Epitome Astronomiae Copernicanae: Liber VI, part 6
How is the monthly argument found? The distance of the monthly apogee (that is, either of the place of the Sun or of its opposite) from the apogee of the Moon of the simple anomaly is to be subtracted from the anomaly of the eccentric, a whole circle being added to it if need be; so the monthly argument remains.
In diagram 89 let AH be the line of the place of the Sun, AD the line of the apogee of the simple inequality, HAD less than a right angle, and therefore AH the line of the monthly apogee, and parallel to it BP toward the same parts HP, and HAD the distance of the Sun, as of the monthly apogee, from the apogee of the Moon. Let AL also be the line of the motion of the Moon first equated, and DL or DBL the anomaly of the eccentric. Take away HAD, that is PBD, from LBD, and there will remain LBP, the monthly argument. Let AM be the line of the motion of the Moon, DBM the anomaly of the eccentric; PBM will be the monthly argument.
By what are the monthly equations of the Moon, belonging to the proposed places of the Moon, represented and measured? By the areas of the triangles standing upon a base, the eccentricity of the monthly point, with the vertex in the proposed point of the Moon’s orbit. These areas are this time reduced to degrees and minutes, of which the area of the whole orbit is worth 360 degrees.
As, since C is the monthly point and CA its eccentricity from the centre of the Earth A, if at some moment there is such a disposition, the Moon being in L, then the area of the triangle CLA measures the monthly equation. But if at another time the same disposition has returned, that is the same technical month, the Moon then being in M, again the area CAM gives the measure of the monthly equation.
How is the area of the triangle computed whose base is the eccentricity CA of the monthly point C? Each of these triangles has a companion, whose vertex is the same, but whose base is BZ, the monthly eccentricity of the centre of the eccentric B; the difference between the two is the odd particle. First, therefore, these companion triangles and the odd particle are to be computed; then the latter added to the former makes up the areas of the equatorial triangles in that monthly semicircle which has the apogee of the simple inequality; but the odd particle taken from the companion triangles leaves the quantity of the area of the equatorial triangles standing in the other monthly semicircle, which includes the perigee. The semicircles are to be determined by the parallel to the line of the conjunctions drawn through the centre of the eccentric.
As if the quantity of the area CLA is to be sought in the semicircle PLQ, which includes the perigee F, the companion triangle BLZ is to be computed first, which is greater than CLA by the odd particle, so that, this particle being taken from the area BLZ, the area CLA is left. Conversely, if the quantity of the area CMA is to be sought in the semicircle QMP, which has the apogee D, BMZ is sought first and the odd particle is added to it; so the quantity of the area CMA is made up.
Teach, then, how to compute the area of each triangle standing upon the monthly eccentricity of the centre of the eccentric. The geometrical method is the same by which we were taught in book V to compute the physical part of the equation. For the sine of the monthly argument is multiplied into the value of the area of the triangle which is greatest in any proposed month, reduced to seconds of degrees, and the ciphers being cast away, the value of the area of the triangle in question is left.
How the greatest triangle in any technical month is to be computed, we have already set forth.
As if we wish to know the area of the triangle BLZ, which is companion to the triangle CLA, then the area of the triangle BNZ, the greatest in this month, is multiplied into TL, the sine of the argument PL, and the product divided by the whole BN, and the area BLZ comes out. By logistic the value of the area BLZ is taken from the full month, just as if the base BZ equalled EA; this value is multiplied into the monthly minutes, and the true value of the area BLZ comes out; so the matter comes to the same.
Here, however, a short way is to be observed. For because the greatest monthly equation does not exceed 2° 30′, it can therefore be treated as a straight line. And since in any month the area BLZ grows from P to N in proportion to the sines LT and so on, while their sequence is the same as that of the monthly minutes (since those too are from sines, though built up in the contrary order), the multiplication of the monthly minutes into the minutes of the contrary order, adapted to the monthly argument, can therefore be made: the two logarithms being added, the sum will at once show the value of the area here sought from a separate little table adapted to this.
Let the monthly minutes be 48′ 33″, logarithm 21193; let the argument be 45 degrees 42′ 24″, logarithm 34700.
The sum 55893 shows from the special table 1° 25′ 45″, the area of the companion triangle, or the proper part of the equation, to be leavened with the odd particle.
