Astronomical Lore in Chaucer

APPENDIX

I. Most of the terms at present used to describe the movements of the heavenly bodies were used in Chaucer’s time and occur very frequently in his writings. The significance of Chaucer’s references will then be perfectly clear, if we keep in mind that the modern astronomer’s description of the apparent movements of the star-sphere and of the heavenly bodies individually would have been to Chaucer a description of real movements.

When we look up into the sky on a clear night the stars and planets appear to be a host of bright dots on the concave surface, unimaginably distant, of a vast hollow sphere at the canter of which we seem to be. Astronomers call this expanse of the heavens with its myriad bright stars the celestial sphere or the star sphere, and have imagined upon its surface various systems of circles. In descriptions of the earth’s relation to the celestial sphere it is customary to disregard altogether the earth’s diameter which is comparatively infinitesimal.

If we stand on a high spot in the open country and look about us in all directions the earth seems to meet the sky in a circle which we call the terrestrial horizon. Now if we imagine a plane passing through the center of the earth and parallel to the plane in which the terrestrial horizon lies, and if we imagine this plane through the earth’s center extended outward in all directions to an infinite distance, it would cut the celestial sphere in a great circle which astronomers call the celestial horizon. On the celestial horizon are the north, east, south and west points. The plane of the celestial horizon is, of course, different for different positions of the observer on the earth.

If we watch the sky for some time, or make several observations on the same night, we notice, by observing the changing positions of the constellations, that the stars move very slowly across the blue dome above us. The stars that rise due east of us do not, in crossing the dome of the sky, pass directly over our heads but, from the moment that we first see them, curve some distance to the south, and, after passing their highest point in the heavens, turn toward the north and set due west. A star rising due east appears to move more rapidly than one rising some distance to the north or south of the east point, because it crosses a higher point in the heavens and has, therefore, a greater distance to traverse in the same length of time. When we observe the stars in the northern sky, we discover that many of them never set but seem to be moving around an apparently fixed point at somewhat more than an angle of 40°[194] above the northern horizon and very near the north star. These are called circum-polar stars. The whole celestial sphere, in other words, appears to be revolving about an imaginary axis passing through this fixed point, which is called the north pole of the heavens, through the center of the earth and through an invisible pole (the south pole of the heavens) exactly opposite the visible one. This apparent revolution of the whole star sphere, as we know, is caused by the earth’s rotation on its axis once every twenty-four hours from west to east. Chaucer and his contemporaries believed it to be the actual revolution of the nine spheres from east to west about the earth as a center.

Fig. 1.

For determining accurately the position of stars on the celestial sphere astronomers make use of various circles which can be made clear by a few simple diagrams. In Figure 1, the observer is imagined to be at O. Then the circle NESW is the celestial horizon, which we have described above. Z, the point immediately above the observer is called the zenith, and Z', the point immediately underneath, as indicated by a plumb line at rest, is the nadir. The line POP' is the imaginary axis about which the star-sphere appears to revolve, and P and P' are the poles of the heavens. The north pole P is elevated, for our latitude, at an angle of approximately 40° from the north point on the horizon. PP' is called the polar axis and it is evident that the earth’s axis extended infinitely would coincide with this axis of the heavens.

In measuring positions of stars with reference to the horizon astronomers use the following circles: Any great circle of the celestial sphere whose plane passes through the zenith and nadir is called a vertical circle. The verticle circle SPNZ', passing through the poles and meeting the horizon in the north and south points, N and S, is called the meridian circle, because the sun is on this circle at true mid-day. The meridian is the plane in which this circle lies. The vertical circle, EZ'WZ, whose plane is at right angles to the meridian, is called the prime vertical and it intersects the horizon at the east and west points, E and W. These circles, and the measurements of positions of heavenly bodies which involve their use, were all employed in Chaucer’s time and are referred to in his writings.[195]

The distance of a star from the horizon, measured on a vertical circle, toward the zenith is called the star’s altitude. A star reaches its greatest altitude when on the part of the meridional circle between the south point of the horizon, S, and the north pole, P. A star seen between the north pole and the north point on the horizon, that is, on the arc PN, must obviously be a circum-polar star and would have its highest altitude when between the pole and the zenith, or on the arc PZ. When a star reaches the meridian in its course across the celestial sphere it is said to culminate or reach its culmination. The highest altitude of any star would therefore be represented by the arc of the meridional circle between the star and the south point of the horizon. This is called the star’s meridian altitude.

