Epitome Astronomiae Copernicanae: Liber VI, part 5
Part IV. Of the Moon
What is the arrangement of the Moon’s orb, what is its proportion to the other orbs, and what is the principle of its motion? The heaven of the Moon, if you conceive solid orbs in the mind, is set into the great orb like a star, or rather like some epicycle, and holds the body of the Earth in its midst, and is carried round the Sun with one motion common to it and the Earth, changing place after place. See diagram No. 72, and the indubitable example in the satellites of Jupiter. The Moon herself meanwhile circles the Earth in the space of a month, marking out an orbit eccentric to the Earth’s centre, and that upon a plane which is imagined to be drawn out from the centre of the Earth as far as the zodiac, so that it is attached to a line drawn through the centre of the Earth and through some point of the zodiac. The greatest distance of the Moon from the Earth is the 59th part of the greatest distance of the Sun from the same Earth, and contains as many, namely 59, semidiameters of the globe of the Earth. See book IV, from page 327 to 328. The apparent diameter of the globe of the Moon is the 720th part of that circle in which the Moon circles the Earth, and equals in angle of vision the diameter of the body of the Sun when farthest from the Earth. These things were established above in book IV, from page 325, and the observations agree with them. See the Optical Part of Astronomy, chapter XI (II. 340 ff.).
In what way and in what number do the inequalities of the Moon correspond to the inequalities of the others? Whereas the primary planets appear to advance with two inequalities mingled together, of which the first is proper and real in each of them singly, and the second happens to all five in common from without, from the conditions of sight, that is, on account of the great orb, in the Moon, on the other hand, that first and real inequality of motions is not one but threefold, while the second, the accidental or apparent one, is none at all in her. For although the whole heaven of the Moon shares in the circuit of the great orb, which is for the five planets the cause of the second inequality, as has been said, yet this heaven of the Moon and the Earth, the dwelling-place of our eyes, are carried along together; and so no diversity from this motion, true as it is, overflows into the observations of the Moon; she is always seen to advance directly, never to stand still, never to go backward, as was explained in part II of this sixth book, and therefore this motion of the Moon (for the sake of lunar astronomy) is reckoned as mere rest. Yet this lack of an accidental inequality is made up, first by the tripling, already mentioned, of the true inequality; second by the equation of time, explained in the doctrine of the sphere on pages 251 and 252 and in part I of this sixth book; third by the diurnal parallax (of which we shall treat in the doctrine of eclipses), which is also an accident of sight. For while in the other planets, on account of their immense distances, it cannot be perceived, in the Moon, as a near body, it is easily sensible.
Of the simple inequality of the Moon.
By what names and by what difference of things are the three said real inequalities of the Moon distinguished from one another? In respect of the first of them the motion of the Moon is similar to the motions of the primary planets, explained in book V; in respect of the second and third it is dissimilar. The first is in a manner its own master, observing its own proper period; the other two are bound to the configuration of the three bodies, Sun, Moon and Earth, or to the apparent meetings of the Sun and Moon. The first, therefore, is called periodic, the others synodic; the first simple, the others monthly, that is, bound to the phases of the months.
On what occasions is the motion of the Moon in longitude made threefold unequal? The Moon is carried round the Earth by two powers: first, by the species which emanates from the body of the Earth set in rotation; second, by the force of the light of the Sun, as was argued in book IV, page 359. Of these causes, although the latter degenerates into the conditions of the former, since it is nothing else than a strengthening of it, as on pages 360 and 365, yet its effect is distinguished by an express quantity from that of the former: for whenever the Moon completes a quarter of her orbit from the conjunction, that is 90°, these 90° of longitude are to be shared between the two moving causes mentioned, and 87° 51′ are to be credited to the Earth, but the remaining 2° 9′ to the light of the Sun, as we shall hear below in the explanation of the variation. But the species of the Earth, so far as in it lies, exerts its force equally, whereas the effect of the light is dispensed unequally, for the causes stated on page 364. This, then, is one inequality, arising from the moving cause itself in longitude, which will be treated below in the last place and will be called the variation. There now supervenes upon both moving causes the eccentricity of the Moon, making the effects of each motive cause unequal of themselves also. And behold, three inequalities, each of which will now be explained in its own order from this place.