By what is the odd particle measured? The odd particle is measured and represented by the little area of the triangle whose base is the eccentricity of the monthly point, and whose vertex is the centre of the eccentric. The double of this little area is elsewhere called the rectangle of the quadrant. And the little area is reduced to minutes of a degree, of which the area of the whole eccentric is 360°.
In the diagram set before, if C is the monthly point and CA its eccentricity, then in all moments at which this disposition of the points C, B, A is found, and this quantity of the angle CAB, always the little area CBA (that is, half of the rectangle of the quadrant CBZA) is the odd particle, in whatever point of the orbit the Moon is found. This little area, therefore, if it be added to the area BMZ, makes up the area CMA. The same little area CBA, if it be taken from the area BLZ, leaves the area CLA. This is plain thus: for CA and BZ are equal, wherefore the triangles on CA and BZ are as their altitudes above each base, and as VL to TL, so CLA to BLZ. And as TL, the altitude of the triangle BLZ, to CB or VT, the altitude of the triangle CBA, so is the area of the former to the area of the latter. But TV is the difference of the altitudes VL and LT, therefore the area CBA also is the difference of the areas CLA and BLZ.
Teach how to compute the odd particle. The method of computing is the same as was above in book V for the rectangle of the quadrant. The altitude CB is multiplied there into the whole base CA, here into half of it. But that we may know briefly the value of each little area, coming out in degrees and minutes, of which the whole area of the eccentric is worth 360°, we must compute the greatest of all, namely at the angle CAB of 45°, and its value.
The eccentricity AE, 4362, being multiplied into itself, makes the double of the greatest rectangle, namely 19027044, therefore the rectangle CBZA, when it is greatest, is 9513522. But if the area of the circle is worth 360°, this little area will be worth 0° 6′ 49″, and the odd particle, half of this, namely 3′ 25″ or 205″. Now in book V the proportion of the rectangles of the quadrant (and so also of these triangles) to one another was found in numbers, of which the greatest is 1000. If then 1000 is worth 205″, it is easily computed how much any of the lesser is worth.
On account of this smallness the odd particle can either be neglected altogether, or be picked out among the variations, of which below: for it is nearly an ounce of the corresponding variation.
Why did you give the name of odd particle to it? And what is the manner of using it, that the monthly equation may come out? Because since the equatorial triangle itself, for example CLA, is made up of the companion BLZ and of this particle or little area CBA, the former BLZ yields to its sine LT the due lot from the greatest of this month, BNZ, but this little area CBA, outside the lot, is taken away from the triangle BLZ in one monthly semicircle, outside the lot is added to it in the other, by the same quantity through the whole semicircles PFQ, QDP, whether the sine LT receiving its lot be small or great. But if the proposed point of the Moon’s orbit be in the semicircle which includes the perigee, and its triangle on BZ, say BHZ, be less than CBA, the subtrahend itself, then BHZ is in turn subtracted from CBA, and the remainder is the area of the equatorial triangle sought and takes its denomination from the semicircle GDH, which has the apogee D.
By what other names is this temporary monthly equation called? Tycho Brahe called it the prosthaphaeresis of the eccentricity, Copernicus the prosthaphaeresis of the second epicycle, Ptolemy the prosneusis of the epicycle, as it were a nodding, the Arabs from it the diversity of aspect of the epicycle: each uses a word suited to his own form of hypothesis, by which he supposes something geometrical and optical, whereas to me it is from merely physical causes.
How great is this monthly equation when it is greatest or full? Tycho Brahe fixed it equal to half the greatest equation in the conjunctions, which according to me is the part of the physical equation in the simple inequality, namely 2° 30′, which comes out very conveniently for the diagrams, so that there is the same base AE of the triangle for both, as before we supposed in its definition. But Ptolemy makes it a little greater, namely 2° 41′, as was said in book IV, page 390, which quantity serves very well the speculative connection of this equation with the following business of the variation: let this therefore remain in doubt until observations decide sufficiently. In use let us meanwhile follow Tycho.
What is the difference of these monthly equations from the earlier ones of the simple anomaly? In the primary planets and in the simple anomaly the physical parts of the equations at equal distance of the points of the orbit from the apogee, both before and behind, are equal. But in the monthly anomaly, except in the full month, there is always a difference: of two points of the orbit removed to contrary sides at equal angles at the Earth from the monthly apogee, that is from the Sun or its opposite, the monthly equations are unequal, and that has the greater monthly equation which is nearer the apogee of the simple inequality; the doubled odd particle shows the difference.
