the degree of thy sonne up-on thyn est orisonte, and ley thy label on the degree of the sonne, and at the poynt of thy label in the bordure set a prikke. Turn thanne thy riet aboute til the degree of the sonne sit up-on the west orisonte, and ley thy label up-on the same degree of the sonne, and at the point of thy label set a-nother prikke. Rekne thanne the quantitee of tyme in the bordure by-twixe bothe prikkes, and tak ther thyn ark of the day. The remenant of the bordure under the orisonte is the ark of the night. Thus maistow rekne bothe arches, or every porcion, of whether that thee lyketh. And by this manere of wyrking maistow see how longe that any sterre fix dwelleth above the erthe, fro tyme that he ryseth til he go to reste. But the day natural, that is to seyn 24 houres, is the revolucioun of the equinoxial with as moche partie of the zodiak as the sonne of his propre moevinge passeth in the mene whyle. And for the more declaracioun, lo here thy figure.

8. To turn the houres in-equales in houres equales.

[Ad convertendum horas inequales in horas equales.]

Knowe the nombre of the degrees in the houres in-equales, and departe hem by 15, and tak ther thyn houres equales. And for the more declaracioun, lo here thy figure.

9. To knowe the quantitee of the day vulgare, that is to seyen, from spring of the day un-to verrey night.

'''[Ad cognoscendum quantitatem diei vulgaris, viz. ab ortu diei usque''' ad noctem.]

Know the quantitee of thy crepusculis, as I have taught in the chapitre bi-forn, and adde hem to the arch of thy day artificial; and tak ther the space of alle the hole day vulgar, un-to verrey night. The same manere maystow worke, to knowe the quantitee of the vulgar night. And for the more declaracioun, lo here the figure.

10. To knowe the quantite of houres in-equales by day.

[Ad cognoscendum horas inequales in die.]

Understond wel, that thise houres in-equales ben cleped houres of planetes, and understond wel that som-tyme ben they lengere by day than by night, and som-tyme the contrarie. But understond wel, that evermo, generaly, the hour in-equal of the day with the houre in-equal of the night contenen 30 degrees of the bordure, whiche bordure is ever-mo answering to the degrees of the equinoxial; wher-for departe the arch of the day artificial in 12, and tak ther the quantitee of the houre in-equal by day. And yif thow abate the quantitee of the houre in-equal by daye out of 30, than shal the remenant that leveth performe the houre inequal by night. And for the more declaracioun, lo here the figure.

11. To knowe the quantite of houres equales.

[Ad cognoscendum quantitatem horarum inequalium.]

The quantitee of houres equales, that is to seyn, the houres of the clokke, ben departed by 15 degrees al-redy in the bordure of thyn Astrolabie, as wel by night as by day, generaly for evere. What nedeth more declaracioun? Wher-for, whan thee list to know how manye houres of the clokke ben passed, or any part of any of thise houres that ben passed, or elles how many houres or partie of houres ben to come, fro swich a tyme to swich a tyme, by day or by nighte, knowe the degree of thy sonne, and ley thy label on it; turne thy riet aboute ioyntly with thy label, and with the point of it rekne in the bordure fro the sonne aryse un-to the same place ther thou desirest, by day as by nighte. This conclusioun wol I declare in the laste chapitre of the 4 partie of this tretis so openly, that ther shal lakke no worde that nedeth to the declaracioun. And for the more declaracioun, lo here the figure.

12. Special declaracioun of the houres of planetes.

[Specialis declaracio de horis planetarum.]

Understond wel, that evere-mo, fro the arysing of the sonne til it go to reste, the nadir of the sonne shal shewe the houre of the planete, and fro that tyme forward al the night til the sonne aryse; than shal the verrey degree of the sonne shewe the houre of the planete. Ensample as thus. The 13 day of March fil up-on a Saterday per aventure, and, at the arising of the sonne, I fond the secounde degree of Aries sitting up-on myn est orisonte, al-be-it that it was but lite; than fond I the 2 degree of Libra, nadir of my sonne, dessending on my west orisonte, up-on which west orisonte every day generally, at the sonne ariste, entreth the houre of any planete, after which planete the day bereth his name; and endeth in the nexte stryk of the plate under the forseide west orisonte; and evere, as the sonne climbeth uppere and uppere, so goth his nadir dounere and dounere, teching by swich strykes the houres of planetes by ordre as they sitten in the hevene. The first houre inequal of every Satterday is to Saturne; and the secounde, to Iupiter; the 3, to Mars; the 4, to the Sonne; the 5, to Venus; the 6, to Mercurius; the 7, to the Mone; and thanne agayn, the 8 is to Saturne; the 9, to Iupiter; the 10, to Mars; the 11, to the Sonne; the 12, to Venus; and now is my sonne gon to reste as for that Setterday. Thanne sheweth the verrey degree of the sonne the houre of Mercurie entring under my west orisonte at eve; and next him succedeth the Mone; and so forth by ordre, planete after planete, in houre after houre, al the night longe til the sonne aryse. Now ryseth the sonne that Sonday by the morwe; and the nadir of the sonne, up-on the west orizonte, sheweth me the entring of the houre of the forseide sonne. And in this maner succedeth planete under planete, fro Saturne un-to the Mone, and fro the Mone up a-gayn to Saturne, houre after houre generaly. And thus knowe I this conclusioun. And for the more declaracioun, lo here the figure.