How could so many real inequalities and their circuits be distinguished from one another by observation? The monthly inequalities are bound to the Sun and its opposite, or to the quadrature places, so that they begin from these points and end in them; astronomers can therefore easily lie in wait for the simple inequality, when it stands alone, at the very moments of conjunctions, oppositions and quadratures. But once the simple inequality has been explored, the place of the Moon is computed according to it for any intermediate moment of the month and compared with the observed place; so from the difference between the two the quantity of the monthly inequalities also becomes manifest.
How can the simple inequality be observed even at the conjunction of the Moon with the Sun, when the Moon then lies hidden under the Sun’s rays? Although for the most part the Moon lies hidden under the Sun, yet when she covers the Sun she is then seen most of all and most readily, namely on the very disc of the solar body.
In what respect does the simple inequality of the Moon’s motion agree with the inequality of the primary planets called the first, and in what respect does it differ? 1. The same tract, namely of the zodiac in the sphere of the fixed stars, overhangs the orbit of the Moon as well as the other six orbits of the primary planets. 2. As the primary planets and the Earth herself circle the body of the Sun in orbits eccentric to the Sun and with a speed adapted to the variable intervals from the Sun, so the Moon, in her own proper motion, circles the globe of the Earth in an orbit eccentric to the Earth and with a speed adapted to the variable intervals of the Moon and the Earth.
What circles are needed to teach the simple inequality of the Moon? A single eccentric orbit, of a figure very nearly circular, that is, elliptical, inclined to the plane of the ecliptic upon a line which passes through the centre of the Earth, such orbits as the primary planets also have used.
How are the lines of the apsides, limits and nodes, and the motion of the Moon and the places of each, to be described according to Copernicus? Copernicus did not assume a sensible distance between the centre of the Earth and the centre of the zodiac, compared with the immense height of the fixed stars; therefore these lines are simply understood to be drawn from the centre of the Earth through the points of the apsis, limit, node, or any points whatever in the orbit of the Moon, as far as the fixed stars, and there to mark the places of the said points or of the Moon. But because in hand instruments of the theories that immensity of the zodiac cannot be expressed, these lines (no less in the Moon than in part II of this sixth book in the other planets) are to be defined technically in this way, that they be drawn from the centre of the solar body, as from the centre of the fixed stars, and proceed parallel to the lines already described. For by this parallelism it comes about that, although the ecliptic or zodiac in the hand theory is not made much larger than the orb of Saturn, nevertheless these lines drawn from the Sun coincide with those previously defined in the same places of the zodiac.
How great is the period of time within which the Moon completes and finishes the simple inequality of her orbit? The centre of the body of the Moon is carried round the centre of the body of the Earth by a motion truly unequal in its parts, in the order of the signs, and having departed from the line of the apsides returns to it when the circuit is complete in 27 days 13 hours 18′ 35″ in mean space; for if we look to the true moments at which the Moon falls upon the apsides, this time, on account of the monthly inequalities, does not remain wholly unvaried in all periods.
The mean diurnal motion of the simple or periodic anomaly is therefore 13° 3′ 54″, the hourly 32′ 40″, and it is numbered either in the equant circle, as if this were described from F (Fig. 77), or more properly in the segments PAH, PAE of the plane PRE which is enclosed by the orbit PHER, as was explained in book V.
How great is the eccentricity of this orbit, how great the greatest equation, how great the variety of the hourly motions? The eccentricity of the Moon AB is 4362, of such parts as the semidiameter BP is 100000; wherefore the breadth ED of the elliptical lune, after the pattern of the others, is 190 parts. The greatest equation, therefore, composed (as declared in book V) of its two elements, the physical, of the area EBA (or the equivalent DBA), and the optical, of the angle BEA, is 5° 0′, so great being it then when the quadratures fall in the apsides, as conversely the greatest equation occurs in the conjunctions. Hence the hourly motion is made, at the least 29′ 58″, at the greatest 35′ 42″, if this inequality alone were ever present on any day of the lunation, the Moon being at the same time either farthest from or nearest to the Earth.
Is this magnitude of the daily motions, the greatest and the least, therefore inconstant? In the conjunctions it is for the most part greater and larger or swifter, in the quadratures smaller or slower; in the octants of the month the quantity is also disturbed by the monthly equation, as will be taught later.