In what proportion to one another are two such equations, removed at equal angles from the Sun to contrary quarters? They are to one another in the proportion of the distances of the Moon and the Earth. As if DAH, HAL are equal angles, the area CDA will be to the area CLA as DA to LA.
How do we use this monthly equation, or what is the anomaly equated first or second, and what likewise the mean anomaly, with respect to the monthly temporary equation applied? A certain place of the Moon in her orbit being proposed, and a certain line of the true place of the Moon under the ecliptic, for example AL, the area LDA alone, consisting of the sector LDB and the triangle LBA, is the mean anomaly corresponding to that place and to the anomaly of the eccentric DL and the anomaly first coequated DAL. But if now in the semicircle HNG, begun from H, the monthly apogee, and in this situation of the points B, C, A, we add the area of the monthly triangle CLA to the area DLA, then the angle DAL in respect of this increase is the anomaly second equated, and there corresponds to it now the larger mean anomaly, composed namely of DAL and LCA; understand again, of such parts as the simple area of the circle is 360°. The same holds in the latter semicircle GOH, begun from the monthly perigee G, if we subtract, for example in M, the area CAM from the area HAMGN. For the area HAMGNH is the mean anomaly, corresponding to the anomaly of the eccentric DNGM and to the first coequated, namely the complement of the angle DAM; and the area HAMGNH, diminished by the area CAM, is the mean anomaly corresponding to the same anomaly DAM second equated: for so Tycho Brahe calls it, when two equations have been applied to the mean anomaly, the first from the simple inequality, the second the monthly temporary. He, with the other astronomers, fits two coequated anomalies to one mean anomaly, subtracting the equations in the first semicircle and changing the seen places; but here two mean anomalies are fitted to one coequated and one seen place, adding the equations in the first semicircle, so that the delays of time are changed, because the causes of so many equations are physical, and not all of them are yet explained.
In the parts of this semicircle between D, H, the apogees, or F, G, the perigees, there is this caution. Let P be an intermediate point, a place of the orbit, and AP the line of the true place of the Moon, and DP the anomaly of the eccentric, and the angle DAP the coequated anomaly; if therefore this is understood to be first coequated, then there corresponds to it the mean anomaly, the area DPA, consisting of the sector DPB and the simple triangle PBA; but if DAP is the anomaly second coequated, then the measure of the mean anomaly corresponding to it will be the area DAP, diminished by the little area PAC. For here the area PAB is in the descending semicircle DPF of the simple inequality, but the area PAC is in the ascending semicircle GPH of the monthly.
You teach, then, to find for a proposed anomaly second equated its mean anomaly: but it happens oftener that the mean anomaly is proposed from the knowledge of time, and I would wish to find for it its second coequated. Again for this, no less than above in book V, there is no way a priori, but we must use the rule of false position, or tables made for this use.
Yet if we need not proceed most scrupulously, then the monthly temporary equation found, considered as an angle, can also by the contrary way be subtracted in the first semicircle and added in the second to the anomaly first coequated, so that it becomes the second coequated, answering to the proposed mean remaining unchanged; or the angle of the triangle CLA can also, instead of the value of the area CAL, be subtracted from the angle DAL, and the angle CMA or CPA be added to the sum of DAF, FAM and to DAP, that the anomaly second coequated may be represented in angles thus corrected, from which, however, the reasoning of the diagram does not lightly differ, unless you draw from C the secants CA, CM, CP themselves, indicating the place of the Moon thus secondly equated, which seems more intricate (see book V). And in respect of the use of this latter, since it agrees with the usual rule of the equations in the planets, the first semicircle, begun from the apogean conjunction according to the usual rule of astronomy, is held to have a subtractive affection, the second, begun from the perigean conjunction, an additive affection.
What is the place of the Moon called when affected by the two foregoing equations? In respect of the third little equation that follows, the place of the Moon is called nearly true in Tycho’s Progymnasmata.
By what laws are the equations of both kinds, of the simple and of the monthly temporary anomaly, mingled together into one compound?
