thanne turne thy rete aboute joyntlie with thy label and with the point of it recken in the bordure fro the sonne arysyng unto the same place there thou desirest by daye as by nyghte. This conclusion woll I declare in the fowerthe partye of the laste chapiter of this tretyse so openly that there shal lacke no word that nedeth declaracion.

XIII. Speciall declaracion of the houres of the planettes.

U nderstonde wel that evermore fro the arysying of the sonne tyl it go to rest, the nadyr of the sonne shall shewe the houre of the planet; and fro that tyme forward al the nyght tyl the sonne aryse; then shall the very degre of the sonne shewe the howre of the planet. Ensample as thus: the 13 daye of March fel upon a Satyrday peraventure, and at the arysyng of the sonne I founde the seconde degre of Aries syttynge upon myn Est orizonte al be it was but lytel. Then founde I the seconde degre of Libra nadyre of my sonne, discendynge on my West orizonte, upon which West orizont every daye generally at the sonne arysyng entreth the hour of any planet undyr the foresaied West orizonte, after the whiche planete the daye bereth his name, and endeth in the nexte strike of the planete undyr the foresaied Weste orizonte: and ever as the sonne clymbeth upper and upper, so goth hys nadyre downer and downer, and echinge* fro such strikes the houres of the planettes by order, as they sytten in heven. The fyrst hour inequale of every satyrday is to Saturne, and the second to Jupiter, the thyrd to Mars, the fourth to the Sonne, the fyfth to Venus, the syxt to Mercurius, the seventh to the Mone, and then ayen the eyghth to Saturne, the nyneth to Jupiter, the tenth to Mars, the eleventh to the Sonne, the twelfth to Venus. And nowe is my sonne gon to reste as for that satyrdaye; Than sheweth the very degre of the sonne the hour of Mercury entryng under my west orizont at even: And nexte hym succedeth the Mone, and so forth by order, planete after planete, in hour after hour al the nyght longe tyl the sonne aryse. Nowe riseth the sonne the Sunday by the morowe, and the nadyr of the sonne upon the west orizont sheweth me the entryng of the hour of the foresaied sonne. And in this maner succedeth planete undyr planete fro Saturn unto the Mone,

* Stevins, not understanding this word echinge, changed it in his MS. to teachinge: but it has every appearance of being a genuine word, and the verb "to eche" is found in such a comparatively modern authority as Bailey's Dictionary Here it seems to mean adding one by one, or dealing out.

and fro the Mone up ageyn to Saturn, hour aftyr hour generally, and thus knew I this conclusion.

XIV. To knowe with which degre of the Zodiake any sterre fyxe, in thyn astrolabye, aryseth upon the Est orizonte, although the orizonte be in anothyr sygne.

Sette

ette the centure of the sterre upon the Est orizonte and loke what degre of any signe that sytteth upon the same orizonte at the same tyme and understonde wel that with the same degre ariseth the same sterre.

And this mervaylous arysing with a straunge degre in another sygne is bycause the latitude of the sterre fyxe is eyther north or south fro the Ecliptike.* For sothely the latitudes of planetes ben comenly rekenyd fro the ecliptike bycause that none of hem declineth but fewe degrees fro the brede of the Zodiake. And take gode kepe of thys chapiter of arysing of celestiall bodyes, for trusteth wel that neyther Mone, neyther sterre, in oure embolyfe orizonte that aryseth with the same degre of his longitude save in one case, and that is when thei have no latitude fro the eclyptike lyne. But nevertheles somtyme is everiche of these planetes undyr the same lyne.

XV. To knowe the declinacyon of any degre in the Zodiake fro the equinoccial cercle.

Sett

ette the degre of any signe upon the lyne meridional and reken his altytude in the almicanteras fro the Est orizonte up to the same degre sette in the forsaied lyne, and sette there a prycke: turne up thy rete and sette the hed of Aries or Libra in that same merydyonal lyne and sette there another prycke; and when this is done consyder the altitudes of hem bothe: for sothely the difference of thilke altytude is the declynacyon of thilke degre fro the equinoctial. And if it so be that thilke degree be northward fro the equinoctiall than is hys declynacyon north, and if it be southward than it is south.

XVI. To knowe for what latytude in any region the almicanteras in thy tables ben compowned.

* In all MSS. and printed copies that I have examined this word is " Equinoctial," and the next word "But." These I have altered into Ecliptike and For, respectively not so much because of the expression "latitude," which Chaucer often uses in the sense of declination; but because the problem necessarily requires the star's divergence from the ecliptic, while with the equinoctial it might coincide, and still be within the conditions of the problem.

Reken how many degrees of almicanteras, in the meridionall

line, be from the circle equinoctiale unto the signet ;* or els from the pole artike unto the north orizont: and for so grete a a latitude, or so smale a latitude is the table compouned.

XVII. To knowe the altytude of the sonne in the myddes of the daye, that is cleped the altytude meridian.

Sette the degre of thy sonne upon the lyne meridionale and reken how many degrees of almicanteras ben betwixe thyn Est orizonte and the degre of thy sonne and take there thin altitude meridian; that is to sain, the highest degre of the sonne as for that daye. So mayest thou knowe in the same lyne the highest degret that any sterre fyxe clymbeth by nyght: this is to saine that when any sterre fyxe is passed the lyne meridionall, than begynneth it to discende; and so doth the sonne.

