[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 (α 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 ♈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 (α 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.