1. They are always accumulated in the greater part of the semicircles. This follows from the fact that it was said: at the time when the semicircles, in the simple the ascending and descending, in the monthly those of the waxing and waning Moon, as it were bisect one another by the application of the terms, the monthly equation is wholly nothing. As soon, therefore, as there is any monthly inequality, there is already more than a quadrant in the semicircle of the waxing, communicating its affection to it.
2. In the remaining part of the semicircles, where the affections are contrary to one another, a subtraction of the lesser equation from the greater is made, and the right of denomination or affection stands from the greater element.
3. When, therefore, the apsides are in the conjunctions, then, the Moon coming to the quadratures, not only the monthly equation but also one composed of the two elements is made.
4. When the apsides fall in the quadratures, then throughout the whole technical month (and in so far as this is true of the whole natural month) the equations come out simple, by the law of the simple inequality; nor is there any monthly equation with which it may be mingled.
5. When the apsides are in intermediate places, or in the octants, in that month the Moon in the conjunctions indeed has a simple equation, by the laws of the simple anomaly, but being in the quadratures, beyond what belongs to her by the laws of the simple, she receives also a particle of the monthly equation, as great as it can become in that whole half of the technical month; finally, falling in the apsides when gibbous or horned, in which points she ought to have lacked an equation by the laws of the simple, she has nevertheless some monthly equation; conversely, coming into the mean longitudes when gibbous or horned, where she acquires the greatest equation by the law of the simple, she does not add the greatest monthly.
What inequality of the parts of the month follows from this? One half of the month, from quadrature to quadrature, can run out to nearly sixteen days, the other half be contracted to thirteen days and a half, a little more.
Of the third inequality of the motion of the Moon in longitude, or of the variation.
How do the monthly inequalities, the temporary one explained so far and the fixed or perpetual one now following, differ from one another in form? 1. The temporary, as was said, depends jointly both on the apogee and on the line of the conjunctions; the perpetual depends on the line of the conjunctions alone. 2. The former arises from comparing the eccentricity of the Moon with the plane of the circle of illumination, the latter exists through the same circle of illumination, but without regard to the eccentricity. 3. That equation is dispersed through whole semicircles, so that it is greatest in the quadratures; this is dispersed through quadrants, vanishes both in the quadratures and in the conjunctions, and is greatest about the octants. 4. Therefore on account of the temporary the Moon becomes once in the month slow, once swift, but on account of this one she becomes twice slow, namely in each quadrature, twice swift, namely in each conjunction; and the swifter in any place, the nearer it is to the conjunctions.
What name has the little equation which arises through this anomaly? Tycho Brahe, the discoverer, called it the variation. Keep that name, that from the very prerogative of the name we may be reminded that it arises otherwise than the equations hitherto spoken of, being namely an accident of the mean motion, which it varies even without the help of eccentricity.
What is the variation? It is the angle contained between two lines from the centre of the Earth, indicating the nearly true place of the Moon and the true and absolute place: but because, to avoid confusion, these lines are not drawn in the diagram, the variation is more rightly defined from its cause, as the unequal and real excess of the promotion made by the light over the equal or fictitious one. Where by the word promotion we understand the effect in degrees and minutes of the elongation of the Moon from the Sun, nearly true.
Teach how to find the variation geometrically, according to the physical principles delivered in book IV, page 364 and following. If from the centre of the Earth, at any interval, a quadrant be described, begun from the line of the conjunctions and carried across the line of the nearly true place of the Moon, and in that quadrant a rectangle about the little portion cut off of the line of the said place of the Moon, as about a diagonal: the area of this rectangle, reduced to minutes, of which the greatest rectangle is worth 40′ 30″ according to Tycho (or 51′ according to the reasoning of book IV), will give the value of the variation belonging to the proposed nearly true place of the Moon.
In diagram 89 let the quadrant EY be described with the centre of the Earth A, from the line of the conjunctions HA, across the line of the place of the Moon AL, which must cut the quadrant in R; let the rectangle CX be about AR as diagonal; the area CX will be the measure of the variation which belongs to the nearly true place of the Moon, indicated by the line AL.
In this way the little circle which is described from the centre of the eccentric B about the centre of the Earth A applies its area, which before served for leavening the monthly equation, now also for despatching the variation, so that we have no need of greater apparatus. But according to Tycho Brahe, from the nearly true place of the Moon in the ecliptic there is to be drawn, on the surface of the fixed stars, a little circle with semidiameter 40′ 30″; the little part of it stretched under the ecliptic, corresponding to the sine of the doubled elongation of the Moon, nearly true, from the Sun, will be the variation belonging to it.
