[Margin: Opinion concerning the magnitude of the Fixed stars on the Copernican hypothesis.]

[I.] Much more absurd seemed to most the huge Magnitude of the Fixed [stars] which follows from their distance on the Copernican hypothesis, than the bulk of the whole sphere of the Fixed [stars] — whether the Fixed [stars] themselves be compared with the globe of the Earth, or with the Sun, or with the whole Annual Orb. But [this Magnitude], greater than just, was believed by the Copernicans to have come out [so] for Tycho, Longomontanus, Scheiner, Claramonti, and some others, because they assumed too great an apparent diameter of the Fixed [stars]. For Tycho, in the letter of the year 1589, written Nov. 24 to Rothmann, p. 167, says: Nay, then too the Fixed stars of the third magnitude, which have one Minute in diameter, will necessarily be equal to this whole Annual Orb — that is, will comprehend in diameter 2284 semidiameters of the Earth, for they will be distant 7,850,000 [by] the same semidiameters very nearly. What shall we say of the stars of the first magnitude, some of which occupy two, some three Minutes in visible diameter? Which he repeats on p. 191. But he assumes from Copernicus the semidiameter of the annual orb [as] 1142 terrestrial semidiameters, and so the diameter [as] 2284.

[Margin: Method of investigating the true magnitude of a Star.]

[II.] Longomontanus likewise (bk. 1 of the Theorica, ch. 1, p. 1), by an added diagram, investigates the magnitude of the fixed [stars] — in the way we too did (bk. 6, ch. 9, no. 9). For let ABC be a portion of the circumference of the supreme heaven, described from center G; and let DHEF be some Fixed star, whose center [is] B; and let there be drawn from G the straight lines GD and GE, touching the globe of the star at D and E; for, by [proposition] 18 of the third [book] of Euclid, they will make the angle D and E right with the star’s semidiameter (on this side BD, on that BE), and the angle DGE will subtend and measure the apparent diameter of the star. And the angle G being bisected by the straight line BG, there will arise a double triangle, right-angled at D and at E. Therefore, if in either triangle — say in DBG — the base BG be given (that is, the distance of the Fixed star from the World’s center G), and the angle DGB (measuring the apparent semidiameter of the star), then by the rules of right-angled rectilinear triangles the true quantity of the semidiameter DB will at once be found, in parts of which GB is known; and BD doubled, the diameter will be known, which multiplied by itself will give the square number, and this again multiplied by the number of the diameter will give the cubic number, indicating how many times the solidity of the earth is contained in the solidity of the star — provided the true diameter of the star be known in diameters of the earth.

[Engraved figure (#43) — Longomontanus’s star-measuring construction. A circle (the Fixed star) has center B, with H at its top and F at its bottom; a horizontal arc A–D–…–E–C (a portion of the supreme heaven’s circumference) crosses through the star near B. From the apex G far below, two straight lines rise tangent to the star-circle at D and at E, and the vertical line H–B–F–G bisects the apex-angle — forming a tall narrow “ice-cream-cone” of the star atop a long thin isosceles triangle, whose apex-angle DGE at the World-center G is the star’s tiny apparent diameter.]

Therefore Longomontanus, having found, from the Copernican hypothesis, the distance of the Fixed [stars] BG [to be] 7,906,818 terrestrial semidiameters, and having assumed the apparent semidiameter of a Fixed star of the first magnitude (that is, the angle BGD) [to be] 1 Minute, makes [it]: as the Radius BG of 10,000,000 parts to the distance BG of 7,906,818 terrestrial semidiameters, so the angle BGD of 1 Minute [is to] the Tangent 2909, to BD, 2300 terrestrial semidiameters; and so he gathers the diameter of such a star [to be] 2300 terrestrial diameters; of which number, multiplied into itself, the square number is 5,290,000; and this again multiplied by 2300 makes the Cube 12,167,000,000. So many times, then, he says, a star of the first magnitude contains the Earth, on the Copernican hypothesis — the cube of the terrestrial diameter, taken as 1, being 1. But since the Sun contains the Earth 140 times (as Longomontanus thinks), if 12,167,000,000 be divided by 140, the same star will be found greater than the Sun 86,907,143 times. Finally, since the diameter of the Annual Orb is 1142 Earth-diameters for Copernicus, but the diameter of the aforesaid Fixed [star] is 2300 Earth-diameters — that is, about double, and is to it as 2 to 1, and the cube of the number 2 is 8, and the cube of unity is 1 — it follows that a Fixed star of the first rank comes out eightfold greater than the whole annual orb. Wherefore Longomontanus concludes, in the same place: Since, therefore, both from the immense removal of the fixed stars from the earth, and the gap between the last of the Planets and the orb of those same Fixed [stars] — [and even more] than from the incredible [magnitude] which hence follows for the annual orb of the earth, likewise the Sun, and much more for the earth, [in] the magnitude of a fixed [star] of the first rank — all the well-constituted symmetry of the mundane parts is easily destroyed, besides [other] absurdity, etc.: therefore I judge that the Copernican hypothesis — especially concerning the annual motion of the earth, and concerning its libration upon its own poles — must rightly be rejected.

