Epitome Astronomiae Copernicanae: Liber VI, part 9
How many forms of lunar eclipses are there? Three: partial, total without delay in the darkness, and total with delay.
In the former diagram (92) the partial ones are pictured; if the points V of the Moon and N of the shadow coincided, the eclipse would be total without delay. In the other the eclipse is total with delay.
What is called the arc between the centres, or the latitudinal arc? It is the arc perpendicular from the centre of the shadow to the way or orbit of the Moon, very nearly less than the latitude of the Moon and almost coinciding with it.
As if A be the centre of the shadow, FAG the ecliptic, and QTA at right angles to it the circle of latitude, BCE the way of the centre of the Moon, oblique to the ecliptic FAG, AC the perpendicular from A to BE (let diagram 92 be corrected so that AC is perpendicular to BE), then AT would indeed be the latitude of the Moon, if her centre were in T in true opposition; but AC, nearest to AT itself, is called the arc between the centres.
How great is this arc required to be for any one form of eclipse?
In a total eclipse without delay this arc added to the semidiameter of the Moon makes a sum equal to the semidiameter of the shadow; in a total one with delay, less than the semidiameter of the shadow; in a partial one, greater indeed than the semidiameter of the shadow, yet less than the sum made up of the semidiameters of the shadow and of the Moon.
As AE is the sum of the semidiameters AO, OE, but AY in the former diagram is indeed less than that, yet greater than AN, the semidiameter of the shadow; the eclipse is therefore not total, but partial. In the other diagram AC is less than AD, AK, the difference of the semidiameters, and so AY is less than AN; the eclipse is therefore total.
What are the minutes 1) of half duration, 2) of incidence or fall, 3) of emersion, 4) of half delay? They are arcs of the Moon’s way, in which, while the centre of the Moon is in them, either 1) some part of the Moon fails, the Moon 2) either falling into the shadow, 3) or emerging, 4) or also being wholly darkened.
As BC, CE are the minutes of half duration, BK of incidence, DE of emersion, KC or CD of half delay. Although incidence or fall and emersion are properly used only of total eclipses, yet also in the partial one of the former diagram BC may be called incidence, CE emersion.
How do duration and delay differ? Duration is to be understood of every kind of defect and of all eclipses; delay only of the full defects of certain ones, for it is understood: in darkness or shadow.
What is the middle of the eclipse, or greatest obscuration? When the centre of the Moon is either joined to the centre of the shadow, or in that perpendicular from the centre of the shadow to the Moon’s way. As in fig. 93, when the centre C falls in AN.
Why do you not use the circle of latitude AQ, at right angles rather to the ecliptic, as in the old astronomy? 1. Because, properly speaking, the Moon falling upon this circle does not come deepest into the shadow. 2. Because the times and minutes of incidence BT and emersion TE would be made unequal when compared with this circle ATQ. 3) Because so the duration and delay in the darkness are computed more briefly.
How are the minutes of half duration, half delay, incidence and emersion to be computed? From the antilogarithms of the sum and of the difference of the semidiameters of the Moon and of the shadow the antilogarithm of the arc between the centres, or latitudinal arc, is taken away; the remainders sought among the antilogarithms show among the arcs the minutes, there of half duration, here of half delay: but the latter, of delay, being taken from the former, of duration, there remain the minutes of incidence and emersion.
As BC is computed from AB the sum and AC, because BCA is right, so KC from AK the difference of the semidiameters and from AC; but KC being taken from BC, BK remains.
In what do the places of true conjunction and of greatest obscuration differ?
They differ by a very small arc, the double of the reduction of the Moon’s place to the ecliptic, by which arc the Moon at greatest obscuration is always nearer the node than the centre of the shadow.
As if T were the place of conjunction, C is the place of greatest obscuration, TC the double of the reduction to the ecliptic.
How great is the angle of latitude, or of the intersection of the ways of the Sun or shadow FG and of the Moon BE in eclipses? A little above it was said that in the conjunctions this angle is greatest, 5° 18′. The old astronomy uses 5° 0′, to the harm of the calculation.