XVIII. To knowe the degre of the sonne by the rete for a maner curyosyte.

Seke busely with thy rule the highest of the sonne in myddes of the daye; tourne than thyn astrolabie, and with a prycke of ynk marke the nombre of the same altitude in the lyne meridionale. Tourne than thy rete about tyl thou finde a degre of thy zodiake accordyng with the prycke; that is to sain, syttyng on the prycke-and in sothe thou shalt finde but two degrees in al the zodiake of that condycyon, and yet thilke two degrees ben in divers sygnes. Than mayst thou lightly by the seson of the yere knowe the signe in whiche is the sonne.

XIX. To knowe which day is like to other in lengthe throughout the yere.

Loke whiche degrees ben lyke [far] from the hedes of Cancer and Capricorn; and loke when the sonne is in any of thilke degrees; than ben the dayes lyke of length; that is to sain-that as long is that day in that moneth as was soche a daye in soche a moneth, there varieth but lyttel. Also if thou take two dayes naturelles in the yere ylike farre from eyther point of the equinoctial in the opposite partyes; than as long is the day artificial on that one day, as on that other; and eke the contrarie.

*Signet, i.e., Zenith; pronounced with the French silent g, and sometimes written synet or sinet by Chaucer (See page 45).

In all the copies this word is "lyne." It ought manifestly to be "degre."

XX. This chapter is a

maner declaracion to conclusions that followeth.

Understande wel that thy zodiake is departed in two halfe

circles, from the hed of Capricorn unto the hed of Cancer, and ayenwarde from the hed of Cancer unto the hed of Capricorn. The hed of Capricorn is the lowest poinct where as the sonne goth in winter, and the hed of Cancer is the highest poinct in which the sonne goth in sommer. And therefore understande wel that any two degrees that ben ylike far from any of these two hedes truste wel that thilke two degrees ben of lyke declinacion, be it southward or northward, and the dayes of hem ben lyke of length, and the nyghtes also; and shadowes ylyke, and the altitudes ylyke at mydday for ever.

XXI. To knowe the verrey degre of any maner sterre strange, after his altitude.† Though he be indeterminate in thyn astrolabye sothely to the trouth thus he shal be knowe.

Take

ake the altitude of thy sterre when he is on the Est syde of the lyne meridional as nygh as thou mayest gesse, and take that ascendant anone right by some maner sterre fyxe which thou know

*Sic in MSS. The printed copies have "into halfe circles."

+ This word is latitude in all the copies-an obvious error, since the chief object of the problem is longitude. But as the transposition of the first two letters is the commonest of all errors, and as altitude makes very good sense as one of the principal elements of the problem, I have substituted it as the most probable original. This is one of the Conclusions that Stevins "cleane put out for utterly false and untrew," possibly because he could not understand it, or because it was one of those denounced by Stoeffler. It is true that it is not strictly correct in theory—because a mean of two ascendants is not necessarily the ascendant of the mean of their relative culminating points. But the method is sufficiently correct for practical purposes with an instrument so imperfect in itself as the Astrolabe: provided that the interval between the equal altitudes be short, a condition which Chaucer seems to have had in view when he directs the first altitude to be taken on the east side of the meridian as nygh as thou may'st gesse." If the interval were not more than an hour the greatest error in the result could not exceed a degree of longitude-much less than might arise from the other sources of error to which such an observation would be exposed. It must be observed that the longitude of which Chaucer speaks was not, as at present, referred to the Pole of the ecliptic; but was that degree of the ecliptic that came to the meridian with the star. This was "Longitudo secundum cœli mediationem," and was still in use long after Chaucer's time. Also it must be observed that Chaucer makes no distinction between latitude and declination but treats both these terms as synonymous.

est; and forget not the altitude of the firste sterre ne thyn ascendente. And when that thys is done aspye dilygently when this first sterre passeth anythyng to the south westward and cacche him anone ryght in the same nombre of the altitude on the west syde of thys lyne meridional as he was caught on the est syde, and take newe ascendente anone right by some maner sterre fyxe the which that thou knowest, and forgete not this second ascendant. And when this is done, reken thou howe many degrees ben bitwixe the first ascendent and the second ascendent, and reken wel the myddel degre betwix bothe ascendentes; and sette thilke myddel degre upon thyn Est orizonte: and then loke what degre sitte upon the lyne meridional and take there the very degre of the ecliptike in whiche the sterre standeth for the tyme. For in the ecliptike is the longitude of a celestiale body, rekoned even fro the hed of Aries unto the ende of Pisces; and his latitude is rekened after the quantyte of his declinacion north or south toward the poles of this worlde. As thus: if it be of the sonne, or any fyxe sterre, reken his latitude or his declynacion fro the equinoctial circle; and if it be of a planete reken than the quantite of his latitude from the ecliptike lyne. Albeit so that from the equinoctial maye the declinacion or the latitude of any body celestiale be rekened after the sight,* north or south, and after the quantite of his declinacion; and yet so maye the latitude or declynacion of any body celestiall, save only of the sonne, after his sight, and after the quantite of his declinacion, be rekened from the ecliptike lyne, fro which lyne all planetes sometyme decline north or south save only the forsaied sonne.

XXII. To knowe the degrees of longitude of fyxe sterres after that thei be determinate in thin astrolabye if it so be that they ben trewly sette.

It is difficult to interpret "after the sight." It is just possible it may mean according to visual observation: or it may be that sight is a copy-error for height culmen = meridian altitude. The passage in which it occurs, unless some part of it has been lost, merely declares that latitude or declination may be reckoned from either the equinoctial or the ecliptic; except in the case of the sun, when it must be from the equinoctial only. Then, reading" declinacion" in that place as simply deviation, the sense might be :-The latitude of any celestial body may be reckoned from the equinoctial, north or south, after its height, and after the quantity of its deviation, &c.