Epitome Astronomiae Copernicanae: Liber V, part 4
2. Let the planet be, second, below the dihelios MAN, say in S, and the anomaly of the eccentric PW, and its excess over a quadrant DW. As, therefore, above, the libration, sought through BZ, the sine of that arc, had to be subtracted from the radius, so as to give the true interval AS, so now the eccentricity BA is to be subtracted from the sine BZ, so that AZ, the other side of the right-angled triangle, remains. Again, therefore, the number of the side AZ, extended by 5 ciphers, being divided by the side AS, there comes out the sine of the angle ASZ, to which MAS is equal, the excess of the angle sought, PAS, over the right angle PAM, or quadrant.
3. Let the planet be, third, between DBT and MAN, as if the anomaly of the eccentric is PX and its excess over a quadrant DX, with sine BY, by which the libration is indeed computed as subtractive, since it is stretched below B; but since it is itself less than the eccentricity EA, it is now to be taken from this, so that YA remains. With this, then, and with the true interval, we must proceed as in the first case.
What do you call the eccentric place of the planet? That point in the zodiac on which falls the straight line drawn from the centre of the Sun through the body of the planet.
What is the equation or prosthaphaeresis, and what is the reason of the name? It is the difference of the number of degrees and minutes of the mean anomaly from the degrees and minutes of the equated anomaly, or, according to the old form of astronomy, it is the angle at the centre of the Sun and its measure, the arc of a great circle under the fixed stars, intercepted between the lines of the mean and of the eccentric motion of the planet. Since this has to be taken away in one semicircle and added in the other to the mean, to make the equated, it was called from this a compound word, prosthaphaeresis; and equation, because by its addition to or subtraction from the equated anomaly, which obtains unequal arcs and times, the mean anomaly becomes uniform, in equal portions.
What do you call the name or title of the equation? Two words, or their index syllables or letters: A (Add.), S (Subt.).
How many parts has the equation, and what is the measure of each? There are two parts, one called physical, the other optical. The former is on account of the inequality which truly befalls the planetary motion from physical causes, the latter on account of an inequality only apparent or quasi-apparent, that is, on account of the greater or lesser remoteness of the arc of the true orbit from the Sun. Both are in a manner distinguished in the same triangle, which is hence called the equatorial, for the ends of the eccentricity A, B being joined with the body of the planet C, the physical part of the equation finds its measure in the area BAC (or by equivalence in the area BAK), but the optical part of the equation would be equal to the angle BCA, if that were computed; the angle BKA, which is always a little less than it, being easier to compute.
What is the use of this equation, and of its titles? In this renewed form of astronomy the use of the whole equation, compounded of both elements, is neither necessary nor very great. For the anomalies are not constituted by this equation, but contrariwise, by comparing the equated anomaly (which we first compute) with the mean anomaly, we elicit the equation, if at any time we wish to use it.
In tables, however, three distinct anomalies are set down: first, the anomaly of the eccentric is placed at the left, by whole degrees from 1 to 180 in order, and for this reason, that from this given one the computation of the rest begins, and of the distance, or interval, of the planet and the Sun itself; second, under this anomaly of the eccentric, in the same column, is placed the physical part of the equation, or the value of the area of the equatorial triangle in degrees, minutes and seconds; from which we understand that the anomaly of the eccentric with the physical part of the equation, in the same cell, added together, constitute the corresponding mean anomaly. Third, at the side of this, in a peculiar column, is placed the equated anomaly, answering to the arc. If anyone now wishes to know the compound equation, let him subtract the equated anomaly from the mean placed beside it, or from the sum of the anomaly of the eccentric and the physical part of the equation, and the equation sought will remain, which in the descending semicircle bears the title subtractive, in the ascending additive.
Say, however, how these parts of the equation, compared with one another, stand to each other. The less the eccentricity, the more they approach equality with each other; in the upper semicircle, however, above the diacentros, the optical part is a little less than the physical, in the lower, below the diacentros, a little greater.
As in the adjoined diagram, if A is the Sun, PAR the line of apsides, DBT, MAN at right angles to it, the upper semicircle, or nearly, is DPT, the lower DRT.
