[I.] From Parallax we conveniently make a step to the Refraction of the Fixed [stars], whence the following arguments seem able to be deduced against the excessive distance of the Fixed [stars] consequent upon the annual motion of the Earth.

I. Argument, from the Destruction of the sensible Refraction of the Fixed stars in our Air.

[Form.] If the Earth were turned in the annual orb, there would be no sensible Refractions of the Fixed [stars] in the air more closely surrounding the earth. But the Consequent is false. Therefore also that from which it follows.

[Margin: Proof of the Major.]

The Major is proved: since, every sensible inclination [angle] of the ray above the surface of our air being taken away, every sensible refraction would be taken away — since refraction, by the common opinion of the writers on Dioptrics (to be referred [to] bk. 10, sect. 6, ch. 2), is always less than the inclination of the ray in the same transparent [medium]; but if the Earth were turned in the annual orb, it would be necessary, to avoid the sensible parallax of the Fixed [stars], to remove them so far from us that the whole diameter of the annual orb — much less the orb of the Earth clothed with air — would be insensible with respect to the distance of the Fixed [stars]; or [the Earth] would be as a point escaping all visual angle, or incapable of founding an angle of a sensible inclination.

[Margin: A sign of the Refraction of the Fixed stars.]

The Minor is clear from the observations of Tycho and others, and ours: by which, comparing the true altitude of the Fixed [stars] (foreknown from calculation) with the seen or observed [altitude] — especially near the horizon — this [observed altitude] is found greater than the true; which cannot come about from any other [cause] than from Refraction, whose property is to raise the stars apparently. And Kepler himself, the keenest follower of the Copernicans, in the Epitome of Copernican Astronomy (bk. 1, p. 61), to confirm the Refractions of the Fixed [stars], adduces this from Tycho’s frequent observation: that the star of the Lion’s Tail [Denebola] and the Spike of Virgo [Spica], when they are near the Meridian, were found by instruments to be distant from each other 35 [degrees] and nearly two minutes; but when, in the eastern part, the Lion’s Tail reaches the altitude of 34° 30′, the Spike of Virgo, in nearly the same vertical, now begins to appear on the physical horizon — which could not happen unless a refraction of about 30 minutes diminished Spica’s distance from the Lion’s Tail, and raised it above either horizon.

[Margin: Response.]

It is responded, nevertheless, by denying the Major; and to its proof — the rest being conceded — it is denied that, from the smallness of the orb of the earth (taken with the refractive air) compared to the orb and distance of the Fixed [stars], it follows that there is no angle of inclination of the rays, or that it is insensible with respect to the eye on the surface of the Earth, or to the angle existing on the surface of the air; [even] granted that the whole sphere of the elements would appear, from the Fixed stars, as a point — nay, would not even appear [at all]: for this angle and its quantity we measure from the given refraction, and the altitude of the refractive air, and the semidiameter of the earth, as we shall teach in bk. 10, sect. 6, problem 45. But see here the response to the 3rd argument.

II. Argument, from the Excess of the Refraction over the Inclination.

[Margin: Form of the 2nd Argument.]

[II.] If the Earth were moved in the Annual Orb, the Refraction of the ray of the Fixed [stars] would come out greater than the Inclination of the same above the surface of the refractive air. But this is unfitting. Therefore, etc.

[Margin: Proof of the Major, and of the Minor.]

The Major is urged: because, on account of the huge distance of the Fixed [stars], the inclination would come out insensible, or much less than 30 minutes; and yet the horizontal Refraction of the Fixed [stars], from Tycho (vol. 1 of the Progymnasmata, p. 280), is 30′. The Minor is proved from the laws of the Opticians in Dioptrics: because the refraction is always less than the inclination — since from the angle of Refraction and the refracted angle the Inclination itself is composed (that is, an angle equal to the inclination), as we shall say in bk. 10, sect. 6, ch. 2, axiom 8.

It is responded by denying the Major; and to its proof the consequence is denied: for even so the horizontal refraction of the Fixed [stars] would be much less than the inclination of the ray, as will presently be clear from the response to the 3rd argument — however great a distance of the Fixed [stars] be imagined by the Copernicans.

III. Argument, from the Excess of the Refraction of the Fixed stars over half the Inclination.

[III.] If the Earth were moved in the Annual Orb, the Refraction of the Fixed [stars] would be greater than half the angle of Inclination of the ray of the same. But this is unfitting. Therefore, etc.