Because in book V the demonstration of the equivalence of this plane with the physical causes introduced in book IV was left suspended, despatch it here. Remember, therefore, that this was laid down in book IV, that the light of the Sun helps the moving species of the Earth in the doubled proportion of that which the sines of the complements of the angles hold, by which the species of the light of the Sun, as a luminous spherical surface stretched about the Sun, and the spherical species of the body of the Earth stretched about the Earth, are applied to each other.
As, because in the conjunction H the angle is none, being merely a contact of the moving species, and the complement of the angle 0° is 90°, therefore the sine EA of the arc EY, 90°, is the measure of the help from the light in H. Conversely in I, the quadrature, the surfaces of the species cut one another at right angles, but the complement of a right angle, or 90°, is 0°, and therefore the sine of this too is nothing: no help of motion, therefore, from the light of the Sun does the Moon feel when placed in I. And in L, supposing that the elongation HAL or EAR (because AL and AR must be one straight line) is 30°, the sine RX of the complement RY, 60°, will measure the help of the motion of the Moon placed in L. These sines, I say, will measure this not by themselves, but by their squares, since their proportion is double the proportion of the sines themselves. Now if you cut the quadrant EY into very many equal parts by RX, parallel to EA itself, which parts are now ER, RY, as EA stands to RX, the sine of the next division, so very nearly stands the distance of the sines AX to the distance of the next, as XY; this is always the truer, the finer the division of the quadrant. This is used in book V and demonstrated. Wherefore in the area of the quadrant segments are constituted (such as EAXR) all of proportional sides. For as the length EA to the breadth AX, so also the length RX to the breadth XY, by force of the infinite division. The segments are therefore similar in power, because in the infinite division we disregard the upper curvatures ER, RY, and use the segments, being very small, for perfect parallelograms. If the segments between pairs of sines are similar to one another, the proportion of them is therefore double the proportion of the sines themselves to which they are bounded. But the light of the Sun also, in promoting the motion of the Moon, uses for this the double proportion of these sines. The promotion of the light, therefore, is dispensed in the proportion of those segments, and consequently, as the truncated figure of the quadrant AERX grows successively, until the quadrant becomes entire in Y, so also grows, in the same measure, this promotion of the light, and beginning from E becomes in Y 2° 9′ according to Tycho’s quantity, or 2° 41′ according to the quantity derived a priori.
But if this promotion of the light were equable, then it would grow with the arc ER itself and its sector EAR; but because it grows with the truncated figure AERX, the excess of AERX itself over EAR, that is the triangle RAX, measures the excess of the promotion from the light, and therefore measures the variation. But the area of the rectangle CX is double the area of the triangle RAX, and of doubles the proportion is the same as of the simples among themselves; therefore the areas of the rectangles of the quadrant also measure the variation.
Demonstrate also the equivalence with Tycho’s little circle. Tycho fixed the sine of the arc which is double of HAL as the measure of the variation. Now the rectangles of the quadrant, such as CRXA, also grow in proportion to the sines of the arc which is double of the angle ER or HAL. For the lesser arc ER is doubled and its sine halved, and five ciphers being added, the rectangle CX comes out. This process is demonstrated from the artifices of the abridged trigonometry, three terms, the whole sine, the sine of the arc and the sine of the complement, being applied to the spherical rectangle. But the proportion of the halves is the same as of the wholes: the equivalence is therefore most perfect.
Give an example of the method of computing the variation. Let the elongation of the Moon, nearly true, be 30°. Therefore the sine 86603 of the doubled arc 60° is reduced to minutes, of which 100000 are 40½ or 51′, which value multiplied into 86603 (it can be done by logarithms) shows the variation.
The sum is the logarithm 53687, namely 35′ 5″ of what was sought. Or in the other demonstrative quantity 51′ 0″, logarithm 16251.
The sum is the logarithm 30635, namely 44′ 10″ of what was sought.
How is the variation used, or what is the true and absolute place of the Moon? The variation is added in the quadrants begun from the line of the conjunctions, and in the rest taken away from the nearly true place of the Moon: so by the third operation the true and exact or absolute place of the Moon is at length made, namely by the equation of the simple inequality, the monthly equation and the variation.
