Epitome Astronomiae Copernicanae: Liber V, part 2
What are the proper properties of the ellipse as identical? 1. It is established from the Conics of Apollonius of Perga that an ellipse, about which a circle is circumscribed on the common diameter (which is the longer diameter of the ellipse), cuts all the ordinates applied to that diameter in the same proportion of their segments.
So if KX, GF, DB, NA, SY are lines applied as ordinates to PR, then, if the curved line PLHEOTR is an ellipse, it must be that as DB is to BE, so is GF to FH and KX to XL, and so also NA to AO and SY to YT.
2. The ellipse has two points from which it is described as from centres, which I am accustomed to call foci. The lines, therefore, drawn from the two foci to any point of the ellipse, or even from one focus to the opposite points drawn from the centre of the ellipse, are always together equal to the longer diameter. Hence it follows that when they are drawn to those points of the ellipse which lie on the shorter diameter midway between the vertices, each of them equals the semidiameter of the circle.
So if A is a focus and B the centre of the circle, with AB and BF equal, F will be the other focus, and AH and HF together will be equal to the diameter PR; so also AL, LF, and AO, OF; and so, since BE is the shorter semidiameter and E a point on it, AE and EF will be equal, and each equal to the semidiameter BP, BR or BD.
This is applied to the planets thus: we have said that observation testifies that the planets are then distant from the Sun (the other focus of this ellipse) by the semidiameter of the eccentric circle, when they have completed exactly a quadrant of their orbit from the apsis P.
Demonstrate that these elliptical properties are represented in the orbit of the planet which arises from those librations. Let a new figure, then, be described by the laws hitherto delivered: with centre B, the circle PDR, which the ellipse is to touch, the longer diameter of which is PR, and on it A the focus, or place of the Sun. Draw through B a line perpendicular to PR, which shall be DT; the shorter diameter will lie on it. And since the eccentricity BA is half the libration, so much therefore belongs to a perfect quadrant; the planet, therefore, falling on the line DB, will be less distant from the Sun than in P by the difference BA; it will therefore be distant by the quantity BP. So let an interval equal to BP be extended from A to DB, and let its end be E. The orbit of the planet will therefore cut DB in E. Again, let an arc of the circle PG be taken, together with its sine or ordinate GFZ and its versed sine PF. Make it so that as BP is to PF, so BA, the half libration, is to the part belonging to PG; and let this part, taken from AP, leave a remainder which is extended from A to GF, its end falling at H. I say that as DB is to BE, so also is GF to FH. For let squares be drawn, GIOF upon GF, and HK upon HF, so that HIK is a gnomon; then let G be joined with A and with B, and from A let a perpendicular fall upon GB produced, which shall be AC.
I say first that the square on AC is equal to the gnomon HIK.
For since it was so made that as BP is to PF, so BA is to the difference of the lines AP, AH, therefore also as PB is to BF, so BA is to the excess by which AH still exceeds BP. But also as PB or GB is to BF, so AB is to BC, because the right-angled triangles GFB and ACB have equal angles GBF and ABC at the vertex. Therefore BC equals the portion by which AH exceeds BP; but CG also exceeds BP, that is BG, by the same portion BC, and so GC and HA are equal. But the square on the straight line GC together with the square on the perpendicular AC equals the square on GA. And on the other side the square on AF together with the square on FG equals the square on the same GA. Therefore the two squares on GF and FA together are equal to the squares on GC and CA together. Let equals be taken away: on one side the square on GC, on the other the square on the equal line AH, that is the two squares on AF and on FH, namely HK; there remains here the square on AC, there the gnomon HIK.
From this the rest of the proposed demonstration is now easily completed.
For as one sine GF is to its perpendicular AC, so are all the others to theirs from A. As, therefore, the square of the sine GO is to the square on AC, that is to the gnomon HIK, so are the squares of all the sines to their gnomons; and so, the gnomons being taken away, as the square GO of the one sine GF is to the square HK of FH, determined by the distance HA of the planet from the Sun, so is the square of each sine to the square of the lesser line determined by its own distance. But those whose squares are proportional to one another are themselves, as sides, proportional to one another. Therefore, as GF is to FH, the portion bounded by AH, so is any sine, as DB to BE, to the portion bounded by its own AE: which ratio is the true one of the ellipse.