The azimuth of a star is its angular distance from the south point, measured westward on the horizon, to a vertical circle passing through the star. The amplitude of a star is its distance from the prime vertical, measured on the horizon, north or south.

For the other measurements used by astronomers in observations of the stars still other circles on the celestial sphere must be imagined. We know that the earth’s surface is divided into halves, called the northern and southern hemispheres, by an imaginary circle called the equator, whose plane passes through the center of the earth and is perpendicular to the earth’s axis. If the plane of the earth’s equator were infinitely extended it would describe upon the celestial sphere a great circle which would divide that sphere into two hemispheres, just as the plane of the terrestrial equator divides the earth into two hemispheres. This great circle on the celestial sphere is called the celestial equator, or, by an older name, the equatorial, the significance of which we shall see presently. A star rising due east would traverse this great circle of the celestial sphere and set due west. The path of such a star is represented in Figure 2 by the great circle EMWM', which also represents the celestial equator. All stars rise and set following circles whose planes are parallel to that of the celestial equator and these circles of the celestial sphere are smaller and smaller the nearer they are to the pole, so that stars very near the pole appear to be encircling it in very small concentric circles. Stars in an area around the north celestial pole, whose limits vary with the position of the observer never set for an observer in the northern hemisphere. There is a similar group of stars around the south pole for an observer in the southern hemisphere.

Fig. 2.

The angle of elevation of the celestial equator to the horizon varies according to the position of the observer. If, for example, the observer were at the north pole of the earth, the north celestial pole would be directly above him and would therefore coincide with the zenith; this would obviously make the celestial equator and the horizon also coincide. If the observer should pass slowly from the pole to the terrestrial equator it is clear that the two circles would no longer coincide and that the angle between them would gradually widen until it reached 90°. Then the zenith would be on the celestial equator and the north and south poles of the heavens would be on the horizon.

We have still to define a great circle of the celestial sphere that is of equal importance with the celestial equator and the celestial horizon. This is the sun’s apparent yearly path, or the ecliptic. We know that the earth revolves about the sun once yearly in an orbit that is not entirely round but somewhat eliptical. Now since the earth, the sun, and the earth’s orbit around the sun are always in one plane, it follows that to an observer on the earth the sun would appear to be moving around the earth instead of the earth around the sun. The sun’s apparent path, moreover, would be in the plane of the earth’s orbit and when projected against the celestial sphere, which is infinite in extent, would appear as a great circle of that sphere. This great circle of the celestial sphere is the ecliptic. The sun must always appear to be on this circle, not only at all times of the year but at all hours of the day; for as the sun rises and sets, the ecliptic rises and sets also, since the earth’s rotation causes an apparent daily revolution not only of the sun, moon, and planets but also of the fixed stars and so of the whole celestial sphere and of all the circles whose positions upon it do not vary. The ecliptic is inclined to the celestial equator approximately 23½°, an angle which obviously measures the inclination of the plane of the earth’s equator to the plane of its orbit, since the celestial equator and the ecliptic are great circles on the celestial sphere formed by extending the planes of the earth’s equator and its orbit to an infinite distance. Since both the celestial equator and the ecliptic are great circles of the celestial sphere each dividing it into equal parts, it is evident that these two circles must intersect at points exactly opposite each other on the celestial sphere. These points are called the vernal and the autumnal equinoxes.

We shall next define the astronomical measurements that correspond to terrestrial latitude and longitude. For some reason astronomers have not, as we might expect, applied to these measurements the terms ‘celestial longitude’ and ‘celestial latitude.’ These two terms are now practically obsolete, having been used formerly to denote angular distance north or south of the ecliptic and angular distance measured east and west along circles parallel to the ecliptic. The measurements that correspond in astronomy to terrestrial latitude and longitude are called declination and right ascension and are obviously made with reference to the celestial equator, not the ecliptic. For taking these measurements astronomers employ circles on the celestial sphere perpendicular to the plane of the celestial equator and passing through the poles of the heavens. These are called hour circles. The hour circle of any star is the great circle passing through it and perpendicular to the plane of the equator. The angular distance of a star from the equator measured along its hour circle, is called the star’s declination and is northern or southern according as the star is in the northern or southern of the two hemispheres into which the plane of the equator divides the celestial sphere. It is evident that declination corresponds exactly to terrestrial latitude. Right ascension, corresponding to terrestrial longitude, is the angular distance of a heavenly body from the vernal equinox measured on the celestial equator eastward to the hour circle passing through the body.