The Major is proved by the same reasoning by which the Major of the 1st and 2nd Argument was proved. The Minor is proved from the laws of the Opticians: for the refractions themselves in glass — which are greater than the refractions in air — up to the thirtieth degree of Inclination, are nearly the third part of the inclination (as we shall teach in bk. 10, sect. 6, ch. 3, prop. 3); but the sensible refractions of the Fixed [stars] cease before one reaches the thirtieth degree of inclination; therefore, before that [degree], they ought always to be less than half the inclination.

[Margin: Response.]

I respond by denying the Major: since the horizontal refractions of the Fixed [stars], which are the greatest, can never be equated to half the angle of inclination, however great a distance of the Fixed [stars] be posited by the Copernicans — which it will here be worth the trouble to demonstrate; for I know that a man otherwise most skilled in Optics, and most friendly to me, in letters sent to me, presumed from this matter a great argument against the Copernicans — but in vain.

[Margin: Exposition of the figure.]

From the center of the Earth A, let there be described three more-than-quadrant arcs of vertical circles: one, BCD, in the Heaven of the Fixed [stars]; another, EFG, of the refractive air; and a third, HIK, of the terrestrial surface; and let the true horizon be AL, and the physical [horizon] HC, and the vertical line AB, and the eye at H. But let a Fixed star be at D, distant from the center A by the straight line AD, and propagating the ray DF to the point F of the air, in such a way that it enters the air and does not tend straight to R, but from the point of incidence F is refracted to the eye H through the horizontal line HF, and so appears to the eye H as if it were at C; and let there be drawn from A, through F,…

[Engraved figure (#44) — the stellar-refraction construction. From the Earth’s center A spring three more-than-quadrant arcs: the innermost HIK (terrestrial surface), the middle EFG (top of the refractive air), and the outermost BCD (heaven of the Fixed stars). The vertical AB rises at left; the true horizon AL extends right (through L, N to the outer arc at the far-right point); the physical horizon HC runs from the eye H (on the surface arc) horizontally to C (on the outer arc). A star sits at D (lower right on the outer arc); its ray DF strikes the air at F — un-refracted it would continue to R (upper left, near T/E/Y), but it bends at F along the horizontal HF to the eye H, so the star appears at C. Lines AD, AF (produced to P), and the points I, K, G, O, 9 complete the geometry, illustrating that the horizontal refraction stays far below half the inclination angle.]

[…continues on p. 464 (PDF 499) with the catchword “per F” — “…and let there be drawn from A, through F, [a line],” continuing the demonstration of the refraction angles.]

(printed p. 464 — within Chapter XXXI: the figure-#44 demonstration is completed, showing that stellar refraction can never reach half the inclination however great the distance, so the objection collapses. Argument IV is then set out: if the stars’ horizontal refraction is 30 minutes (Tycho, Kepler), their distance must be far less than Copernicus requires — proved first geometrically from the figure, then from the proportion of the Moon’s, Sun’s, and stars’ refractions, capping the stars far below the Copernican minimum.)

[Header: BOOK IX. SECTION IV. — 464]

…[and let there be drawn from A, through F,] the perpendicular AP; for the refraction will be the angle HFR; and the refracted angle HFA [is] equal to the angle PFC, by [proposition] 15 of the first [book] of the Elements; and the angle of inclination PFD — by which, namely, the incident ray DF declines from the perpendicular PF — which angle, by the definitions of the Opticians, is equal to the refracted angle PFC (that is, AFH) and the angle of refraction CFD (that is, HFR) taken together; and [the inclination] is always, at least in air, more than double the angle of refraction.

[Margin: 1st Progression, for the Inclination.]

[IV.] These [things] being posited, the inclination PFD must be inquired — which will easily be done, the semidiameter of the earth AH being given (from bk. 2, ch. 7) [as] 4139 Bolognese Miles, and AF consisting of AI (which equals AH itself) and the altitude IF of the refractive air, which I shall show somewhere to be 20 Miles (bk. 10, sect. 6, problem 51); wherefore the whole AF is sometimes 4159 Miles. Therefore in the triangle AHF, right-angled at H (for the horizon HC makes right angles with the vertical AB), by the rules of triangles the angle AFH is found [to be] 84° 22′; and just so great is PFC also, to which if you add the horizontal refraction of the Fixed [stars] from Tycho, RFH (that is, CFD), 30′, the angle of Inclination PFD comes out [to be] 84° 52′.

[Margin: 2nd Progression, for the Parallax.]