[Margin: Scheiner’s opinion on the magnitude of the Fixed stars on the Copernican hypothesis.]

[III.] But Christoph Scheiner, in the Mathematical Disquisitions, p. 26, having derived from the Copernican hypothesis the distance of the Fixed [stars] [as] 13,133,376 terrestrial semidiameters, assumes the Sun’s distance [as] 1208 terrestrial semidiameters, and its true diameter [as] 11 terrestrial semidiameters; multiplying 13,133,376 by these [11], he gathers the sum 144,467,136; which divided by 1208, he finds 119,592 terrestrial semidiameters, which the Sun’s diameter would occupy if it were in the Firmament. Then, the visual diameter of the Sun’s apogee being posited [as] 30 minutes, and the visual diameter of a fixed star of the first magnitude [as] 2 minutes, he makes [it]: as 30 to 119,592, so 2 to 7972 terrestrial semidiameters, included in the diameter of a fixed star; and by the same reasoning, for a star of one apparent minute (or of the third magnitude), he gathers 3986 semidiameters. Hence he deduces that the diameter of the annual orb, which is 2416 terrestrial semidiameters, is contained in the diameter of a star of the first rank 3 181/604 times, and in the diameter of a star of the third rank 1 [+ a fraction] times, and in a star of the sixth order [magnitude] nearly once. Finally, since the cube of the diameter of the annual orb, taken as 1, is 1, and the cube of the diameter of a Fixed star of the first rank, taken as 3 18/[..], is nearly 32, he gathers that a star such as is the Dog-star [Sirius] contains in itself the annual orb 32 times. Therefore (p. 28) he concludes thus: The Copernican Motion of the Earth makes any first[-magnitude] star thirtyfold, and so much ampler than the whole heaven of the Sun; and by the same [token makes] any little star seen in the starry sphere greater [than it appears]. Therefore it does not seem to be admitted.

[Margin: Claramonti’s opinion on the magnitude of the Fixed stars on the Copernican hypothesis. And Lansberg’s.]

[IV.] By what method, moreover, Claramonti (in the defense of the Anti-Tycho, part 3, ch. 13) gathers that a fixed star of the first magnitude, on the Copernican hypothesis, contains the globe of the earth 1,349,232,625 times, and the Sun 8,079,237 [times] — has already been said in the preceding chapter, number 12. But among the Copernicans, Lansberg (in the Uranometria, bk. 3, from element 20), assigning to a Fixed star of the first magnitude a diameter of 1 minute, and to the distance of the Fixed [stars] 41,958,000 terrestrial semidiameters, gathers (by the method of number 2) the true diameter of a Fixed star, and establishes it [as] 40,712 terrestrial diameters; and that in its solidity the Great Orb is contained 20,053 times — for the diameter of the annual orb is for him 1498½ Earth-diameters, whose cube is 3,364,884,747, but the cube of the number 40,712 is 67,478,794,224,128, which contains the former cube 20,053 times. But for a star of the sixth rank he makes the apparent semidiameter 2½″, and the true [semidiameter] 3388 terrestrial semidiameters; whence he deduces that its diameter is to the diameter of the Annual Orb as 3388 to 1498½, and that its solidity (or cube) is greater than the solidity of the annual orb 11½ times. Concerning the remaining stars, moreover, see what I related in bk. 6, ch. 9, at the end of Scholium 4.