Tell what variety of hourly motions arises hence. The hourly motion of the Moon from the equinox, by the Tychonic variation, comes out in the conjunctions, in apogee 29′ 41″, in perigee 38′ 32″; in the quadratures, in apogee 29′ 35″, in perigee 35′ 11″. But by the increased variation these will be: in the conjunctions, in apogee 29′ 50″, in perigee 38′ 45″; in the quadratures, in apogee 29′ 26″, in perigee 34′ 39″; and the mean hourly motion of the Sun, 2′ 27″, being taken away, the motion of the Moon from the Sun comes out, in perigee in conjunction 36′ 5″, in apogee in quadrature 27′ 8″, between which is precisely a diatessaron, or the proportion 3:4. So the apogean motion of the Moon from the Sun in conjunction is 27′ 14″, the perigean in quadrature 32′ 44″, between which is precisely a soft third, or the proportion 5:6; so also between the purely conjunctional there is a diatessaron, between the purely quadratural a lesser ditone, and the difference of the conjunctions from the quadratures gives a lesser tone.
How could the value of the greatest variation at the 45th degree be known, that it is 40′ 30″ or 51′? Tycho derived his quantity of 40′ 30″ from a comparison of the observed places of the Moon in the octants with those places of the Moon at the moments of observation which were computed by the prescription of the two earlier equations. For the difference which was found between these and those won the name of variation. And although not all the observations showed the same quantity of the variation, but were found to disagree with one another by as much as 10 minutes, yet Brahe judged that he must keep that quantity which was the mean between the disagreeing. But the other quantity, 51′, was deduced in book IV, pages 367 and 390, from a probable doctrine, by which it was asserted that whatever more than 12 lunations accrues to the motion of the Moon in the sidereal year (and it is 132° 45′ of the thirteenth revolution) must be credited to the light of the Sun; so that 12 pure lunations remain to be completed by the species of the body of the Earth. This supposed, we easily inquired what then of any quadrant of the elongation of the Moon from the Sun was to be transferred to the light. For if of 12 revolutions with a fragment the light of the Sun claims for itself that fragment, namely 132° 45′, then of one quadrant of the revolution, or of 90°, the light of the Sun will claim for itself in the same proportion 2° 41′.
The quantity of the effect of the promotion from the light in one whole quadrant being thus fixed, there is now also had the greatest variation, which results from this promotion in the octants.
For because the light does not distribute its 2° 41′ equally, not, I say, in the proportion of the sectors EAR, but unequally in the proportion of the truncated figures or parallelograms AERX, which area exceeds the sector EAR, most of all at 45° (where RX, XA are equal), by the excess of the area RAX, therefore as the area of the whole quadrant YEA is to the area RAX, so also is 2° 41′, the effect through one whole quadrant, to that excess of this unequal and true effect over the equable and imaginary in the octant: which was the definition of the variation. And since geometers have found the area of the quadrant 7853981634, but the area of the rectangle of the quadrant RCAX, when it is greatest, is 5000000000, of which the half, 2500000000, is the excess of the triangle RAX: if therefore 78540 is worth 2° 41′. And since for the appendix of 12 lunations, which is 132° 45′, we elicited on page 367 (again a priori) 127° 10′, which is almost a twenty-fourth part less: if therefore we also begin hence and fix the greatest variation, it will come out about 49′ and so still nearer to the Tychonic.
If the monthly equation and the variation are fused into one, what name will it have? Since both arise from the same illumination of the Sun, though in diverse ways, the compound of both will most rightly be called the equation of light.
Of the monthly latitude of the Moon.
By what geometrical positions is the monthly latitude of the Moon demonstrated?
Of the physical causes indeed, and what is in the thing itself, it was said in book IV, page 392. At present the question is not of the real apparatus, but the imagination is helped.