What inequality of the parts of the period follows from this simple equation? A conjunction and opposition nearest one another, when each is ecliptic, are observed to stand at unequal distances from one another; for there may be more than 15½ days between them, or not more than 14.
What name have the apsides of the eccentric of the Moon, the highest P and the lowest B? What we called aphelion and perihelion in the primary planets, which are carried round the Sun, that in the Moon, which revolves round the Earth, is to be called apogee and perigee.
How is the apogee of this eccentric orbit of the Moon moved, and how great is its period of time? It is moved in the order of the signs with an equable motion and is restored to the same point of longitude of the zodiac in 8 Egyptian years, 311 days, 6 hours. By this motion B, the centre of the Moon’s eccentric, upon the plane of the Moon’s orbit (if we think of it as attached to a line drawn from the centre of the Earth to some one of the fixed stars), describes, with respect to longitude, a little circle BERY (Fig. 89), concentric about the centre of the Earth A. The diurnal motion of the apogee under the zodiac is 6′.
How do the latitudes of the points of this orbit stand to the inclinations? Since in the Moon the parallax of the orb, which befalls the primary planets, has no place, the angles of inclination at the centre of the Earth of the points of the eccentric are the same as the angles of apparent latitude of the Moon when set in those points. In the Moon, therefore, the word inclination is used only of the limits, never of the intermediate points.
How great is this inclination of the eccentric orbit, or the greatest latitude of the Moon when set in its limits? The greatest inclination of the limits is as great as the greatest equation, composed of the optical and physical parts, of the periodic or simple inequality just set forth, namely 5 degrees; and that only when both the limit and the mean longitude stand in the conjunctions. For apart from the conjunctions each becomes greater, both the greatest latitude of the limit and the greatest equation; but then each by its own quantity, so that, considered thus apart from the conjunctions, they no longer remain equal to one another.
If the latitude or inclination of the limits to the plane of the ecliptic is not always the same, will not the circle standing over the Moon’s orbit under the fixed stars become tortuous? Throughout that whole month in which the limits remain in the conjunctions (so far as they do remain), all the latitudes of the Moon are arranged under very nearly the same great circle; so that the angle by which the plane of the orbit is inclined to the plane of the ecliptic remains very nearly constant throughout that month. But as the limits depart from the conjunctions this angle of inclination is increased, and then the latitudes of the Moon through a whole month agree less and less with one great circle of the sphere, until the limits approach the quadratures. Then, if the limits are in the quadratures themselves and so the nodes in the conjunctions, the orbit of the Moon again agrees very nearly with a great circle, but a more inclined one. So arises this slow, half-yearly libration, of which below; but within a single monthly circuit of the Moon it is scarcely felt.
How are the nodes and limits of the Moon peculiarly named? Ptolemy gave them the same names in the Moon as in the other primary planets. But the Arabs call the nodes of the Moon, the ascending one the head of the dragon, the descending the tail, and the limit the belly of the dragon; they seem to have meant of the serpent, on account of the shape or figure of the space on the surface of the sphere of the fixed stars, intercepted between the ecliptic and the circle standing over the lunar orbit; for this space begins from a point, as from the snout of a serpent, and is not broader in the middle than 5°, though it is 180° long, and ends in another point, as in the tail of a serpent. Look at the diagram in book IV, No. 71, but imagine it to be on the spherical surface.
How are the nodes situated in the Moon? As in the other primary planets both nodes are situated in the same straight line with the centre of the Sun, so in the theory of the Moon the same nodes fall in the same straight line with the centre of the Earth.
Of what kind and how great is the motion of the nodes or limits? The lines of the nodes under the ecliptic, or of the limits under the parallels of the ecliptic, are moved contrary to the order of the signs and are restored to the same point of longitude of the zodiac in 18 Egyptian years, 228 days, 3 hours, 50′; in almost as many years as the node completes degrees in a single year, because 19 times 19 make 361. The mean diurnal motion is therefore 3′ 10″ 38‴, to be reckoned backward under the ecliptic. This motion arrives at the conjunctions and quadratures at equal intervals of time, but in the places intermediate between conjunctions and quadratures another motion is mingled with it, by which mixture it becomes unequal, of which below.