13. To knowe the altitude of the sonne in middes of the day, that is cleped the altitude meridian.

[Ad cognoscendum altitudinem solis in medio diei, que vocatur altitudo meridiana.]

Set the degree of the sonne up-on the lyne meridional, and rikene how many degrees of almikanteras ben by-twixe thyn est orisonte and the degree of the sonne. And tak ther thyn altitude meridian; this is to seyne, the heyest of the sonne as for that day. So maystow knowe in the same lyne, the heyest cours that any sterre fix climbeth by night; this is to seyn, that whan any sterre fix is passed the lyne meridional, than by-ginneth it to descende, and so doth the sonne. And for the more declaracioun, lo here thy figure.

14. To knowe the degree of the sonne by thy riet, for a maner curiositee, &c.

[Ad cognoscendum gradum solis curiose.]

Sek bysily with thy rewle the heyest of the sonne in midde of the day; turne thanne thyn Astrolabie, and with a prikke of ink marke the nombre of that same altitude in the lyne meridional. Turne thanne thy riet a-boute til thou fynde a degree of thy zodiak acording with the prikke, this is to seyn, sittinge on the prikke; and in sooth, thou shalt finde but two degrees in al the zodiak of that condicioun; and yit thilke two degrees ben in diverse signes; than maistow lightly by the sesoun of the yere knowe the signe in whiche that is the sonne. And for the more declaracioun, lo here thy figure.

15. To know which day is lyk to which day as of lengthe, &c.

[Ad cognoscendum quales dies in longitudine sunt similes.]

Loke whiche degrees ben y-lyke fer fro the hevedes of Cancer and Capricorn; and lok, whan the sonne is in any of thilke degrees, than ben the dayes y-lyke of lengthe. This is to seyn, that as long is that day in that monthe, as was swich a day in swich a month; ther varieth but lite. Also, yif thou take two dayes naturaly in the yer y-lyke fer fro eyther pointe of the equinoxial in the opposit parties, than as long is the day artificial of that on day as is the night of that othere, and the contrarie. And for the more declaracioun, lo here thy figure.

16. This chapitre is a maner declaracioun to conclusiouns that folwen.

[Illud capitulum est quedam declaracio ad certas conclusiones sequentes.]

Understond wel that thy zodiak is departid in two halfe cercles, as fro the heved of Capricorne un-to the heved of Cancer; and agaynward fro the heved of Cancer un-to the heved of Capricorne. The heved of Capricorne is the lowest point, wher-as the sonne goth in winter; and the heved of Cancer is the heyest point, in whiche the sonne goth in somer. And ther-for understond wel, that any two degrees that ben y-lyke fer fro any of thise two hevedes, truste wel that thilke two degrees ben of y-lyke declinacioun, be it southward or northward; and the dayes of hem ben y-lyke of lengthe, and the nightes also; and the shadwes y-lyke, and the altitudes y-lyke at midday for evere. And for more declaracioun, lo here thy figure.

17. To knowe the verrey degree of any maner sterre straunge or unstraunge after his longitude, though he be indeterminat in thyn Astrolabie; sothly to the trowthe, thus he shal be knowe.

[Ad cognoscendum verum gradum alicuius stelle aliene secundum eius longitudinem, quamvis sit indeterminata in astrolabio; veraciter isto modo.]

Tak the altitude of this sterre whan he is on the est side of the lyne meridional, as ney as thou mayst gesse; and tak an assendent a-non right by som maner sterre fix which that thou knowest; and for-get nat the altitude of the firste sterre, ne thyn assendent. And whan that this is don, espye diligently whan this same firste sterre passeth any-thing the south westward, and hath him a-non right in the same noumbre of altitude on the west side of this lyne meridional as he was caught on the est side; and tak a newe assendent a-non right by som maner sterre fixe which that thou knowest; and for-get nat this secounde assendent. And whan that this is don, rikne thanne how manye degrees ben by-twixe the firste assendent and the seconde assendent, and rikne wel the middel degree by-twene bothe assendentes, and set thilke middel degree up-on thin est orisonte; and waite thanne what degree that sit up-on the lyne meridional, and tak ther the verrey degree of the ecliptik in which the sterre stondeth for the tyme. For in the ecliptik is the longitude of a celestial body rekened, evene fro the heved of Aries un-to the ende of Pisces. And his latitude is rikned after the quantite of his declinacion, north or south to-warde the poles of this world; as thus. Yif it be of the sonne or of any fix sterre, rekene his latitude or his declinacioun fro the equinoxial cercle; and yif it be of a planete, rekne than the quantitee of his latitude fro the ecliptik lyne. Al-be-it so that fro the equinoxial may the declinacion or the latitude of any body celestial be rikned, after the site north or south, and after the quantitee of his declinacion. And right so may the latitude or the declinacion of any body celestial, save only of the sonne, after his site north or south, and after the quantitee of his declinacioun, be rekned fro the ecliptik lyne; fro which lyne alle planetes som tyme declynen north or south, save only the for-seide sonne. And for the more declaracioun, lo here thy figure.

