[I.] From number 4 of the preceding chapter it is established that, if the parallax of the Fixed [stars] arising from the annual Orb of the Earth must be unobservable — or at least not exceeding 10 Seconds — it is necessary that the distance of the Fixed [stars] be greater than 47,000,000 terrestrial semidiameters, as is gathered from the table set down in that same place. But from so great a distance it follows that between Saturn and the Fixed [stars] there is an incredible space, and that the sphere of the Fixed [stars] is of an immense bulk, exceeding almost all belief. But because, in reckoning the magnitude of each, different [authors] suppose different things, I judge that the chief opinions are here to be rehearsed — both those of the Copernicans themselves, and of others, and our own.

Opinions of the Copernicans.

[Margin: 1. Opinion of Aristarchus of Samos.]

[II.] The first and most ancient Opinion was that of Aristarchus of Samos, who, although he said that the annual orb (through which he supposed the Earth to be carried round) is, with respect to the sphere of the Fixed [stars], as a point or center — yet is to be taken in that sense in which the Cosmographers are wont to say that the Earth, with respect to the heaven, is as a point (namely, to sense); and so Archimedes kindly interpreted him at the beginning of the Sand-Reckoner: where, having said that it is impossible for the annual orb of the Earth to be as a center with respect to the sphere of the Fixed [stars] — since a center is nothing, or has no proportion at all to a sphere — he subjoined: But it must be granted that Aristarchus understood this: namely, since we think the Earth to be established about the center of the world, he judged it must be laid down (the demonstrations being taken from the Phenomena) that what proportion the Earth has to the World [as we] have called it [i.e. to the annual orb], the same analogy the sphere of the circle carrying the Earth round has to the sphere of the Fixed stars. Moreover the “World,” which Archimedes had a little before described, is the annual orb of the Sun itself; for he had said: But you are not ignorant that “World” is called by many Astrologers indeed a sphere, whose center is the center of the Earth, and [whose] Radius is equal to the straight line which lies between the center of the Sun and the center of the Earth.

[Margin: Aristarchus’s view on the distance of the Sun.]

Wherefore, by Archimedes’ interpretation, Aristarchus thought the proportion between the annual orb and the orb of the Fixed [stars] to be that which is between the Earth and the annual orb. It is needful, then, to find this proportion according to Aristarchus’s view. Now Aristarchus, in the book On the Sizes and Distances of the Sun and Moon, proposition 7, demonstrated this Theorem: The distance by which the Sun is distant from the Earth is greater indeed than eighteen-fold the Lunar distance from the Earth, but less than twenty-fold — which is to say that it is to the Lunar distance as 19, very nearly, to 1. To which Plutarch (bk. 2 On the Opinions of the Philosophers, ch. 31) and Pliny (bk. 2, ch. 21) allude, saying: Many have tried also to track out the intervals of the heavenly bodies from the Earth, and [have] given out that the Sun is distant from the Moon nineteen parts [of] as much as the Moon itself [is] from the Earth. Now the Moon is distant from the earth, according to the Ancients, [by] 64 terrestrial semidiameters, which, multiplied by 19, make the Sun’s distance — that is, the semidiameter of the Annual Orb — 1216 terrestrial semidiameters; and just so many are the diameters of the earth in the diameter of the annual orb. But of this number 1216, multiplied into itself, the Square number is 1,478,656; which, multiplied again by 1216, makes the Cubic number 1,788,045,696; but one cube of the diameter of the Earth is unity, or 1. Aristarchus thinks, therefore, that the sphere of the Fixed [stars] contains the annual orb nearly 1,800,000,000 times; and that the semidiameter of the annual orb is to the semidiameter of the sphere of the Fixed [stars] as the semidiameter of the Earth is to the semidiameter of the annual orb. Let it be made, then, as 1 to 1216, so 1216 to another, and there will come forth the semidiameter of the orb of the Fixed [stars] [as] 1,478,656 terrestrial semidiameters. Which proportion Mästlin too admits, among others, as we shall presently see.

[Margin: 2. Opinion of Mästlin.]

[III.] The second Opinion is that of Michael Mästlin, in the additions to the First Narration of Georg Joachim Rheticus, p. 114, where he does not hesitate to admit, in the Fixed stars rising or setting at about midnight, a horizontal parallax of two or three Minutes — which, however, in that situation he denies can be observed, the horizontal refractions absorbing it; and he adds that, for the Copernican hypothesis, it is not required that the distance of the Fixed [stars], compared with the distance of the Sun from the Earth, vanish utterly, but only that it vanish to the sense and judgment of the eyes — that is, that the parallax cannot be observed; and that it neither contradicts Copernicus, nor [does] any absurdity follow, if the distance of the Fixed [stars] from the center of the universe contain the supreme distance of Saturn from the same center seven-hundred-fold and more — Saturn’s distance being assumed from Tycho [as] 12,900 terrestrial semidiameters — and so [if] the distance of the Fixed [stars] contain 9,030,000 terrestrial semidiameters, namely 12,900 multiplied by 700. But a little after he says: No one, therefore, can be convicted of any absurdity, who does not hesitate to assert that the proportion of the distance of the fixed [stars] from the World’s center, to the [proportion of the distance of Saturn], at the least equals the proportion of the distance of the Sun from the Earth — that is, [equals] the semidiameter of the Great Orb — and that on that account the Orb of the Fixed [stars] can even thus be called like the infinite: which very thing we taught that Aristarchus held, at number 2. Now the mean distance of the Sun from the Earth, by what was said in bk. 3, ch. 7, is for Mästlin 1160 terrestrial semidiameters; wherefore, if it be made as 1 to 1160, so 1160 to another, there will come forth the distance of the Fixed [stars] [as] 1,345,600 terrestrial semidiameters. But below, on the same page, he has thus: Therefore it is beyond doubt that Copernicus looked to the judgment of the eyes, not to the universal exclusion of all parallax: since, for the demonstrations of the motions, that removal of the starry orb suffices on account of which — and up to which — the magnitude of the Great Orb cannot be discerned by the eyes. But that removal, indicated in the said manner, and weighed according to all the circumstances, it is altogether most probable [to be such] that, even if it do not exceed the supreme altitude of Saturn seven-hundred-fold, yet it equals [it] a hundred-fold or two-hundred-fold. Since, therefore, he there says that the supreme distance of Saturn from the world’s center is raised to 12,900 terrestrial semidiameters, and that so great [a one], or not less, is established from the Astronomical demonstrations, it follows that the most probable distance of the Fixed [stars] from the World’s center, according to his opinion, is either 1,290,000 or even 2,580,000 terrestrial semidiameters. Finally, elsewhere he sets nearly the same distance that Galileo [sets], namely 13,046,400 terrestrial semidiameters — so inconstantly did he bear himself in this matter, as the numbers below written show; yet among these he seems to have rested in the mean…

[…continues on p. 455 (PDF 490) with the catchword “quæ” — “…which, namely, contains Saturn’s distance two-hundred-fold; that is, the [distance] which has 2,580,000 terrestrial semidiameters.”]