The hour angle of a star is the angular distance measured on the celestial equator from the meridian to the foot of the hour circle passing through the star.

Fig. 3.

It remains to describe in greater detail the apparent movements of the sun and the sun’s effect upon the seasons. In Figure 3, the great circle MWM'E represents the equinoctial and XVX'A the ecliptic. The point X represents the farthest point south that the sun reaches in its apparent journey around the earth, and this point is called the winter solstice, because, for the northern hemisphere the sun reaches this point in mid-winter. When the sun is south of the celestial equator its apparent daily path is the same as it would be for a star so situated. Thus its daily path at the time of the winter solstice, about December 21, can be represented by the circle Xmn'. The arc gXh represents the part of the sun’s path that would be above the horizon, showing that night would last much longer than day and the rays of the sun would strike the northern hemisphere of the earth more indirectly than when the sun is north of the equator. As the sun passes along the ecliptic from X toward V, the part of its daily path that is above the horizon gradually increases until at V, the vernal equinox, the sun’s path would, roughly speaking, coincide with the celestial equator so that half of it would be above the horizon and half below and day and night would be of equal length. This explains why the celestial equator was formerly called the equinoctial (Chaucer’s term for it). As the sun passes on toward X' its daily arc continues to increase and the days to grow longer until at X' it reaches its greatest declination north of the equator and we have the longest day, June 21, the summer solstice. When the sun reaches this point, its rays strike the northern hemisphere more directly than at any other time causing the hot or summer season in this hemisphere. Next the sun’s daily arc begins to decrease, day and night to become more nearly equal, at A the autumnal equinox[196] is reached and the sun again shapes its course towards the point of maximum declination south of the equator. The two points of maximum declination are called solstices.

The two small circles of the celestial sphere, parallel to the equator, which pass through the two points where the sun’s declination is greatest, are called Tropics; the one in the northern hemisphere is called the Tropic of Cancer, that in the southern hemisphere, the Tropic of Capricorn. They correspond to circles on the earth’s surface having the same names.

II. By “artificial day” Chaucer means the time during which the sun is above the horizon, the period from sunrise to sunset. The arc of the artificial day may mean the extent or duration of it, as measured on the rim of an astrolabe, or it may mean (as here), the arc extending from the point of sunrise to that of sunset. See Astrolabe ii.7.

There has been some controversy among editors as to the correctness of the date occurring in this passage, some giving it as the 28th instead of the 18th. In discussing the accuracy of the reading “eightetethe” Skeat throws light also upon the accuracy of the rest of the passage considered from an astronomical point of view. He says (vol. 5, p. 133):

“The key to the whole matter is given by a passage in Chaucer’s ‘Astrolabe,’ pt. ii, ch. 29, where it is clear that Chaucer (who, however merely translates from Messahala) actually confuses the hour-angle with the azimuthal arc (see Appendix I); that is, he considered it correct to find the hour of the day by noting the point of the horizon over which the sun appears to stand, and supposing this point to advance, with a uniform, not a variable, motion. The host’s method of proceeding was this. Wanting to know the hour, he observed how far the sun had moved southward along the horizon since it rose, and saw that it had gone more than half-way from the point of sunrise to the exact southern point. Now the 18th of April in Chaucer’s time answers to the 26th of April at present. On April 26, 1874, the sun rose at 4 hr. 43 m., and set at 7 hr. 12 m., giving a day of about 14 hr. 30 m., the fourth part of which is at 8 hr. 20 m., or, with sufficient exactness, at half past eight. This would leave a whole hour and a half to signify Chaucer’s ‘half an houre and more’, showing that further explanation is still necessary. The fact is, however, that the host reckoned, as has been said, in another way, viz. by observing the sun’s position with reference to the horizon. On April 18 the sun was in the 6th degree of Taurus at that date, as we again learn from Chaucer’s treatise. Set this 6th degree of Taurus on the east horizon on a globe, and it is found to be 22 degrees to the north of the east point, or 112 degrees from the south. The half of this at 56 degrees from the south; and the sun would seem to stand above this 56th degree, as may be seen even upon a globe, at about a quarter past nine; but Mr. Brae has made the calculation, and shows that it was at twenty minutes past nine. This makes Chaucer’s ‘half an houre and more’ to stand for half an hour and ten minutes; an extremely neat result. But this we can check again by help of the host’s other observation. He also took note, that the lengths of a shadow and its object were equal, whence the sun’s altitude must have been 45 degrees. Even a globe will shew that the sun’s altitude, when in the 6th degree of Taurus, and at 10 o’clock in the morning, is somewhere about 45 or 46 degrees. But Mr. Brae has calculated it exactly, and his result is, that the sun attained its altitude of 45 degrees at two minutes to ten exactly. This is even a closer approximation than we might expect, and leaves no doubt about the right date being the eighteenth of April.”