Whence do we learn the quantity AC of the arc between the centres, or latitudinal arc? From the distance of A, the centre of the shadow, from the place of the node, or the arc of the ecliptic intercepted, which differs only by the reduction to the ecliptic, by half of TC itself, from the argument of latitude, or its complement to a semicircle.
What are the ecliptic limits of the Moon and how great are they? The ecliptic limits are arcs of the ecliptic, begun from the node, in which arcs, the arc between the centres lying in them, at least some particle of the body can be eclipsed; they end in those points before and after the node in which, the arc between the centres lying there, the Moon in opposition to the Sun touches the shadow and does not graze it; the least is 10° 40′ in the apogee of the Moon, the greatest 12° 0′ in the perigee of the Moon.
If the shadow in the perigee of the Moon is deeper or thicker and the Moon’s passage longer, is not the duration of the eclipse then the longest? On the contrary, in a shorter space of time then the Moon labours, other things equal, and conversely in the apogee, by a shorter passage, she delays longer. For the hourly motion of the perigean Moon to the hourly motion of the apogean is in greater proportion than the perigean passage to the apogean.
Of how many elements does the true hourly motion of the Moon, useful in eclipses, consist? Of these five: first, the equable or mean periodic hourly motion of the Moon; next its simple equation from the simple anomaly; third, close to the very hour of the syzygy the monthly equation is added, by which two elements that mean is diminished in apogee, increased in perigee; fourth, the variation accelerates both and all the intermediate ones, each in its own proportion; fifth, finally, from the hourly motion of the Moon so formed the hourly motion of the Sun, whatever it be at the time, is to be taken away, that the true hourly excess may be had.
How great, then, is the variety of the true hourly motions in eclipses? The hourly motion from the equinox, least in apogee, comes out by the Tychonic variation 29′ 46″, greatest in perigee 38′ 30″, but by the increased variation the former 29′ 56″, the latter 38′ 43″; from these the mean hourly motion of the Sun being taken, there remain 27′ 18″ or 27′ 28″, and 36′ 2″ or 36′ 15″. Where it is to be noted that it is more right to take away the hourly motion of the Sun which is at any time.
How great is the greatest duration hence made of lunar eclipses, how great the greatest delay of total ones, and how is it to be computed? The sum or difference of the semidiameters of the Moon and shadow being divided by the hourly motion, the hours and minutes of the longest duration or delay come out. By logistic logarithms thus:
In the perigee of the Sun and apogee of the Moon.
Logarithms 180922, 189200. These show the minutes 9′ 50″ or 9′ 3″ above 2 hours.
The whole duration is therefore 4 hours 19′ 40″ or 4 hours 18′ 6″.
If we proceed in this way also in the apogee of the Sun and apogee of the Moon, the greatest duration will come out 4 hours 20′ 4″ or 4 hours 18′ 18″. In the apogee of the Sun and perigee of the Moon the greatest duration 3 hours 39′ 22″ or 3 hours 38′ 46″. In the perigee of the Sun and perigee of the Moon the greatest duration 3 hours 38′ 40″ or 3 hours 37′ 26″.
But the difference of the delays is less than that of the durations of every kind, because the perigean Moon has a smaller proportion to her passage than the apogean to hers: for the apparent semidiameter of the Moon is increased in perigee on account of one cause only, but the semidiameter of the shadow on account of two.
Therefore the longest delay, in the perigee of the Sun and apogee of the Moon, is 2 hours 7′ 20″ or 2 hours 6′ 38″.
How is the time of incidence or emersion known? By subtraction of the half delay from the half duration: there remains the time of incidence or of emersion, for these are equal.
Whence are the minutes of defect had in a partial eclipse or any not central? By subtraction of the latitudinal arc from the sum of the semidiameters of the Moon and shadow the minutes of defect are had.
What are the ecliptic digits and how are they computed? They are twelfths of the semidiameter of the Moon. The minutes of defect are multiplied by 12, and the sum divided by the semidiameter of the Moon. Or from the logistic logarithm of half the minutes of defect the logarithm of the semidiameter of the Moon is taken away: what is left shows in its column the digits failing.
By how many digits can the Moon fail? Properly speaking not by more than she has in her diameter, but astronomers, abusing the word digit, turn also the remaining minutes in a total eclipse with delay into digits, reckoning how much the Moon could fail if her semidiameter also were greater, extended in that direction which the centre of the Moon holds.