Let us now posit the distance of the Fixed [stars] AD to be, first, as great as Tycho assumed — namely 14,000 terrestrial semidiameters, that is, 5,794,600 Bolognese Miles (namely 4139, [the value] of AH itself, multiplied by 14,000); for in the triangle AFD, AD being given [as] 5,794,600 — of which [units] AF is 4159 — and the angle AFD (which is the complement of the found inclination PFD, and so is 95° 8′), it follows that the parallactic angle FDA is 15 Seconds: which parallax is not of the orb of the earth taken bare, but of [the earth] increased by the thickness of the refractive air.

[Margin: 3rd Progression, for the decrement of the Inclination.]

[V.] Let the Heaven of the Fixed [stars] now be not the Tychonic BCD, but the Copernican ZOQ — which, on account of the huge distance, is just as if it were described from the center of the annual orb — so that the star which was supposed at D is [now] at Q, and that which appeared at C appears at O, through the refracted and horizontal ray HF; and let the ray incident at F be QF, and the inclination be PFQ. It is now proposed to find how much smaller the inclination PFQ can come out than PFD, the distance AD being increased and pushed forward up to Q, as far as the Copernicans wish — that is, how great is the angle DFQ, which is the difference of the aforesaid inclinations. But briefly I say that, even if AD be increased to infinity, the star Q remaining at the same true distance from the vertex Z (that is, [at] the angle ZAQ), the angle DFQ can never come out greater — nay, nor even equal — to the parallactic angle ADF previously posited; because, by [propositions] 16 and 32 of the first [book] of Euclid’s Elements, the external angle ADF is greater than either of the internal [angles] DFQ and DQF — being indeed equal to both of them together. Therefore the inclination PFD, just now (number 4) found [to be] 84° 52′, cannot come out less than 84° 51′ 45″ — even if you increase the distance AQ to infinity, because [no more] than 15″ can be subtracted from it. And the same I say of any other parallax ADF, than which it is certain that in the Fixed [stars] one of a [whole] degree cannot be imagined — nay, nor of 4 minutes; for it would exceed the Sun’s parallax. Just as it is not permitted to imagine the altitude of the refractive air greater than 100 Italian Miles, as we shall teach in bk. 10, sect. 6, problem 51. And so, all [things] being computed, the Inclination of the ray of the Fixed [stars] above the refractive air is greater than 84 degrees, nor can it ever so decrease but that it be always greater than double the horizontal refraction of the Fixed [stars] (if this be posited one degree less); nor, conversely, can the Refraction of the Fixed [stars] ever reach half of this inclination — that is, [reach] 42 degrees — since even if the air came out as dense as water, the refraction from air to water, corresponding to an inclination of 84 degrees, does not exceed 35 degrees, as is clear from Witelo (bk. 10, prop. 8) and from the anaclastic tables to be set at the end of bk. 10.

But someone will say that, with the distance of the Fixed [stars] increasing, the angle OFQ would be diminished, so that at last it would wholly perish, together with the refraction. For this argument will seem to have a greater appearance [of force], since it is easier — on account of the star’s distance freely increased — for the inclination of the ray so to decrease that it coincides with the horizontal line OF, than that it reach the perpendicular PF itself. It is responded, nevertheless, by denying the sequel: because, from what has been said, that angle could not decrease beyond 15″.

IV. Argument, from the Distance of the Fixed stars deduced from their Horizontal Refraction.

[Margin: Form of the 4th Argument.]

[VI.] If the horizontal Refraction of the Fixed [stars] is not less than 30 Minutes, it follows that their distance is much less than the hypothesis of the Copernicans (who turn the Earth in the annual orb) requires. But it is not less than 30′ Minutes; therefore the Earth is not turned in the annual orb.

The Minor is certain from Tycho’s observations (vol. 1 of the Progymnasmata, p. 280), nay from Kepler himself, who — although a Copernican — nevertheless in the Rudolphine [Tables], in the last precept and in the table of refractions which he sets there on the last page of the tables, admits the horizontal refraction of the Fixed [stars] [as] 30′; and so [do] most of the more recent Astronomers.

[Margin: 1st proof of the Major.]

