letter X in the border (fig. 2). Turn the _rete_ till the first point of

Aries lies under the label, which is made to point to X, and the label shews at the same moment that the degree of the sun is very nearly at the point where the equinoctial circle crosses the azimuthal circle which lies 50° to the E. of the meridian. Hence the conjunction takes place at a point of which the azimuth is 50° to the E. of the S. point, or 5° to the eastward of the S.E. point. The proposition merely amounts to finding the sun's azimuth at a given time. Fig. 11 shews the position of the _rete_ in this case.

33. Here 'senyth' is again used to mean azimuth, and the proposition is, to find the sun's azimuth by taking his altitude, and setting his degree at the right altitude on the almicanteras. Of course the two co-ordinates, altitude and azimuth, readily indicate the sun's exact position; and the same for any star or planet.

34. The moon's latitude is never more than 5¼° from the ecliptic, and this small distance is, 'in common treatises of Astrolabie,' altogether neglected; so that it is supposed to move in the ecliptic. First, then, take the moon's altitude, say 30°. Next take the altitude of some bright star 'on the moon's side,' i.e. nearly in the same azimuth as the moon, taking care to choose a star which is represented upon the _Rete_ by a pointed tongue. Bring this tongue's point to the right altitude among the almicanteras, and then see which degree of the ecliptic lies on the almicantera which denotes an altitude of 30°. This will give the moon's place, 'if the stars in the Astrolabe be set after the truth,' i.e. if the point of the tongue is exactly where it should be.

35. The motion of a planet is called _direct_, when it moves in the direction of the succession of the zodiacal signs; _retrograde_, when in the contrary direction. When a planet is on the right or east side of the Meridional line, and is moving forward along the signs, without increase of declination, its altitude will be less on the second occasion than on the first at the moment when the altitude of the fixed star is the same as before. The same is true if the planet be retrograde, and on the western side. The contrary results occur when the second altitude is greater than the first. But the great defect of this method is that it may be rendered fallacious by a change in the planet's declination.

36. See fig. 14, Plate VI. If the equinoctial circle in this figure be supposed to be superposed upon that in fig. 5, Plate III, and be further supposed to revolve backwards through an angle of about 60° till the point 1 (fig. 14) rests upon the point where the 8th hour-line crosses the equinoctial, the beginning of the 2nd house will then be found to be on the line of midnight. Similarly, all the other results mentioned follow. For it is easily seen that each 'house' occupies a space equal to 2 hours, so that the bringing of the 3rd house to the midnight line brings 1 to the 10th hour-line, and a similar placing of the 4th house brings 1 to the 12th hour-line, which is the _horizon obliquus_ itself. Moving onward 2 more hours, the point 7 (the nadir of 1) comes to the end of the 2nd hour, whilst the 5th house comes to the north; and lastly, when 7 is at the end of the 4th hour, the 6th house is so placed. To find the nadir of a house, we have only to add 6; so that the 7th, 8th, 9th, 10th, 11th, and 12th houses are the nadirs of the 1st, 2nd, 3rd, 4th, 5th, and 6th houses respectively.

37. Again see fig. 14, Plate VI. Here the 10th house is at once seen to be on the meridional line. In the quadrant from 1 to 10, the even division of the quadrant into 3 parts shews the 12th and 11th houses. Working downwards from 1, we get the 2nd and 3rd houses, and the 4th house beginning with the north line. The rest are easily found from their nadirs.

38. This problem is discussed in arts. 144 and 145 of Hymes's Astronomy, 2nd ed. 1840, p. 84. The words 'for warping' mean 'to prevent the errors which may arise from the plate becoming warped.' The 'broader' of course means 'the larger.' See fig. 15, Plate VI. If the shadow of the sun be observed at a time _before_ midday when its extremity just enters within the circle, and again at a time _after_ midday when it is just passing beyond the circle, the altitude of the sun at these two observations must be the same, and the south line must lie half-way between the two shadows. In the figure, S and S' are the 2 positions of the sun, OT the rod, Ot and Ot' the shadows, and OR the direction of the south line. Ott' is the metal disc.

