Epitome Astronomiae Copernicanae: Liber V, part 3
Second Part of the Fifth Book. Of the astronomical terms arising from the calculus and the eccentric orbit.
How is the orbit of each planet called? It is called by the old word the eccentric, understanding “circle.” For although the orbits are elliptical, such as this PERI, which have two quasi-centres A, L, which in physics we call foci, and in one of them, A, the Sun himself, as the centre of the world, is placed: yet the point midway between the foci, as B, is also by the writers on conics called by a peculiar right the centre of the figure; and besides, a perfect circle PDR is circumscribed about the figure for the sake of measuring, with its centre B different from A, the centre of the world.
What name has the longer diameter PE of the ellipse in astronomy? It is called the line of apsides, because, being drawn through A, the centre of the world, and B, the centre of the orbit, it shows by its intersections with the orbit P the highest apsis and R the lowest.
Whence are the highest and lowest apsis so called, and what other name have they? The word apsis is drawn from wheels, for they are points of the eccentric, P the most remote from the Sun A, R the nearest to him. But in geometry the reason of the signification is more evident. For the word apsis is derived from touching, and indeed in the points P, R the measuring circle touches the elliptical orbit. The Greek word apsis, apsides, the Latin versions of Arabic books render by the words aux, auges; as if the Arabs had changed the Greek Psi into Xi. A certain man, however, boasting a knowledge of the Arabic tongue, assured me that by the word augh height is signified.
In the sixth book these points will be called, in the primary planets, aphelion and perihelion, in the Moon apogee and perigee.
What necessity compels us to suppose, in place of the circular course of the planet believed by the ancients, an elliptical one, that is, one falling short of the circle, and in it to set the longer diameter and on that very diameter the Sun? Both of these have been demonstrated by observations and by most certain demonstration in the Commentaries on the motions of the star Mars, and made use of in book IV, in diagrams 51, 52 and folios 354, 372, and also in this fifth book, part one. Unless, therefore, we supposed these things, we should never represent the observations.
By what names are the halves of the eccentric distinguished from one another, from this line set up? The one half PER or PDR is called the descending or former semicircle, the other RIP or RTP the ascending or latter.
What is the eccentricity? By the Greek word ekkentrotes it is the line connecting the centres, A of the world (or of the body about which the motion is ordered) and B of the eccentric; namely AB, part of the line of apsides PR.
What name have the lines drawn from the centre of the body about which the motion is ordered to the eccentric orbit? In Greek they are called apostemata, in Latin intervals or distances; in the versions of the Arabic books they are called longitudes, as AP, AC, AE, AM, AS, AF, AR, AN, AQ, and so on.
Which are the more notable of these longitudes? The longer longitude in the Arabic books, or the aphelion or apogee distance AP; the shorter longitude, or perihelion distance, or in the Moon perigee, AR; and the mean longitude, which is the arithmetical mean between the longer and the shorter; of which that which is in the descending semicircle, namely AE, is called the first mean longitude, and that in the ascending, as AI, the second.
What else does the mean longitude signify? By metonymy it is taken for those points of the orbit which have a mean distance from the Sun, as E, I, which are distant by a quadrant, or 90 degrees, from the apsides, before or behind. Sometimes it is also taken for the point of the zodiac itself which is a quadrant distant, before or behind, from the place of the zodiac on which the line of apsides produced falls.
Where it is to be noted and guarded against that on that degree of the zodiac which is called the mean longitude, the line AE of the same name, when produced, does not fall, but rather BE from the centre, or AM parallel to it, as lines which form right angles with the line of apsides PR.
What is the name of the difference between the mean longitude or distance and any other? This difference is called the libration of the planet, because the whole libration, as in the motion of the pans of a balance, is slow at the beginning, when the planet is at its greatest distance from the Sun, and at the end, when it becomes nearest to the Sun; swift in the middle.
In the diagram, because AP is the greatest distance and AR the least, let AR be transferred to the line AP and extended from A to G, so that the whole libration may be set before the eyes in the single line AP as if at rest, which will be PG, double the eccentricity BA. This libration is therefore slow about P and G, namely when the planet is either in P or in R, and swift about H, when the planet, or the line AH, is transferred to AE or AI.
You have said that a circle is circumscribed about the orbit for the sake of measuring; say in how many ways it serves for measuring this orbit. In four ways: 1. This circle names and distinguishes the arcs of the elliptical orbit. As the arc PC receives both its name and its determination from the arc PK. 2. The circle yields the measures of the planet’s librations and so forms the lengths of the intervals between the planet and the Sun. As AC or AO is determined by the arc PK or its complement KD, because this teaches the quantity of the libration HO to be added to the semidiameter AH. 3. The circle also exhibits the measure of the time which the planet consumes in any arc of its elliptical orbit. As by the arc PK we learn how long the planet is moving in the arc PC. 4. These things being found, the angle at the Sun can also be investigated, which an arc of the orbit subtends. As, without the arc PK known, AC being unknown, the angle CAP cannot be found.