What is the argument or anomaly of the motion in latitude of the Moon, and how great is its period? This anomaly is the arc of the ecliptic intercepted between the place of the ascending node (or, for Copernicus, also the place of the northern limit) and the true eccentric place of the centre of the body of the Moon, reduced to the ecliptic. It is completed in 27 days 5 hours 5′ 36″. The mean diurnal motion of the anomaly of latitude is therefore 13° 13′, the hourly 33′ 3″.
Why are these motions, the eccentricity and the inclinations, thus delivered separately, when other motions, whose circuit is monthly, are mingled with them? Because these almost suffice for the doctrine of the months and of eclipses, the chief and very conspicuous accident of the revolution of the luminaries, of which below in part V.
How many kinds of month do astronomers recognise? Two, the periodic and the synodic; the former word signifies a circuit, that is, under the fixed stars or the zodiac; the latter a coming together or meeting, that is, with the Sun.
What is the periodic month, and what other name has it? It is the space of time in which the line of the Moon’s motion, having departed from the same circle of latitude, fixed immovably to a certain point of the ecliptic, returns to it when the circuit is completed. It may be called the critical month of the physicians. It is, by equable measure, 27 days 7 hours 43′ 5″ 8‴, a little shorter than the period of the simple anomaly. The daily motion is 13° 10′ 35″, the mean hourly 32′ 56″ 30‴. But by the simple anomaly alone it is slowest 30′ 15″, swiftest 36′ 0″.
What is the name of these hourly motions, and what is their use? They are called in the tables fictitious, from the thing itself, and we use them, compared with the true hourly motion of the Sun, to find out briefly the moments of the conjunctions and quadratures.
What is the synodic month, and how long is it? It is the space of time within which the line of the Moon’s motion, having departed from the circle of latitude in which is the line of the Sun’s motion (or of its opposite point), returns to the same when the circuit is completed. It is also called a lunation, because within this time the orb of the Moon is successively filled with light and in turn emptied. One mean period is completed in 29 days 12 hours 44′ 3″ 11‴; the Moon is separated from the Sun daily by an angle of 12° 11′ 27″, in an hour 30′ 26″ 37‴ 12⁗ at the mean rate. Therefore in a year there are 12 lunations and more than 132° 45′ of a thirteenth.
What makes the synodic months unequal, and by how much? The inequality of the motion of each star, the Sun as well as the Moon. For by reason of the Sun, in summer, when its motion is slow about its apogee, the months come out shorter, because the Moon overtakes the Sun sooner; in winter about the Sun’s perigee the months are longer, because the Moon overtakes the swift Sun more slowly. Again by reason of the Moon, since she is slow in her apogee and swift in her perigee, she completes more slowly there than here, other things equal, that remainder which is left to her to reach the Sun beyond the anomalistic return already completed. When the causes are put together, therefore, when the Sun is in perigee and the Moon in apogee, the month is almost 30 whole days, for only 4 hours 23′ are lacking. Conversely when the Sun is in apogee and the Moon in perigee, the month has only 29 days 6 hours 42′. But if we count from quadrature to quadrature, the inequality of the months can be greater. The luminaries in apogee make a month 29 days 15 hours 7′ long, those in perigee 29 days 10 hours 24′ long.
Of the monthly inequalities.
By what names do you distinguish the two remaining monthly inequalities? Let the first be called the temporary, the second the perpetual. For the first is not called monthly because it returns every month, but because it arises only from the illumination which makes the month; it does not therefore remain of the same quantity in all lunations, but gradually vanishes in the following months; the other keeps the same quantity, constant through all lunations, and so is called monthly by a double title, both because it comes from the illumination which makes the month, and because it returns the same every month.
In what order are they to be delivered? Although, as we argued in book IV, page 364, the perpetual one is by nature prior and akin to the mean motion of the Moon and the cause of the temporary, and on that account we explained its natural causes in the first place, from page 364 to 368, but the causes of the temporary later, from page 388 to 393, yet now in astronomical practice we must begin with the temporary, because this, in so far as it has in any month its certain measured amount, is most like the simple anomaly and concerns, as that does, both longitude and latitude. But the perpetual is of another kind and varies longitude alone; therefore Brahe’s calculus also applies it last.