The other property of the ellipse is evident of itself.
For according to the prescript of the laws of libration (because, namely, in the quadrant of the orbit PE half the libration, equal to BA, must be used up) we extend from A to DB a line equal to the remainder BP, namely AE. For since A is one focus, if BA is made equal to it on BP extended from B, the other focus will be marked out, whose distance from E will be equal to AE, and the two together will equal the diameter: which happens in the ellipse.
What is the proportion of DE, the breadth of the lune cut off from the circle by the ellipse, to the eccentricity BA? The eccentricity BA is a mean proportional between DE and ET. In the same way every perpendicular, such as AC, is a mean proportional between GH and HZ, the remainder of the subtense.
For the rectangle under GH and HZ is equal to the gnomon HIK. But this gnomon is equal to the square on AC, and so the rectangle GHZ is equal to the same square on AC. GH, AC, HZ are therefore in continued proportion.
What am I to hold concerning the length of this elliptical orbit and of its parts? The figures of the circle and the ellipse being cut by infinitely many ordinates GF, DB, the first portions ending in P (as GP to PH) will be as GF to FH; the last, ending in D, E (as GD to HE), will be equal to one another; so the proportion DB to BE, beginning from P, is gradually effaced, and in D, E vanishes into a mere proportion of equality. But whole arcs begun from P have to one another a proportion compounded of all the proportions of all the smallest particles, and so they never wholly lose the whole proportion DB to BE. For the quadrants DP to PE, and so also the whole circular line to the whole elliptical, are as DB to the arithmetical mean between DB, BE, which is a little longer than the proportional mean.
Since the elliptical plane will also be of use, I ask in what proportion the plane of the ellipse stands to the plane of the circle, and likewise the plane of any segment of the semicircle to the plane of the segment of the semiellipse made by the same ordinate? Apollonius demonstrates in the Conics that the proportion of the longer diameter to the shorter holds everywhere. So if DB, GF are ordinates: as DB is to BE, so is the area of the semicircle PDR to the area of the semiellipse PER, and as GF is to FH, that is DB to BE, so also is the segment of the semicircle GPF to the segment of the semiellipse HPF, and so also the greater segment of the semicircle GRF to the greater segment of the semiellipse HRF.
Now let the semicircle be cut by the straight line GA, and the semiellipse by the straight line HA: the triangles HAF, GAF will be of the same altitude FA, and so as the base GF is to the base FH, so is the area GAF to the area FAH. But as GF is to FH, so also is the area GPF to the area FPH; and so as GF to FH, or as DB to BE, so also is the composite area PGA to the composite area PHA.
Finally, I should like to know also the proportion of the lines from the centre of the figure to the circumference of the ellipse to the semidiameter of the circle. The shortest, indeed, as BE, is less than the semidiameter BD by the whole breadth of the lune DE. But all the rest, as BH, fall short of the semidiameter BG by less than the breadth of the lune at any place, as GH.
For the two sides GH, BH of the triangle GHB together must exceed the third, GB. The proportion, therefore, of the defect in E to the defect in H is greater than DE to GH: but this is that of the sines DB to GF. Greater, therefore, is the proportion of the defect in E to the defect in H than that of the sines DB to GF.
Conversely, the proportion of the squares of GF and HF is the double of that of GF to HF. But when the squares of BF are added to the squares of GF and HF, the sums of the squares make a smaller proportion; and so their sides GB, BH will make a smaller proportion than GF, FH. The greater, therefore, BF is, the more the proportion GB to BH is diminished, so that it does not equal GF to FH. And conversely, the more PF increases, the more the proportion GB, BH also increases, approaching the proportion GF, FH. But PF increases slowly from P, and swiftly near DB. Therefore, if GH remained everywhere of the same size, it would vary the defect HB slowly about P and swiftly about D. But GH does not remain, but increases swiftly about P and slowly about E, namely with the sines GF, DB themselves. Again, therefore, the defect HB increases swiftly about P and slowly about E. Smaller, therefore, is the proportion of the defect EB to the defect HB than that of the versed sine PB to the versed sine PF. But also the proportion of the arc DP to PG is indeed greater than that of the sine DB to the sine GF, and less than that of the versed sine BP to the versed sine FP. Therefore the proportion of the defect of the lines BH approaches the proportion of the degrees PG. Yet it tends toward D to the proportion of the sines DB to GF, but toward P to the proportion of the versed sines BP to FP. (Compare p. 77.)