How does the circle name and distinguish the arcs of the ellipse, and by what means and why? Since the elliptical circumference cannot of itself be divided geometrically into equal parts, nor the parts so made be named by number, the circle is therefore divided, in place of the ellipse, into equal parts, beginning from the apsides, and from the points of division perpendiculars are drawn to the line of apsides, cutting the ellipse. The arc of the circle, therefore, between the aphelion and each perpendicular gives its name to the elliptical arc intercepted between the same limits, imparting to it its own number of degrees and minutes.
Let PK be 50° 0′, KL a perpendicular to PR, cutting the ellipse in C: then the arc of the ellipse PC also is said to be 50° 0′.
But the name is false, since the arc of the ellipse is not so great, either in respect of the circle or in respect of its whole elliptical orbit. This disturbs nothing; for it is nothing, at present at least, but a name, and indeed the name not of an apparent measure but of a geometrical determination and cutting off; nor is it necessary to know the genuine length of the elliptical arc itself, as if measured out with a ten-foot rod, provided that we afterwards know what angle this very arc of the ellipse, so determined, makes at the centre of the Sun and how long the planet stays in it. What of this? For in the first part of this fifth book I demonstrate that this arc of the ellipse is equal to it, if not in length, at least in power.
How are these perpendicular cutting lines of the ellipse called? In the circle they are called the sines of the arcs of the circle, begun from the aphelion; in the ellipse, by the generic word, ordinates, namely to the axis. As here KL is the sine of the arc KP, CL is the ordinate.
But in particular that which is drawn through the centre of the figure, as EBI, is called the shorter diameter, or the latus rectum of the figure. We may use the Greek word diacentros. That which, finally, passes through the centre of the Sun, as MAN, lacks a name, though it is among the principal. Let it be called by a new word, dihelios.
What is the office of those perpendiculars, the diacentros and the dihelios? They divide the orbit into parts, upper and lower; the former into equal parts, but unequal in time and in appearance; the latter into parts unequal both in time and in length, but which nevertheless appear equal, as from the Sun.
As EPI, which is constituted by EBI, is indeed of 180 degrees, but appears under an angle EAI less than 180°. But MPN, the greater segment, cut off by the line MAN, and MRN, the lesser segment, each appears equal in quantity to 180 degrees.
Teach how to measure and compute the librations and determine the intervals. Let PK be an arc of the eccentric less than a quadrant, for example 46° 18′ 51″; its complement KD will therefore be 43° 41′ 9″ and its sine BL 69070, and let the eccentricity AB, or half the libration PH, be 9265, of which BP is 100000. Multiplying 69070 by 9265 and cutting off the last 5 figures, the libration OH comes out 6399, to be added to BP or AH in the upper semicircle EPI, and AO, or AC equal to it, namely the distance of the planet from the Sun, will be 106399, belonging to the arc PK or PC, of which the semidiameter is 100000.
If the arc of the eccentric is 313° 41′ 9″, the excess over three quadrants, or 270°, will also be 43° 41′ 9″, giving the same sine to be multiplied; with which, when the libration 6399 is constructed, it is likewise to be added, since it is in the upper semicircle, but ascending. But if the semidiameter BP has taken another measure, for example 152342, we shall multiply this also into AC 106399, cutting off the last 5 figures, and AC will come out in this measure. By the Napierian art this whole operation is accomplished most expeditiously by a single addition. For the logarithm of the sine of the arc KD is added to the logarithms of the eccentricity 9265 and of the proposed measure 152342: the sum sought, as a logarithm, exhibits the libration 9748, to be added to the measure 152342.
Let the arc PW be then greater than a quadrant, namely 133° 39′ 7″. The excess over the quadrant DW is 43° 39′ 7″, and its sine or logarithm with the two said principles yields the libration 9777, to be subtracted from 152342, since it is in the lower semicircle of the diacentros, so that the corresponding interval AS comes out 142565.
It will be the same if the arc of the eccentric has 226° 20′ 53″. For its complement to three quadrants will be 43° 39′ 7″, as great in the ascending as DW in the descending semicircle.
Review the principal cases of libration. 1. When the planet begins to depart from the apsis, then the libration begins at the same time, the planet beginning, namely, to descend toward the Sun, having a little before ended its departure from the Sun. 2. When the planet has 60 degrees from the apsis, then the libration equals half the eccentricity. 3. When the planet has completed a quadrant of its orbit from the apsis, then half of the libration is accomplished, so that the planet is distant from the Sun by the semidiameter of the eccentric. As if PD is 90, then AE equals BD. 4. When the planet has completed 120 degrees from the apsis, three quarters of the libration are accomplished. 5. When the planet is in the lowest apsis, then, having become nearest to the Sun, it completes the whole libration. The order is the contrary through the ascending half. 6. Any two distances from the Sun, the planet being at equal arcs of the eccentric removed, in the one case from the aphelion, in the other from the perihelion, together equal the diameter. As if from C a straight line is drawn through B to Q, CA and AQ together equal RP.