18. To knowe the degrees of the longitudes of fixe sterres after that they ben determinat in thin Astrolabie, yif so be that they ben trewly set.

[Ad cognoscendum gradus longitudinis de stellis fixis que determinantur in astrolabio, sicut in suis locis recte locentur.]

Set the centre of the sterre up-on the lyne meridional, and tak keep of thy zodiak, and loke what degree of any signe that sit on the same lyne meridional at that same tyme, and tak the degree in which the sterre standeth; and with that same degree comth that same sterre un-to that same lyne fro the orisonte. And for more declaracioun, lo here thy figure.

19. To knowe with which degree of the zodiak any sterre fixe in thyn Astrolabie aryseth up-on the est orisonte, althogh his dwelling be in a-nother signe.

[Ad cognoscendum cum quibus gradibus zodiaci que stella fixa in astrolabio ascendit super orizontem orientalem, quamvis eius statio sit in alio signo.]

Set the centre of the sterre up-on the est orisonte, and loke what degree of any signe that sit up-on the same orisonte at that same tyme. And understond wel, that with that same degree aryseth that same sterre; and this merveyllous arysing with a strange degree in another signe is by-cause that the latitude of the sterre fix is either north or south fro the equinoxial. But sothly the latitudes of planetes ben comunly rekned fro the ecliptik, bi-cause that non of hem declineth but fewe degrees out fro the brede of the zodiak. And tak good keep of this chapitre of arysing of the celestial bodies; for truste wel, that neyther mone ne sterre as in oure embelif orisonte aryseth with that same degree of his longitude, save in o cas; and that is, whan they have no latitude fro the ecliptik lyne. But natheles, som tyme is everiche of thise planetes under the same lyne. And for more declaracioun, lo here thy figure.

20. To knowe the declinacioun of any degree in the zodiak fro the equinoxial cercle, &c.

[Ad cognoscendum declinacionem alicuius gradus in zodiaco a circulo equinoctiali.]

Set the degree of any signe up-on the lyne meridional, and rikne his altitude in almikanteras fro the est orizonte up to the same degree set in the forseide lyne, and set ther a prikke. Turne up thanne thy riet, and set the heved of Aries or Libra in the same meridional lyne, and set ther a-nother prikke. And whan that this is don, considere the altitudes of hem bothe; for sothly the difference of thilke altitudes is the declinacion of thilke degree fro the equinoxial. And yif so be that thilke degree be northward fro the equinoxial, than is his declinacion north; yif it be southward, than is it south. And for the more declaracioun, lo here thy figure.

21. To knowe for what latitude in any regioun the almikanteras of any table ben compouned.

[Ad cognoscendum pro qua latitudine in aliqua regione almicantre tabule mee sunt composite.]

Rikne how manye degrees of almikanteras, in the meridional lyne, be fro the cercle equinoxial un-to the senith; or elles fro the pool artik un-to the north orisonte; and for so gret a latitude or for so smal a latitude is the table compouned. And for more declaracion, lo here thy figure.

22. To knowe in special the latitude of oure countray, I mene after the latitude of Oxenford, and the heighte of oure pol.

[Ad cognoscendum specialiter latitudinem nostri regionis, scilicet latitudinem Oxonie, et altitudinem poli nostri.]

Understond wel, that as fer is the heved of Aries or Libra in the equinoxial from oure orisonte as is the senith from the pole artik; and as hey is the pol artik fro the orisonte, as the equinoxial is fer fro the senith. I prove it thus by the latitude of Oxenford. Understond wel, that the heyghte of oure pool artik fro oure north orisonte is 51 degrees and 50 minutes; than is the senith from oure pool artik 38 degrees and 10 minutes; than is the equinoxial from oure senith 51 degrees and 50 minutes; than is oure south orisonte from oure equinoxial 38 degrees and 10 minutes. Understond wel this rekning. Also for-get nat that the senith is 90 degrees of heyghte fro the orisonte, and oure equinoxial is 90 degrees from oure pool artik. Also this shorte rewle is soth, that the latitude of any place in a regioun is the distance fro the senith unto the equinoxial. And for more declaracioun, lo here thy figure.

23. To prove evidently the latitude of any place in a regioun, by the preve of the heyghte of the pol artik in that same place.

[Ad probandum evidenter latitudinem alicuius loci in aliqua regione, per probacionem altitudinis de polo artico in eodem loco.]