The Major is proved, first, Geometrically, in the figure exhibited at number 3 [fig. #44], the triangles AHF and AFD being considered. For AH, the semidiameter of the Earth, is 4139 Bolognese Miles (from bk. 2, ch. 7); and AF cannot be greater than 4239 Miles, nor less than 4141 Miles, because the altitude IF of the refractive air cannot be imagined greater than 100 Miles, or less than 2 — nay, from what will be said (bk. 10, sect. 6, problem 51) it is about 20 — so that the whole AF is 4159 Miles. But from the side AH and the base AF, by the laws of triangles, the angle AFH is made manifest, to which is equal both the [angle] PFC at the vertex (by 15 of the first [book] of Euclid), and the alternate [angle] FAL, between the two parallel horizons (by 29 of the same book). Now if you add the refraction of 30′ (that is, the angle HFR, to which CFD is equal by 15 of the first [book] of Euclid) to the angle PFC, the inclination PFD will be known; and by 13 of the first [book] of Euclid the angle AFD will be manifest, as the complement to two right [angles]. Finally, the angle of the depth of the Fixed star which has undergone horizontal refraction — that is, LAD — is known: for it is as great as the horizontal refraction, if in computing that Tycho neglected the parallax of the Fixed [stars] (which he reckoned [to be] none); or, if he computed it, [LAD] is a little less than 30′, yet greater than 29′. But if it is greater than 29′ and less than 30′, and be added to the angle FAL (now known, as I said), the angle FAD will be known, in the triangle AFD, in which the angle AFD and the side AF are manifest; wherefore by the laws of triangles the side AD will be known — that is, the distance of the Fixed [stars] from the Earth; which, if LAD be greater than 29′ and less than 30′ (or even exactly 30′), will be much less than the Copernican hypothesis requires.

[Margin: 2nd proof of the Major.]

[VII.] It is proved, secondly, because as the difference of the horizontal refractions of the Sun and Moon is to the difference of their distances from the earth, so [is] the difference of the horizontal refractions of the Fixed [stars] and either Luminary, to the difference of the distance of the Fixed [stars] compared with the distance of either the Moon or the Sun. Now the horizontal refraction, from Tycho (vol. 1 of the Progymnasmata, pp. 79, 124, and 280) and from Kepler (at the end of the Rudolphine [Tables]), is: of the Moon 33′, of the Sun 34′, of the Fixed [stars] 30′. But on account of the excessive parallax of the Sun (as Kepler notes in the last precept of the Rudolphine [Tables]), the Solar [refraction] came out greater than the Lunar — when it ought, on account of the smaller Inclination following from the greater distance, to be smaller, and at most 32′. But the distance of the Sun to [that of] the Moon is, for Copernicus, as 1140 to about 60 — that is, twentyfold; but for Kepler, sixtyfold; and for Us, as 140 to 1 (from what is said in bk. 3, ch. 7, and bk. 4, ch. 14). Therefore, if the difference of two minutes between the refractions of the Sun and Moon does not suppose an excess of the Solar distance over the Lunar greater than twentyfold, or sixtyfold, or a-hundred-and-fortyfold; the difference also of two minutes, between the refraction of the Fixed [stars] and of the Sun, does not suppose a greater excess of the distance of the Fixed [stars] from the earth over the Solar distance than either twentyfold, or sixtyfold, or a-hundred-and-fortyfold. But the distance of the Sun, from what is said in bk. 3, ch. 7, is not greater (for Copernicus) than 1180 terrestrial semidiameters, (for Kepler) 3469, (for Us) 7600. Therefore the distance of the Fixed [stars] cannot be greater (for Copernicus) than 23,600 terrestrial semidiameters, (for Kepler) 208,140, (for Us) 1,064,000 — as is clear to one multiplying 1180 by 20, or 3469 by 60, or 7600 by 140. But these distances are much smaller than the Copernican hypothesis requires; since (from what is said in bk. 6, ch. 7, no. 17) among the Copernicans no one requires a smaller distance of the Fixed [stars] than 47,000,000 terrestrial semidiameters.

[…continues on p. 465 (PDF 500) with the catchword “VIII. Re-” (Respondetur) — the response to the fourth Argument.]

(printed p. 465 — Chapter XXXI closes with the responses to Argument IV (both proofs denied) and the brief Arguments V and VI on telescope lenses and the multiplication of parallaxes, likewise answered. Chapter XXXII then presents and answers a single argument from the excessive license of inventing diverse systems. Chapter XXXIII opens the grand Epitome, reducing all the foregoing arguments and solutions on the diurnal and annual motion to a summary, with Riccioli’s warning not to rely on the summary alone.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 465]

[Margin: 1st Response, to the 1st proof of the Major.]