Let the equatorial triangles be, in the upper, BCA, BFA, in the lower, BSA, BQA. Since, then, the areas of the triangles are the measure of the physical part of the equation, but the angles at C, F, S, Q of the optical part: the upper areas are certainly greater portions of the area of the whole circle (360) than their angles are of four right angles, or 360; the lower are lesser. For with centres C, S and intervals the semidiameters CB, SB, let the arcs BL, BH be described, ending in CA and SA continued; these arcs will measure the angles C and S, and the areas CBL, SBH are worth the same as the same arcs. If, therefore, these areas were the parts of the optical equations, both would be equal parts of one equation. But it is not CBL but the greater area CBA that is the measure of the physical part, and so not SBH but the lesser area SBA in the lower. The physical part therefore exceeds in the upper, the optical in the lower.
Where is the compound equation greatest? Of the parts, the former, the physical, is greatest in D, T, the ends of the diacentros, because the altitude of no triangle can be greater than BD or BT itself, which is the semidiameter in the circle, and also the longest of the ordinates in the ellipse. The latter part, the optical, if the orbit were a circle, would be greatest in M, N, the ends of the dihelios; for there the perpendicular drawn from the centre B to the straight line through A would be longest, and that is the sine of the angle BMA, the optical part, of which BM is the whole sine. For in EA, already above, there falls from B a perpendicular shorter than BA. But because the orbit of the planet is elliptical, the greatest part of the optical equation is between M and D, and so between N and T. For first the angle BMA itself is greater than the angle ADB, because both triangles are right-angled on the same base, and the altitude DB is greater than the altitude MA, the shorter diameter being less than any ordinate. Then, the points E, I being made in the middle of the arcs DM and TN or thereabouts, the angles AEB, AIB are again greater than AMB, ANB. For of all the lines from the centre B to the orbit BD is the shortest, and the others are the longer the more remote; BM is therefore sensibly longer than BE; but the perpendicular from B to AM is not sensibly longer than that from B to AE. The proportion MB to BA is therefore greater than EB to its perpendicular. And so the angle BEA is greater than BMA. Therefore, BA being bisected in G and the perpendicular EGI drawn, the greatest optical equation will be about E, I. But the greatest physical was about D, T; the greatest compound will therefore fall midway between DE and TI.
You have taught how to compute from a proposed anomaly of the eccentric the mean anomaly and the equated anomaly; but more frequent use requires, the mean being given, as from a given time, to find the others; teach this also. Here there is no direct way, but for him who wishes to compute this without tables the rule of position is to be used; namely, by supposing the anomaly of the eccentric (in the last but one diagram) PK to be such and such, and computing for it, so taken, its mean anomaly PKA. For if this comes out as great as was proposed, the anomaly of the eccentric PK will have been well assumed. But if it does not come out so great, the position will have to be amended from what comes out, and the labour repeated.
Could you teach by an example a convenient method, so that the unpractised may not go too far astray in vague positions? Let the above example, then, be resumed, and let the mean anomaly, or area PKA, now be given, 50° 9′ 10″; it is manifest that if the area of the triangle KBA were known, the remaining area KBP would have the same number of degrees as its arc PK, and so, the value of KBA being taken from PKA, the anomaly of the eccentric PK would be left. Since, then, PKA is greater than PKB, the sine of the arc PK will be less than the sine of 50° 9′ 10″, and so less than 76775. Let this sine in the first position be 70000, for ease of multiplication.
This sine, then, multiplied into the value DBA of the triangle, which in the above example was 11910″, five figures being cast away, makes BKA 8337″ or 2° 18′ 57″, which add to the arc 44° 25′ of the sine 70000, and the area PKA becomes 46° 44′; this is too small, for it falls short by 3° 25′, whereas it ought to come out 50° 9′, as great as is given. Let a greater sine therefore be put in the second position, the defect 3° 25′ being added to the arc before assumed, 44° 25′, so that PK becomes about 47° 50′, whose sine is very nearly 74000, which again I choose for ease of calculation. This multiplied into 11910 makes BKA now increased by 7′ 56″, namely 2° 26′ 53″, which add to PK as secondly assumed, namely to PKB 47° 44′ 6″, and PKA is made 50° 10′ 59″, and we exceed the due 50° 9′ 10″ by 1′ 49″. We understand, therefore, that this little excess is to be taken away from the second position of PK itself, and the anomaly of the eccentric sought, or PK, will be 47° 42′ 17″. This may be verified. For the sine of this arc is 73979, which claims from 11910″ 2° 26′ 50″ for KBA, and so this being added there is made 50° 9′ 7″, which differs insensibly from the due 50° 9′ 10″.
Of the deflection of the planets from the ecliptic.