Therefore, the semidiameter of the perigean Moon, 16′ 22″, being added to the semidiameter of the perigean shadow, 48′ 40″, there are heaped up 65′ 2″, of whose half, 32′ 31″, the logarithm is 61249.
From this take the logarithm of 32′ 44″, the diameter of the Moon, 60600.
As a logarithm it shows in its column 22 digits 39′.
What is the cause why the light of the Moon, about to fail, is first dulled and grows pale, especially in the part which is near the shadow? The cause is not in the full shadow of the Earth itself, as if it radiated of itself some blackness and breathed it upon the approaching Moon, as some incautiously either write themselves or are understood by readers; but this happens because, before the whole Sun is covered from any particle of the Moon, that is before the Moon begins to graze the total shadow, a greater and greater part of the Sun is intercepted from her by the Earth. But the whole Sun illuminates the Moon more brightly than a part, and a greater part more brightly than a lesser. See Astronomiae Pars Optica, page 239 (II. 280).
Do all eclipses of the Moon agree with these reasonings? It often happens that the lucid substance about the Sun, inflamed by contact with the Sun, irradiates the outermost limits of the shadow, and the Moon, although wholly immersed in the shadow, nevertheless receives these rays at her extreme margin: which false light, since it cannot be compared with the true light glided down from the Sun (the rest of the Moon being wholly extinguished), is itself held for true, so that the Moon is reckoned not to have entered wholly into the shadow. For this reason it comes about that the delays of total eclipses are estimated much shorter. Nay, even those which are total without delay or with a very short delay are reckoned for this cause not total but partial, as if something remained from the north or south. See Astronomiae Pars Optica, page 302 (II. 319). And so Galileo also judges.
Whence is it plain that this excuse is lawful? What if there are causes, not yet sufficiently known to astronomy? 1) Nay, this effect is so evident that, if it be referred to astronomical causes, it involves most open contradictions and overturns the whole reckoning of the dimensions of the shadow and the most certain laws of optics. 2) Testimony is also borne to this physical cause by some solar eclipses, of which below. 3) When we observe the diameter of the Sun by a ray admitted through a very narrow aperture, this fringe of dusky colour, when it is present in the heaven, paints itself most clearly upon the tablet around the disc of the Sun, so that between the disc of the Sun and this fringe no dividing boundary appears, and therefore the diameter of the Sun is then gathered certainly greater than just.
What is the cause why the Moon in total eclipses appears with so great variety, at times is wholly lost, at times shines with a most evident redness, either all over or in part of her body, nor is this always equally turned toward the nearest margin of the shadow, but runs out more widely on the other side? The proper light of the Moon is not the cause, for it would be perpetual, at least in the same part of the body, nor that lucid substance about the Sun in the ether itself, for it would illuminate in order either the whole or only the extremities of the margin nearest the shadow. But it is necessary that the Moon with that part of her body which reddens fall into the secondary rays of the Sun, that is those twice refracted in our air poured around the Earth, once at entry, again at exit: by which very rays refracted once at entry we see the Sun, when he is still below our horizon, and we see him then indeed reddening. For since these rays, refracted once, are still so bright that they make day with us (though not the brightest, as from the bare and clear Sun), they will be able, refracted a second time at exit toward the Moon, to cause in her a brightness not much less. This cause is fit to represent the variety of that effect. For that circle of the Earth, or of the elementary globe, from which the shadow of the Earth rises at any moment, is stretched through all the zones, running out from one polar zone to the opposite, but the constitution of the air through the different zones is different, wherefore the refractions also will not be caused of one quantity. On these see more in Astronomiae Pars Optica, page 271, and especially the diagram on page 279 (II. 300, 304). Likewise in the Dissertation with the Sidereal Messenger, page 20 (II. 498).