[Margin: Hortensius’s opinion on the same.]

[V.] Hortensius, nevertheless — himself also an Astronomer of the Copernican sect — in the dissertation with Gassendi On Mercury seen under the Sun…

[…continues on p. 461 (PDF 496) with the catchword “Sole” — “…and Venus unseen,” where Hortensius gives 1st-magnitude stars only 8″, making them smaller than the Great Orb 422 times.]

(printed p. 461 — within Chapter XXX: the size-survey finishes with Hortensius and Galileo, who make the stars’ apparent diameters (and hence bulks) far smaller. Riccioli then sets out his own measurements with Grimaldi — Sirius 18 seconds, Alcor 4 seconds 24 thirds — and tabulates in four tables the stars’ true sizes and their ratios to the annual orb on the various distances. Argument I of the chapter is then formed: on a moving Earth even the faintest fixed stars would exceed the Sun and the whole annual orb, sealed by Rothmann’s own confession to Tycho.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 461]

…[On Mercury seen under the Sun], and Venus unseen, gives to stars of the first magnitude only 8″ for the apparent diameter, and to [those] of the sixth magnitude 2″; whence, both from the distance of the Fixed [stars] (which for him is 10,312,227 terrestrial semidiameters) and from the semidiameter of the Annual Orb (1498½ terrestrial semidiameters), he gathers that a star of the first magnitude is smaller than the Great Orb 422 times, and a star of the third magnitude 27,826 times — so far is it from being equal to it or greater.

[Margin: Galileo’s opinion on the same magnitude of the Fixed stars.]

But Galileo makes them yet smaller; for in Dialogue 3, On the System of the World (Latin p. 265, Italian 351), he rebukes those who, from Copernicus’s system, gather that some Fixed star is greater than the whole Great Orb; and (p. 267) he teaches a way of measuring the apparent diameter of the Fixed [stars], from which he asserts (p. 268) that the apparent diameter of a star of the first magnitude does not exceed 5″ — from which there follows a bulk far smaller than [that] of the annual orb. But his method I related in bk. 6, ch. 9, no. 3.

[Margin: Our opinion on the Magnitude of the Fixed stars.]

[VI.] But I — what I have accomplished, together with Fr. Francesco Maria Grimaldi, for investigating in a far more certain way the apparent diameters of the Fixed [stars] — I have already narrated (bk. 6, ch. 9, from number 6; bk. 7, sect. 6, ch. 9), and have illustrated with examples and demonstrations (bk. 7, sect. 6, ch. 10 and 11); and I showed that a star of the first magnitude, such as is Sirius, has an apparent diameter of 18″; but the least of the stars visible to the naked eye — of which kind is the little star in the tail of the Great Bear called Alcor — has a diameter of 4″ 24‴. From these, according to the various opinions about the distance of the Fixed [stars], I have deduced the various true Magnitude of these stars, and digested [it] into three tables, exhibited in that chapter [11], of which the last two rest on the Copernican distances — to which I judge the Reader [is] to be referred, for ampler erudition. Yet I shall here repeat what is more necessary to declare the force of this argument which I have in hand: and in the 1st and 3rd table I shall set the cubic Earth-diameters which are contained in the body of Sirius and of Alcor (fixed stars), that from these two — the greatest and least of the Fixed [stars] visible to the naked eye — a judgment may be made about the rest; but in the 2nd and 4th table I shall set how many times stars of this kind contain the Annual Orb, or are contained in it — namely, by dividing the number of cubic Earth-diameters contained in the body of the star, by the number of cubic [Earth-]diameters contained in the annual orb (taken from the preceding chapter, number 15, in the 1st column of table 3).

[Engraved table — TABLE I. The True Magnitude of the greatest of the naked-eye Fixed stars (SIRIUS) and of the least (ALCOR), the apparent diameter of Sirius being posited 18″ and Alcor 4″ 24‴, and the distance of the Fixed stars asserted by the four below-written Copernicans. (Diameters in Earth-diameters; Bodies as multiples of the Earth.)]