In some winters night, whan the firmament is clere and thikke-sterred, waite a tyme til that any sterre fix sit lyne-right perpendiculer over the pol artik, and clepe that sterre A. And wayte a-nother sterre that sit lyne-right under A, and under the pol, and clepe that sterre F. And understond wel, that F is nat considered but only to declare that A sit evene overe the pool. Tak thanne a-non right the altitude of A from the orisonte, and forget it nat. Lat A and F go farwel til agayns the dawening a gret whyle; and come thanne agayn, and abyd til that A is evene under the pol and under F; for sothly, than wol F sitte over the pool, and A wol sitte under the pool. Tak than eft-sones the altitude of A from the orisonte, and note as wel his secounde altitude as his firste altitude; and whan that this is don, rikne how manye degrees that the firste altitude of A excedeth his seconde altitude, and tak half thilke porcioun that is exceded, and adde it to his seconde altitude; and tak ther the elevacioun of thy pool, and eke the latitude of thy regioun. For thise two ben of a nombre; this is to seyn, as many degrees as thy pool is elevat, so michel is the latitude of the regioun. Ensample as thus: par aventure, the altitude of A in the evening is 56 degrees of heyghte. Than wol his seconde altitude or the dawing be 48; that is 8 lasse than 56, that was his firste altitude at even. Take thanne the half of 8, and adde it to 48, that was his seconde altitude, and than hastow 52. Now hastow the heyghte of thy pol, and the latitude of the regioun. But understond wel, that to prove this conclusioun and many a-nother fair conclusioun, thou most have a plomet hanging on a lyne heyer than thin heved on a perche; and thilke lyne mot hange evene perpendiculer by-twixe the pool and thyn eye; and thanne shaltow seen yif A sitte evene over the pool and over F at evene; and also yif F sitte evene over the pool and over A or day. And for more declaracion, lo here thy figure.

24. Another conclusioun to prove the heyghte of the pool artik fro the orisonte.

[Alia conclusio ad probandum altitudinem de polo artico ab orizonte.]

Tak any sterre fixe that nevere dissendeth under the orisonte in thilke regioun, and considere his heyest altitude and his lowest altitude fro the orisonte; and make a nombre of bothe thise altitudes. Tak thanne and abate half that nombre, and tak ther the elevacioun of the pol artik in that same regioun. And for more declaracioun, lo here thy figure.

25. A-nother conclusioun to prove the latitude of the regioun, &c.

[Alia conclusio ad probandum latitudinem regionis.]

Understond wel that the latitude of any place in a regioun is verreyly the space by-twixe the senith of hem that dwellen there and the equinoxial cerkle, north or southe, taking the mesure in the meridional lyne, as sheweth in the almikanteras of thyn Astrolabie. And thilke space is as moche as the pool artik is hey in the same place fro the orisonte. And than is the depressioun of the pol antartik, that is to seyn, than is the pol antartik by-nethe the orisonte, the same quantite of space, neither more ne lasse. Thanne, yif thow desire to knowe this latitude of the regioun, tak the altitude of the sonne in the middel of the day, whan the sonne is in the hevedes of Aries or of Libra; (for thanne moeveth the sonne in the lyne equinoxial); and abate the nombre of that same sonnes altitude out of 90, and thanne is the remenaunt of the noumbre that leveth the latitude of the regioun. As thus: I suppose that the sonne is thilke day at noon 38 degrees and 10 minutes of heyghte. Abate thanne thise degrees and minutes out of 90; so leveth there 51 degrees and 50 minutes, the latitude. I sey nat this but for ensample; for wel I wot the latitude of Oxenforde is certein minutes lasse, as I mighte prove. Now yif so be that thee semeth to long a taryinge, to abyde til that the sonne be in the hevedes of Aries or of Libra, thanne waite whan the sonne is in any other degree of the zodiak, and considere the degree of his declinacion fro the equinoxial lyne; and yif it so be that the sonnes declinacion be northward fro the equinoxial, abate thanne fro the sonnes altitude at noon the nombre of his declinacion, and thanne hastow the heyghte of the hevedes of Aries and Libra. As thus: my sonne is, par aventure, in the firste degre of Leoun, 58 degrees and 10 minutes of heyghte at noon and his declinacion is almost 20 degrees northward fro the equinoxial; abate thanne thilke 20 degrees of declinacion out of the altitude at noon, than leveth thee 38 degrees and odde minutes; lo ther the heved of Aries or Libra, and thyn equinoxial in that regioun. Also yif so be that the sonnes declinacioun be southward fro the equinoxial, adde thanne thilke declinacion to the altitude of the sonne at noon; and tak ther the hevedes of Aries and Libra, and thyn equinoxial. Abate thanne the heyghte of the equinoxial out of 90 degrees, and thanne leveth there the distans of the pole, 51 degrees and 50 minutes, of that regioun fro the equinoxial. Or elles, yif thee lest, take the heyest altitude fro the equinoxial of any sterre fix that thou knowest, and tak his nethere elongacioun lengthing fro the same equinoxial lyne, and wirke in the maner forseid. And for more declaracion, lo here thy figure.

26. Declaracioun of the assensioun of signes, &c.

[Declaracio de ascensione signorum.]