IV. Of the measure of time, or of the delay of the planet in any arc of the orbit.
In what way does the plane of the elliptical segment become fit for measuring the delay of the planet in the arc of that segment? In no other way than if, by the division of the circle into equal parts, unequal arcs of the ellipse are set up, small about the apsides and larger about the mean longitudes, in this manner.
With centre B and interval BP let the circle PDRT be described, whose diameter is PBR, and in it, as in the line of apsides, let A be the Sun, the source of motion toward R, AB the eccentricity, and equal to it BV toward P, so that P, R are the apsides.
Now, A and V being the foci, let an ellipse be described, touching the circle in P, R, which shall be PERI, representing the orbit of the planet; and let its shorter diameter be EI, and that of the circle DT, erected at right angles to PR.
Now let the semicircle PDR be divided into small equal parts, and let P, O, N, D, R be the marks between the divisions, from which let perpendiculars to the line of apsides PR be drawn, as OM, NK, cutting the ellipse in the points C, K.
The points C, K, E, I of the sections being joined with the Sun A, I say that the delay of the planet in the arc PC is measured by the area PCA; so the measure of the delay in the arc PCK is the area PCKA, and of the delay in PE the area PEA, and finally of the delay in PER, the half of the orbit from the apsis P to the apsis R, the area PERP, which is likewise the half of the area of the whole ellipse PERIP.
Show in what proportion, by this division of the planet’s orbit, the middle parts become greater than the parts about the apsides. In the proportion of the longer semidiameter to the shorter.
Let there be in the circle equal parts PO and ND, the former at the apsis P, the latter at the mean longitude D. Since, then, to these there correspond, of the divided ellipse, the arcs PC, KE, it has been said above that KE is equal to ND (the division being supposed very minute); KE will therefore also be equal to PO. It has been said further that as OM is to MC, that is DB to BE, or the longer semidiameter PB to the shorter BE, so is the arc PO of the circle to the arc PC of the ellipse: as, therefore, PB to BE, so also will be the arc KE of the ellipse at the mean longitude to the arc PC at the apsis.
What follows from this division of the elliptical orbit into unequal arcs? This follows: that to the arcs of the orbit about both apsides taken together, which are the lesser, and to the arcs about each mean longitude taken together, which are the greater, equal areas are assigned as the measures of the delays in them; though those taken together are equally distant from the Sun with these taken together.
For let PC and RG be equal as above; then the areas PCB and RGB will also be equal. Let KE and LI again be equal to each other, but greater than the former, as has now been demonstrated: the areas KEB and LIB will also be equal.
Now it has been demonstrated that as PB is to BE, so (in the division of the orbit given) KE is to PC. The triangles BPC and BEK (rectilinear, or nearly so) are therefore reciprocally proportional, because as the altitude BP of the one is to the altitude BE of the other, so is the base KE of the latter to the base PC of the former. Therefore the areas BEK and BPC are equal to each other. And so the areas of BEK, BIL together are equal to the areas of BPC, BRG together. But BPC, BRG together are equal to APC, ARG together, because the altitudes BP, BR together are equal to the altitudes AP, AR together. And BEK, BIL together are equal to AEK, AIL together, because on the bases EK, IL, or their tangents at E, I, the triangles BEK, AEK, and likewise BIL, AIL, have the same altitudes BE, BI and the same bases, EK for the former, IL for the latter. Therefore here the areas EAK, IAL are assigned to the long arcs KE, LI, and to the shorter arcs PC, RG together equal areas APC, ARG are assigned, though the distances of the former, EA, AI together, are equal to those of the latter, PA, AR together, as was demonstrated before.