(printed p. 455 — within Chapter XXIX, the review of opinions on the fixed-stars’ distance continues through the third to ninth opinions. Mästlin’s figures are tabled; then Kepler (settling on 60,000,000 Earth-radii), Lansberg and Hortensius, Galileo, Herigone, and Gassendi (who later piously abandoned Copernicanism) among the Copernicans. Under the non-Copernicans, Magini and Tycho (about 7,850,000 Earth-radii) and Longomontanus, whose computation is begun.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 455]

…the mean — which, namely, contains Saturn’s distance two-hundred-fold; that is, that [distance] which has 2,580,000 terrestrial semidiameters.

[Engraved table:]

[Margin: 3. Opinion of Kepler.]

[IV.] The third Opinion was Kepler’s, in the book On the New Star in Serpentarius, ch. 16, where to the Sun’s semidiameter he gives 6 terrestrial semidiameters, and to the distance of the Earth from the Sun 1432 terrestrial semidiameters, and to the distance of Saturn the tenfold of this, namely 14,320; and he says it is probable that Saturn’s distance is the mean proportional between two immovables — that is, between the Sun, mover of the Planets, and the sphere of the Fixed [stars]; wherefore he makes [it] as 6 to 14,320, so 14,320 to 34,077,066⅔ terrestrial semidiameters, and there he established so great a distance of the Fixed [stars]. But in the Epitome of Copernican Astronomy, bk. 4, pp. 479, 485, and 492, he chooses the Sun’s semidiameter [as] 15 terrestrial semidiameters, and its distance from the Earth [as] 3469 terrestrial semidiameters, and Saturn’s distance [as] 30,000 terrestrial semidiameters. Accordingly, the same proportion (of which above) being retained, he makes [it] as 15 to 30,000, so 30,000 to 60,000,000 terrestrial semidiameters — in which distance of the Fixed [stars] he rests.

[Margin: 4. Opinion of Lansberg and Hortensius.]

[V.] The fourth Opinion was that of Lansberg formerly (in the commentaries On the Motion of the Earth), and of Martin Hortensius (in the dissertation with Gassendi On Mercury seen under the Sun, and Venus), who reckoned that the parallax of the Fixed [stars] made by the annual orb could reach to 30 Seconds, without evident inconvenience to the observations; from which, and from the semidiameter of the annual orb [of] 1498½ terrestrial semidiameters, by the rules of right-angled triangles, they gathered the distance of the Fixed [stars] [to be] 10,312,227 terrestrial semidiameters. But that parallax seemed afterward to Lansberg too great; wherefore (bk. 3 of the Uranometria, element 7) he makes [it] that, as the periodic time of one revolution of the Earth through the annual orb (that is, as one year) to the radius of its orb [of] 10,000, so the periodic time of the revolution of the Fixed [stars] (which he establishes [as] of 28,000 years) to the radius of the sphere of the Fixed [stars] [of] 280,000,000. Wherefore, since the radius of the annual orb is, for Lansberg, 1498½ terrestrial semidiameters, if it be made as 10,000 to 1498½, so 280,000,000 to another, there will arise the distance of the Fixed [stars] [as] 41,958,000 terrestrial semidiameters. Hence he concludes that the ancient and more recent Astronomers erred by the whole heaven, who made one heaven out of two — since between the Planetary heaven and the heaven of the Fixed [stars] there is a vast gap; and, just as the Sun is the center of the Planetary sphere, so the whole sphere of the Planets is, as it were, the center of the sphere of the Fixed [stars].

[Margin: 5. Opinion of Galileo.]

[VI.] The fifth Opinion was Galileo’s, in the 3rd dialogue On the System of the World, in which (Latin p. 244, Italian 323) he does not disapprove of Simplicius saying that it is more probable that not all the Fixed [stars] are equally distant from the World’s center, but that some are higher than others, and that an innumerable multitude of them is contained between two surfaces — one convex, the other concave — distant from each other by a great gap; which very thing he confirms (Latin p. 284). But (Latin p. 270, Italian 357) he determines the distance of the Fixed [stars] from the proportion of the periodic revolution of the Fixed [stars], which he takes from Ptolemy [as] 36,000 years. Therefore, since the conversion of the Sun (or Earth) is completed in one year, and Saturn’s in 30 years — and Saturn’s distance is nine times higher than the Earth, as he thinks — he argues thus: If Saturn’s orb, since it is nine times greater than the Sun’s orb, revolves in a thirty times greater time; then, by the inverse ratio, the orb (or distance) of the Fixed [stars], since it revolves 36,000 [times] more slowly, ought to be [of] 10,800 semidiameters of the Great Orb. By which reckoning this distance turns out fivefold greater than that which he had a little before computed from the hypothesis that a Fixed star of the sixth magnitude equals the Sun — namely (Latin p. 266, Italian 351), where he had deduced the distance of the Fixed [stars], from that supposition, [as] not greater than 2160 semidiameters of the Great Orb; and there he had said that the distance of the Sun from the earth is, by the consent of all, 1208 terrestrial semidiameters. Multiplying, therefore, 10,800 by 1208, there result 13,046,400 for the distance of the Fixed [stars] from the World’s center.

[Margin: 6. Opinion of Herigone.]

