Part II, § 1. [The Latin headings to the propositions are taken from the

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

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

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

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

5. This merely amounts to taking the mean between two results.

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

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

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

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

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

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

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

1642753|16427531642753164275316.

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

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

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

15. This is best done by help of the _back_ of the instrument, fig. 1. Thus May 13 (old style), which lies 30° to the W. of the S. line, is nearly of the same length as July 13, which lies 30° to the E. Secondly, the day of April 2 (old style), 20° above the W. line, is nearly of the same length as the night of Oct. 2, 20° below the E. line, in the opposite point of the circle. This is but an approximation, as the divisions on the instrument are rather minute.

16. This merely expresses the same thing, with the addition, that on days of the same length, the sun has the same meridional altitude, and the same declination from the equator.

17. Here _passeth any-thing the south westward_ means, passes somewhat to the westward of the South line. The problem is, to find the degree of the zodiac which is on the meridian with the star. To do this, find the altitude of the star _before_ it souths, and by help of problem 3, find out the ascending degree of the zodiac; secondly, find the ascending degree at an equal time _after_ it souths, when the star has the same altitude as before, and the mean between these will be the degree that ascends when the star is on the meridian. Set this degree upon the Eastern part of the _horizon obliquus_, and then the degree which is upon the meridional line souths together with the star. Such is the solution given, but it is but a very rough approximation, and by no means always near to the truth. An example will shew why. Let Arcturus have the same altitude at 10 P.M. as at 2 A.M. In the first case the 4th of Sagittarius is ascending, in the second (with sufficient accuracy for our purpose) the 2nd of Aquarius; and the mean between these is the 3rd of Capricorn. Set this on the Eastern horizon upon a globe, and it will be seen that it is 20 min. past midnight, that 10° of Scorpio is on the meridian, and that Arcturus has past the meridian by 5°. At true midnight, the ascendent is the 29° of Sagittarius. The reason of the error is that right ascension and longitude are here not sufficiently distinguished. By observing the degrees of the _equinoctial_, instead of the _ecliptic_, upon the Eastern horizon, we have at the first observation 272°, at the second 332°, and the mean of these is 302°; from this subtract 90°, and the result, 212°, gives the right ascension of Arcturus very nearly, corresponding to which is the beginning of the 5° of Scorpio, which souths along with it. This latter method is correct, because it assumes the motion to take place round the axis of the equator. The error of Chaucer's method is that it identifies the motion of the equator with that of the ecliptic. The amount of the error varies considerably, and may be rather large. But it can easily be diminished, (and no doubt was so in practice), by taking the observations _as near the south line as possible_. Curiously enough, the rest of the section explains the difference between the two methods of reckoning. The modern method is to call the co-ordinates _right ascension_ and _declination_, if reckoned from the equator, and _longitude_ and _latitude_, if from the ecliptic. Motion in _longitude_ is not the same thing as motion in _right ascension_.

18. The 'centre' of the star is the technical name for the extremity of the metal tongue representing it. The 'degree in which the star standeth' is considered to be that degree of the zodiac which souths along with it. Thus Sirius or Alhabor has its true longitude nearly equal to that of 12° of Cancer, but, as it souths with the 9th degree, it would be said to stand in that degree. This may serve for an example; but it must be remembered that its longitude was different in the time of Chaucer.

19. Also it rises with the 19th degree of Leo, as it is at some distance from the zodiac in latitude. The same 'marvellous arising in a strange sign' is hardly because of the latitude being north or south from the _equinoctial_, but rather because it is north or south of the _ecliptic_. For example, Regulus ([alpha] Leonis) is on the ecliptic, and of course rises with that very degree in which it is. Hence the reading _equinoctial_ leaves the case in doubt, and we find a more correct statement just below, where we have 'whan they have no latitude fro the ecliptik lyne.' At all places, however, upon the earth's equator, the stars will rise with the degrees of the zodiac in which they stand.]

20. Here the disc (fig. 5) is supposed to be placed beneath the Rete (fig. 2). The proposition merely tells us that the difference between the meridian altitudes of the given degree of the zodiac and of the 1st point of Aries is the _declination_ of that degree, which follows from the very definition of the term. There is hardly any necessity for setting the second prick, as it is sufficiently marked by being the point where the equinoctial circle crosses the south line. If the given degree lie _outside_ this circle, the declination is _south;_ if _inside_, it is _north_.

21. In fig. 5, the almicanteras, if accurately drawn, ought to shew as many degrees between the south point of the equinoctial circle and the zenith as are equal to the latitude of the place for which they are described. The number of degrees from the pole to the northern point of the _horizon obliquus_ is of course the same. The latitude of the place for which the disc is constructed is thus determined by inspection.