In the diagram set before, let some plane be imagined, drawn through the points A, C, oblique to the plane of the ecliptic, upon a line drawn through the places of the nodes and through A, the centre of the Earth, at a constant angle of 5°. In this plane let there be the line HCAG of the conjunctions, designated by the section of the plane passing through the centres of the Sun and Earth, at right angles to the plane of the ecliptic. On this line HCAG let the eccentric of the Moon itself, DLFM, be inclined to the plane first mentioned, not indeed always, but by the following laws. Let I, K be the places of the quadratures under the fixed stars, and let there go out from these points of the sphere of the fixed stars two arcs, at right angles to the plane first mentioned, each described from the centre A, neither longer than 18 minutes to either side. When, therefore, the limits of the simple anomaly are in the line HCAG, then let the plane of the eccentric DLFM be united with the former plane, so that there is no mutual inclination between them, but only an inclination common to both to the plane of the ecliptic. But when the limit of the simple inequality, say the northern, begins to depart from the opposition H toward the quadrature K, as if it should come to D, as the sine of the angle HAD stands to the sine of the right angle HAK, so let the portion of the libration to the north of the monthly limit K itself stand to the total libration or inclination of the planes, which is 18 minutes, which total inclination comes about in that month in which the northern limit of the simple inequality comes to K, the place of the quadrature or monthly limit. In this way it comes about that the northern limit of the simple inequality rises from the former plane to the north so long as until it falls in the quadrature K; thence it gradually inclines again toward the former plane, and at length, coming in G, falls back into the plane itself. As soon as here the northern limit of the simple inequality has passed beyond the point G toward I, the semicircle HKG becomes, in respect of the monthly latitude, southern, GIH northern, and the northern limit of the simple inequality is again raised to the north, since the whole plane of the eccentric GIH is raised to the north in the arc drawn through I, and the whole HKG is sunk to the south in the arc through K.
What is the method of computing the libration or inclination of the monthly limit?
Let the Sun be distant from the node 30° or 60°.
Sine of the complement 86603 or 50000.
(the last 5 figures cut off) 15′ 35″ or 9′ 0″ is the inclination of the monthly limit.
What do you call the minutes of latitude? They are the sines of the complements to one or three quadrants, or of the excesses over one or three quadrants, of the distance of the Sun from the ascending node of the Moon, reduced to astronomical minutes, of which the whole sine is worth 60′, or they are three and a third times the angle of inclination of the monthly limits, constant in any technical month.
What is the monthly argument of latitude? It is nothing else than the distance of the true and absolute place of the Moon from the true place of the Sun.
What is the monthly latitude of the Moon? It is the arc of a great circle described under the ecliptic, whose plane, passing through the centres of the Earth and Moon, is erected at right angles upon the plane inclined at the constant angle of 5° to the plane of the ecliptic, intercepted between this plane continued and the place of the centre of the body.
How is it computed? By multiplying the monthly minutes of latitude into the inclination of the limit, the monthly argument of latitude being taken from the full month.
What follows from this supposition of a doubled latitude for the motion of the nodes? The true node, or the intersection of the orbit itself with the ecliptic, receives an unequal motion, slow indeed when the Sun passes through the nodes, swift if the Sun goes through the limits; and the greatest prosthaphaeresis of the nodes, the Sun being in the octants, is 1° 39′, which however we do not need for finding out the latitudes.
How do these hypotheses of the monthly inequality of the Moon differ from the hypotheses of Tycho Brahe? In the effect of the calculation of the longitude and latitude of the Moon nothing, or certainly less than could be detected by any diligence of an observer; in the distances of the Moon moderately, in the form of the motions very much. 1) For to Brahe, as to Copernicus, the apogean distance of the Moon from the Earth is small in the conjunctions, great in the quadratures, more so to the latter than to the former; to me it is equal on both sides. 2) To Brahe the change of eccentricity (or what is equivalent to it in Copernicus and Ptolemy) is monthly; to me the real eccentricity remains unvaried, but the eccentricity of the equatorial point, called monthly, undergoes an annual nutation. 3) The increase of latitude Copernicus did not know; Brahe, its discoverer, so dispenses it that both at the beginning, when it is greatest, and at the end, when it vanishes, it is slow, and in the middle swift; but here only when the increments are greatest is their change slow, and swiftest when they vanish. 4) Brahe leads the limits round in the circumference of a little circle, disturbing thereby the motion of the eccentric, which disturbance however he remits to the Moon herself, just as if she were not attached to the eccentric but rubbed against it, nodding hither and thither (I speak according to the mind of the ancients, for Tycho rejected solid orbs); to me the limits librate in an arc or line erected upon the plane of the ecliptic. 5) The nodes with Tycho reciprocate twice in one month and from retrograde become direct; here the motion of the nodes has a half-yearly inequality, or one returning twice a year, nor do they ever become direct, but only like stationaries, twice a year.