[VII.] The sixth Opinion is Pierre Herigone’s, who (vol. 5 of the Mathematical Course, p. 618) assumes the distance of the Sun from the Earth [to be] 1200 terrestrial semidiameters; and (p. 615) supposes the proportion of the semidiameter of the Earth to the semidiameter of the annual orb to be the same as [that] of the semidiameter of the annual orb to the semidiameter of the Firmament; wherefore, if it be made as 1 to 1200, so 1200 to another, the distance of the Fixed [stars] comes out [as] 1,440,000 — from which he gathers a parallax of 2′ or 3′.

[Margin: 7. Opinion of Gassendi.]

[VIII.] The seventh Opinion is Pierre Gassendi’s (2nd letter On impressed motion transferred from a mover), asserting that, on the Copernican hypothesis, such a vastness of the World must be conceived that the whole sphere of the heaven of the Planets, seen from the Fixed [stars], would appear as a single star — in the manner in which to us the cluster of Jupiter and the Jovian stars appears [as] a single Planet; or at least so that the whole annual orb, seen from the Fixed [stars], would appear as a point, or as the least of the tiny little stars. Assuming, therefore, the semidiameter of the annual orb [as] 1150 terrestrial semidiameters (which he does not disapprove in Tycho, bk. 2 of the Astronomical Institution), and the apparent semidiameter of the smallest of the Fixed [stars] visible to the naked eye — which (bk. 6, ch. 11) we established to be not [more than] two Seconds, since the little star Alcor has, in apparent diameter, only 4″ 24‴ — from this angle, I say, and the side opposite to it [of] 1150 semidiameters, by the problem handed down in the preceding chapter, number 5, it follows that the distance of the Fixed [stars] is 118,602,274 terrestrial semidiameters.

[Margin: Gassendi once, but not afterward, a Copernican.]

But I have reckoned Gassendi among the favorers of the Copernican sect, because in truth he long favored it, before he had seen the decrees of the sacred Congregation — which, once seen or heard, he piously and prudently acquiesced in, as is clear from the end of the same letter.

Opinions of non-Copernicans, but [of those] differing from the Copernican Hypothesis.

[Margin: 8. Opinion of Magini. And of Tycho.]

[IX.] The eighth Opinion was that of Giovanni Antonio Magini, who (bk. 11 of the Primum Mobile, Problem 29) has these words: That we may leave aside the opinion of Copernicus asserting that between the sphere of Saturn and the Eighth [sphere] there is so immense a vastness that the Eighth sphere is removed from the Earth by at least 7,850,000 semidiameters. Which number, however, is nowhere found in Copernicus. But whether Tycho drew the number hence, or Magini from Tycho, I do not know. Certainly Tycho, in the letter given to Rothmann in the year 1589, November 24, old style, p. 167, has these words: Concerning the other, Annual motion of the Earth — which would remove the Eighth sphere so greatly that the Orb which this would describe vanishes with respect to it — set forth whether it here seems probable to you that the space which is from the Sun (then the center of the universe) to Saturn is contained more than 700 times between this [Saturn] and the sphere of the Fixed [stars]; and that the whole [is] filled with no stars? But this must necessarily be so, if at least the Annual Orb of the Earth will appear as one Minute. 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. So (vol. 1 of the Progymnasmata, p. 481) he affirms [it] to follow from the Copernican hypothesis that, if the semidiameter of the annual orb must vanish at the distance of the Fixed [stars], this [distance] be 700 times greater than the distance of Saturn from the World’s center.

[Margin: 9. Opinion of Longomontanus.]

[X.] The ninth Opinion, nearest to the Tychonic, is Longomontanus’s (in the Astronomia Danica, bk. 1 of the Theorica, ch. 1, folio — in mine — 159), who from Tycho assumes the semidiameter of the Annual Orb [to be] 1150 terrestrial semidiameters, and [holds] that the annual orb, in order to vanish at the sphere of the Fixed [stars], ought to be of one minute, and accordingly that the semidiameter of the Annual Orb ought to subtend an angle of 30 Seconds. Let there now be repeated hither the figure set in the preceding chapter, number 5 [fig. #41]; and let the semidiameter of the annual orb in it be DP, subtending the angle DCP [of] 30″; and accordingly let the other, acute angle CPD, in the triangle CDP (right-angled at D), be 89° 59′ 30″; and let the distance of the Fixed [stars] CD be inquired by his Method, which is this: As 1,454,441 (the Right Sine of the angle DCP, [of] 30 Seconds) to the side DP [of] 1150 semidiame-…

[…continues on p. 456 (PDF 491) with the catchword “metro-” (semidia-metro-rum) — “…terrestrial semidiameters,” completing Longomontanus’s computation of the fixed-stars’ distance.]

(printed p. 456 — within Chapter XXIX, the opinion-review finishes with Longomontanus (7,906,818 Earth-radii), Scheiner, Claramonti (nearly nine billion), and Riccioli’s own twelfth opinion — a range from 47 to 604 million Earth-radii, required so that no sensible parallax appear. A synoptic Table I of all twelve opinions follows. A new comparison of the fixed sphere’s bulk against the Earth, the annual orb, and Saturn’s orb begins, with Table II of the three-sphere diameters.)

[Header: BOOK IX. SECTION IV. — 456]

…semidiameters; so [is] 9,999,999,894 (the Right Sine of the angle CPD, of 89° 59′ 30″) to CD, 7,906,818 semidiameters of the earth — namely to the distance of the Fixed [stars]; which is somewhat greater than that which Tycho deduced.

[Margin: 10. Opinion of Scheiner.]

[XI.] The tenth Opinion is that of our [own] Christoph Scheiner, in the Mathematical Disquisitions, pp. 25, 26, and 27, who — that he may remove the Fixed [stars] from every suspicion of parallax, after the mind of Copernicus — uses this proportion: As the Earth’s semidiameter, 1, to Saturn’s maximum distance [of] 10,872 terrestrial semidiameters, so the semidiameter of the annual Orb, 1208 terrestrial semidiameters, to the distance of the Fixed [stars], 13,133,376 terrestrial semidiameters; which [distance] he confirms is not excessive, from Tycho (bk. 1 of the Progymnasmata, pp. 480 and 481), who says that the Copernican distance of Saturn from the Earth is 12,900 terrestrial semidiameters, and that this must be multiplied more than seven-hundred-fold to have the Copernican distance of the Fixed [stars]; and so, if it be taken × 800, there comes out for Scheiner 10,320,000 terrestrial semidiameters — that is, not much less than the preceding.