Thus it appears that Chaucer’s method of determining the date was incorrect but his calculations in observing the sun’s position were quite accurate. For fuller particulars see Chaucer’s Astrolabe, ed. Skeat (E. E. T. S.) preface, p. 1.

III. It was customary in ancient times and even as late as Chaucer’s century to determine the position of the sun, moon, or planets at any time by reference to the signs of the zodiac. The zodiac is an imaginary belt of the celestial sphere, extending 8° on each side of the ecliptic, within which the orbits of the sun, moon, and planets appear to lie. The zodiac is divided into twelve equal geometric divisions 30° in extent called signs to each of which a fanciful name is given. The signs were once identical with twelve constellations along the zodiac to which these fanciful names were first applied. Since the signs are purely geometric divisions and are counted from the spring equinox in the direction of the sun’s progress through them, and since through the precession of the equinoxes the whole series of signs shifts westward about one degree in seventy-two years, the signs and constellations no longer coincide. Beginning with the sign in which the vernal equinox lies the names of the zodiacal signs are Aries (Ram), Taurus (Bull), Gemini (Twins), Cancer (Crab), Leo (Lion), Virgo (Virgin), Libra (Scales), Scorpio (Scorpion), Sagittarius (Archer), Aquarius (Water-carrier), and Pisces (Fishes).

In this passage, the line “That in the Ram is four degrees up-ronne” indicates the date March 16. This can be seen by reference to Figure 1 in Skeat’s edition of Chaucer’s Astrolabe (E. E. T. S.) The astrolabe was an instrument for making observations of the heavenly bodies and calculating time from these observations. The most important part of the kind of astrolabe described by Chaucer was a rather heavy circular plate of metal from four to seven inches in diameter, which could be suspended from the thumb by a ring attached loosely enough so as to allow the instrument to assume a perpendicular position. One side of this plate was flat and was called the back, and it is this part that Figure 1 represents. The back of the astrolabe planisphere contained a series of concentric rings representing in order beginning with the outermost ring: the four quadrants of a circle each divided into ninety degrees; the signs of the zodiac divided into thirty degrees each; the days of the year, the circle being divided, for this purpose, into 365¼ equal parts; the names of the months, the number of days in each, and the small divisions which represent each day, which coincide exactly with those representing the days of the year; and lastly the saints’ days, with their Sunday-letters. The purpose of the signs of the zodiac is to show the position of the sun in the ecliptic at different times. Therefore, if we find on the figure the fourth degree of Aries and the day of the month corresponding to it, we have the date March 16 as nearly as we can determine it by observing the intricate divisions in the figure.