[VIII.] It is responded, nevertheless, by denying the Major; and to its first proof — the rest being conceded — it is denied that, from the aforesaid angles and the side AF, there follows a distance of the Fixed [stars] smaller than the Copernican [one]. For if the angle LAD be precisely 30′ (as great as the refraction CFD), then FD and AD would be parallel (by 28 of the first [book] of Euclid), and so the triangle AFD would perish; and the distance AD could be infinite. Or surely, this triangle being retained, there would be in it more than two right angles — against 32 of the first [book] of Euclid. Which is easily shown. For let, from the calculation of number 4, the angle PFC (and so FAL) be 84° 22′; the refraction of 30′ being added to each, both PFD and FAD will come out 84° 52′; and by 13 of the first [book] of Euclid, AFD will be 95° 8′, as the complement to two right [angles]. Therefore AFD (95° 8′) and FAD (84° 52′) taken together will be 180 degrees — that is, two right [angles]; wherefore the angle ADF, [added] with them, however small it be imagined, will make more than two right angles. But if LAD be posited less than 30′ and greater than 29′, the parallactic angle ADF will not indeed be greater than one minute, but it can nevertheless — all the rest posited in the proof of the Major being safe — be of so few Seconds that the distance AD comes out infinitely greater than the Copernicans require, as is clear from what is said in bk. 6, ch. 7, no. 17. Wherefore, unless a Parallax of the Fixed [stars] be established, distinct from the simple Refraction (which raises the star from D to L), the whole Refraction — which raises it from D to C, overcoming the horizontal parallax (which is as great as the arc CL) — cannot be known; but the parallax of the Fixed [stars] itself cannot be known unless their distance be first supposed. Besides that the parallactic angle ADF is not [computed] from the semidiameter of the earth AH or AI, but from the semidiameter AF; and the horizontal Refraction makes the star existing at D appear higher by as much as the angle CFD is — which is somewhat greater than the arc CD, since it is described not from F but from A; wherefore the Refraction, rigorously taken, is greater than the arc CD. But let this be said for good measure.

[Margin: 2nd Response, to the 2nd proof of the Major.]

[IX.] To the 2nd Proof of the Major, the Major is denied — that is, that proportion of the distances and refractions, which is too loosely [rhetorically] feigned, and can be proved neither from the common [principles] of Optics or Dioptrics, nor from the proper principles of Anaclastics; but the decrement of the Refraction follows the decrement of the Inclination, which is unknown so long as the whole Refraction (including the correction of parallax) is unknown; but the parallax of the Fixed [stars] is unknown, so long as their distance is unknown.

V. Argument, from the Destruction of the Refractions of the Fixed stars in the lenses of the Telescope.

[Margin: Form of the 5th Argument.]

[X.] If the Earth were moved in the Annual Orb, the Fixed [stars] would not be represented greater, by the force of refraction, through the lenses of the Telescope, than without them. But they are [so] represented. Therefore, etc.

The Minor is clear. The Major is proved, because for that representation a sensible refraction, made toward the perpendicular in the glass lenses, is needed, as is known from Dioptrics. But if the Earth were turned in the Annual orb, on account of the huge distance of the Fixed [stars] there would be no sensible inclination of the ray of the Fixed [stars] upon the glass lenses; wherefore [there would be] no sensible refraction [either].

[Margin: Response to the Argument.]

I respond by Denying the Major, for the cause once stated in the response to the proof adduced for the major of the 1st argument.

VI. Argument, from the Multiplication, or uncertainty, of the True distance from the vertex [zenith], and of the Parallaxes.

[Margin: Form of the 6th Argument.]

[XI.] If the Earth is moved in the Annual Orb, either we are uncertain what the true distance of a star from the vertex is, or certainly a double distance is multiplied, and a double parallax, without necessity. But this is absurd. Therefore, etc.

The Minor is clear. The Major is proved, because if we are not certain whether the earth is moved or not, we do not know whence we ought to estimate and measure the true distance of a Fixed star from the vertex — whether from the center of the Earth, or from the center of the Annual Orb; but if from the center of the annual orb, then both this [distance] and the parallax of the annual orb above the parallax of the orb of the earth is multiplied without necessity — since, without these, on the hypothesis of a moving earth all the Phenomena are saved.

[Margin: Response to the Argument.]

It is responded by denying the first part of the Major (concerning the uncertainty) and the second (concerning the multiplication made without necessity). For by this very [fact], that the Earth is supposed to be moved in the annual orb, it is certain that the true distance of a Fixed star from the vertex must be estimated principally from the center of the Annual Orb; but the apparent [distance] from the surface of the Earth — granted that secondarily the true [distance] could be estimated from the center of the orb of the earth, provided that the one be not confounded with the other, but they be distinguished. As for the necessity: although it be not a posteriori from the phenomena, the Copernicans will nevertheless say it is a priori from most congruent reasons — lest other entities be multiplied without necessity; and so they will elude the argument.

✠ [Chapter XXXI ends here.]