[Engraved table — TABLE II. Therefore SIRIUS [and] ALCOR [contains / is contained in the annual orb] so many Times:]

[Engraved table — TABLE III. The True Magnitude of SIRIUS and ALCOR, their apparent semidiameter being supposed (Sirius 18″, Alcor 4″ 24‴), and the distance of the Fixed stars [taken] so great that the Annual-orb Parallax does not exceed 10″, and the semidiameter of the Annual Orb [taken] from the below-written Authors:]

[Engraved table — TABLE IV. Therefore SIRIUS [and] ALCOR contains the Annual Orb so many Times:]

[Margin: The Argument in Form.]

[VII.] These [things] being premised, the Argument is thus Formed:

If the Earth were moved through the Annual orb, not only [stars] of the first, but even of the lowest magnitude, Fixed stars would have a bulk too great — that is, greater than the Sun and than the very Annual Orb, or [than] the Solar heaven. But this is absurd and incredible. Therefore, etc.

The Major is proved from the tables just now premised. The Minor is proved from the comparison of the Fixed [stars] with the Sun and the other Planets (which are much smaller), and from the symmetry of the Universe, and from Rothmann himself — although a Copernican, who at length yielded in this to Tycho; as Tycho himself narrates in the letters, p. 192, saying of Rothmann:

[Margin: Rothmann’s confession concerning the absurdity of the Copernican hypothesis.]

For I remember that he once, among other things, said to me: If the Copernican assumption stood firm in the truth of things, it would be necessary that very many of the Affixed stars should surpass the whole annual Orb (or the sphere of the Sun) in their true magnitude, before so inexhaustible a distance — such as the reasoning of Copernicus necessarily attributes to them — would concede to them that visible magnitude which we discern from the Earth: just as [these things were demonstrated] by me…

[…continues on p. 462 (PDF 497) with the catchword “a me” — completing Rothmann’s confession to Tycho.]

(printed p. 462 — within Chapter XXX: the giant-stars argument is answered and rebutted. Galileo’s and Hortensius’s denial of the Major fails on Riccioli’s measurements, and the Copernicans’ threefold denial of the Minor — Divine Magnificence, the parallel of the Sun’s bulk, and the claim that Copernican bulk is more credible than Ptolemaic speed — is refuted in turn. The famous disputation whether the stars’ bulk or their diurnal swiftness is more credible then begins, Riccioli arguing the velocity is not incredible and that Scripture commends the Sun’s speed, never the stars’ bulk.)

[Header: BOOK IX. SECTION IV. — 462]

…[just as it was] objected by me. But he did not deny that this exceeds all belief, and seems utterly absurd; nor did he strive to excuse it further by [appeal to] the Divine Omnipotence; and he not unwillingly admitted that a great asymmetry [disproportion] is also induced from these [things].

[Margin: 1st Response, insufficient.]

Galileo and Hortensius respond first (as we saw at number 5) by denying the Major: namely that Tycho and others, from too great an apparent diameter of the Fixed [stars], deduced their magnitude greater than just. But granted that Tycho, Longomontanus, Scheiner, and some others assumed too great a diameter of them — yet Galileo and Hortensius themselves assumed too small a one. But assuming that [diameter] which must be chosen from our more certain observations, and the Annual Orb asserted by the Copernicans, and the distance of the Fixed [stars] either asserted by the same (as we did in Table 1), or to be asserted, if they wish to avoid the parallax of 10″ which would arise from the whole diameter of the Annual Orb (as we did in Table 3) — these [things], I say, being taken together, Sirius (which is a Fixed star of the first magnitude) and Alcor (which is a star of the sixth and lowest magnitude — of those visible, indeed, without the Telescope) come out too great with respect to the Annual orb (as is clear from Table 2), nay both greater than the annual Orb — Sirius indeed at least 44 times, but Alcor 4 times. Wherefore, treating these [things] sincerely, the Major cannot be eluded by the Copernicans.

[Margin: 2nd Response, and its threefold reason, but insufficient.]