The next passage “Noon hyer was he, whan she redy was” means evidently, ‘he was no higher than this (i. e. four degrees) above the horizon when she was ready’; that is, it was a little past six. The method used in determining the time of day by observation of the sun’s position is explained in the Astrolabe ii, 2 and 3. First the sun’s altitude is found by means of the revolving rule at the back of the astrolabe. The rule, a piece of metal fitted with sights, is moved up and down until the rays of the sun shine directly through the sights. Then, by means of the degrees marked on the back of the astrolabe, the angle of elevation of the rule is determined, giving the altitude of the sun. The rest of the process involves the use of the front of the astrolabe. This side of the circular plate, shown in Fig. 2, had a thick rim with a wide depression in the middle. On the rim were three concentric circles, the first showing the letters A to Z, representing the twenty-four hours of the day, and the two innermost circles giving the degrees of the four quadrants. The depressed central part of the front was marked with three circles, the ‘Tropicus Cancri’, the ‘AEquinoctialis,’ and the ‘Tropicus Capricorni’; and with the cross-lines from North to South, and from East to West. There were besides several thin plates or discs of metal of such a size as exactly to drop into the depression spoken of. The principal one of these was the ‘Rete’ and is shown in Fig. 2. “It consisted of a circular ring marked with the zodiacal signs, subdivided into degrees, with narrow branching limbs both within and without this ring, having smaller branches or tongues terminating in points, each of which denoted the exact position of some well-known star. * * * The ‘Rete’ being thus, as it were, a skeleton plate, allows the ‘Tropicus Cancri,’ etc., marked upon the body of the instrument, to be partially seen below it. * * * But it was more usual to interpose between the ‘Rete’ and the body of the instrument (called the ‘Mother’) another thin plate or disc, such as that in Fig. 5, so that portions of this latter plate could be seen beneath the skeleton-form of the ‘Rete’ (i. 17). These plates were called by Chaucer ‘tables’, and sometimes an instrument was provided with several of them, differently marked, for use in places having different latitudes. The one in Fig. 5 is suitable for the latitude of Oxford (nearly). The upper part, above the Horizon Obliquus, is marked with circles of altitude (i. 18), crossed by incomplete arcs of azimuth tending to a common centre, the zenith (i. 19).” [Skeat, Introduction to the Astrolabe, pp. lxxiv-lxxv.]

Now suppose we have taken the sun’s altitude by §2 (Pt. ii of the Astrolabe) and found it to be 25½°. “As the altitude was taken by the back of the Astrolabe, turn it over, and then let the Rete revolve westward until 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.” [Skeat’s note on the Astrolabe ii. 3, pp. 189-190].

IV. Chaucer would know the altitude of the sun simply by inspection of an astrolabe, without calculation. Skeat has explained this passage in his Preface to Chaucer’s Astrolabe (E. E. T. S.), p. lxiii, as follows:

“Besides saying that the sun was 29° high, Chaucer says that his shadow was to his height in the proportion of 11 to 6. Changing this proportion, we can make it that of 12 to 6⁶/11; that is, the point of the Umbra Versa (which is reckoned by twelfth parts) is 6⁶/11 or 6½ nearly. (Umbra Recta and Umbra Versa were scales on the back of the astrolabe used for computing the altitudes of heavenly bodies from the height and shadows of objects. The umbra recta was used where the angle of elevation of an object was greater than 45°; the umbra versa, where it was less.) This can be verified by Fig. 1; for a straight edge, laid across from the 29th degree above the word ‘Occidens,’ and passing through the center, will cut the scale of Umbra Versa between the 6th and 7th points. The sun’s altitude is thus established as 29° above the western horizon, beyond all doubt.”

V. Herberwe means ‘position.’ Chaucer says here, then, that the sun according to his declination causing his position to be low or high in the heavens, brings about the seasons for all living things. In the Astrolabe, i. 17, there is a very interesting passage explaining in detail, declination, the solstices and equinoxes, and change of seasons. Chaucer is describing the front of the astrolabe. He says: “The plate under thy rite is descryved with 3 principal cercles; of whiche the leste is cleped the cercle of Cancer, by-cause that the heved of Cancer turneth evermor consentrik up-on the same cercle. (This corresponds to the Tropic of Cancer on the celestial sphere, which marks the greatest northern declination of the sun.) In this heved of Cancer is the grettest declinacioun northward of the sonne. And ther-for is he cleped the Solsticioun of Somer; whiche declinacioun, aftur Ptholome, is 23 degrees and 50 minutes, as wel in Cancer as in Capricorne. (The greatest declination of the sun measures the obliquity of the ecliptic, which is slightly variable. In Chaucer’s time it was about 23° 31', and in the time of Ptolemy about 23° 40'. Ptolemy assigns it too high a value.) This signe of Cancre is cleped the Tropik of Somer, of tropos, that is to seyn ‘agaynward’; for thanne by-ginneth the sonne to passe fro us-ward. (See Fig. 2 in Skeat’s Preface to the Astrolabe, vol. iii, or E. E. T. S. vol. 16.)