The excellence of the spere solide, amonges other noble conclusiouns, sheweth manifeste the diverse assenciouns of signes in diverse places, as wel in the righte cercle as in the embelif cercle. Thise auctours wryten that thilke signe is cleped of right ascensioun, with which more part of the cercle equinoxial and lasse part of the zodiak ascendeth; and thilke signe assendeth embelif, with whiche lasse part of the equinoxial and more part of the zodiak assendeth. Ferther-over they seyn, that in thilke cuntrey where as the senith of hem that dwellen there is in the equinoxial lyne, and her orisonte passing by the poles of this worlde, thilke folke han this right cercle and the right orisonte; and evere-mo the arch of the day and the arch of the night is ther y-like long, and the sonne twyes every yeer passinge thorow the senith of her heved; and two someres and two winteres in a yeer han this forseide poeple. And the almikanteras in her Astrolabies ben streighte as a lyne, so as sheweth in this figure. The utilite to knowe the assenciouns in the righte cercle is this: truste wel that by mediacioun of thilke assenciouns thise astrologiens, by hir tables and hir instrumentz, knowen verreyly the assencioun of every degree and minut in al the zodiak, as shal be shewed. And nota, that this forseid righte orisonte, that is cleped orison rectum, divydeth the equinoxial in-to right angles; and the embelif orisonte, wher-as the pol is enhaused up-on the orisonte, overkerveth the equinoxial in embelif angles, as sheweth in the figure. And for the more declaracioun, lo here the figure.

27. This is the conclusioun to knowe the assenciouns of signes in the right cercle, that is, circulus directus, &c.

[Ad cognoscendum ascenciones signorum in recto circulo, qui vocatur circulus directus.]

Set the heved of what signe thee liste to knowe his assending in the right cercle up-on the lyne meridional; and waite wher thyn almury toucheth the bordure, and set ther a prikke. Turne thanne thy riet westward til that the ende of the forseide signe sitte up-on the meridional lyne; and eft-sones waite wher thyn almury toucheth the bordure, and set ther another prikke. Rikne thanne the nombre of degrees in the bordure by-twixe bothe prikkes, and tak the assencioun of the signe in the right cercle. And thus maystow wyrke with every porcioun of thy zodiak, &c. And for the more declaracioun, lo here thy figure.

28. To knowe the assencions of signes in the embelif cercle in every regioun, I mene, in circulo obliquo.

[Ad cognoscendum ascenciones signorum in circulo obliquo, in omni regione.]

Set the heved of the signe which as thee list to knowe his ascensioun up-on the est orisonte, and waite wher thyn almury toucheth the bordure, and set ther a prikke. Turne thanne thy riet upward til that the ende of the same signe sitte up-on the est orisonte, and waite eft-sones wher as thyn almury toucheth the bordure, and set ther a-nother prikke. Rikne thanne the noumbre of degrees in the bordure by-twixe bothe prikkes, and tak ther the assencioun of the signe in the embelif cercle. And understond wel, that alle signes in thy zodiak, fro the heved of Aries unto the ende of Virgo, ben cleped signes of the north fro the equinoxial; and these signes arysen by-twixe the verrey est and the verrey north in oure orisonte generaly for evere. And alle signes fro the heved of Libra un-to the ende of Pisces ben cleped signes of the south fro the equinoxial; and thise signes arysen ever-mo by-twixe the verrey est and the verrey south in oure orisonte. Also every signe by-twixe the heved of Capricorne un-to the ende of Geminis aryseth on oure orisonte in lasse than two houres equales; and thise same signes, fro the heved of Capricorne un-to the ende of Geminis, ben cleped 'tortuos signes' or 'croked signes,' for they arisen embelif on oure orisonte; and thise crokede signes ben obedient to the signes that ben of right assencioun. The signes of right assencioun ben fro the heved of Cancer to the ende of Sagittare; and thise signes arysen more upright, and they ben called eke sovereyn signes; and everich of hem aryseth in more space than in two houres. Of which signes, Gemini obeyeth to Cancer; and Taurus to Leo; Aries to Virgo; Pisces to Libra; Aquarius to Scorpioun; and Capricorne to Sagittare. And thus ever-mo two signes, that ben y-lyke fer fro the heved of Capricorne, obeyen everích of hem til other. And for more declaracioun, lo here the figure.

29. To knowe iustly the foure quarters of the world, as est, west, north, and sowth.

[Ad cognoscendum evidenter quatuor partes mundi, scilicet, orientem, austrum, aquilonem, et occidentem.]

Take the altitude of thy sonne whan thee list, and note wel the quarter of the world in which the sonne is for the tyme by the azimutz. Turne thanne thyn Astrolabie, and set the degree of the sonne in the almikanteras of his altitude, on thilke side that the sonne stant, as is the manere in taking of houres; and ley thy label on the degree of the sonne, and rikene how many degrees of the bordure ben by-twixe the lyne meridional and the point of thy label; and note wel that noumbre. Turne thanne a-gayn thyn Astrolabie, and set the point of thy gret rewle, ther thou takest thyne altitudes, up-on as many degrees in his bordure fro his meridional as was the point of thy label fro the lyne meridional on the wombe-syde. Tak thanne thyn Astrolabie with bothe handes sadly and slely, and lat the sonne shyne thorow bothe holes of thy rewle; and sleyly, in thilke shyninge, lat thyn Astrolabie couch adoun evene up-on a smothe grond, and thanne wol the verrey lyne meridional of thyn Astrolabie lye evene south, and the est lyne wole lye est, and the west lyne west, and north lyne north, so that thou werke softly and avisely in the couching; and thus hastow the 4 quarters of the firmament. And for the more declaracioun, lo here the figure.

