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

Little Lewis my son, I perceive that thou wouldst learn the Conclusions of the Astrolabe; wherefore I have given thee an instrument constructed for the latitude of Oxford, and purpose to teach thee some of these conclusions. I say some, for three reasons; (1) because some of them are unknown in this land; (2) because some are uncertain; or else (3) are too hard. This treatise, divided into five parts, I write for thee in English, just as Greeks, Arabians, Jews, and Romans were accustomed to write such things in their own tongue. I pray all to excuse my shortcomings; and thou, Lewis, shouldst thank me if I teach thee as much in English as most common treatises can do in Latin. I have done no more than compile from old writers on the subject, and I have translated it into English solely for thine instruction; and with this sword shall I slay envy.

The first part gives a description of the instrument itself.

The second teaches the practical working of it.

The third shall contain tables of latitudes and longitudes of fixed stars, declinations of the sun, and the longitudes of certain towns.

The fourth shall shew the motions of the heavenly bodies, and especially of the moon.

The fifth shall teach a great part of the general rules of astronomical theory.

Here begins the first part; i.e. the description of the Astrolabe itself.

1. The Ring. See figs. 1 and 2. The Latin name is Armilla suspensoria; the Arabic name is spelt alhahuacia in MS. Camb. Univ. Ii. 3. 3, but Stöffler says it is Alanthica, Alphantia, or Abalhantica. For the meaning of 'rewle,' see § 13.

2. The Turet. This answers nearly to what we call an eye or a swivel. The metal plate, or loop, to which it is fastened, or in which it turns, is called in Latin Ansa or Armilla Reflexa, in Arabic Alhabos.

3. The Moder. In Latin, Mater or Rotula. This forms the body of the instrument, the back of which is shewn in fig. 1, the front in fig. 2. The 'large hole' is the wide depression sunk in the front of it, into which the various discs are dropped. In the figure, the 'Rete' is shewn fitted into it.

4. See fig. 1; Chaucer describes the 'bak-half' of the instrument first. The centre of the 'large hole amydde' is the centre of the instrument, where a smaller hole is pierced completely through. The Southe lyne (marked Meridies in figs. 1 and 2) is also called Linea Meridiei; the North lyne is also named Linea Mediæ Noctis.

5. The Est lyne is marked with the word Oriens; the West lyne, with Occidens.

6. The rule is the same as in heraldry, the right or dexter side being towards the spectator's left.

7. As the 360 degrees answer to 24 hours of time, 15° answer to an hour, and 5° to twenty minutes, or a Mile-way, as it is the average time for walking a mile. So also 1° answers to 4 minutes of time. See the two outermost circles in fig. 1, and the divisions of the 'border' in fig. 2.

8. See the third and fourth circles (reckoning inwards) in fig. 1.

9. See the fifth and sixth circles in fig. 1.

10. See the seventh, eighth, and ninth circles in fig. 1. The names of the months are all Roman. The month formerly called Quinctilis was first called Julius in B.C. 44; that called Sextilis was named Augustus in B.C. 27. It is a mistake to say that Julius and Augustus made the alterations spoken of in the text; what Julius Cæsar really did, was to add 2 days to the months of January, August (Sextilis), and December, and 1 day to April, June, September, and November. February never had more than 28 days till he introduced bissextile years.

11. See the two inmost circles in fig. 1. The names given are adopted from a comparison of the figures in the Cambridge University and Trinity MSS., neither of which are quite correct. The letters of the 'Abc.' are what we now call the Sunday letters. The festivals marked are those of St. Paul (Jan. 25), The Purification (Feb. 2), The Annunciation (Mar. 25), The Invention of the Holy Cross (May 3), St. John the Baptist (June 24), St. James (July 25), St. Lawrence (Aug. 10), The Nativity of the Blessed Virgin (Sept. 8), St. Luke (Oct. 18), St. Martin of Tours (Nov. 11), and St. Thomas (Dec. 21).

12. The 'scale' is in Latin Quadrans, or Scala Altimetra. It is certain that Chaucer has here made a slip, which cannot be fairly laid to the charge of the scribes, as the MSS. agree in transposing versa and recta. The side-parts of the scale are called Umbra versa, the lower part Umbra recta or extensa. This will appear more clearly at the end of Part II. (I here give a corrected text.)

13. See fig. 3, Plate III. Each plate turns on a hinge, just like the 'sights' of a gun. One is drawn flat down, the other partly elevated. Each plate (tabella vel pinnula) has two holes, the smaller one being the lower. This Rewle is named in Arabic Alhidada or Al´idāda; in Latin Verticulum, from its turning easily on the centre; in Greek Dioptra, as carrying the sights. The straight edge, passing through the centre, is called the Linea Fiduciæ. It is pierced by a hole in the centre, of the same size as that in the Mother.

14. See fig. 4, Plate III. The Pin is also called Axis or Clavus, in Latin-Arabic Alchitot; it occupies the position of the Arctic or North Pole, passing through the centre of the plates that are required to turn round it. The Wedge is called cuneus, or equus restringens, in Arabic Alfaras or the horse, because it was sometimes cut into the shape of a horse, as shewn in fig. 7, Plate IV, which is copied from MS. Univ. Camb. Ii. 3. 3.

15. See fig. 2, Plate II. In the figure, the cross-lines are partly hidden by the Rete, which is separate and removable, and revolves within the border.

16. The Border was also called Margilabrum, Margolabrum, or Limbus. It is marked (as explained) with hour-letters and degrees. Each degree contains 4 minutes of time, and each of these minutes contains 60 seconds of time.