The middel cercle in wydnesse, of thise 3, is cleped the Cercle Equinoxial (the celestial equator of the celestial sphere); up-on whiche turneth evermo the hedes of Aries and Libra. (These are the two signs in which the ecliptic crosses the equinoctial.) And understond wel, that evermo this Cercle Equinoxial turneth iustly fro verrey est to verrey west; as I have shewed thee in the spere solide. (As the earth rotates daily from west to east, the celestial sphere appears to us to revolve about the earth once every twenty-four hours from east to west. Chaucer, of course, means here that the equinoctial actually revolves with the primum mobile instead of only appearing to revolve.) This same cercle is cleped also the Weyere, equator, of the day; for whan the sonne is in the hevedes of Aries and Libra, than ben the dayes and the nightes ilyke of lengthe in al the world. And ther-fore ben thise two signes called Equinoxies.

The wydeste of thise three principal cercles is cleped the Cercle of Capricorne, by-cause that the heved of Capricorne turneth evermo consentrix up-on the same cercle. (That is to say, the Tropic of Capricorn meets the ecliptic in the sign Capricornus, or, in other words, the sun attains its greatest declination southward when in the sign Capricornus.) In the heved of this for-seide Capricorne is the grettest declinacioun southward of the sonne, and ther-for is it cleped the Solsticioun of Winter. This signe of Capricorne is also cleped the Tropik of Winter, for thanne byginneth the sonne to come agayn to us-ward.”

VI. The moon’s orbit around the earth is inclined at an angle of about 5° to the earth’s orbit around the sun. The moon, therefore, appears to an observer on the earth as if traversing a great circle of the celestial sphere just as the sun appears to do; and the moon’s real orbit projected against the celestial sphere appears as a great circle similar to the ecliptic. This great circle in which the moon appears to travel will, therefore, be inclined to the ecliptic at an angle of 5° and the moon will appear in its motion never far from the ecliptic; it will always be within the zodiac which extends eight or nine degrees on either side of the ecliptic.

The angular velocity of the moon’s motion in its projected great circle is much greater than that of the sun in the ecliptic. Both bodies appear to move in the same direction, from west to east; but the solar apparent revolution takes about a year averaging 1° daily, while the moon completes a revolution from any fixed star back to the same star in about 27¼ days, making an average daily angular motion of about 13°. The actual daily angular motion of the moon varies considerably; hence in trying to test out Chaucer’s references to lunar angular velocity it would not be correct to make use only of the average angular velocity since his references apply to specific times and therefore the variation in the moon’s angular velocity must be taken into account.

VII. On the line “In two of Taur,” etc., Skeat has the following note: “Tyrwhitt unluckily altered two to ten, on the plea that ‘the time (four days complete, l. 1893) is not sufficient for the moon to pass from the second degree of Taurus into Cancer? And he then proceeds to shew this, taking the mean daily motion of the moon as being 13 degrees, 10 minutes, and 35 seconds. But, as Mr. Brae has shewn, in his edition of Chaucer’s Astrolabe, p. 93, footnote, it is a mistake to reckon here the moon’s mean motion; we must rather consider her actual motion. The question is simply, can the moon move from the 2nd degree of Taurus to the 1st of Cancer (through 59 degrees) in four days? Mr. Brae says decidedly, that examples of such motion are to be seen ‘in every almanac.’

For example, in the Nautical Almanac, in June, 1886, the moon’s longitude at noon was 30° 22' on the 9th, and 90° 17' on the 13th; i. e., the moon was in the first of Taurus on the former day, and in the first of Cancer on the latter day, at the same hour; which gives (very nearly) a degree more of change of longitude than we here require. The MSS all have two or tuo, and they are quite right. The motion of the moon is so variable that the mean motion affords no safe guide.” [Skeat, Notes to the Canterbury Tales, p. 363.]

VIII. The moon’s “waxing and waning” is due to the fact that the moon is not self-luminous but receives its light from the sun and to the additional fact that it makes a complete revolution around the earth with reference to the sun in 29½ days. When the earth is on the side of the moon that faces the sun we see the full moon, that is, the whole illuminated hemisphere. But when we are on the side of the moon that is turned away from the sun we face its unilluminated hemisphere and we say that we have a ‘new moon.’ Once in every 29½ days the earth is in each of these positions with reference to the moon and, of course, in the interval of time between these two phases we are so placed as to see larger or smaller parts of the illuminating hemisphere of the moon, giving rise to the other visible phases.

