[Margin: Definition of the Parallax of the Annual Orb.]

[I.] The Parallax of the Annual Orb is a diversity of aspect, arising from the different situation of the spectator, [who is] transferred together with the Earth through the annual orb. Which diversity can be either in the apparent magnitude of the Stars, or in their altitude or distance from the zenith, or in their distance from some point or visible and immovable object — of which kind is the Sun, on the Copernican hypothesis. Accordingly, the asserters of the Earth’s immobility contend: if the Earth were transferred through the annual orb, now here, now there, the aforesaid diversities would appear in some stars; which, since they do not appear, the argu[ment]…

[…continues on p. 451 (PDF 486) with the catchword “men-” (argu-men-tum) — “…the argument [concludes] this, that the Earth is not moved through the annual orb.”]

(printed p. 451 — within Chapter XXVIII: the geometric setup of annual parallax by an engraved diagram (#41), with Galileo’s two propositions that the maximum parallax in altitude belongs to stars at the ecliptic poles and the maximum in apparent magnitude to stars in the ecliptic. Riccioli exposes the Copernicans’ lapse of computing only the semiparallax rather than the full angle on the orb’s diameter. A table of maximum annual-orb parallax follows: on close-star distances a glaring 6°–9°, on Copernican distances mere seconds.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 451]

…the argument [concludes] this, that the Earth is not moved through the annual orb.

[Margin: Exposition of the Diagram for the Parallax of the Annual Orb.]

Now, that the force of the Arguments and of the Solutions may be perceived: let there be, in the following diagram, at A, the center of the Sun and of the World, about which let the semicircle of the Ecliptic (or annual orb) be BDC, so that the diameter of the Annual Orb is BAC, and the semidiameter AB; and let the center of the Earth be now at B (where the nocturnal hemisphere of the earth is BK, and the diurnal, or [the part] illumined by the Sun, is BS), now after 6 months at C (where the nocturnal hemisphere is CN, and the diurnal CR) — as happens, for example, when it is transferred from one to the other of the Solstitial points, or to the other of the Equinoctial. And from the same center A let a great semicircle EFG be described, in the sphere of the Fixed [stars], whose diameter EAG coincides with the diameter of the annual orb; which semicircle is to be understood [as] erected orthogonally to the plane of the Ecliptic, and let it be (for example) the Solstitial Colure, and at the same time hold the place of the Meridian; and on it let the Pole of the Ecliptic be F, at which let some star be imagined to be, to which let the straight lines BF and CF be drawn from the centers B and C; and let the eye on the earth’s surface be once at I and once at O, through which points let the straight lines AIM and AOP be drawn, touching the earth at I and at O, and serving in the place of the Physical Horizon — through which the eye I sees the Sun A setting for itself (since the diurnal motion goes from S to I, toward K, etc.), but at O sees the Sun rising for itself. Let there be, again, another star L, outside the Ecliptic and the poles of the Ecliptic, to which let the straight lines BL and CL be drawn from the Earth’s centers B and C; and — the parallax of the earth’s orb being now neglected — let the eye B be imagined to see L through the straight line BL; for it will see it raised by the angle GBL, much greater than when, transferred to C, it sees it through CL, raised by the angle GCL. Wherefore the parallax of the annual orb, in this case, will be the angle BLC, arising from the diameter BAC — just as, in the prior case, [the parallax] would be the angle BFC.

Which Parallax we shall call per se, whether it then be observable, or per accidens not observable — because the star L (or F), once seen when the earth is at B, cannot be seen after six months, when it is at C, the day or the brightness of the Sun obstructing. For afterward we shall distinguish these aspects, and shall teach what can be observed when the Earth is transferred.

[Engraved figure (#41) — the annual-parallax construction. At the center A (Sun and World-center) stands the vertical diameter B–A–C of the annual orb (semicircle BDC, with D on the orb at right); the Earth is drawn as a small globe at the top (B, its night-side BK shaded, with surface-points S and I) and at the bottom (C, night-side CN shaded, with O, and E below). A large arc E–F–G (the fixed-star semicircle / Solstitial Colure) rises at the right, its horizontal diameter E–A–G through the center, with the Ecliptic Pole F on the arc (H at far left). From the two Earth-positions, lines run to the polar star F (BF, CF) and to the off-ecliptic star L at lower right (BL, CL); tangent lines A–I–M and A–O–P (the physical horizons) and points K, M, P, R, V complete the figure. The narrowing of the angle as the star moves from F toward G (where it vanishes) shows the parallax greatest at the ecliptic pole.]

[Margin: Galileo’s 2nd Proposition.]