[Margin: 11. Opinion of Claramonti.]

[XII.] The eleventh Opinion is that of Scipione Claramonti, in the Italian defense of his Anti-Tycho, part 4, ch. 13, where he assumes from Copernicus that the semidiameter of the Annual Orb is as a point to the semidiameter of the Firmament — so that, through a sight-line parallel to the horizon, the same Fixed [star] can be seen whether the eye be at the center of the annual orb, or on its circumference, or on the surface of the Earth — and [that] the chord subtended to the diameter of a Fixed star is at least 1105 terrestrial semidiameters (as great as is the least distance of the Sun from the earth in Copernicus). Multiplying, therefore, 1105 cubically, [it] makes 1,349,232,625 — namely the proportion of the solid area (or solidity) of such a star to the solidity of the terrestrial globe (he ought, however, to have said: to the eighth part of the Earth’s solidity). But, says he, the Sun contains the Earth only 166 or 167 times; therefore such a star would contain the Sun 8,079,237 times. And since Galileo, in the dialogues (Italian p. 265), had said that the Fixed stars are not less splendid than the Sun, he concludes: if one Fixed [star] is 8,079,237 times greater than the Sun, and yet ought to illuminate only as much as the Sun, then — so that the distance be proportional to the magnitude — the distance of the Sun, which is 1105 semidiameters of the earth, must be multiplied by 8,079,237; that is, it becomes 8,927,556,885 terrestrial semidiameters.

[Margin: 12. Our Opinion.]

[XIII.] The twelfth Opinion is our own: namely that the distance of the Fixed [stars] must be so great that, the Earth being transferred through the annual orb from one point to the opposite point of the annual circumference, no sensible parallax can be observed in any Fixed stars — that is, not even in those placed in the Ecliptic itself — [no parallax of] 10 Seconds: which being posited, from the various semidiameters of the annual orb established by the Copernicans (or quasi-Copernicans) we elicited the various distance of the Fixed [stars] by the Problem handed down in the preceding chapter, number 4, where we also exhibited a table for these distances; which nevertheless we shall here repeat, that it may be compared at the same time with the distances collected from others up to this point. Behold, therefore, in the following Synopsis the aforesaid opinions.

[Engraved table — TABLE I:]

On the Bulk and Distance of the Sphere of the Fixed stars, compared to the bulk of the Earth, of the annual Orb, and of the Orb of Saturn, and to the Distance of the Sun and of Saturn.

[XIV.] Since an argument against the Copernican hypothesis can be taken not only from the distance, but also from the bulk of the sphere of the Fixed [stars] — and this bulk seems comparable chiefly to three bodies: namely to the globe of the Earth, to the Great or annual Orb (which the Ptolemaic hypothesis would call the sphere of the Solar heaven), and to the Orb of Saturn (that is, to the whole sphere including the whole system of the elements and Planets) — it is needful to draw it out from the distance of the Fixed [stars] (that is, from the semidiameter of the sphere of the Fixed [stars]), by proposition 18 of bk. 12 of Euclid’s Elements (where it is shown that spheres are to one another in the triplicate ratio of their diameters), or by proposition 12 of bk. 8 of the same Elements (by which it is shown that spheres are to one another as the cubes of their diameters) — concerning which [see] more [in] us, bk. 7, sect. 6, ch. 8, problem 8. But it is not worth the trouble to employ all the distances of the Fixed [stars] just rehearsed, or their precise numbers; but it suffices to use some selected and more notable ones — partly extreme, partly middle — and their round numbers in Millions of terrestrial semidiameters. Moreover, the semidiameter of Saturn’s orb we shall always make tenfold the semidiameter of the annual orb; but how great the semidiameter of the annual orb is, we have already said in the preceding chapter, in the table of number 4, and in bk. 3, at the end of ch. 7. These [things] being posited, let us exhibit in the following table the diameters of the three spheres — namely of the annual orb, of the sphere of Saturn (or the Planetary), and of the sphere of the Fixed [stars] — in parts of which the diameter of the terrestrial globe is one, that hence we may afterward draw out the proportion of the spheres.

[Engraved table — TABLE II:]

[…continues on p. 457 (PDF 492) with the catchword “XV. Iam” — “Now the cubic numbers of the aforesaid numbers must be investigated.”]

(printed p. 457 — within Chapter XXIX: Tables III and IV complete the bulk-computation, giving the cubes of the three-sphere diameters and how many times the fixed sphere contains the planetary sphere and annual orb on each author’s figures. Riccioli notes that authors differing wildly in the orb’s size still agree on these ratios. The first of the chapter’s three arguments then begins — Scheiner’s argument from the immensity of the fixed-star sphere, supported by a faithful rendering of Archimedes’ Sand-Reckoner account of Aristarchus.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 457]

[XV.] Now the cubic numbers of the aforesaid numbers must be investigated, which is done by multiplying them by themselves, and again multiplying the product by the first number. So 1142, multiplied by 1142, make this square number, 1,304,164; which, again multiplied by 1142, makes the cube or cubic number, 1,489,355,288.

[Margin: A warning for the following table.]

For which, nevertheless, we shall put in the following table the two leading initial [digits] — namely 15 — with 8 ciphers, that from these round numbers a sufficient judgment may more quickly be made concerning the bulk of the Sphere of the Fixed [stars] compared with the bulk of the spheres of the annual Orb and of Saturn. Moreover, to the initial numbers we shall not add the ciphers themselves spread out, but [only] the number of ciphers to be added to the initial numbers — lest we offer the Typographers occasion of erring. But when you see, for example, to the initial number 15 [are] to be added 8 ciphers, know it to be the same as if you read 1,500,000,000; and so of the rest.

[Engraved table — TABLE III. Translator’s note: each cell gives the leading digits and the number of ciphers (zeros) to append, per ¶XV — e.g. “18 + 8z” means 18 followed by 8 zeros = 1,800,000,000.]