17. We may place under the Rete any plates we please. If only the Mother be under it, without any plate, we may suppose the Mother marked as in fig. 2. The plate or disc (tympanum) which was usually dropped in under the Rete is that shewn in fig. 5, Plate III, and which Chaucer now describes. Any number of these, marked differently for different latitudes, could be provided for the Astrolabe. The greatest declination of the sun measures the obliquity of the ecliptic, the true value of which is slightly variable, but was about 23° 31′ in Chaucer's time, and about 23° 40′ in the time of Ptolemy, who certainly assigns to it too large a value. The value of it must be known before the three circles can be drawn. The method of finding their relative magnitudes is very simple. Let ABCD (fig. 8, Pl. IV) be the tropic of Capricorn, BO the South line, OC the West line. Make the angle EOB equal to the obliquity (say 23½°), and join EA, meeting BO in F. Then OF is the radius of the Equatorial circle, and if GH be drawn parallel to EF, OH is the radius of the Tropic of Cancer. In the phrase angulus primi motus, angulus must be taken to mean angular motion. The 'first moving' (primus motus) has its name of 'moving' (motus) from its denoting motion due to the primum mobile or 'first moveable.' This primum mobile (usually considered as the ninth sphere) causes the rotation of the eighth sphere, or sphæra stellarum fixarum. See the fig. in MS. Camb. Univ. Ii. 3. 3 (copied in fig. 10, Pl. V). Some authors make 12 heavens, viz. those of the 7 planets, the firmamentum (stellarum fixarum), the nonum cœlum, decimum cœlum, primum mobile, and cœlum empyræum.

18. See fig. 5, Pl. III. This is made upon the alt-azimuth system, and the plates are marked according to the latitude. The circles, called in Latin circuli progressionum, in Arabic Almucantarāt, are circles of altitude, the largest imperfect one representing the horizon (horizon obliquus), and the central dot being the zenith, or pole of the horizon. In my figure, they are 'compounded by' 5 and 5, but Chaucer's shewed every second degree, i.e. it possessed 45 such circles. For the method of drawing them, see Stöffler, leaf 5, back.

19. Some Astrolabes shew 18 of these azimuthal circles, as in my figure (fig. 5, Pl. III). See Stöffler, leaf 13, where will be found also the rules for drawing them.

20. If accurately drawn, these embelife or oblique lines should divide the portions of the three circles below the horizon obliquus into twelve equal parts. Thus each arc is determined by having to pass through three known points. They are called arcus horarum inequalium, as they shew the 'houres inequales.'

21. In fig. 2, Pl. II, the Rete is shewn as it appears when dropped into the depression in the front of the instrument. The shape of it varied much, and another drawing of one (copied from Camb. Univ. MS. Ii. 3. 3, fol. 66 b) is given in fig. 9, Pl. IV. The positions of the stars are marked by the extreme points of the metal tongues. Fig. 2 is taken from the figures in the Cambridge MSS., but the positions of the stars have been corrected by the list of latitudes and longitudes given by Stöffler, whom I have followed, not because he is correct, but because he probably represents their positions as they were supposed to be in Chaucer's time very nearly indeed. There was not room to inscribe the names of all the stars on the Rete, and to have written them on the plate below would have conveyed a false impression. A list of the stars marked in fig. 2 is given in the note to § 21, l. 4. The Ecliptic is the circle which crosses the Equinoctial at its East and West points (fig. 2). In Chaucer's description of the zodiac, carefully note the distinction between the Zodiac of the Astrolabe and the Zodiac of Heaven. The former is only six degrees broad, and shews only the northern half of the heavenly zodiac, the breadth of which is imagined to be 12 degrees. Chaucer's zodiac only shewed every other degree in the divisions round its border. This border is divided by help of a table of right ascensions of the various degrees of the ecliptic, which is by no means easily done. See Note on l. 4 of this section. I may add that the Rete is also called Aranea or Volvellum; in Arabic, Al´ancabūt (the spider).

22. The Label. See fig. 6, Pl. III. The label is more usually used on the front of the instrument, where the Rete and other plates revolve. The rule is used on the back, for taking altitudes by help of the scale.

23. The Almury; called also denticulus, ostensor, or 'calculer.' In fig. 2, it may be seen that the edge of the Rete is cut away near the head of Capricorn, leaving only a small pointed projecting tongue, which is the almury or denticle, or (as we should now say) pointer. As the Rete revolves, it points to the different degrees of the border. See also fig. 9, where the almury is plainly marked.

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.

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 ( 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&apos; 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 ♈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♈ and ZOS are both right angles. Hence also Chaucer's second statement, that the arcs PN and ♈Z are equal. In his numerical example, PN is 51° 50′; and therefore ZP is the complement, or 38° 10′. So also ♈Z is 51° 50′; and ♈S is 38° 10′. Briefly, ♈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 ( 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♈ 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 ♈A′, the highest altitude of a star A′ above the equinoctial, and also ♈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°.

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 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 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 heyȝt 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 Eb = 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 Eb = 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.

41a: 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.

41b: 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.]

42a 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.]

43a: 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. Ea, which = AB.

42b.: Here, 'by the craft of Umbra Recta&apos; 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 bE be called nm, and mn, Ec be produced to meet in o, then mo : mE :: bE : 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. 41a and 41b are mere repetitions of 41; 42a and 42b, of 42; and 43a, of 43.