What, then, is understood under the name of orbit? Properly, that line which the planet truly describes about the Sun with the centre of its body. As in diagram 82, if ECGD is part of the plane of the ecliptic, HCFD will be the orbit. Secondarily, however, there is understood also that great circle in which the plane of the orbit, continued, cuts the sphere of the fixed stars. As here the section MN, touched by the plane CAK continued.
What do you call the inclination of a planet, or of any point in its orbit, and what the circle of inclination? Inclination properly belongs not to planets or points, but to lines or planes among themselves; but because those planes are circumscribed about the orbits of the planets, and because the lines of the planets’ motions are understood to be described in the planes, it has been received in use that these words be simply transferred to the planets themselves, for brevity of speech.
Since, then, that which below, in book VI, will be called latitude partakes also of the adventitious or optical inequality, which we call the second, therefore, that things diverse may be distinguished also by names, let the true wandering of the planet from the ecliptic be called not latitude but inclination; and let it be defined thus: it is the arc of a great circle in the sphere of the fixed stars, described from the centre of the Sun, at right angles to the ecliptic, which circle may be called the circle of inclination, intercepted between the ecliptic and the eccentric place of the planet. Or it is the angle at the Sun which this arc measures.
In the diagram, if A is the Sun, FKDHC the orbit, MLO the ecliptic, the inclination of the point K will be the angle KAI or NAL, or its arc NL described from the Sun A.
What do you call nodes, what limits? The nodes are two points of the ecliptic in which it is cut by the plane of the orbit continued. In Greek anthebroi, because by them the diverse courses, of the apparent Sun and of the planet, are connected; one ascending, in which the planet, leaving the southern hemisphere, turns aside toward the north, the other descending, which carries the planet over to the south; the words ascending and descending being adapted to our hemisphere, as that in which the first discoverers of astronomy lived. As if the plane of the orbit and the plane of the ecliptic meet in the line CAD, showing the section, that line continued under the ecliptic will show the nodes M, O.
Limits are called the points of the ecliptic which are distant a quadrant from the nodes; the northern, from which the planet is distant toward the north, the southern, from which it is distant toward the south. They are called limits because the planet, coming to those points, wanders no further into those regions, but, turning thence, begins to return to the ecliptic. As in the diagram the points E, G of the ecliptic are called limits. But H, F also, points of the true orbit, and the points standing above them in the sphere of the fixed stars, come under the same name, and this more frequently.
What do you call the argument of inclination? It is the arc of the orbit of the planet under the fixed stars, intercepted between the ascending node and the eccentric place of the planet, numbered in the order of the signs. As if O is the ascending node, N the eccentric place of the planet, OMN will be the argument of the inclination LN. Copernicus takes for the ascending node the northern limit.
Is the greatest inclination of the limit in any planet the same in all ages? According to the physical principles used in book IV it is of itself immutable, but on account of the dislocation of the ecliptic itself, of which in book VII, it can accidentally be changed.
How is the inclination of the planet computed? In no other way than in book III the declination of a point of the ecliptic: the sine of the greatest inclination being multiplied into the sine of the argument of inclination, and 5 final figures cut off from the product, the sine of the inclination appears. See the process at folio 230 ff. If for the sines of the arcs you use their logarithms, the multiplication will be converted into a simple addition.
What is the eccentric place of the planet in the ecliptic? That point of the ecliptic in which it is cut by the circle of inclination drawn through the eccentric place, simply so called. As if the planet is in K, its eccentric place (so simply called) is N, and NL the circle of inclination, the angles NLM, NLO being right, L will be the eccentric place of the planet in the ecliptic. It is not called the ecliptic place simply, because this involves also the second inequality, the matter of book VI; but the word eccentric is added, that we may understand that the place is meant which is determined under the ecliptic by the eccentric alone, the concurrence of the great orb, of which in book VI, being now removed.
What is held to be the eccentric longitude of the planet? The arc of the ecliptic, numbered in the order of the signs from the beginning of Aries to the circle of inclination of the planet, or the eccentric place in the ecliptic. It is called eccentric not because it is numbered in the eccentric, but because the eccentric causes it.
What is called the reduction to the ecliptic? The small arc by which two arcs differ from each other, one of the orbit, the other of the ecliptic, begun from the common node and ended at the circle of inclination. As here the difference between MN and ML.
How is it computed? In no other way than in book III, folio 234 ff., the difference of the right ascension and the corresponding arc of the ecliptic. The sine of the complement of the greatest inclination is multiplied into the tangent of the argument of inclination, and 5 last figures being cut off from the product, the tangent of the reduced argument appears. Or the antilogarithm of the greatest inclination is added to the mesologarithm of the argument, and the mesologarithm of the reduced argument is thus heaped up.