[XVI.] Dividing, therefore, the cube of the diameter of the sphere of the Fixed [stars] (placed in the last column of the preceding table) by the cube of the diameter of the sphere of Saturn (in the penultimate), or by the cube of the diameter of the annual orb (placed in the 2nd column), you will find how many times — according to the hypotheses of the aforesaid authors — the sphere of the Fixed [stars] contains the sphere either of Saturn or of the Annual orb. The cubic number itself, moreover, placed in the last column, teaches — without any other division — how many times the sphere of the Fixed [stars] contains the globe of the Earth: for example, according to Copernicus’s hypotheses the sphere of the Fixed [stars] contains the Earth 100,000,000,000,000,000,000,000 times (that is, the number 10 taken with 22 ciphers); and so of the rest. But by dividing the same number by 15 with 11 ciphers, the sphere of the Fixed [stars] contains the sphere of Saturn 66,666,666,666⅔ times. Finally, by dividing the same number by 15 with eight ciphers, the sphere of the Fixed [stars] contains the annual orb 66,666,666,666,666⅔ times; and so of the rest. Let there be, then, Table IV, in which likewise we have chosen round numbers, and two initial [digits] of them, ciphers being substituted in place of the remaining [digits].

[Engraved table — TABLE IV:]

But let no one wonder if, according to certain Authors who greatly disagree among themselves in the semidiameter of the Annual Orb, nevertheless the sum [bulk] of the sphere of the Fixed [stars], compared with the annual orb or the Planetary sphere, does not so disagree; for, the semidiameter of the annual orb being increased, the semidiameter of Saturn’s orb (and [so] their orbs) is increased proportionally — whence it comes about that these orbs are contained fewer times in the sphere of the Fixed [stars], for this too must grow proportionally with the increase of the semidiameter of the annual orb, in order to avoid a sensible parallax (which parallax would be the greater, the greater the semidiameter of the annual orb is). Wherefore [it is] no wonder if, among the measures of Copernicus, Lansberg, Wendelin, and Ours, set in the 4th table, there is so great an affinity — namely of the numbers 67, 65, 69, 61, with 9 ciphers in the former column and 12 in the latter. These [things] being sufficiently — nay, abundantly — prepared for the proof of the following Arguments, now the arguments themselves must be proposed.

I. Argument, from the Immensity of the sphere of the Fixed stars.

[Margin: Form of the 1st Argument.]

[XVII.] If the Earth were moved through the Annual Orb, the sphere of the Fixed stars would come out Immense, and of an altogether incredible — or improbable — bulk. Therefore the Earth is not moved through the Annual Orb.

[Margin: Scheiner’s argument from the vastness of the sphere of the Fixed stars. And [the words] of Archimedes.]

This is the sum of the Argument, which Scheiner urges with so great a weight of words in the Mathematical Disquisitions, p. 27, saying: This so great immensity of the Firmament, and vastness of the stars, Copernicus himself (bk. 1 of the Revolutions, ch. 6) sufficiently noted, nor do his followers dare to dissemble it — as Rheticus, in the First Narration, p. 118 — but thence they go on to magnify the majesty of the Creator: ridiculously, since they (that is, the stars) appear so tiny, and every most learned [man] scarcely comprehends this hugeness. But the wiser Astronomers, both formerly and now, would rightly bear ill this superfluous bulk. In which place he also brings in Archimedes himself as accounting this bulk among the impossibles. The words of Archimedes from the Sand-Reckoner (that is, from the book on the number of the sand) are these, faithfully translated from the Greek: These [things], therefore, which are found written among the Astrologers, Aristarchus of Samos — refuting and altering [them] — handed down certain writings, filled with certain hypotheses, from which it is evident that the World is much greater than the World hitherto described. For with him it is supposed that the fixed stars and the Sun remain immovable; but that the Earth is carried about the Sun in the circumference of a circle, which is situated in the middle of the course of the Planets; and that the sphere of the fixed stars is situated about the same center as the Sun, but is held [to be] of such a magnitude that the circle about which the Earth is supposed to be carried has the same proportion to the interval of the fixed stars as the center of any sphere has to its surface. But this is manifestly impossible; for since the center of a sphere has no magnitude, neither must it be supposed to have any proportion to the surface of the sphere. But it must be granted that Aristarchus understood this: namely, since we think the Earth established about the center of the world, he judged it must be laid down — by demonstrations taken from the phenomena — that what proportion the Earth has to the World [as] called by us, the same [proportion]…

[…continues on p. 458 (PDF 493) with the catchword “habe-” — “…the sphere of the circle carrying the Earth round has to the sphere of the fixed stars.”]

(printed p. 458 — within Chapter XXIX: the Sand-Reckoner quotation closes and the Antecedent of Argument I is proved from Table IV. The Copernicans’ fourfold response follows: the parallax need vanish only to sense (Mästlin); a vast motionless sphere is more probable than a small whirled one (Kepler, Galileo); other accepted spheres are already comparably vast (Gassendi); and the bulk commends Divine Omnipotence and Magnificence, sealed with Baruch 3. Argument II then opens — the immense star-void, idle interval between Saturn and the fixed stars, urged chiefly by Tycho.)

[Header: BOOK IX. SECTION IV. — 458]

…[that what proportion the Earth has to the World as called by us] the sphere of the circle carrying the Earth round has [the same] to the sphere of the fixed stars. Moreover, the World which Archimedes had before spoken of extends as far as the annual Orb of the Sun; for he had said, at the beginning of the Sand-Reckoner: But you are not ignorant that “World” is called by many Astrologers a sphere, whose center is the center of the Earth, and [whose] Radius is equal to the straight line which lies midway from the center of the Sun to the center of the Earth. Although, therefore, Archimedes thought it absurd that the Annual orb should be as a point to the sphere of the Fixed [stars], yet — interpreting Aristarchus more kindly — he gathers from his hypothesis that the proportion of the annual orb to the orb of the Fixed [stars] is [the same] as [that] of the Earth to the Annual Orb, or to the World commonly understood: so that, as we are wont to say that the Earth is as a point to the whole heaven, so the annual orb would be as a point to the orb of the Fixed [stars] in Aristarchus’s hypothesis — that is, not absolutely a point and nothing, but comparatively, or to sense. But whatever be [the truth] about Aristarchus’s mind, the Antecedent is now proved from Table 4, exhibited at number 16; for in it, if you take the least measure raked out from Aristarchus, the sphere of the Fixed [stars] contains the Annual Orb 130,000,000 times. But if [you take] the Copernican measure, [it contains it] 67,000,000,000,000 times; if the Lansbergian, 65,000,000,000,000 times; if ours, 69,000,000,000,000 times; nor are the other measures deduced from others’ hypotheses so small but that they contain an incredible vastness of the orb of the Fixed [stars], as will be clear to one surveying that Table.