The Copernicans respond secondly by denying the Minor — [first] in view of the Divine Magnificence (which grows the greater from the vastness of these stars); [secondly] by comparing other bodies of the World among themselves, which nevertheless must be admitted by all Cosmographers — for the Sun must be admitted (to be silent about the rest), although it appears a foot wide, yet to be greater than the terrestrial globe not only 140 or 160 times, but 38,600 times (if its distance be true, which we elicited from the Lunar dichotomies), nay 262,144 times (if the Sun’s distance asserted by Wendelin be true, as I taught in bk. 3, ch. 11, problem 1); [thirdly] finally, by comparing the Copernican bulk of the Fixed [stars] with the Ptolemaic swiftness of the same: for that [bulk] is much more credible than this [swiftness].

But none of these reasons satisfies. Not the first: because if God had wished his Magnificence to be recognized from the vastness of the Fixed [stars] in this life by mortals, there was no reason why he should remove them so far an interval from our eyes that they appear so tiny to us, and scarcely equal the hundredth part of the apparent diameter of the Sun by their apparent diameter; or surely he would have permitted us, by some other sensible indication, to come to a certain knowledge of this distance and magnitude. But that he has by no means done this is clear from what has hitherto been said, since all Astronomical phenomena are saved without the Copernican hypothesis; whereas the physical experiments of heavy [bodies] and the impacts of projectiles evidently convict that hypothesis of falsity. Nor does the Second reason avail: because for the Sun to be of so great a bulk is not incredible, if its services in the Universe be regarded — since the Sun is the fount of Light of all the Planets, and the universal second cause of all natural sublunary effects: [it is] no wonder, therefore, if it be so much greater than the Earth. But it seems a wonder, and little credible, that any little fixed star be greater not only than the Earth and the Sun — and that far more than the Sun is — but even greater so many times than the whole heaven of the Sun itself, when no reason for so great a bulk appears; for if the reason had been that it might operate more copiously and strongly upon these lower [things] than the Sun, there was no cause why it should be removed so far from the Planetary system. But it is established from Sacred Scripture — namely from Deuteronomy — that all the stars were created for the ministry [service] of men, and that on this account too their worship is prohibited, since they themselves serve, [and] do not lord it over us. The Third reason, finally — concerning the greater likelihood of the huge bulk of the Fixed [stars] without motion, than of a moderate magnitude with a notable velocity — how invalid it is, will become clear from what is about to be said.

It is discussed: Whether the Bulk of the Fixed stars on the Copernican hypothesis is more credible than the Swiftness of the same on the Ptolemaic hypothesis.

[VIII.] That the velocity of the Fixed [stars] on the Ptolemaic — nay, nor on our [hypothesis] — is not incredible, we showed in chapter 6, where from number 1 to 7 we enumerated the opinions and calculations of the Copernicans against this swiftness, but at numbers 7 and 8 we dissolved their twofold argument, full of equivocation and Sophistical fallacy: wherefore thither we refer the Reader, certain that, if he be willing to judge without prejudice, he will pronounce sentence against the Copernicans in this part. And, to gather summarily the chief reasons there adduced: I said and say that the velocity of the Fixed [stars] does not grow beyond the measure of the subject, as Galileo (with Kepler) pretends; for it grows [only] as much as their distance [does] — for it is certain that, in any wheel or movable sphere, the parts of the circumference are moved with the common motion (for the comparison is to be made in this) the more swiftly, the more they are distant from the center; and then, that the velocity of the Fixed [stars] seems excessive because we express it by miles of our paces — just as the velocity of a Giant would seem excessive, if someone were to express one of his paces, chopped up into a thousand paces of a little ant.

But comparing the Ptolemaic velocity of the Fixed [stars] with the Copernican Magnitude of the same:

[Margin: 1st Reason against the Copernicans.]

First, I say this comparison does not militate against those semi-Copernicans who attribute only the diurnal motion to the Earth, not to the Fixed [stars] — as do Longomontanus, Origanus, Argoli, etc.; for to them the Copernican magnitude of the Fixed [stars] seemed absurd; nor yet can its absurdity be removed by retorting upon them the diurnal rapidity of the Fixed [stars], since they transfer it from the Fixed [stars] to the Earth.

[Margin: 2nd Reason against the Copernicans.]