30. To knowe the altitude of planetes fro the wey of the sonne, whether so they be north or south fro the forseide wey.

[Ad cognoscendum altitudinem planetarum a cursu solis, utrum sint in parte australi vel boreali a cursu supra dicto.]

Lok whan that a planete is in the lyne meridional, yif that hir altitude be of the same heyghte that is the degree of the sonne for that day, and than is the planete in the verrey wey of the sonne, and hath no latitude. And yif the altitude of the planete be heyere than the degree of the sonne, than is the planete north fro the wey of the sonne swich a quantite of latitude as sheweth by thyn almikanteras. And yif the altitude of the planete be lasse than the degree of the sonne, thanne is the planete south fro the wey of the sonne swich a quantite of latitude as sheweth by thyn almikanteras. This is to seyn, fro the wey wher-as the sonne wente thilke day, but nat from the wey of the sonne in every place of the zodiak. And for the more declaracioun, lo here the figure.

31. To knowe the senith of the arysing of the sonne, this is to seyn, the partie of the orisonte in which that the sonne aryseth.

[Ad cognoscendum signum de ortu solis, scilicet, illam partem orientis in qua oritur sol.]

Thou most first considere that the sonne aryseth nat al-wey verrey est, but some tyme by north the est, and som tyme by southe the est. Sothly, the sonne aryseth never-mo verrey est in oure orisonte, but he be in the heved of Aries or Libra. Now is thyn orisonte departed in 24 parties by thy azimutz, in significacion of 24 partiez of the world; al-be-it so that shipmen rikne thilke partiez in 32. Thanne is ther no more but waite in which azimut that thy sonne entreth at his arysing; and take ther the senith of the arysing of the sonne. The manere of the devisioun of thyn Astrolabie is this; I mene, as in this cas. First is it devided in 4 plages principalx with the lyne that goth from est to west, and than with a-nother lyne that goth fro south to north. Than is it devided in smale partiez of azimutz, as est, and est by southe, whereas is the firste azimut above the est lyne; and so forth, fro partie to partie, til that thou come agayn un-to the est lyne. Thus maistow understond also the senith of any sterre, in which partie he ryseth, &c. And for the more declaracion, lo here the figure.

32. To knowe in which partie of the firmament is the coniunccioun.

[Ad cognoscendum in qua parte firmamenti sunt coniuncciones solis et lune.]

Considere the tyme of the coniunccion by thy kalender, as thus; lok how many houres thilke coniunccion is fro the midday of the day precedent, as sheweth by the canoun of thy kalender. Rikne thanne thilke nombre of houres in the bordure of thyn Astrolabie, as thou art wont to do in knowing of the houres of the day or of the night; and ley thy label over the degree of the sonne; and thanne wol the point of thy label sitte up-on the hour of the coniunccion. Loke thanne in which azimut the degree of thy sonne sitteth, and in that partie of the firmament is the coniunccioun. And for the more declaracioun, lo here thy figure.

33. To knowe the senith of the altitude of the sonne, &c.

[Ad cognoscendum signa de altitudine solis.]

This is no more to seyn but any tyme of the day tak the altitude of the sonne; and by the azimut in which he stondeth, maystou seen in which partie of the firmament he is. And in the same wyse maystou seen, by the night, of any sterre, whether the sterre sitte est or west or north, or any partie by-twene, after the name of the azimut in which is the sterre. And for the more declaracioun, lo here the figure.

34. To knowe sothly the degree of the longitude of the mone, or of any planete that hath no latitude for the tyme fro the ecliptik lyne.

[Ad cognoscendum veraciter gradum de longitudine lune, vel alicuius planete qui non habet longitudinem pro tempore causanto linea ecliptica.]

Tak the altitude of the mone, and rikne thyn altitude up among thyne almikanteras on which syde that the mone stande; and set there a prikke. Tak thenne anon-right, up-on the mones syde, the altitude of any sterre fix which that thou knowest, and set his centre up-on his altitude among thyn almikanteras ther the sterre is founde. Waite thanne which degree of the zodiak toucheth the prikke of the altitude of the mone, and tak ther the degree in which the mone standeth. This conclusioun is verrey soth, yif the sterres in thyn Astrolabie stonden after the trowthe; of comune, tretis of Astrolabie ne make non excepcioun whether the mone have latitude, or non; ne on whether syde of the mone the altitude of the sterre fix be taken. And nota, that yif the mone shewe himself by light of day, than maystow wyrke this same conclusioun by the sonne, as wel as by the fix sterre. And for the more declaracioun, lo here thy figure.

35. This is the workinge of the conclusioun, to knowe yif that any planete be directe or retrograde.

[Hec conclusio operatur ad cognoscendum si aliqua planeta sit directa vel retrograda.]