How is the synodic month divided on account of the monthly inequalities? It is divided into two halves, one of the waxing Moon, GKC (Fig. 66), the other of the waning, CFG, whose middles the quadratures IK, EF hold, and whose ends are the conjunctions CD, GH. Further, other phases L, M, N, O bisect these four quadrants; Brahe called them octants, because by them the month is divided into eight parts.
Does the word month or lunation in this place take on some technical meaning, and what is it? Entirely: for although the apsides and nodes depart from the conjunctions and quadratures with a continuous motion, so that no part of the month is like another part in the disposition of the apogee or node to the phases, yet we suppose that this disposition remains throughout the month such as it is found at that moment for which the place of the Moon is to be examined.
What then do you call octants, what quadratures and what conjunctions? Generally and technically, at any proposed moment or place of the Moon, they are those points of the orbit at which, if the Moon were then there, she would be configured with the Sun in an octile, or quadrate, or trioctile aspect, or be conjoined with it from the same or the opposite quarter.
As in diagram 89, in which A is the Earth, B the centre of the eccentric D: if the Sun is in the line AH and the Moon in D or N or any other point of the orbit, then the points H and G take the place of the conjunctions, I and K that of the quadratures, and Ij that of the octant.
Specially and more rarely, the places themselves of the true phases of the Moon, each of which appears at its own moment, come under these names. For the Moon in the conjunction farther from the Sun appears full, in the octants next to it gibbous, in the quadratures halved, in the octants nearer the Sun sickle-shaped or horned, in Greek a crescent, in the nearer conjunction she is extinguished. But then only by abuse of the word can the intervening arcs be called quadrants, octants or semicircles, since in the ecliptic they are a little smaller than these names imply, but in the eccentric orbit of the Moon larger by about a twelfth part, as will appear from what follows.
Of the temporary inequality.
In what situation of the Moon does the temporary inequality appear, and what kind of appearances does it make, and in what order? In every situation indeed outside the conjunctions, but chiefly in the quadratures and in the octants. For when the apogee or node is in the quadratures, the equations or latitudes come out simple throughout the month, as was said of the simple inequality. This month might be called empty in respect of the monthly equation or latitude. In the next month, when the Sun’s quadrature has left the apogee or node of the Moon, so that those points are now toward the octant, there now come out equations, or latitudes, some little monthly ones also, and that nevertheless, even if the Moon were in her very apogee by equable motion or in the very node, where by the laws of the simple inequality she ought to lack all equation or latitude. Again, in succeeding months these inequalities come out greater, until the conjunctions have overtaken the apogee or node; in such a month the greatest monthly equations or latitudes occur and join themselves to those from the simple inequality, so that both become greatest at the same moments: and so throughout almost the whole month the equation or latitude, heaped up from both parts, comes out regular and by the usual laws. This might be called the full month, and that by a double title, if these affections of longitude and of latitude coincided in one month. In the following months this monthly inequality decreases again by the same degrees until it is wholly extinguished, from which time a new one arises through the contrary configurations of the Sun. And in the contrary quadratures or conjunctions the affections of the equations or latitudes are also contrary. For if the half of the waning Moon shall have intercepted that mean longitude of the simple anomaly of which the equation is subtractive, or that limit which raises the Moon farthest to the north, then also the monthly equations throughout that whole half of the waning are subtractive, or the monthly latitudes throughout that whole half are northern, even in those points of the semicircle in which the equation of the simple inequality is additive, or the latitude of the simple inequality southern; and the opposite holds in the semicircle of the waxing. But after some months, when the former mean longitude of the simple inequality or the northern limit has departed from the half of the waning and the other half, of the waxing, has begun to claim for itself these points of the simple inequality, either both or one alone, the subtractive affection or northern character, either both or one, passes into this other half, namely into all the equations or latitudes of the whole of it, called monthly; the contrary affections, additive or northern, into the former. These things, thus brought together under one view, will now be unfolded more distinctly.