When the moon is between the earth and the sun she is said to be in conjunction, and is invisible to us for a few nights. This is the phase called new moon. As she emerges from conjunction we see the moon as a delicate crescent in the west just after sunset and she soon sets below the horizon. Half of the moon’s surface is illuminated, but we can see only a slender edge with the horns turned away from the sun. The crescent appears a little wider each night, and, as the moon recedes 13° further from the sun each night, she sets correspondingly later, until in her first quarter half of the illuminated hemisphere is turned toward us. As the moon continues her progress around the earth she gradually becomes gibbous and finally reaches a point in the heavens directly opposite the sun when she is said to be in opposition, her whole illumined hemisphere faces us and we have full moon. She then rises in the east as the sun sets in the west and is on the meridian at midnight. As the moon passes from opposition, the portion of her illuminated hemisphere visible to us gradually decreases, she rises nearly an hour later each evening and in the morning is seen high in the western sky after sunrise. At her third quarter she again presents half of her illuminated surface to us and continues to decrease until we see her in crescent form again. But now her position with reference to the sun is exactly the reverse of her position as a waxing crescent, so that her horns are now turned toward the west away from the sun, and she appears in the eastern sky just before sunrise. The moon again comes into conjunction and is lost in the sun’s rays and from this point the whole process is repeated.

IX. That the apparent motions of the sun and moon are not so complicated as those of the planets will be clear at once if we remember that the sun’s apparent motion is caused by our seeing the sun projected against the celestial sphere in the ecliptic, the path cut out by the plane of the earth’s orbit, while in the case of the moon, what we see is the moon’s actual motion around the earth projected against the celestial sphere in the great circle traced by the moon’s own orbital plane produced to an indefinite extent. These motions are further complicated by the rotation of the earth on its own axis, causing the rising and setting of the sun and the moon. These two bodies, however, always appear to be moving directly on in their courses, each completing a revolution around the earth in a definite time, the sun in a year, the moon in 29½ days. What we see in the case of the planets, on the other hand, is a complex motion compounded of the effects of the earth’s daily rotation, its yearly revolution around the sun, and the planets’ own revolutions in different periods of time in elliptical orbits around the sun. These complex planetary motions are characterized by the peculiar oscillations known as ‘direct’ and ‘retrograde’ movements.

Fig. 4.

The motion of a planet is said to be direct when it moves in the direction of the succession of the zodiacal signs; retrograde when in the contrary direction. All of the planets have periods of retrograde and direct motion, though their usual direction is direct, from west to east. Retrograde motion can be explained by reference to the accompanying diagrams. In Fig. 4, the outer circle represents the path of the zodiac on the celestial sphere. Let the two inner circles represent the orbits of the earth and an inferior planet, Venus, around the sun, at S. (An inferior planet is one whose orbit around the sun is within that of the earth. A superior planet is one whose orbit is outside that of the earth.) V, V' and V, and E, E', and E are successive positions of the two planets in their orbits, the arc VV being longer than the arc EE because the nearer a planet is to the sun, the greater is its velocity. Then when Venus is at V and the earth at E, we shall see Venus projected on the celestial sphere at V₁. When Venus has passed on to V' the earth will have passed to E' and we shall see Venus on the celestial sphere at V₂. The apparent motion of the planet thus far will have been direct, from west to east in the order of the signs. But when Venus is at V and the earth at E Venus will be seen at V₃ having apparently moved back about two signs in a direction the reverse of that taken at first. This is called the planet’s retrograde motion. At some point beyond V, the planet will appear to stop moving for a very short period and then resume its direct motion. In Fig. 5, the outer arc again represents the path of the zodiac on the celestial sphere. The smaller arcs represent the orbits of the superior planet, Mars, and the earth around the sun, S. At the point of opposition of Mars (when Mars and the sun are at opposite points in the heavens to an observer on the earth) we should see Mars projected on the zodiac at M₁. After a month Mars will be at M' and the earth at E', so that in its apparent motion Mars will have retrograded to M₂. After three months from opposition Mars will be at M and the earth at E, making Mars appear at M₃ on the celestial sphere, its motion having changed from retrograde to direct.

Fig. 5.

Both Figures 4 and 5 take no account of the fact that the earth’s orbit and those of the planets are not in exactly the same planes. Remembering this fact we see at once that the apparent oscillations of the planets are not back and forth in a straight line but in curves and spirals. It is easy to see why the apparent motions of the planets were accounted for by deferents and epicycles, before the Copernican system revealed the true nature of the solar system as heliocentric and not geocentric.

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