If the Moon cannot be deprived of the light of the Sun outside the line of opposition, drawn through the Sun, the eye and the Moon, how does it happen that at times the Sun and the eclipsed Moon appear together above the horizon and so not in one straight line? The blame of this matter is borne by the air, with which the Earth, the dwelling of the eyes, is clothed. For the rays, from one side of the Sun, from the other of the Moon, both being below the horizon, when they touch the surface of the air, as of a denser medium, are refracted downward to the eye lurking in the depth of the air. From what quarters, therefore, the last rays of the Sun and Moon glide down, namely from some altitude on each side above the horizon toward those quarters, in those altitudes the deceived eye places the luminaries, because of the refraction of the rays that has been made nothing is known to it; it supposes, therefore, that there is a single line from the eye to the luminary. See above, book I, page 148, the diagram and explanation.
Are other stars also so obscured as the Moon? By the shadow of the Earth the nearest planets, Venus and Mars, suffer nothing; she, because, even if she were lower than she is, she is not opposed to the Sun, he, because he is higher than the last point of the terrestrial shadow, even when he approaches nearest of all. Much less, therefore, can the stars higher than Mars be covered by the shadow. But by the shadow which the globe of Jupiter casts of itself on high, the four Jovian planets, that is the secondary planets, are obscured and extinguished, as Galileo the Italian and Marius the German prove by no idle documents.
VIII. Of the eclipse of the Sun.
In how many ways can eclipses of the Sun be considered? In two ways: either in respect of that which the Earth suffers during the occultation of the Sun, or in respect of that which the Sun seems to suffer. In the former way the eclipse of the Sun is considered universally, in respect of all places of the Earth in which it can be seen; in the latter the consideration is particular and is bound to one certain place of the Earth.
What, then, does the Earth suffer in an eclipse of the Sun? The new Moon, interposed between the Sun and some parts of the disc of the Earth, deprives them either of all the light of the Sun, or of some part of the solar light.
What do you call the disc of the Earth? It is the plane of the circle of illumination of the Earth, so far as that appears from some point in the body of the new Moon, overshadowing the Earth.
It is called a disc because we imagine the illuminated surface of the Earth to be projected into the plane of this circle.
In diagram 94 let it be understood under VE.
How great is the semidiameter of this disc of the Earth as it appears from the Moon, so to speak? If from the horizontal parallax of the Moon you take away one and a half times the horizontal parallax of the Sun, there remains the semidiameter of the disc of the Earth, or the angle of this apparent semidiameter in measure, of which a right angle is 90 degrees. I demonstrate it in my Hipparchus.
What is understood by the words umbra and penumbra of the Moon? By the word lunar umbra is understood all that circular space of the disc of the Earth, or the corresponding spherical space of the surface projected into the disc, in which the whole Sun is covered by the Moon. The penumbra is all that space of the disc or surface of the Earth in which some particle of the solar body is covered at that moment.
As if VE be the disc of the Earth, NO the umbra, MN, OP the penumbra: although that is drawn too large in respect of the disc of the Earth, since the penumbra is less than the disc of the Earth, as in the diagram next following.
How great is the semidiameter of the umbra of the Moon and how is it found? If from the apparent semidiameter of the Moon you take away the apparent semidiameter of the Sun, there remains very nearly the apparent semidiameter of the umbra of the Moon.
In the diagram let IBL be the semidiameter of the Moon, ABD or ARD the semidiameter of the Sun (for they differ little), BLR, that is BLO, will be the semidiameter of the umbra of the Moon.
If the subtraction cannot be made, then there is no full umbra of the Moon.
How great is the semidiameter of the penumbra of the Moon, whose heart is the umbra, and how is it found? If you throw into one sum the apparent semidiameters of the Sun and Moon and about half the parallax of the Sun, the semidiameter of the penumbra of the Moon will be heaped up, namely from the centre of the full umbra, if there is one, to the extremity of the penumbra.
As if ABD be 15′ 0″ and IBL 16′ 20″ and the parallax of the Sun BDE 1′ 0″, half 0′ 30″, BLP, the semidiameter of the umbra and penumbra joined, will be 31′ 50″. The demonstration of this and of what adheres to it is in my Hipparchus. (Vol. III, p. 525.)
