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
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_.