If equal areas are assigned to unequal arcs equally distant from the Sun, while the times, or delays, of unequal arcs equally distant from the Sun must also be unequal, by the axiom made use of above: how then will equal areas measure unequal delays? Although in this way the pairs of arcs are in truth unequal to one another, they are nevertheless equivalent to equals in sharing the periodic time.
It was said above that, the orbit being divided into equal minute particles, the delays of the planet through them increase in the proportion of the intervals between them and the Sun. But this is to be understood not of complete equality of the portions, but chiefly of those which are directly presented to the Sun, as PC, RG, where the angles APC, ARG are right; in the rest, which are obliquely presented, it is to be understood only of that part of each of those portions which belongs to the motion about the Sun. For since the orbit of the planet is eccentric, two elements of motion are mingled to form it, as has been demonstrated hitherto: one of circulation about the Sun by the Sun’s one power, the other of libration toward the Sun by another power of the Sun, distinct from the former. As in IL, the ends I and L have unequal distances from A, the source of motion; AL being therefore continued to Q, so that AQ is a mean in quantity between AL and AI, and the arc QS being described with centre A and interval AQ, cutting the longer AI in S, the arc QS belongs to the former element of the compound motion, but the difference between AL and AI, that is LQ and SI together, are portions of the latter element of motion, which must now be separated in the mind: for nothing of the periodic time is owed to it, since it has already received its portion above, where libration was treated, by other laws. But this second element of motion cannot be separated in any other way than by that division of the orbit into unequal parts which we gave above. For by as much as KE, LI together exceed PC, RG together, the whole of this belongs to the latter element of motion, and that excess being separated, there remains for the former element something equal to PC, RG together, which I demonstrate thus.
For since AE, AI have been demonstrated above to be equal to BP, BR, therefore, arcs being described through the points E, I, of which the one cuts off and excludes toward K from the area AEK as much as the other adds to AIL above L, so that the triangles (more truly, sectors) acquire these new straight bases in place of the oblique bases KE, LI, it follows that the area equal to PCB, RGB together, applied to AE, AI, will make the bases, or arcs, described through E, I equal to the bases described through P, R. But it was demonstrated before that the areas KEA, LIA together are equal to PCB, RGB together. What, therefore, belongs of the oblique bases KE, LI to the circulation about the Sun is equal to the arcs PC, RG together, where almost no libration toward the Sun is mingled with it, because AP, AC differ insensibly, and so do AR, AG.
The same will be demonstrated also of other particles of the orbit: as if CF be taken, and, CB, FB being continued to G and H, the corresponding GH be added, and the four points be joined with A, the source of motion. For it was demonstrated above that CA, AG together, and also FA, AH together, are equal to PA, AR together, that is to the longer diameter PR; so also, as before, the areas ACF, AGH together will be equal to BCF, BGH together, and through these to APC, ARG together, although CF, by the manner of division adopted, has turned out a little longer than PC and GH longer than RG. For the new arcs described with centre A and intervals AC, AG, and cutting AF, AH, taken together will equal the arcs PC, RG, because by as much as the former is an arc of a greater circle than the latter, by so much the smaller angle CAF it measures, and so much the greater the other measures, GAH, so that the angles CAF, GAH together always remain equal to PAC, RAG together.
Since, then, the equality of the other element in the planet’s motion, namely of its advance about the Sun, consists in the equality of the angles about the Sun A, namely of pairs of them taken together, and since the area of the ellipse is distributed equally among the arcs which subtend these angles, namely pairs of areas always equal to other pairs: area is rightly established as the measure of time (so far at least, and as far as we treat of pairs of arcs); since the delays of time, at the same distance from the Sun, are owed to equal advances about the Sun, not to arcs wholly equal.
Let the area of the ellipse, then, be rightly distributed in this way among the pairs of opposite arcs: demonstrate now that the single triangles, taken separately, are the most exact measures of the single delays. The demonstration is easy from what has been premised.