[Margin: Response to Argument 1.]

[XVIII.] The Copernicans, nevertheless, respond by denying the Antecedent — namely [denying] that from the motion of the Earth through the annual orb there follows so great a Bulk of the sphere of the Fixed [stars] that it is incredible. The reasons for which they deny it to be incredible or improbable are four.

[Margin: 1st Reason of the Response.]

The first is Mästlin’s, denying that, on the mind of Copernicus, so great a distance of the Fixed [stars] — or [so great] a bulk of the sphere of the Fixed [stars] — is required that, compared to this, the distance of the Earth from the Sun (or the annual orb) should utterly vanish, but [that] it suffices if [it vanish] to sense. See his words already related at number 3, in the 2nd opinion; to which the 1st opinion of Aristarchus is favorable, not requiring so great a distance of the Fixed [stars] on the hypothesis of a moving Earth.

[Margin: 2nd Reason of the Response.]

The second reason is [the one] which Kepler and Galileo insinuate — namely the comparison of this distance and Magnitude of the sphere of the Fixed [stars] on the Copernican hypothesis, with the velocity of the same sphere on the Ptolemaic hypothesis; for they say that that Copernican Magnitude, but immovable, is much more probable than the Ptolemaic smallness, but movable with so rapid a swiftness. But about this very comparison [we] must treat more calmly below, in chapter 30.

[Margin: 3rd Reason of the Response.]

The third reason is the comparison of other spheres among themselves, whose magnitude must be admitted — and is admitted — on the hypothesis of an immovable Earth. Does not Gassendi say (2nd letter On impressed motion, p. 124) that, by the common opinion, the heaven of the Fixed [stars] is so vast that, compared to it, our Earth — however great it be — is nevertheless only as a point? Just as, therefore, this, which is commonly reckoned a Paradox, is nevertheless most true in Astronomical rigor, why should it seem incredible that the whole Annual Orb, with respect to the sphere of the Fixed [stars], is as a point? Who would believe that the Sun — which is commonly esteemed a foot wide — is greater than the Earth 38,600 times, as we taught in bk. 3, ch. 11? Besides, the excess of the Planetary sphere over the sphere of the land-and-water globe [the Earth] is greater than the excess of the sphere of the Fixed [stars] over the Planetary sphere, or over the sphere of the annual orb, if you follow the opinion of many Copernicans; for, as above in Table 3, the Planetary sphere, on Kepler’s hypothesis, contains the Earth very nearly 42,000,000,000,000 times; but in Table 4, the sphere of the Fixed [stars], on the same Kepler’s hypothesis, contains the Planetary sphere only 520,000,000 times, and the Annual orb 520,000,000,000 times; and so of not a few others. As, therefore, so great a vastness of Saturn’s sphere — and [its] excess with respect to our Earth — is not incredible to the wise, why should the smaller excess of the sphere of the Fixed [stars] over the sphere of Saturn or of the annual orb seem incredible? For by this reasoning, says Galileo (Dialogue 3), Elephants would seem incredible to ants, and Whales to little fish.

[Margin: 4th Reason of the Response.]

The fourth and most powerful reason is the commendation of the Divine Omnipotence and Magnificence, to which esteem is conciliated by so great a bulk of the work. And so Copernicus, having said (bk. 1, ch. 10) that from the supreme of the wandering [stars], Saturn, to the sphere of the Fixed [stars] there is a very great distance, and that between the moved (that is, the Planets) and the non-moved (that is, the Fixed [stars]) there intervenes a maximum interval — substituted admiration for incredulity, exclaiming: So great, indeed, is this Fabric of the Best and Greatest [God]!

[Margin: Rothmann’s opinion for the immensity of the heaven, for the commendation of the Divine Majesty.]

So Christoph Rothmann, in a Letter given to Tycho in the year 1590, on the 18th day of April (which is held among the Tychonic [letters]), p. 186, thus responds to Tycho: Why should it not seem probable to me that the space from the Sun to Saturn is contained so many times between Saturn and the removal of the affixed stars? Or what absurdity follows, if a star of the third magnitude equals the whole annual orb? But does that either fight with the divine will, or is it impossible to the divine Nature, or does it not befit an infinite Nature? These [things] must altogether be demonstrated by you, if you wish to gather anything absurd from this. Things are not so easily convicted of absurdity which are commonly seen as absurd at first sight. But far greater is the Divine Wisdom and Majesty; and however great a vastness and magnitude of the World you concede, it will nevertheless have no proportion to the infinite Creator. The greater the King, the greater and ampler a palace he thinks befits his Majesty. What will you think of GOD?

[Margin: The opinions of Rheticus and Pliny for the same.]

And indeed Tycho himself, in the same Letters, p. 191, conceded that the World is of almost infinite bulk, when he said: And since the Universe is finite indeed in place and time, yet [is] like the infinite in incredible magnitude and perennity: why, in the same place and p. 167, does he call incredible that Magnitude which the Copernicans ascribe to the sphere of the Fixed [stars]? More rightly, surely, did Rheticus, in the First Narration, commending this Copernican magnificence, confirm [it] by that Plinian sentence (bk. 2, ch. 1): To search out the things outside of the World or heaven — by whose enfolding all things are covered, etc. — neither concerns men, nor does the conjecture of the human mind grasp [them]. It is sacred, etc., immense, etc., finite and like the infinite.

[Margin: Mästlin’s opinion for the same. Lansberg’s opinion for the same.]