22. In the _first_ place where '_orisonte_' occurs, it means the _South_ point of the horizon; in the _second_ place, the _North_ point. By referring to fig. 13, Plate V, it is clear that the arc [Aries]S, representing the distance between the equinoctial and the S. point, is equal to the arc ZP, which measures the distance from the pole to the zenith; since PO[Aries] and ZOS are both right angles. Hence also Chaucer's second statement, that the arcs PN and [Aries]Z are equal. In his numerical example, PN is 51° 50'; and therefore ZP is the complement, or 38° 10'. So also [Aries]Z is 51° 50'; and [Aries]S is 38° 10'. Briefly, [Aries]Z measures the latitude.

23. Here the altitude of a star (A) is to be taken twice; firstly, when it is on the meridian in the most _southern_ point of its course, and secondly, when on the meridian in the most _northern_ point, which would be the case twelve hours later. The mean of these altitudes is the altitude of the pole, or the latitude of the place. In the example given, the star A is only 4° from the pole, which shews that it is the Pole-star, then farther from the Pole than it is now. The star F is, according to Chaucer, any convenient star having a right ascension differing from that of the Pole-star by 180°; though one having the _same_ right ascension would serve as well. If then, at the first observation, the altitude of A be 56, and at the second be 48, the altitude of the pole must be 52. See fig. 13, Plate V.

24. This comes to much the same thing. The _lowest_ or northern altitude of Dubhe ([alpha] Ursæ Majoris) may be supposed to be observed to be 25°, and his _highest_ or southern altitude to be 79°. Add these; the sum is 104; 'abate' or subtract half of that number, and the result is 52°; the latitude.

25. Here, as in § 22, Chaucer says that the latitude can be measured by the arc Z[Aries] or PN; he adds that the depression of the Antarctic pole, viz. the arc SP' (where P' is the S. pole), is another measure of the latitude. He explains that an obvious way of finding the latitude is by finding the altitude of the sun at noon at the time of an equinox. If this altitude be 38° 10', then the latitude is the complement, or 51° 50'. But this observation can only be made on two days in the year. If then this seems to be too long a tarrying, observe his midday altitude, and allow for his declination. Thus, if the sun's altitude be 58° 10' at noon when he is in the first degree of Leo, subtract his declination, viz. 20°, and the result is 38° 10', the complement of the latitude. If, however, the sun's declination be _south_, the amount of it must be added instead of subtracted. Or else we may find [Aries]A', the highest altitude of a star A' above the equinoctial, and also [Aries]A, its nether elongation extending from the same, and take the mean of the two.

26. The 'Sphere Solid' answers nearly to what we now call a globe. By help of a globe it is easy to find the ascensions of signs for _any latitude_, whereas by the astrolabe we can only tell them for those latitudes for which the plates bearing the almicanteras are constructed. The signs which Chaucer calls 'of right (i.e. direct) ascension' are those signs of the zodiac which rise more directly, i.e. at a greater angle to the horizon than the rest. In latitude 52°, Libra rises so directly that the whole sign takes more than 2¾ hours before it is wholly above the horizon, during which time nearly 43° of the equinoctial circle have arisen; or, in Chaucer's words, 'the more part' (i.e. a larger portion) of the equinoctial ascends with it. On the other hand, the sign of Aries ascends so obliquely that the whole of it appears above the horizon in less than an hour, so that a 'less part' (a smaller portion) of the equinoctial ascends with it. The following is a rough table of Direct and Oblique Signs, shewing approximately how long each sign takes to ascend, and how many degrees of the equinoctial ascend with it, in lat. 52°.

_Oblique Degrees of the Time of | _Direct Degrees of the Time of Signs._ Equinoctial. ascending. | Signs._ Equinoctial. ascending. Capricornus 26° 1 h. 44 m. | Cancer 39° 2 h. 36 m. Aquarius 16° 1 h. 4 m. | Leo 42° 2 h. 48 m. Pisces 14° 0 h. 56 m. | Virgo 43° 2 h. 52 m. Aries 14° 0 h. 56 m. | Libra 43° 2 h. 52 m. Taurus 16° 1 h. 4 m. | Scorpio 42° 2 h. 48 m. Gemini 26° 1 h. 44 m. | Sagittarius 39° 2 h. 36 m.

These numbers are sufficiently accurate for the present purpose.