39. This begins with an explanation of the terms 'meridian' and 'longitude.' 'They chaungen her Almikanteras' means that they differ in latitude. But, when Chaucer speaks of the longitude and latitude of a 'climate,' he means the length and breadth of it. A 'climate' (_clima_) is a belt of the earth included between two fixed parallels of latitude. The ancients reckoned _seven_ climates; in the sixteenth century there were _nine_. The 'latitude of the climate' is the breadth of this belt; the 'longitude' of it he seems to consider as measured along lines lying equidistant between the parallels of latitude 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. 7 _b_. 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.

5. Central line through Rome (lat. 41°); from 39° to 43°, nearly.

6. Central line through Borysthenes (lat. 45°); from 43° to 47°.

7. Through the Riphæan mountains (lat. 48°); from 47° to 50°. But Chaucer must have included an _eighth_ climate (called _ultra Mæotides 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. Ii. 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, 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 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--'þou shalt do wel y-now'--from the MS. in St. John's College, Cambridge; they also occur in the old editions.

41. Sections 41-43 and 41_a_-42_b_ 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_ : E_b_ :: EB : BC. But E_b_ 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-p_ar_tyes þe hey[gh]t 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 = 5/12 BC. Conversely, BC is 12/5 EB = 48, if EB = 20.

42. See fig. 1, Plate I. See also fig. 17, Plate VI. Let E_b_ = 12, _bc_ = 1; also E'_b'_ = 12, _b'c'_ = 2; then EB = 12 BC, E'B = 6 BC; therefore EE' = 6 BC. If EE' = 60 feet, then BC = 1/6 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 E_b_ = E'_b'_ = 12; _b'c'_ = 1, _bc_ = 2. Hence E'B = 1/12 BC, EB = 2/12 BC. whence EE' = 1/12 BC. Or again, if _bc_ become = 3, 4, 5, &c., successively, whilst _b'c'_ remains = 1, then EE' is successively = 2/12 or 1/6, 3/12 or 1/4, 5/12, &c. Afterwards, add in the height of E.

44. Sections 44 and 45 are from MS. Digby 72. This long explanation of the method of finding a planet's place depends upon the tables which were constructed for that purpose from observation. The general idea is this. The figures shewing a planet's position for the last day of December, 1397, give what is called the _root_, and afford us, in fact, a _starting-point_ from which to measure. An 'argument' is the angle upon which the tabulated quantity depends; for example, a very important 'argument' is the planet's _longitude_, upon which its _declination_ may be made to depend, so as to admit of tabulation. The planet's longitude for the given above-mentioned date being taken as the _root_, the planet's longitude at a second date can be found from the tables. If this second date be less than 20 years afterwards, the increase of motion is set down separately for each year, viz. so much in 1 year, so much in 2 years, and so on. These separate years are called _anni expansi_. But when the increase during a large round number of years (such as 20, 40, or 60 years at once) is allowed for, such years are called _anni collecti_. For example, a period of 27 years includes 20 years _taken together_, and 7 separate or _expanse_ years. The mean motion during smaller periods of time, such as months, days, and hours, is added in afterwards.

45. Here the author enters a little more into particulars. If the mean motion be required for the year 1400, 3 years later than the starting-point, look for 3 in the table of expanse years, and add the result to the number already corresponding to the 'root,' which is calculated for the last day of December, 1397. Allow for months and days afterwards. For a date earlier than 1397 the process is just reversed, involving subtraction instead of addition.

46. This article is probably not Chaucer's. It is found in MS. Bodley 619, and in MS. Addit. 29250. The text is from the former of these, collated with the latter. What it asserts comes to this. Suppose it be noted, that at a given place, there is a full flood when the moon is in a certain quarter; say, e.g. when the moon is due east. And suppose that, at the time of observation, the moon's actual longitude is such that it is in the first point of Cancer. Make the label point due east; then bring the first point of Cancer to the east by turning the _Rete_ a quarter of the way round. Let the sun at the time be in the first point of Leo, and bring the label over this point by the motion of the label only, keeping the _Rete_ fixed. The label then points nearly to the 32nd degree near the letter Q, or about S.E. by E.; shewing that the sun is S.E. by E. (and the moon consequently due E.) at about 4 A.M. In fact, the article merely asserts that the moon's place in the sky is known from the sun's place, if the difference of their longitudes be known. At the time of conjunction, the moon and sun are together, and the difference of their longitudes is zero, which much simplifies the problem. If there is a flood tide when the moon is in the E., there is another when it comes to the W., so that there is high water _twice_ a day. It may be doubted whether this proposition is of much practical utility.