What apparatus of circles do we need to demonstrate this temporary monthly inequality and set it before the eyes? Although we must use neither a new real eccentricity, as it were in an eccentric of the eccentric, as was guarded against in book IV, page 389, nor altogether new circles beyond those which we used in the demonstration of the simple inequality and in all the primary planets also, yet truly and really the Moon is doubly accelerated or retarded in that one and the same orbit of hers; wherefore we must also conceive two sources of the Moon’s motion, from which the Moon in the upper semicircles of the orbit, which have the apogean conjunction, runs out farther, in the lower more briefly, and in respect of those sources a double eccentricity also of one and the same centre of the single orbit of the Moon, a double apogee, two lines of the apsides, and consequently also two equatorial triangles, in whose areas lie the measures of the retardations and accelerations. So also, for the cause of latitude, the Moon truly runs out doubly to the sides of the ecliptic, wherefore we must conceive one and the same orbit of the Moon to be librating to the plane of the ecliptic, that is, the inclination of the planes, which was simple in the simple inequality and is such in all conjunctions, we must conceive variable outside the conjunctions, and thence double limits and nodes.
What then is this other, monthly line of the apsides, line of the nodes, what is the apogee, which the monthly ascending node, and of what kind finally is their motion? In the primary planets indeed, as in the simple anomalies of the Moon, the line of the apsides and the line of the nodes were separated from one another by contrary motions through whole quadrants in succession; but here, in the monthly anomalies of the Moon, the line of the monthly apsides and the line of the monthly nodes perpetually coincide in the section of the plane of the Moon’s orbit with the plane through the centres of the Sun and Earth, at right angles to the plane of the ecliptic. For the apogee is perpetually at one or other of the conjunctions, the ascending node alternately either in the same conjunction or in the opposite, nor do they go out from thence, passing by circular motion into the opposite conjunctions, but in its own conjunction each both arises and dies, and again, after perishing in one, rises in the other. Each is excited or extinguished by the circular departures of the simple apsides and nodes from one monthly semicircle into another, that is, from the waxing Moon into the semicircle of the waning, which departure takes place contrary to the order of the signs, the phases of the Moon, on account of the common motion of the Earth and the heaven of the Moon, being transposed in the order of the signs; as, for example, the full moon in diagram 72 transposed from B to C, by which transposition the slowest apogee of the Moon and the node of the Moon are left behind, and moreover are retrograde: so that the places of both present the appearance of receding from the phases in the contrary order, which apparent recession unrolls the whole circle of lunations in about one year. These things, indicated here in general, will be explained more clearly in what follows by geometrical signs.
Since, then, the line of the conjunctions itself serves the office both of the line of the apsides and of the line of the nodes, each monthly, it comes about that we do not always use even the names of the monthly apogee or monthly node, that there may be so much the less confusion.
That conjunction takes the place of the monthly apogee to which the apogee of the simple inequality is nearer within a quadrant of the ecliptic, before or behind. Thus, because in diagram 89 D is the apogee of the simple inequality, HAG the line of the conjunctions, HAD less than a right angle, DAG greater than a right angle, therefore the conjunction H, which is nearer to the apogee D itself, takes the place of the apogee, G of the monthly perigee. So that conjunction provides the place of the ascending node from which the ascending node of the simple inequality is distant by less than a quadrant, before or behind.
In what order does one come to the knowledge of the monthly equation or latitude, and by the use of what terms? The sum of the process and the list of terms is this: at the beginning, by the distance of the Sun from the apogee of the Moon, the monthly argument and the proportional minutes of longitude are sought, with the odd particle. So, with the distance of the Sun from the ascending node, the minutes of latitude are sought. Then by the monthly argument of longitude or latitude there is to be taken out from the table there the equation, here the monthly latitude, as from a full month. This equation or latitude, multiplied into its minutes, gives the proportional portions, of latitude indeed the absolute portion, but of the equation one still to be leavened with the odd particle, that the just monthly equation may result. The definitions and uses of these terms will now be delivered singly and separately, more clearly.
How are those distances of the Sun from the apogee or node of the Moon numbered? They are numbered in the ecliptic, or also in any circle described from the centre of the Earth; they are arcs either of the ecliptic or of such a circle, beginning from the line or place of the apogee or ascending node, and proceed in the order of the signs and end in the true place of the Sun, or the line of the true place of the Sun. And since the ecliptic cannot conveniently be expressed in diagrams, it is enough, instead of the arcs, to show the angles at the Earth, or the complements of the angles to four right angles, which angles those arcs measure.
Therefore in the added diagram, in which DBE is the line of the apsides, D of the apogee, F of the perigee, HAG the line of the true place of the Sun and of its opposite, the angle DAG is the complement to four right angles of the distance of the Sun, marked by the line AG, from the apogee of the Moon D, understood by the line AD.
