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, 20 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 25 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 knowo 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 5 zodiak. The degree of the longitude, par aventure, of Venus or of another planete, was 6 of Capricorne, and the latitude of him of the places from which the climates are named. See Stöffler, fol. 20 b; and Petri Apiani Cosmographia, per Gemmam Phrysium restituta, ed. 1574, fol. 76. The seven climates were as follows: 1. That whose central line passes through Meroë (lat. 17°); from nearly 13° to nearly 20°. 2. Central line, through Syene (lat. 24°); from 20° to 27°, nearly. 3. Central line through Alexandria (lat. 31°); from 27° to 34°, nearly. 4. Central line through Rhodes (lat. 36°); from 34° to 39', nearly. 7. Through the Riphæan mountains (lat. 48°); from 47° to 50°. But Chaucer must have included an eighth climate (called ultra Meotides paludes) from 50° to 56°; and a ninth, from 56° to the pole. The part of the earth to the north of the 7th climate was considered by the ancients to be uninhabitable. A rough drawing of these climates is given in MS. Camb. Univ. Lib. li. 3. 3, fol. 33 b. 40. The longitude and latitude of a planet being ascertained from an almanac, we can find with what degree it ascends. For example, was northward 2 degrees fro the ecliptik lyne. I tok a subtil compas, and cleped that oon poynt of my compas A, and that 10 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 15 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 20 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 25 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 30 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 35 it may. given that the longitude of Venus is 6° of Capricorn, and her N. latitude 2°. Set the one leg of a compass upon the degree of longitude, and extend the other till the distance between the two legs is 2° of latitude, from that point inward, i. e. northward. The 6th degree of Capricorn is now to be set on the horizon, the label (slightly coated with wax) to be made to point to the same degree, and the north latitude is set off upon the wax by help of the compass. The spot thus marking the planet's position is, by a very slight movement of the Rete, to be brought upon the horizon, and it will be found that the planet (situated 2° N. of the 6th degree) ascends together with the head (or beginning of the sign) of Capricorn. This result, which is not quite exact, is easily tested by a globe. When the latitude of 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 40 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 45 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 50 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 55 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 shalt 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. * the planet is south, its place cannot well be found when in Capricorn for want of space at the edge of the Astrolabe. As a second example, it will be found that, when Jupiter's longitude is at the end of 1° of Pisces, and his latitude 3° south, he ascends together with the 14th of Pisces, nearly. This is easily verified by a globe, which solves all such problems very readily. It is a singular fact that most of the best MSS. leave off at the word 'houre,' leaving the last sentence incomplete. I quote the last five words—'pou shalt do wel y-now'- from the MS. in St. John's College, Cambridge; they also occur in the old editions. SUPPLEMENTARY PROPOSITIONS. 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 5 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 betwise 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 10 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 umbra versa. Yif so be that thou may nat come to the bas of the tour, I see him thorw the nombre of 1; I sette ther a prikke at my fote ; than go I neer to the tour, and I see him thorw at the poynt of 2, and there I sette 41. Sections 41-43 and 41a-42b are from the MS. in St. John's College, Cambridge. For the scale of umbra recta, see fig. 1, Plate I. Observe that the umbra recta is used where the angle of elevation of an object is greater than 45°; the umbra versa, where it is less. See also fig. 16, Plate VI; where, if AC be the height of the tower, BC the same height minus the height of the observer's eye (supposed to be placed at E), and EB the distance of the observer from the tower, then bc: Eb::EB: BC. But Eb is reckoned as 12, and if bc be 4, we find that BC is 3 EB, i. e. 60 feet, when EB is 20. Hence AC is 60 feet, plus the height of the observer's eye. The last sentence is to be read thus—And if thy “rewle” fall upon 5, then are 5-12ths of the height equivalent to the space between thee and the tower (with addition of thine own height).' The MS. reads ‘5 12-partyes þe heyzt of þe space,' &c.; but the word of must be transposed, in order to make sense. It is clear that, if bc=5, then 5:12 :: EB : BC, which is the same as saying that EB=i. BC. Conversely, BC is a EB=48, if EB=20. 42. See fig. 1, Plate I. See also fig. 17, Plate VI. Let Eb=12, bc a-nother prikke; and I beholde how i hath him to 12, and ther 5 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 10 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 ; 15 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 20 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 between 1 and 2, and thou shalt 5 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, 10 adde the stature of thyn heyghte to thyn eye. =1; also E'V=12, b'd=2; then EB = 12 BC, E'B=6 BC; therefore EE'=6 BC. If EE'=60 feet, then BC=* EE'=10 feet. To get the whole height, add the height of the eye. The last part of the article, beginning 'For other poyntis,' is altogether corrupt in the MS. 43. Here versa (in M.) is certainly miswritten for recta, as in L. See fig. 18, Plate VI. Here Eb=E'V=12; b'c=1, bc=2. Hence E'B= 1 BC, EB = i BC, whence EE' = ' BC. Or again, if bc become = 3, 4, 5, &c., successively, whilst Wc remains = 1, then EE' is successively = 1 or }, it or }, 12, &c. Afterwards, add in the height of E. Q |