Tak the altitude of any sterre that is cleped a planete, and note it wel. And tak eek anon the altitude of any sterre fix that thou knowest, and note it wel also. Come thanne agayn the thridde or the ferthe night next folwing; for thanne shaltow aperceyve wel the moeving of a planete, whether so he moeve forthward or bakward. Awaite wel thanne whan that thy sterre fix is in the same altitude that she was whan thou toke hir firste altitude; and tak than eftsones the altitude of the forseide planete, and note it wel. For trust wel, yif so be that the planete be on the right syde of the meridional lyne, so that his seconde altitude be lasse than his firste altitude was, thanne is the planete directe. And yif he be on the west syde in that condicion, thanne is he retrograd. And yif so be that this planete be up-on the est syde whan his altitude is taken, so that his secounde altitude be more than his firste altitude, thanne is he retrograde, and yif he be on the west syde, than is he directe. But the contrarie of thise parties is of the cours of the mone; for sothly, the mone moeveth the contrarie from othere planetes as in hir episicle, but in non other manere. And for the more declaracioun, lo here thy figure.

36. The conclusiouns of equaciouns of houses, after the Astrolabie, &c.

[Conclusio de equacione domorum.]

Set the by-ginning of the degree that assendeth up-on the ende of the 8 houre inequal; thanne wol the by-ginning of the 2 hous sitte up-on the lyne of midnight. Remove thanne the degree that assendeth, and set him on the ende of the 10 hour inequal; and thanne wol the byginning of the 3 hous sitte up-on the midnight lyne. Bring up agayn the same degree that assendeth first, and set him up-on the orisonte; and thanne wol the be-ginning of the 4 hous sitte up-on the lyne of midnight. Tak thanne the nadir of the degree that first assendeth, and set him on the ende of the 2 houre inequal; and thanne wol the by-ginning of the 5 hous sitte up-on the lyne of midnight; set thanne the nadir of the assendent on the ende of the 4 houre, than wol the byginning of the 6 house sitte on the midnight lyne. The byginning of the 7 hous is nadir of the assendent, and the byginning of the 8 hous is nadir of the 2; and the by-ginning of the 9 hous is nadir of the 3; and the by-ginning of the 10 hous is the nadir of the 4; and the byginning of the 11 hous is nadir of the 5; and the byginning of the 12 hous is nadir of the 6. And for the more declaracion, lo here the figure.

37. A-nother manere of equaciouns of houses by the Astrolabie.

[De aliqua forma equacionis domorum secundum astrolabium.]

Tak thyn assendent, and thanne hastow thy 4 angles; for wel thou wost that the opposit of thyn assendent, that is to seyn, thy by-ginning of the 7 hous, sit up-on the west orizonte; and the byginning of the 10 hous sit up-on the lyne meridional; and his opposit up-on the lyne of midnight. Thanne ley thy label over the degree that assendeth, and rekne fro the point of thy label alle the degrees in the bordure, til thou come to the meridional lyne; and departe alle thilke degrees in 3 evene parties, and take the evene equacion of 3; for ley thy label over everich of 3 parties, and than maistow see by thy label in which degree of the zodiak is the by-ginning of everich of thise same houses fro the assendent: that is to seyn, the beginning of the 12 house next above thyn assendent; and thanne the beginning of the 11 house; and thanne the 10, up-on the meridional lyne; as I first seide. The same wyse wirke thou fro the assendent doun to the lyne of midnight; and thanne thus hastow other 3 houses, that is to seyn, the byginning of the 2, and the 3, and the 4 houses; thanne is the nadir of thise 3 houses the by-ginning of the 3 houses that folwen. And for the more declaracioun, lo here thy figure.

38. To finde the lyne merydional to dwelle fix in any certein place.

[Ad inveniendum lineam meridionalem per subtiles operaciones.]

Tak a rond plate of metal; for warping, the brodere the bettre; and make ther-upon a iust compas, a lite with-in the bordure; and ley this ronde plate up-on an evene grond, or on an evene ston, or on an evene stok fix in the gronde; and ley it even by a level. And in centre of the compas stike an evene pin or a wyr upright; the smallere the betere. Set thy pin by a plom-rewle evene upright; and let this pin be no lengere than a quarter of the diametre of thy compas, fro the centre. And waite bisily, aboute 10 or 11 of the clokke and whan the sonne shyneth, whan the shadwe of the pin entreth any-thing with-in the cercle of thy plate an heer-mele, and mark ther a prikke with inke. Abyde thanne stille waiting on the sonne after 1 of the clokke, til that the schadwe of the wyr or of the pin passe ony-thing out of the cercle of the compas, be it never so lyte; and set ther a-nother prikke of inke. Take than a compas, and mesure evene the middel by-twixe bothe prikkes; and set ther a prikke. Take thanne a rewle, and draw a stryke, evene a-lyne fro the pin un-to the middel prikke; and tak ther thy lyne meridional for evere-mo, as in that same place. And yif thow drawe a cros-lyne over-thwart the compas, iustly over the lyne meridional, than hastow est and west and south; and, par consequence, than the nadir of the south lyne is the north lyne. And for more declaracioun, lo here thy figure.

39. Descripcion of the meridional lyne, of longitudes, and latitudes of citees and townes from on to a-nother of clymatz.