So also it is, if D were the ascending node.
The distance of the Sun from the apogee of the Moon can also be numbered in the eccentric circle of the Moon: and then it will be determined thus, that it begin from the point of the apogee, be numbered in the order of the signs and end in the straight line which is drawn from the centre of the eccentric parallel to the line of the true motion of the Sun, toward the same quarter in which the Sun stands.
As here, the Sun being in the line AG, if the parallel BQ be drawn to it from B, then DNQ will be the distance of the Sun from the apogee of the Moon.
Teach how to find the monthly equatorial point (and with it the source of motion) and its eccentricity. From the centre of the Moon’s orbit B let a perpendicular BC be drawn to the line of the conjunctions HAG, cutting the line of the conjunctions AH in C. Let there also be drawn through A the line IAK parallel to CB, cutting the parallel to HG through B in the point Z. There are therefore two sources of the Moon’s motion, the point A and the line IAK through the circle of illumination of the Earth. There are also this time two eccentricities of the centre B, namely AB and ZB, the former the ordinary and perpetual one of the simple inequality, ZB however only in the present month or moment, variable through the remaining months, and equal to it is AC, the temporary (rather the momentary) monthly eccentricity of the monthly equatorial point C.
If the centre of the orbit B is in E, that is, in the line of the conjunctions itself, then in that technical month the point E takes the place of the two points B and C, and AE is applied twice for accelerating or retarding the motion.
But if the perpendicular BC falls upon the centre of the Earth A itself, then in that (technical) month there is no temporary monthly eccentricity, except in so far as that perpendicular does not fall on A throughout the whole natural month, since this is momentary. Let the apogee of the Moon D be distant from the Sun’s opposite by 36° 0′. Antilogarithm 21193.
The eccentricity AB is 4362, taken as the number 43620. Logarithm 82965; the quantity sought as a logarithm shows AC, the eccentricity of the point C, to be 3529.
What are the monthly minutes of longitude? They are nothing else than the value of the area of the monthly equatorial triangle on the monthly eccentricity of the centre of the eccentric, standing in any technical month, the greatest of all being, in numbers of such parts, in the full month, 60′.
In the diagram, BC being continued to the points of the orbit O, N, and BZ let fall parallel and equal to CA, and the points N, O being joined with Z, the value of the area ZBN or ZBO gives the minutes of longitude, which hold whenever in any month that disposition of the points A, C, B occurs, or that angle DAH between the line of the apsides AD and the line of the place of the Sun AH, in whatever point of her orbit the Moon is found.
Teach how to compute this triangle and from it the monthly minutes. The manner of computing remains the same as was in book V, in the part on the greatest equation of the physical kind, except that the area produced is converted into minutes, of which in the full month it itself makes 60′.
As if the monthly eccentricity BZ of the centre B be 3529, half of this, 176472, multiplied into the semidiameter BO, creates the area BZO 176450000. But in the full month, when B is in E and the monthly eccentricity AE 4362, this area becomes 218100000. If then 218100000 is worth 60′, the present area 176450000 will be worth 48′ 33″, the monthly minutes for this technical month.
It is to be noted here that if we insist most accurately on the figure of the orbit itself, considering that it is elliptical after the example of the other planets, then the semidiameters BO, BN are a little shorter than the semidiameters BD, BF, and that the more, the straighter DBQ is, about the mean longitudes. Nor are there lacking foundations for computing these shortened semidiameters from book V, either geometrically from the diagram, or briefly and near the truth from the rule there delivered. But since there is no need of such subtlety, it is better to omit it than to render the apprehension of the newly formed equations still more difficult. The greatest of all the shortening, or breadth of the lune, drawn into the total eccentricity and turned into astronomical denomination, does not make 7 seconds, which drawn into 2° 30′, the greatest monthly equation, makes 17″, nothing more. This happens when B falls in E. But the farther B departs from E, the smaller always is this little error, by a double title, both because the shortening is less and because the monthly eccentricity is less.
What is the monthly argument of longitude? It is the arc of the eccentric of the Moon, begun from the parallel of the line of the monthly apogee (that is, either of the place of the Sun or of its opposite) drawn toward the same parts and numbered in the order of the signs as far as the line of the place of the Moon first equated, or affected by the solitary equation of the simple inequality.