III. Of the delay of the planet in any arc.
What does the word anomaly mean? Although properly anomaly (inequality) is an affection of the planet’s motion, astronomers nevertheless take this word for the motion itself, in which this inequality inheres. And since these three measurable things concur in motion: the space to be traversed, the delay of time in the space, and the apparent magnitude of the space, the word anomaly is to be applied to all three. And in respect of time the use of the word is again twofold. For first Ptolemy uses it for the whole time which the planet consumes while all its inequality is restored to its beginning, counting as many anomalies as often as this happens. Secondly, the parts of this whole time are commonly called anomalies, because Ptolemy spoke of the motion of the anomaly, understanding the part completed of the whole.
How many anomalies, then, are taken as a part of the whole? Three are named in any position of the planet: 1) the mean anomaly, 2) the anomaly of the eccentric, and 3) the equated anomaly.
What is the mean anomaly? It is the space of time which the planet consumes in any arc of its orbit, begun from the apsis, reduced into parts and minutes, of which the whole anomaly is worth 360°, by logistic or astronomical numeration.
Whence is it called mean? Not as though it were of a mean quantity among its companions, as will be guarded against a little later: but it is called mean in imitation of the old astronomy, which is wont to call the mean anomaly that for the mean motion of the anomaly, that is the uniform, because the time so reduced into logistic denomination indicates, with its number of degrees and minutes, how great an arc of the circle the planet would have completed, if during all that time, which we call the mean anomaly, it had advanced with a uniform motion, mean between the slowest and the swiftest.
How would the mean anomaly be defined or measured in these diagrams according to the old astronomy? The line BL (Fig. 80) being made equal to the eccentricity AB, on the line of apsides BP, as said in the first part of this fifth book, the mean anomaly after the manner of the old astronomy would be the arc of the equant circle described from L, in the order of the signs, comprehended between two lines from L, one drawn through the apsis P, the other through the body of the planet C. Or it would be the angle of those lines at L, or its complement to 4 right angles. As here, if C were the planet, the angle PLC might serve nearly in place of the mean anomaly.
Define the line of mean motion and the mean place of the planet according to this old hypothesis of the equant. It would be a line led from the centre of the Sun into the sphere of the fixed stars, parallel to the line which is drawn from the centre of the equant, or from the other focus of the ellipse, through the body of the planet, and either of these would show under the fixed stars the mean place of the planet. In the diagram, if C is the planet, and AM parallel to LC, AM would be the line of its mean motion.
If, then, in this new form of astronomy no equant circle is expressed, in what other quantity will the mean anomaly be numbered or measured? In the area comprehended between the arc of the circle which names and determines the proposed arc of the orbit, and between the two straight lines which connect the ends of the arc with the centre of the Sun. As if the place of the planet C is proposed, a perpendicular to PR being drawn from C, cutting the circle PD in K, and P, K being joined with A, the area PKA is the measure of the mean anomaly, of which the area of the whole circle is worth 360°.
Teach how to compute the mean anomaly, or the delay of time which the planet consumes in the proposed arc. Let the eccentricity AB again be 9265, of which the semidiameter BP is 100000. Before all things the area of the greatest triangle is to be sought, which has a right angle at B and the altitude BD; this multiplied into half of AB itself gives 463250000. The value of this area DAB is to be expressed in the number of second minutes, of which the whole area of the circle PDT is 360 degrees, or 21600 first minutes, or 1296000 seconds.
Since, then, BP being 100000, the area of the circle is given by geometers as 31415926536, the area DAB will be 19110 seconds. Let the arc PC now be given by its denominator PK, which is 46° 18′ 51″. The sine of PK itself, namely KL, the altitude of the triangle BKA, multiplied into the value of the greatest triangle, five figures being rejected at the end of the product, will yield the value of the triangle AKB, 13819″, which are 3° 50′ 19″. And the sector KBP is worth as many degrees as are given in the arc PK, namely 46° 18′ 51″; the areas being added, PKA becomes 50° 9′ 10″; so great is the mean anomaly.
In this way the area of the equatorial triangle is to be added, so long as the sector or arc is less than a semicircle; if it exceeds the semicircle, that area is to be subtracted.
Give the rule concerning the relation of these triangles to one another. Any two triangles equally distant from the vertices, one from the highest apsis, the other from the lowest, are of equal magnitude. As if the arcs PK and RW are equal, the areas BKA, BWA will also be equal.