A more useful compendium, for ascension also, is this. The greatest reduction, about the 45th degree from the node, multiplied into the sine of any arc doubled, 5 last figures being cut off, constitutes the reduction due to the proposed single arc.
How is this reduction to be used, and to what end? When the planet proceeds from the nodes to the limits, the reduction is to be subtracted from the argument of inclination, added when it proceeds from the limits to the nodes; and what is made up in this way, added to the place of the ascending node, constitutes the eccentric longitude of the place of the planet.
What do you call curtation? It is a small portion of the distance of the planet from the centre of the Sun, answering to the versed sine of the inclination of the planet, in that proportion in which the whole interval answers to the whole sine.
Let A be the Sun, P, Q the poles of the ecliptic, TAX represent the plane of the ecliptic, EAG the plane of the orbit; let the planet be now in E or G, and with centre A and intervals AE, AG let the arcs HE, GF be described, and from E, G let perpendiculars be dropped to TX, which are ER, GS; HR and SF will be the curtations.
What is the curtate distance? It is the straight line in the plane of the ecliptic between the centre of the Sun and the perpendicular from the centre of the body of the planet. In this diagram, the planet being in E or G, AR or AS is the curtate distance.
The proposed distance, expressed, is multiplied into the sine of the complement of the inclination answering to the proposed distance, and the last 5 figures are cast away from the product. Or the logarithm of the distance is added to the antilogarithms of the corresponding inclination and the index of the logarithm of the curtate distance is made.
Where is the distance most curtailed? About the limits, and more about that which is nearer the aphelion. As if V, Y are the limits, and so Z, W the poles of the orbit, and V nearer the aphelion than Y, HR will be longer than FS and longest of all.
Of the motion of the apsides and of the nodes.
How do you define the motion of the apsis in the primary planets? It is the arc of the orbit under the fixed stars, intercepted between that point of it which is equally distant from the ascending node with a certain point of the ecliptic (say with the beginning of Aries or even with the first star of Aries), and the place of the highest apsis, numbered in the order of the signs.
What kind of motion is this motion of the apsis? It is set down as uniform, 1) on account of the unobservable slowness, by which astronomers are hindered from considering this motion more exactly by single parts, 2) because we have an example of uniformity in one planet in which the period of the apsis is short, namely in the Moon. The physical principles of this motion, therefore, which we touched upon in book IV, folio 382, as resting on mere conjectures, can prejudice nothing against this uniformity, although by them this motion might seem capable of being made non-uniform. But of this more in book VI, under the several planets.
What is to be understood by the motion of the nodes in the primary planets, or what is the longitude of the node? The motion of the node is the arc of the ecliptic, numbered against the order of the signs from a certain point of it (say either from the beginning of Aries, or from the place of the first star of Aries) to the place of the ascending node. But if the numbering is made in the order of the signs, then this arc also can be called the longitude of the node.
What kind of motion is this of the nodes? Although it is reasonable that the motion of this point also is in itself uniform, yet some inequality seems to inhere in it by accident, on account of the dislocation of the ecliptic, of which in book VII.
What figures do the nodes and limits describe by their motions? The nodes move under the great circle of the ecliptic, but the limits of the orbit, in so far as their inclination is supposed to remain immutable, move in circles parallel to the ecliptic, or to that circle in respect of which the inclination is immutable.
To aid comprehension their motion can be set forth by a not inapt imagination of poles, provided we hold this, that physically speaking poles are not at all needed. As in the next diagram let the orbit be VY (the continuation of the plane transposed under the fixed stars), its poles Z, W move in small circles about the poles P, Q of the ecliptic TX. In whatever direction, then, Z verges from P at any time, in the same also will the limit V verge from the part T of the ecliptic and the limit Y from the part X, and, in the circuit of Z itself in a small circle which is parallel to TX, in the same direction will the limit V also follow in the northern parallel, so much the greater as it is nearer to TX itself, and so Y in the southern parallel. For these six points will always be in the same great circle of inclination: the poles of the orbit Z, W, the poles of the ecliptic P, Q, and the limits of the orbit V, Y.
So far, then, concerning the definitions of the terms of the planetary orbit and of the eccentric circle circumscribed about it, which, because they are common to all the planets, had to be premised in this fifth book. The use of these in the several planets we shall deliver in the following book VI.