How are these terms, hourly motion and latitudinal arc, to be understood in this general consideration of the ecliptic, or of the eclipsed Earth? They are to be understood of straight lines in the disc of the Earth. For the ecliptic is this time nothing else than the section of the plane of the ecliptic with the plane of the disc of the Earth, made through its centre. The hourly motion is this time a straight line in the plane of the disc of the Earth, continued if need be, designated by the motion of the centre of the shadow or penumbra of the Moon, inclined to the ecliptic of the disc at the constant angle of 5° 18′. Finally the latitudinal arc is the straight line erected from the centre of the shadow or penumbra perpendicular to the way of its centre and in In the diagram of the eclipse of the Sun of the year 1621, A is the centre of the disc of the Earth HQK, FAG represents the ecliptic, BE the way of the centre of the shadow or penumbra, BL or EP, AC the latitudinal arc, AT the latitude itself of the Moon (or here of the shadow) in the true conjunction itself.
How is the quantity of these lines found? Of whatever size each of them appears in the sphere of the Moon itself, viewed from the Earth, of so much it is reckoned to be already in the disc of the Earth; except that, if we proceed most accurately, to each there is to be added a portion of itself as great as the portion of the parallax of the Moon from the Sun is the parallax of the Sun itself.
When is the greatest obscuration of the Earth reckoned to be? Then, when the perpendicular drawn from the centre of the disc of the Earth to the way of the shadow falls in the centre of the penumbra or umbra, which happens very near the moment of the true conjunction of the Moon and Sun.
In the diagram, the centre of the shadow being in T, which is a point of the arc AT, at right angles to the ecliptic, the conjunction is reckoned exact. But the centre of the shadow having passed into the line AC, perpendicular to the way of the Moon, the greatest obscuration comes about; for then the centres of the disc and of the shadow are nearest.
If the illuminated face of the Earth is projected into the plane of the disc, what places of the Earth are then represented by the centre of the disc? Not one and the same place always through the whole duration, but at each moment of the duration that place of the Earth is represented by the centre of the disc A which then has the Sun at the zenith.
the ecliptic of the disc continued.
What do you call the duration of the total eclipse, and what the duration of the eclipse of every kind? All that time in which some point of the shadow of the Moon, especially its centre, is in the disc of the Earth is attributed to the duration of the total eclipse. For that is an indication that some part of the surface of the Earth, whatever it be, is then wholly deprived of the sight of the Sun in the daytime. But to the eclipse of every kind is reckoned also that time when not the shadow itself, but at least some point of the penumbra, is found in the disc of the Earth. For it is indicated by this that there is some place of the surface of the Earth in which at least some particle of the Sun is covered.
What place of the Earth is indicated by the point of contact of the disc and the penumbra or umbra? That place at which the Sun then, in the case of the umbra wholly eclipsed, rises or sets, but in the case of the penumbra, rising begins to fail at the extreme margin, or setting ceases to be eclipsed.
In the diagram the points of contact of the umbra are R, S, of the penumbra M, O.
Wherefore these points are indices of the places of the Earth in which the Sun rises or sets at the beginning and end of each eclipse. But if the contact of the umbra or penumbra came about in the point N, it indicates a place to which the pole of the ecliptic, or rather of the Moon’s orbit, is at the zenith, and the Sun at the very meridian is in the horizon; there wholly eclipsed, here grazing the edge with the extreme margin of the Moon.
Which parts of the Earth perceive the Sun obscured first, which later, either wholly or in part? The shadow of the Moon always invades the Earth from the more western parts and leaves it at the more eastern. And so if by fiction the eye be placed in the sphere of the Moon or in the Moon, and the north be upward, the face being turned to the Earth: this eclipse of the Earth will seem to begin from the left and end at the right, and so it is also painted in our diagrams.
This, then, is the order of appearance: the earlier in time the eclipse of every kind occupies those parts of the Earth M which, in respect of those which last feel the defect, as O, are western, but in respect of those which a little afterward will see the whole Sun covered, as R, are eastern. For the later total obscuration (if it can become total) touches the more western parts of the Earth, R, which from the beginning of the duration of every kind have meanwhile insinuated themselves into the disc. Conversely the total obscuration by a continuous way runs out farthest into the eastern parts of the Earth S; but while the eclipse of every kind is extended further in respect of time, the parts of the Earth O, more western than the S just mentioned, having gained space for following the full shadow, fall into the penumbra, the eastern parts S having now slipped out of the illuminated disc and been received into the other, nocturnal, hemisphere. So the total duration RS, of shorter time, runs out through the longest tracts of the Earth from west R to east S of the Earth; but of the duration of every kind BE, which is of longer time, the extreme limits of the Earth M, O are confined within shorter spaces.