This lyne meridional is but a maner descripcion of lyne imagined, that passeth upon the poles of this world and by the senith of oure heved. And hit is y-cleped the lyne meridional; for in what place that any maner man is at any tyme of the yeer, whan that the sonne by moeving of the firmament cometh to his verrey meridian place, than is hit verrey midday, that we clepen oure noon, as to thilke man; and therfore is it cleped the lyne of midday. And nota, for evermo, of 2 citees or of 2 tounes, of whiche that o toun aprocheth more toward the est than doth that other toun, truste wel that thilke tounes ban diverse meridians. Nota also, that the arch of the equinoxial, that is conteyned or bounded by-twixe the 2 meridians, is cleped the longitude of the toun. And yif so be that two tounes have y-lyke meridian, or oon meridian, than is the distance of hem bothe y-lyke fer fro the est; and the contrarie. And in this manere they chaunge nat her meridian, but sothly they chaungen her almikanteras; for the enhausing of the pool and the distance of the sonne. The longitude of a clymat is a lyne imagined fro est to west, y-lyke distant by-twene them alle. The latitude of a clymat is a lyne imagined from north to south the space of the erthe, fro the byginning of the firste clymat unto the verrey ende of the same climat, evene directe agayns the pole artik. Thus seyn some auctours; and somme of hem seyn that yif men clepen the latitude, thay mene the arch meridian that is contiened or intercept by-twixe the senith and the equinoxial. Thanne sey they that the distaunce fro the equinoxial unto the ende of a clymat, evene agayns the pole artyk, is the latitude of a clymat for sothe. And for more declaracioun, lo here thy figure.

40. To knowe with which degree of the zodiak that any planete assendith on the orisonte, whether so that his latitude be north or south.

Knowe by thyn almenak the degree of the ecliptik of any signe in which that the planete is rekned for to be, and that is cleped the degree of his longitude; and knowe also the degree of his latitude fro the ecliptik, north or south. And by thise samples folwinge in special, maystow wirke for sothe in every signe of the zodiak. The degree of the longitude, par aventure, of Venus or of another planete, was 6 of Capricorne, and the latitude of him was northward 2 degrees fro the ecliptik lyne. I tok a subtil compas, and cleped that oon poynt of my compas A, and that other poynt F. Than tok I the point of A, and set it in the ecliptik lyne evene in my zodiak, in the degree of the longitude of Venus, that is to seyn, in the 6 degree of Capricorne; and thanne sette I the point of F upward in the same signe, bycause that the latitude was north, up-on the latitude of Venus, that is to seyn, in the 6 degree fro the heved of Capricorne; and thus have I 2 degrees by-twixe my two prikkes. Than leide I doun softely my compas, and sette the degree of the longitude up-on the orisonte; tho tok I and wexede my label in maner of a peyre tables to resceyve distinctly the prikkes of my compas. Tho tok I this forseide label, and leide it fix over the degree of my longitude; tho tok I up my compas, and sette the point of A in the wex on my label, as evene as I coude gesse over the ecliptik lyne, in the ende of the longitude; and sette the point of F endlang in my label up-on the space of the latitude, inwarde and over the zodiak, that is to seyn, north-ward fro the ecliptik. Than leide I doun my compas, and lokede wel in the wey upon the prikke of A and of F; tho turned I my riet til that the prikke of F sat up-on the orisonte; than saw I wel that the body of Venus, in hir latitude of 2 degrees septentrionalis, assended, in the ende of the 6 degree, in the heved of Capricorne. And nota, that in the same maner maistow wirke with any latitude septentrional in alle signes; but sothly the latitude meridional of a planete in Capricorne may not be take, by-cause of the litel space by-twixe the ecliptik and the bordure of the Astrolabie; but sothly, in alle other signes it may.

Also the degree, par aventure, of Iuppiter or of a-nother planete, was in the first degree of Pisces in longitude, and his latitude was 3 degrees meridional; tho tok I the point of A, and sette it in the firste degree of Pisces on the ecliptik, and thanne sette I the point of F dounward in the same signe, by-cause that the latitude was south 3 degrees, that is to seyn, fro the heved of Pisces; and thus have I 3 degrees by-twixe bothe prikkes; thanne sette I the degree of the longitude up-on the orisonte. Tho tok I my label, and leide it fix upon the degree of the longitude; tho sette I the point of A on my label, evene over the ecliptik lyne, in the ende evene of the degree of the longitude, and sette the point of F endlang in my label the space of 3 degrees of the latitude fro the zodiak, this is to seyn, southward fro the ecliptik, toward the bordure; and turned my riet til the prikke of F sat up-on the orisonte; thanne saw I wel that the body of Iuppiter, in his latitude of 3 degrees meridional, ascended with 14 degrees of Pisces in horoscopo. And in this maner maistow wirke with any latitude meridional, as I first seide, save in Capricorne. And yif thou wolt pleye this craft with the arysing of the mone, loke thou rekne wel hir cours houre by houre; for she ne dwelleth nat in a degree of hir longitude but a litel whyle, as thou wel knowest; but natheles, yif thou rekne hir verreye moeving by thy tables houre after houre, [thou shall do wel y-now].

Explicit tractatus de Conclusionibus Astrolabii, compilatus per Galfridum Chauciers ad Filium suum Lodewicum, scolarem tunc temporis Oxonie, ac sub tutela illius nobilissimi philosophi Magistri N. Strode, etc.

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.