What is the anomaly of the eccentric? It is the arc of the eccentric circle, numbered in the order of the signs and intercepted between the line of apsides and the perpendicular to it drawn through the body of the planet, or through any proposed point of the orbit. As, a point C of the orbit being proposed, or the planet being in it, if through C a perpendicular KCL to PAR is drawn, cutting the circle in K, the arc PK will be the anomaly of the eccentric.
In what sense is it called the anomaly of the eccentric? Here too the word motion is understood. For although in the arc PK of the circle itself, according to its figure, no inequality or anomaly appears, yet the motion of the planet in the orbit PC is truly anomalous, unequal, in three respects: first by reason of its elliptical figure, which is bent according to its various parts with unequal curvature and is at unequal distances from the centre of the figure; next by reason of speed, which is not the same in all particles of the orbit; third by reason of appearance as from the Sun, because equal parts of the orbit subtend unequal angles at the Sun. Since, then, the arc PK concurs to determine all these, as said before, by what right the old astronomy introduced the equant circle and numbered the mean anomaly in it, by no worse right do we circumscribe about the real orbit PC the eccentric circle PK and number in it the anomaly of the eccentric, using something uniform to measure that which is non-uniform.
And in the old astronomy, indeed, the equant circle seduced the natural philosophers into imagining for themselves a real circle or at least a real motion; but here no one can be seduced, since it appears to the eye that the true orbit PC of the planet meets this technical circle PK only in the two points of the apsides P, R, and in all the rest of its course withdraws within its embrace toward the centre of the figure.
What is the equated anomaly? It is the arc of a great circle in the latitude of the zodiac, designated by the continuation of the plane of the planetary orbit, numbered in the order of the signs from the place of the apsis to the apparent place of the planet itself or of any point of the orbit; or, what comes to the same, it is the angle which any arc of the true planetary orbit subtends, or which the said two lines form at the centre of the Sun, or the complement of that angle to 4 right angles.
As if the planet is in C, the equated anomaly is the angle PAG; and if the planet is in Q, then the equated anomaly consists of these parts: PAM, MAR two right angles, and besides the angle RAQ. But if with centre A a circle of any size is described, and so also a circle in the sphere of the fixed stars, the arc of this circle, numbered from AP in the order of the signs to AC or AQ continued, will also be called the equated anomaly.
Why is it called equated? Authors have been accustomed to say equated motion of the anomaly (or simply equated anomaly), not as though from a proposed unequal motion an equal motion had been elicited, but on a plainly contrary ground: that, since at the beginning there is proposed a time, or portion of the periodic time, and since this time (reduced into astronomical denomination) indicates how great an arc of the circle the planet would have completed within this space of time, had it advanced with uniform motion, it is next the duty of the astronomer to show how much of the truly unequal apparent motion of the planet answers to this time and the fictitious uniform motion. Equated motion therefore means the same as motion affected by equation and converted into the apparent, clothed, namely, with that inequality which appearance procures for it, from which inequality the whole period is called anomaly.
Since, then, you have both distinguished and formed these three anomalies by the fictitious eccentric circle circumscribed about the orbit, I ask whether the true orbit of the planet cannot serve the same use. Although there is no need, it can nevertheless by equivalence. For as said in the first part of this fifth book, time, and so the mean anomaly, is also measured by the area PCA, and the anomaly of the eccentric anyone who wishes may understand also by the arc PC. The angle PAC was also before called the equated anomaly.
How are these three companion anomalies distinguished in quantity? The number of degrees and minutes of the anomaly of the eccentric is always the mean among the others. But that which is called mean, before it fills a semicircle, is always the greatest of the three, the equated the least; after the semicircle, however, the so-called mean is the least in quantity, the equated the greatest.
IV. Of the angle at the Sun.
Teach how to compute the equated anomaly or the angle at the Sun. There are various ways, but the most compendious is that which uses the interval of the planet and the Sun. For we need that also for other uses.
Of this kind there are three cases; for either the planet is above the diacentros, or below the dihelios, or between the diacentros and the dihelios.
1. Let the planet first be above the diacentros DBT, say in C, and the anomaly of the eccentric PK 47° 42′ 20″, and let the libration of the planet, 6233, be found through the sine LB, 67277, of its complement KD, and this being added to BP, let AC, the true interval of the planet and the Sun, be constituted, 106233, in the measure in which BP is 100000. Let the same sine LB of the complement be added to the eccentricity BA, 9265, to obtain the other side LA, 76542, of the right-angled triangle CAL. LA, therefore, being divided, with 5 ciphers added, by CA, the quotient 72051, as a sine, shows the arc 46° 5′ 48″, which is the angle LCA, whose complement 43° 54′ 12″ is the angle sought, LAC or PAC. If you take the logarithm of half the divisor from the logarithm of half the dividend, there remains the logarithm of the same, whether sine or arc.