Why is each semidiameter of the penumbra, both that which projects on this side of the umbra and that beyond it, divided into any twelve parts, twice-twelve parallels to the way of the Moon or shadow being drawn through the whole disc? Because the diameter of the Sun, like an as, is wont to be divided into twelve ounces or digits: wherefore, parallels to the lunar way of the shadow being drawn through the points of division of the semidiameters of the penumbra, whatever places of the Earth at any moment fall upon one of those parallels, in those places the Sun will be seen to fail by as many digits as is the number of the parallel from the outermost one touching the penumbra, exclusive. As in the places of the Earth which fall on MO at the time of the duration, the defect of the Sun reaches 7 digits, because MO is the seventh from HIK. For the outermost, HIK, touching the penumbra, is the index of no defect, but rather the limit from which the defect begins to be felt. And the northern parallels argue a southern defect in the Sun, the southern on the contrary a northern. It will be more evident thus: as many as the number of any parallel from the way of the full umbra RS, the middle one among the parallels, toward either quarter, just so many lucid digits remain from that quarter of the solar body. As, because M is the fifth after RS toward the south, five digits remain in the Sun from the south in all the places marked from M.
Define the limits of eclipses of the Sun, or rather of the Earth, both total and partial. Let the luminaries be in apogee. Since, therefore, the semidiameters of the disc and of the penumbra equal the parallax of the Moon from the Sun and the semidiameters of the luminaries joined, the sum of the semidiameters of the disc and of the penumbra will be 87′ 23″. But as the parallax of the Moon 58′ 22″ to the parallax of the Moon from the Sun 57′ 23″, so this latitudinal arc between the centres of the disc and of the umbra of the Moon, 87′ 23″, to the latitudinal arc of the centre of the Moon, 86′ 5″. To this latitude (for it differs insensibly) there answers a distance from the node of 15° 43′ for the limit of the partial eclipses. In the perigee of the Sun 6 minutes will be added. And since the semidiameter of the disc is in this situation of the luminaries 56′ 54″, to which answers the latitudinal arc of the Moon 56′ 0″, to this therefore the distance from the node answers of 10° 0′ for the limit of the total eclipses of the Sun through the whole Earth. In the perigee of the Sun the difference is almost nothing.
But let the Moon be in perigee, the Sun in apogee, the first sum will be 94′ 4″, which in the orb of the Moon will become 92′ 37″, which a distance from the node of 16° 57′ represents for the limit of the partial; in the perigee of the Sun 17° 3′.
And since in this situation the semidiameter of the disc is 62′ 12″, to which answers the latitudinal arc of the Moon 61′ 14″, to this therefore answers a distance from the node of 11° 7′ for the limit of the total eclipses of the Sun. Nor much otherwise in the perigee of the Sun.
What follows from the comparison of the limits of solar with those of lunar eclipses about the number of both kinds of eclipses? On the Earth the partial eclipses of the Sun are more numerous, but the total fewer, than the lunar of every kind. But if you compare the total eclipses of the Moon, more often the Moon takes the whole Sun away from the Earth than the Earth takes the whole away from the Moon. For the body of the Earth, broader than the body of the Moon, receives more strokes from the shadow of the Moon than the Moon from the shadow of the Earth. Finally the number of the total eclipses of the Sun through the whole Earth, and of the Moon of every kind, is very nearly the same.
Tell the rules of the number of eclipses in a year, both of the Sun, through the whole Earth, and of the Moon. 1. When the full moon is central or nearest the node, that is when the centre of the Moon passes through the centre of the shadow of the Earth, or very near it, the new moons most nearly surrounding it are sometimes both defective, with a least defect, very rarely and perhaps never neither; namely if the Sun is before and after perigee and the Moon at the full is in apogee.