41_a_: This comes to precisely the same as Art. 41, but is expressed with a slight difference. See fig. 16, where, if _bc_ = 8, then BC = 12/8 EB.

41_b_: Merely another repetition of Art. 41. It is hard to see why it should be thus repeated in almost the same words. If _bc_ = 8 in fig. 16, then EB = 8/12 BC = 2/3 BC. The only difference is that it inverts the equation in the last article.]

42_a_ This is only a particular case of Art. 42. If we can get _bc_ = 3, and _b'c'_ = 4, the equations become EB = 4BC, E'B = 3BC; whence EE' = BC, a very convenient result. See fig. 17.]

43_a_: The reading _versam_ (as in the MS.) is absurd. We must also read '_nat_ come,' as, if the base were approachable, no such trouble need be taken; see Art. 41. In fact, the present article is a mere repetition of Art. 43, with different numbers, and with a slight difference in the method of expressing the result. In fig. 18, if _b'c'_ = 3, _bc_ = 4, we have E'B = 3/12 BC, EB = 4/12 BC; or, subtracting, EE' = (4-3)/12 BC; or BC = 12 EE'. Then add the height of E, viz. E_a_, which = AB.

42_b._: Here, 'by the craft of _Umbra Recta_' signifies, by a method similar to that in the last article, for which purpose the numbers must be adapted for computation by the _umbra recta_. Moreover, it is clear, from fig. 17, that the numbers 4 and 3 (in lines 2 and 4) must be transposed. If the side parallel to _b_E be called _nm_, and _mn_, E_c_ be produced to meet in _o_, then _mo_ : _m_E :: _b_E : _bc_; or _mo_ : 12 :: 12 : _bc_; or _mo_ = 144, divided by _bc_ (= 3) = 48. Similarly, _m'o'_ = 144, divided by _b'c'_ (= 4) = 36. And, as in the last article, the difference of these is to 12, as the space EE' is to the altitude. This is nothing but Art. 42 in a rather clumsier shape.

Hence it appears that there are here but 3 independent propositions, viz. those in articles 41, 42, and 43, corresponding to figs. 16, 17, and 18 respectively. Arts. 41_a_ and 41_b_ are mere repetitions of 41; 42_a_ and 42_b_, of 42; and 43_a_, of 43.

CRITICAL NOTES.

As, in the preceding pages which contain the text, the lower portion of each page is occupied with a running commentary, such Critical Notes upon the text as seem to be most necessary are here subjoined.

TITLE. Tractatus, &c.; adopted from the colophon. MS. F has 'tractatus astrolabii.' A second title, 'Bred and mylk for childeren,' is in MSS. B. and E.

[The MSS. are as follows:--A. Cambridge Univ. Lib. Dd. 3. 53.--B. Bodley, E Museo 54.--C. Rawlinson 1370.--D. Ashmole 391.--E. Bodley 619.--F. Corpus 424.--G. Trin. Coll. Cam. R. 15. 18.--H. Sloane 314.--I. Sloane 261.--K. Rawlinson Misc. 3.--L. Addit. 23002. (B. M.)--M. St. John's Coll. Cam.--N. Digby 72.--O. Ashmole 360.--P. Camb. Univ. Lib. Dd. 12. 51.--Q. Ashmole 393.--R. Egerton 2622 (B. M.).--S. Addit. 29250 (B. M.) See the descriptions of them in the Introduction.]

PROLOGUE. l. 26. thise B; þese C; _miswritten_ this A; see above, ll. 21, 22.

32. curious BC; _miswritten_ curios A.

Many similar very slight alterations of spelling have been silently made in the text, and are not worth specifying here. A complete list of them is given in my edition of this treatise for the Early English Text Society. I give, however, the real variations of reading. Thus, in l. 58, A. has _som_ for _sonne_; and in l. 64 omits the second _the_.