But Mästlin, in the additions to the First Narration of Rheticus, p. 114, judges [it] harsh and almost injurious to the Divine Omnipotence and Wisdom to say that the sphere of the Fixed [stars] — which Tycho gathered from the Copernican hypothesis — is too vast and useless. So Lansberg (bk. 3 of the Uranometria, element 7) concludes: This is our opinion concerning the distance of the fixed stars from the Earth; which, as it is not foreign to Geometry, so also it detracts nothing from the Majesty and infinite Power of GOD, the Best and Greatest, but much more magnifies and illustrates it. For it teaches that, since the sphere of the fixed stars is so vast, we ought to be astonished at the infinite Power of GOD the Architect, and humbly to worship and venerate it.

[Margin: Commendation of the Divine Majesty.]

So, finally, Gassendi, in the 2nd letter On impressed motion, retorting the argument, says: just as the Ptolemaics argue that there is no reason why we should think God wished to make the World so vast and august, so the Copernicans argue that there is no reason why we should think God wished to make the World so narrow; and since neither can demonstrate what pleased God, the latter opinion — which amplifies the majesty of things — seems more to commend the Magnificence of the supreme Maker. Finally, what if God wished to adumbrate some vestige of [his] infinity and immensity in so vast a bulk? What if, by reason of the innumerable number of the Fixed [stars], he granted them so much ampler a region than the seven Planets? What if, for the sake of the Saints, he wished to contrive so great an argument [proof] of his works, that they might see how great [things] he wrought for them and for the throne of Christ the Lord, and how much greater [things] are to be expected from him above the Empyrean?

[Margin: Baruch 3.]

Let the holy Prophet Baruch close this response with that exclamation (Baruch 3): O Israel, how great is the house of God, and how vast the place of his possession! It is great, and has no end; lofty, and immense.

II. Argument, from the Immense Interval between Saturn and the Fixed stars, and that [interval being] void of Stars and idle.

[Margin: Form of the 2nd Argument.]

[XIX.] If the Earth were moved through the Annual Orb, there would be between Saturn and the Fixed [stars] an immense space void of stars, and altogether idle. But this is absurd. Therefore, etc.

This, rather than the preceding argument, they make much of — chiefly Tycho, in the letters, p. 167 (whose words I have already related at number 9), but most of all p. 192, where [he]…

[…continues on p. 459 (PDF 494) with the catchword “con-” — Tycho, against Rothmann, contending that no star-void interval of more than 7,000,000 Earth-radii should be admitted.]

(printed p. 459 — within Chapter XXIX: Argument II is elaborated from Tycho, Scheiner, and Longomontanus — God contrives nothing void or vain, yet the Copernican system interposes an immense idle gap between Saturn and the fixed stars. Four responses are examined and faulted: Copernicus’s demand for a maximum gap between moved and unmoved, the appeal to Divine Magnificence, Descartes’s liberty in placing the stars, and the claim that the gap is filled with telescopic stars. Argument III then opens: the Sun, a mere point at such distance, could not illuminate the fixed stars.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 459]

[Margin: Tycho’s opinion against the immensity [of the heaven] — the Copernicans singing a recantation.]

…[Tycho], against Rothmann (who attributes so great a vastness to the divine Omnipotence), says that God is indeed Omnipotent, but does nothing in a disordered way — nay, disposes all things in weight, number, and measure; and contrives nothing void or vain, nothing not answering to itself by a certain harmony; and that accordingly between Saturn and the Fixed [stars] there ought not to be admitted an interval of more than 7,000,000 terrestrial semidiameters, void of stars and destined for no use of the Earth-dwellers — since nevertheless those Fixed [stars] higher than it [Saturn] are established for their use; and he adds: Away with this un-geometry and a-symmetry, and disordered manner of Philosophizing, most foreign to Divine Wisdom and Providence — which even ROTHMANN himself, when he was with me, dared no longer to defend, but, in the course of comparing, at length not unwillingly confessed that a great absurdity, joined with equal incredibility, underlies these [things]. And after a few [words] he confesses that the world is indeed finite, and has no proportion to the infinite Creator; but that, as regards its parts, it is most highly proportioned to the whole — not otherwise than one may discern in the parts of animals, and especially in the Microcosm, that is, in the human body, whose singular symmetry Dürer demonstrated. The same absurdity he urges in vol. 1 of the Progymnasmata, p. 481.

[Margin: Scheiner’s argument against the Copernicans.]

Nor less sharply does Scheiner press (p. 28 of the Mathematical Disquisitions) against the Copernicans, saying: To what good [is] so great a machine — was it produced for so tiny a little point of Earth? Why, then, [are they] so remote that they appear so tiny, and can do almost nothing upon the Earth? To what [end] that insane abyss between them and Saturn? In vain, forsooth, are all [these] things. And so, at the end of the same page, he concludes thus: The Copernican Motion of the Earth interjects, between Saturn and the Firmament, a certain immense vastness of distance, whose use they themselves are ignorant of: therefore it does not seem to be admitted. Finally, Longomontanus (bk. 1 of the Theorica, ch. 1): To what [end] the empty gap between Saturn and the fixed stars, which from the premises is elicited [as] 7,894,818 terrestrial semidiameters?

[Margin: Proof of the Major.]

The Major of our argument is moreover proved from Table II, exhibited at number 14, if namely you subtract the numbers of the second column from [those of] the third (or last) column. For example, if from the Keplerian distance of the Fixed [stars], which is 60,000,000, you subtract the distance of Saturn, which is nearly 35,000, there will be left between Saturn and the Fixed [stars] a gap of 59,965,000 terrestrial semidiameters; and so on the hypotheses of the rest, from which that interval is always gathered [to be] huge.

[Margin: Proof of the Minor.]

The Minor is proved from an axiom often used by the Copernicans, while they busy themselves to exterminate the Ptolemaic Epicycles and the like; for nothing do they more frequently have in [their] mouth than that GOD and Nature do nothing in vain, nor attempt by more [means] what can be done by fewer: therefore they act absurdly, while they interpose so vast a space — in vain and without any use — between the sphere of the Planets and [that] of the Fixed [stars].

[Margin: 1st Response, of Copernicus, insufficient.]

[XX.] Copernicus would respond, first (bk. 1 of the Revolutions, ch. 10), by denying the Minor: because between the moved and the non-moved there had to be a maximum difference — that is, between the Planets, which are moved, and the Fixed [stars], which he establishes [as] immovable. But since the Sun itself is also immovable, and yet is not distant from movable Mercury (its innermost and nearest [planet]) by more than 561 terrestrial semidiameters (as we taught from Copernicus, bk. 7, sect. 6, ch. 2, in Table 4), [then] between Saturn, the outermost of the movables, and the Fixed [stars], a little more terrestrial semidiameters would have sufficed.