For since, according to our axiom, the delay of the planet in the arc PC is to the delay in the equal arc RG as the distance of the former from the source of motion, AP, is to the distance of the latter, AR; and since the area of the triangle PCA is also to the area of the triangle RGA (which has the base RG equal to the base PC of the former) as the altitude PA of the former is to the altitude RA of the latter: the delay of the planet in the arc PC is therefore to the delay in the equal arc RG as the area of the triangle PCA is to the area of the triangle RGA.
In the same way it will be demonstrated that the delay of the planet in CF, equal in power to CP, is to its delay in GH as the area ACF is to the area AGH, where the sum of both areas is equal to the sum of the former, and so on. The whole area of the ellipse, therefore, cut from A into triangles, is distributed among the arcs in the same proportion in which the whole periodic time is distributed among them. The single triangles are therefore in most exact proportion the measures of their single arcs.
The demonstration of this complete equivalence is given in the Commentaries on Mars, chapter LIX, folio 291 (III. 404), in the thirteenth line from the end of which folio a single little word has introduced great obscurity; if you change it to computaretur, all will be plainer. Though I confess that it is there delivered more obscurely, and with more labour, because the distances are there considered not as triangles but as numbers and lines.
V. Of the equivalence of the circular plane and the elliptical plane in measuring the delays of arcs.
It seems a hard, unusual, even intricate business that the computer should be reduced, in the computation of time, to the plane of the elliptical figure. Yet by the use of the circular plane in place of the elliptical, it becomes, in the opinion of all, easier, so that the old calculus is in no way comparable in facility with this new one.
Demonstrate the equivalence of the planes, for the sake of measuring time. Let figure 77, page 408, be repeated, by which we demonstrated the generation of the elliptical plane.
And since it has so far been demonstrated that as the half of the periodic time, in which the planet traverses the half orbit PER, is to the time which the planet consumes in PH or in PE, so exactly is the area PER to the area PHA or PEA; and since it was also demonstrated above that the area PDR is to PER as PGA to PHA and as PDA to PEA (for the proportion of all was the same, namely DB to BE), and so also alternately: as the area PER is to PHA or PEA, so is the area PDR to PGA or PDA; as, therefore, the half of the periodic time of the arc PER is to the times of the arc PH or PE, so is the area PDR to PGA or PDA. Therefore in these segments of the semicircular plane there lies a most exact measure of the delays which the planet makes in any arc whatever of the ellipse.
Show now also the convenience of this measurement. The segment PGA being taken, let a straight line be drawn from G to the centre B. The proportion of the sector GBP to the whole plane of the circle is therefore given from the given quantity of the arc PG, so that there is no need of computation. For the whole periodic time and the whole plane of the circle are divided into 360 parts in the astronomical manner. There remains, then, the other part of the segment, GBA. But the computation of this is easy. For as DB, the whole sine, is to GF, the sine of the given arc PG, so is DBA to the area GBA. Once, therefore, the area of the greatest triangle DBA has been established, namely by multiplying half the eccentricity into the whole sine and converting the product into astronomical denominations, it will afterwards always be serviceable.
Is there, moreover, any other use of the circular plane? There is a peculiar use of it in the theory of the Moon, for demonstrating one of its inequalities, which the Moon has singularly beyond the other planets. But because this fifth book is given only to those properties which are common to all the planets, what remains of geometrical apparatus for completing the demonstration of this singular use is rightly deferred to part 4 of book VI, namely to the theory of the Moon itself.
By what method does the old Ptolemaic astronomy measure the delays of a planet in any arc of its eccentric, or what has it in place of the circular plane? It uses for this a peculiar circle, to which the name equant has been given, whose centre would be, in our figures, the other of the foci, in the next diagram V, in the antepenultimate F, because it is as far distant from the centre of the eccentric B toward the highest apsis P as the Sun A is distant from the same centre of the eccentric toward the lowest R. For a line being drawn from the centre of the equant V through the body of the planet, the arc of this equant, intercepted between this line and VP, the line of apsides, is set up as the measure of the time which the planet consumes in the arc of its orbit.