Secondly, I said and say that the velocity of the Fixed [stars] arises from the magnitude of the circumference of the eighth (as they call it) sphere — compared by the Copernicans with the circumference of the terrestrial globe, and to be revolved in the time of 24 hours; but the immensity of the Fixed [stars] arises from the solidity or corpulence of the Fixed [stars], which (that the parity of the comparison be just) must indeed be compared with the globe of the Earth. But far greater is the excess of the corpulence of one Fixed [star] over the globe of the earth, on the Copernican hypothesis, than [the excess] of the circumference of the Eighth sphere (to be revolved in 24 hours, on the Ptolemaic and Our hypothesis) over the circumference of the terrestrial globe: for since circumferences are to one another as [their] diameters (by Theorem 5 of bk. 11 of the Collections of Pappus of Alexandria), the diameter of the Eighth sphere contains the diameter of the earth only 20,220 times (if you follow Ptolemy), or 40,440 (if Alfraganus), or 100,000 — or 210,000 — (if you follow one of our [own] opinions, according to what is said in bk. 6, ch. 7, no. 6, and in Table 1); but not only Sirius, the greatest of the stars, but [even] Alcor, the least, on the Copernican hypothesis comes out greater than the Earth by far more times, as will at once be clear to one running over Tables 1 and 3, exhibited in this chapter, in those columns in which is set how many times these stars contain the Earth. But if the Copernicans wish to draw us back to the comparison of a Fixed star with the annual orb, that its excess may appear smaller, we shall not refuse — provided that they too (as is fit and just) compare the circumference of the Fixed [stars], taken from the Ptolemaic or Our hypothesis, with the circumference of the annual orb (Ptolemaic or Ours), and from both estimate the velocity of the diurnal motion. Now Sirius must contain the annual orb at least 44 times (as is clear from Table 4 of this chapter, and its most solid foundations); but the Ptolemaic circumference of the Fixed [stars] is to the circumference of the annual orb as the diameter 20,220 to the diameter 1210 — that is, as 16 690/1210 to 1. But Our circumference of the Fixed [stars] is to the annual orb as the diameter 100,000 to 7300 — that is, as 13 510/730 to 1; or as 210,000 to 7300 — that is, as 28 56/73 to 1. Wherefore far greater is the excess of Sirius over the Earth — nay, even over the Annual Orb itself, Copernican-wise — than [the excess] of the diurnal velocity and circumference of the sphere of the Fixed [stars] (Ptolemaic or Ours) over the diurnal velocity and circumference of the terrestrial globe, or of the Annual Orb.

[Margin: 3rd Reason against the Copernicans.]

Thirdly, as I said in the same place: other things being equal, more credible is a moderate magnitude of a body in the World — commensurate with its other parts — together with some swift motion, than an immoderate [magnitude], but inert and without motion; and God himself, in the sacred letters, commends himself by the velocity of the Sun (which, [as] he revealed to us through the Psalmist, exulted as a giant to run the way of the diurnal motion), rather than by the magnitude of the bulk of the Fixed stars — of which he nowhere made mention, but rather of their innumerable multitude; and be-…

[…continues on p. 463 (PDF 498) with the catchword “inter” — “…among the stars [God] commended by magnitude the Sun alone, calling it a Giant (Psalm 18).”]

(printed p. 463 — the bulk-vs-speed disputation reaches its conclusion: more credible is the diurnal velocity of the fixed stars on a resting Earth than their insane bulk on a moving one, and Chapter XXX ends. Chapter XXXI opens with six arguments against the annual motion from the refraction of the fixed stars. Arguments I–III — that Copernican distances would destroy sensible refraction, make refraction exceed the inclination, or exceed half of it — are stated with their responses, and a refraction-geometry figure (#44) is set up.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 463]

…among the stars [God] commended by magnitude the Sun alone, calling it a Giant (Psalm 18), and a wondrous Vessel, the work of the Most High (in Ecclesiasticus); which [Sun], nevertheless, if the Copernicans be heard, would come out a pygmy, and a little vessel contemptible beside the Fixed stars. Let the Conclusion, therefore, stand.

CONCLUSION.

More credible is the Diurnal Velocity of the Fixed stars on the Hypothesis of a resting Earth, than the insane Bulk of the same on the Hypothesis of an Earth going around through the Annual Orb.

✠ [Chapter XXX ends here.]