[Margin: 2nd Response, of the Copernicans, insufficient.]

They would respond, secondly — Rheticus, Mästlin, Galileo, Lansberg, Gassendi — by recurring to the Divine Omnipotence and Magnificence, according to those [things] which we rehearsed at number 18, in the 4th reason of the response; and so by denying the Minor. But granted that, from the huge bulk of the sphere of the Fixed [stars], the divine Omnipotence and Majesty be commended — inasmuch as that is the sphere of the Fixed [stars], or from that part of the bulk which the innumerable people of the Fixed [stars] inhabits — yet it is not commended from that insane and useless space between the Fixed [stars] and Saturn (whether it be supposed void of every other body, or filled only with ethereal air), unless its use appear. Galileo, nevertheless (Dialogue 3, p. 272), persists, and says: That it is rash to call all that vain and superfluous which does not serve our uses in the Universe.

[Margin: 3rd Response, of René Descartes.]

He would respond, therefore, thirdly — René Descartes, in the Principles of Philosophy, part 3 — by denying rather the Major: because (at number 20) he says: We are not yet certain how far the Fixed stars are distant from us, nor can we imagine them so remote that this conflicts with the Phenomena — lest we be content to suppose them to be above Saturn (as commonly all admit), but [rather] let us take the liberty of esteeming [them] however much higher. For if we wished to compare their altitude with the distances known to us here above the earth, that [altitude] which is now conceded to them by all would be no less incredible than any greater [one]. But if we look to the Omnipotence of God the Creator, none [so great] can be thought of as to be on that account less credible than any smaller [one]. But [Descartes also holds] that, for explaining conveniently not only the Phenomena of the Planets but also of the Comets, a maximum space must be placed between them and the sphere of Saturn — [as] I shall show below. Yet from number 119, where he begins to treat of Comets, nowhere does he show this, but rather supposes, from his own principles, immense vortices, through which the Fixed [stars] are moved, and changed sometimes into Comets, sometimes into new Planets. Nonetheless Seneca favors [him] (bk. 7 of the Natural Questions, ch. 24), where he esteems that innumerable stars unknown to us run through the immense space of the heavens, and so that [that] is not void of stars which seems to us void. But we have now shown (bk. 8) that it has not yet been demonstrated by anyone that Comets or new stars were above Saturn; nor, for the sake of one or another such phenomenon, is it fitting to interpose so great a vastness between Saturn and the Fixed [stars] — since for these Phenomena there abundantly suffices, partly the very region of air or fire, partly the fluid heaven of the Planets, partly the heaven of the Fixed [stars] (especially if it too be admitted [to be] liquid); or at least there suffices some moderate space between the Fixed [stars] and Saturn — [say] as much as is from the center of the World to Saturn.

[Margin: 4th Response, of the Copernicans.]

The Copernicans — not a few — respond, fourthly, by implicitly denying the Major, while they indicate that that space which is above Saturn is not altogether void of stars — granted [that] the stars which can be observed by us with the naked eye, and whose sensible parallax can be none, are removed by a vast interval from Saturn’s sphere. They could therefore say that there are innumerable stars which can be distinguished only by the Telescope, and which cannot be so observed that their parallax (if they have any a little smaller than the Saturnine parallax) could be observed; and that very many stars of this kind are scattered within the aforesaid interval, and that it is not idle and void, as Tycho, Scheiner, and others thought. Which response I had myself fashioned, when I came upon another akin to it in Galileo (Dialogue 3, On the System of the World, Latin p. 284), where he says that some of the fixed stars are twice or thrice more remote than others, and that it can come about that sometimes, by the benefit of the Telescope, in some of these is observed some diversity like the diversity of the Planets. So Mästlin, in the additions to the First Narration of Rheticus, p. 113, denies that the gap must be so vast, since not all the Fixed [stars] are equally distant from the center — Tycho himself conceding this (vol. 1 of the Progymnasmata, [pp.] 470 and 481). But although this response can be convicted of falsity by no one, yet it appears to be arbitrarily fabricated to defend the hypothesis of Copernicus; for who would believe that God so removed from us all the stars which appear greater to us (and whose parallax, if there were any, we could observe), that we could not observe that parallax; but made those others — which are otherwise liable to parallax, as regards their distance — so small that they could not be observed by us with the naked eye? Surely it would be necessary (if it be permitted so to speak) that God, in distributing the Fixed stars, was from the very beginning of the World a Copernican — that is, a favorer of this sect, which nevertheless he foresaw would one day be condemned by the sacred decrees.

III. Argument, from the slenderness of the Sun, [as] not able to illuminate the Fixed stars on account of [their] huge distance.

[Margin: Form of the 3rd Argument.]

[XXI.] If the Earth were moved through the Annual orb about the Sun, it would be necessary to remove the Fixed stars so far from the Sun that the Sun, beheld from them, would be as a point, and therefore could not illuminate the stars with its light. The Consequent is false. Therefore also [is false] that from which it follows.

[…continues on p. 460 (PDF 495) with the catchword “Re-” (Respondetur) — the response to the third Argument.]

(printed p. 460 — Chapter XXIX closes with Gassendi’s response to Argument III (the fixed stars shine by their own light). Chapter XXX opens on the argument from the huge bulk the fixed stars must have on Copernican distances, rehearsing each author’s computation: Tycho, Longomontanus (a first-magnitude star eight times the annual orb, whence he rejects Copernicus), Scheiner (Sirius containing the orb thirty-two times), Claramonti and Lansberg, and the beginning of Hortensius’s much smaller estimate.)

[Header: BOOK IX. SECTION IV. — 460]

Gassendi responds (2nd letter On impressed motion, p. 126), first by denying the Major: for [the Sun] can diffuse its rays even thither, even if it were as a point; secondly by denying the Minor: for the Fixed [stars] have proper light from themselves, not from the Sun — which we too taught in bk. 6, ch. 2.

✠ [Chapter XXIX ends here.]