In ll. 8-11, there is a gap in the sense in nearly all the MSS., but the Bodley MS. 619 fortunately supplies what is wanting, to the effect that, at places situated on the equator, the poles are in the horizon. At such places, the days and nights are always equal. Chaucer's next statement is true for _all_ places _within the tropics_, the peculiarity of them being that they have the sun vertical twice in a year. The statement about the 'two summer and winters' is best explained by the following. 'In the tropical climates, ... seasons are caused more by the effect of the winds (which are very regular, and depend mainly on the sun's position) than by changes in the direct action of the sun's light and heat. The seasons are not a summer and winter, so much as recurrences of wet and dry periods, _two in each year_.'--English Cyclopædia; _Seasons, Change of_. Lastly, Chaucer reverts to places on the equator, where the stars all seem to move in vertical circles, and the almicanteras are therefore straight lines. The line marked _Horizon Rectus_ is shewn in fig. 5, where the _Horizon Obliquus_ is also shewn, cutting the equinoctial circle obliquely.

27. The real object in this section is to find how many degrees of the equinoctial circle pass the meridian together with a given zodiacal sign. Without even turning the _rete_, it is clear that the sign Aries, for instance, extends through 28° of the equinoctial; for a line drawn from the centre, in fig. 2, through the end of Aries will (if the figure be correct) pass through the end of the 28th degree below the word _Oriens_.

28. To do this accurately requires a very carefully marked Astrolabe, on as large a scale as is convenient. It is done by observing where the ends of the given sign, estimated along the _outer_ rim of the zodiacal circle in fig. 2, cross the _horizon obliquus_ as the _rete_ is turned about. Thus, the beginning of Aries lies on the _horizon obliquus_, and as the _rete_ revolves to the right, the end of it, on the outer rim, will at last lie exactly on the same curved line. When this is the case, the _rete_ ought to have moved through an angle of about 14°, as explained in § 26. By far the best way is to tabulate the results once for all, as I have there done. It is readily seen, from fig. 2, that the signs from Aries to Virgo are _northern_, and from Libra to Pisces are _southern_ signs. The signs from Capricorn to Gemini are the _oblique_ signs, or as Chaucer calls them, 'tortuous,' and ascend in less than 2 hours; whilst the _direct_ signs, from Cancer to Sagittarius, take more than 2 hours to ascend; as shewn in the table on p. 209. The _eastern_ signs in fig. 2 are said to _obey to_ the corresponding _western_ ones.

29. Here _both_ sides of the Astrolabe are used, the 'rewle' being made to revolve at the _back_, and the 'label' in _front_, as usual. First, by the back of the instrument and the 'rewle,' take the sun's altitude. Turn the Astrolabe round, and set the sun's degree at the right altitude among the almicanteras, and then observe, by help of the label, how far the sun is from the meridian. Again turn the instrument round, and set the 'rewle' as far from the meridian as the label was. Then, holding the instrument as near the ground and as horizontal as possible, let the sun shine through the holes of the 'rewle,' and immediately after lay the Astrolabe down, without altering the azimuthal direction of the meridional line. It is clear that this line will then point southwards, and the other points of the compass will also be known.

30. This turns upon the definition of the phrase 'the wey of the sonne.' It does not mean the zodiacal circle, but the sun's apparent path on a given day of the year. The sun's altitude changes but little in one day, and is supposed here to remain the same throughout the time that he is, on that day, visible. Thus, if the sun's altitude be 61½°, the _way of the sun_ is a small circle, viz. the tropic of Cancer. If the planet be then on the zodiac, in the 1st degree of Capricorn, it is 47° S. from the way of the sun, and so on.

31. The word 'senith' is here used in a peculiar sense; it does not mean, as it should, the _zenith_ point, or point directly overhead, but is made to imply the point on the horizon, (either falling upon an azimuthal line, or lying between two azimuths), which denotes the point of sunrise. In the Latin rubric, it is called _signum_. This point is found by actual observation of the sun at the time of rising. Chaucer's azimuths divide the horizon into 24 parts; but it is interesting to observe his remark, that 'shipmen' divide the horizon into 32 parts, exactly as a compass is divided now-a-days. The reason for the division into 32 parts is obviously because this is the easiest way of reckoning the direction of the wind. For this purpose, the horizon is first divided into 4 parts; each of these is halved, and each half-part is halved again. It is easy to observe if the wind lies half-way between S. and E., or half-way between S. and S.E., or again half-way between S. and S.S.E.; but the division into 24 parts would be unsuitable, because _third-parts_ are much more difficult to estimate.

32. The Latin rubric interprets the conjunction to mean that of the sun and moon. The time of this conjunction is to be ascertained from a calendar. If, e.g. the calendar indicates 9 A.M. as the time of conjunction on the 12th day of March, when the sun is in the first point of Aries, as in § 3, the number of hours after the preceding midday is 21, which answers to the