This hypothesis seems more convenient for manual demonstrations by instruments, called theorics: why do you not retain it, when you have yourself already twice employed vicarious quantities in place of the true? 1. Because the equant never speaks perfectly true, unless we wish to make its centre librate with an unequal motion, by which course we should depart from the simplicity of the hypotheses and constitute an astronomy far more perplexed and laborious in use than it is in these two books, the fourth and fifth, in the explanation of causes: since, these causes once perceived, or even not believed but at least posited, the use afterwards, in the second part of book V and in book VI, is easy. 2. Because the principle of this equant is with Ptolemy one thing in the superior planets, another in the two inferior, another in the Moon, and would now be another again in the Sun; but the plane of the eccentric circle serves with us in all the planets for the same use in the same way. 3. Because the equant circle departs furthest from the genuine causes of the motions, which the plane of the circle represents closely, being in the same genus with the plane of the ellipse.
Let the same be understood as said also against the other equivalents which the wonderful power of human ingenuity is wont to produce, such as that which David Fabricius gives: by a single libration (though needing two equal circles with contrary motions) of the centre of the eccentric on the shorter diameter of our ellipse, he both saves the planet’s entries at the sides of our immovable eccentric circle, and at the same time librates the apsis, so that now the eccentric circle itself, the count being carried on from the librating apsis to the body of the planet, furnishes us the measure of time. For neither is mere uniformity of motions obtained, nor complete exactness, nor is there any saving of labour, and the causes of the motions are hidden and denied.
The Copernican contrivance is repudiated in every way, which carries two epicycles about on a concentric in the double proportion of the motions: for since observations testify that the planet enters at the sides, at places midway between the apsides, this Copernican hypothesis makes it rather wander outward in the contrary manner. This part of Copernicus’s hypotheses must be wholly amended, saving however his universal hypothesis of the annual motion of the Earth, from which this doctrine takes its name.
VI. Of the regularity of the excursions to the sides.
Is the calculus of latitude also certain, if there are no solid orbs, and if the peculiar filaments in the body of the planet supply this? Granting what is laid down in book IV, folio 383 ff., which is wholly both possible and consistent: it is altogether necessary that a perfect plane of an ellipse arise.
For let TZX in the present diagram be a circle through the poles of the ecliptic, and let A, or the Sun, be in it, if TZX is a plane, or, if TAX is a hemisphere, let A be first the place of the lower section in the hollow, of the ecliptic TX with the orbit EG of the planet, so that its poles are beneath Z, W. Let the fibres of latitude be directed along GA and have the faculty of deflecting the motion XVT imparted by the Sun by the angle GAX; and let the fibres remain parallel throughout the whole circuit. It is manifest that, the planet being in A, the lower section, the fibres stretched along GA will direct the planet through the whole angle, and the planet will come in a perfect plane as far as G, ascending until it reaches the plane drawn through the poles. And since the fibre from G is now directed at the Sun A itself, not across the ecliptic, therefore the planet will no longer wander out here, but G will be the limit; thence, raised gradually above the plane ZXW, it will direct the fibre along the line drawn from the section A through the Sun A, until it comes to A, the upper section now, of the convex surface.
As, therefore, in A the angle of inclination of the fibre to the ecliptic TX is greatest, and decreases swiftly, while in G, E there is no angle of inclination of the fibre to the length of the ecliptic, and this smallness of inclination persists for a long time: so also, if a whole plane is made of the circuit EAG, the parts of it at A are most inclined to the ecliptic TX and the inclination decreases quickly. But about G, E the margin of the plane, understood as extended downward into the depth of the sphere or upward, runs for a long time almost parallel to the plane of the ecliptic. Therefore if in place of the operation of the fibre we use the work itself, namely EAG, as a perfect plane, the calculus will be wholly in agreement with the principles.
Conclusion of the first part of book V
Let these things, then, so far written be for geometers endowed with a keen mind, who deign to admit nothing into calculation which is not fortified by the most exact demonstration and deduced from the natural principles of the motions themselves.