2. But when the new moon is nearly central, that is when the centre of the penumbra of the Moon passes through the centre of the disc of the Earth or very near, the surrounding full moons pass without eclipse, and conversely.
3. It may happen that two central or nearly central new moons fall at the interval of one half-year, one before the apogee of the Sun, the other after it, in which case in the whole lunar year no ecliptic full moon will fall.
4. Every ecliptic full moon not nearly central has a neighbouring ecliptic new moon.
5. If in one half of the year an eclipse of the Moon, central or nearly so, happens, the opposite half of the year cannot lack its eclipse of the Moon.
6. In nearly all half-years eclipses of the Sun happen, oftener two in succession, very rarely and perhaps never none.
7. In nearly all simple lunar years (of 354 days, namely) either two or three eclipses of the Sun happen, but of the Moon either none or one or two, and so in all either two, or three, or four, or five eclipses.
8. Sometimes a sixth comes, but in a year not lunar but solar, of 365 days, namely when the eclipses happen about the beginning of the year.
But far fewer eclipses are seen, and of the Sun much fewer than of the Moon. That in any one certain place of the Earth fewer eclipses of the Moon than has been said, and much fewer of the Sun, occur, the causes are in the lunar one, in the solar two. For first the horizon of every place bisects the heaven, and so conceals half of the eclipses, both lunar and solar, so that those lunar ones which occur in the hours of day, the solar in those of night, cannot be seen. Then, as to the solar, many of those which happen even in the daytime hours of one place nevertheless occur in other climates, because the parallax of the Moon is different in different climates, or, what comes to the same, because the disc of the Earth AN is greater than the circle of the penumbra CV, nor is it ever wholly covered by the penumbra of the Moon. For no eclipse of the Sun is seen outside the bounds of the penumbra, but an eclipse of the Moon is seen at any moment from the whole hemisphere of the Earth.
If now the eclipse of the Sun be considered in respect of one place of the Earth, what new comes to be considered besides what has been so far? These three: 1) the difference between true and visible conjunction, 2) the reduction of the conjunction to the circle of latitude itself, at right angles to the ecliptic, 3) the drawing out of the horizontal parallax of the Moon from the Sun in length, breadth and height.
What do you call the true, what the visible conjunction of the luminaries, or place of the Moon? Hitherto indeed the true place of the star and the apparent place, that is the visible, were the same. For the true place was so called in respect of the mean, which is feigned, the apparent because the line of the true motion was drawn from the centre of the Earth, from which the sight on the surface of the Earth, in respect of the immense interval of the planets, was not distant by any sensible interval. The place of the star, therefore, did not sensibly appear other to sight set on the surface than the true place of the line drawn from the centre of the Earth. But now the nearness of the Moon to the Earth is so great that we cannot disregard the distance of sight set on the surface of the Earth from the centre of the Earth. The true place, therefore, is shown, as hitherto, by the straight line drawn from the centre of the Earth through the centre of the star to the sphere of the fixed stars, the apparent or visible by the straight line drawn from the proposed place of the surface through the centre of the star to the surface of the fixed stars. The difference of both places, the true namely and the apparent or visible, is called parallax of altitude. When, therefore, the line of the true place of the Moon meets the true place of the Sun, the conjunction will be true; when the line of the seen place of the Moon, visible.
What, then, is the parallax of altitude? It is the arc of the vertical circle drawn through the true place of the centre of the body of the Moon, intercepted between this and the visible place.
In diagram 96 let V be the zenith, VO the vertical or azimuthal circle, and in it R the true place of the Moon, S the visible: RS will be the parallax of altitude.
What are the parallax of longitude and the parallax of latitude? They are arcs, the former of the parallel of the ecliptic drawn through the visible place, the latter of the circle of latitude, the former intercepted between the circles of latitude of the true and of the seen or apparent place, the latter between the true place of the Moon and the parallel of the ecliptic drawn through the visible place.
As if the pole of the ecliptic be P, and SI a parallel of the ecliptic, nearest to that, or even the ecliptic itself, passing through S the seen place, PR, PS circles of latitude, that through the true place R, this through the visible S, SI will be the parallax of longitude between PRI and PS, but RI will be the parallax of latitude between R and SI.