[II.] These [things] premised, Galileo (Dialogue 3, On the System of the World, Italian p. 376, Latin 385) demonstrates that the Maximum parallax of the annual orb, which the stars can undergo in altitude or distance from the zenith, is [the parallax] of those which are at the Poles of the Ecliptic; but none [is the parallax] of those which are in the Ecliptic; and [it is] the smaller, the nearer they are to the Ecliptic, [and] the more remote from the poles of the Ecliptic. And so, in the preceding figure, the maximum parallax is of the star F, none of the star G or E; but [the parallax] of the star L is smaller than that of the star F, because the parallactic angle BFC is greater than the parallactic [angle] BLC — nay, [it is] the greatest of all those which are contained on the same base BAC, within the same periphery EFG, as he shows in the same place; and it is gathered even hence, that that angle from F, tending toward G, is so diminished that at last, at G, it vanishes. Wherefore the Fixed stars placed in the Ecliptic are never raised or depressed, more or less, on account of the earth’s motion (though they become now nearer to the eye, now more remote); and therefore (Italian p. 373, Latin 382, of the same dialogue) he demonstrated this other proposition, which is, for him, prior: The maximum parallax of the annual orb, which the stars can undergo in apparent magnitude, is [the parallax] of those which are in the Ecliptic; but none [is the parallax] of those which are at the poles of the Ecliptic; and [it is] the smaller, the nearer they are to the poles of the Ecliptic, and the more remote from the Ecliptic. Yet the first proposition holds in the case which is imagined in the description of the figure, the immense distance of the Fixed [stars] G and E being posited; otherwise a case can be given in which some parallax of the stars G and E too occurs from the annual orb, according to what was said in ch. 11, no. 14.

[III.] It is now worth the trouble to inquire how great the maximum parallax of the annual orb would be — supposing both the mean distance of the Sun and of the Fixed [stars] from the Earth, which some asserters of the Earth’s immobility posit, and that which the more distinguished Copernicans posit; which we shall easily obtain in the triangle ABF (right-angled at A), in which, from the hypotheses of the aforesaid, the side AB is given (the semidiameter of the annual orb, namely the mean distance of the Sun from the Earth) and the side AF (the distance of the Fixed [stars] from the World’s center); wherefore, by the Fifth [proposition] of Plane Right-angled Triangles (on which [see] bk. 10, sect. 1, ch. 2), the angle AFB will become known; which doubled, we shall have the whole parallactic angle BFC, standing on the diameter BC.

[Margin: The Copernicans’ lapse in the Parallax of the Annual Orb.]

[Granted] that all the Copernicans hitherto have absolutely obtruded upon us the angle AFB for the parallax of the annual orb (as we showed in bk. 6, ch. 7, from no. 10 to 15) — whether they used dissimulation, to diminish this parallax, or rather through inadvertence, arising from the habit of the parallax of the earth’s orb [the diurnal], which the semidiameter of the Earth generates. For just as, in the preceding figure, the star being placed at L, the eye rising at K, [the star] seen through the tangent HKL, and the straight line BL drawn from the center B, the horizontal parallax of the star is the angle BLK, standing on the Earth’s semidiameter BK; so they estimated the parallax of the annual orb from the semidiameter alone of the annual orb AB, nor [reckoned it] greater than the angle AFB. But just as, if someone — the earth standing still — were transferred from K to S, the place of his Antipodes, and could see the star L from both [positions], the whole parallax would be the angle SLK, arising from the diameter KS; so, the Earth being transferred from B to C, on account of the whole diameter BC, the Parallax of the annual orb, adequately taken, comes out [to be] BFC. Yet, for the distinction of either opinion, we shall set down in the following Table both the angle AFB and the angle CFB, and its quantity elicited from the following given sides AB and AF — the first of which we took from bk. 3, ch. 7, and the second from bk. 6, ch. 7, where also we adduced the authors of the distances and their places.

[Translator’s note — the engraved table is full-width. “Distances in Earth-semidiameters”: AB = the Sun’s distance from the Earth (= the semidiameter of the annual orb); AF = the Fixed stars’ distance from the World’s center. “Semiparallax (angle AFB)” and “Parallax (angle CFB = 2×AFB)” are given in degrees (G.), minutes (′), and seconds (″). The upper group are asserters of the Earth’s immobility; the lower (bracketed “Copernicans”) posit vastly greater stellar distances.]

TABLE OF THE MAXIMUM PARALLAX OF THE ANNUAL ORB

[…continues on p. 452 (PDF 487) with the catchword “Aduer-” (Aduertendum) — an observation on the Table, and the first of the three parallax-arguments in form.]

(printed p. 452 — within Chapter XXVIII: after noting Lansberg’s shifting parallax figures, Riccioli tabulates how far each author must place the fixed stars so the full annual parallax stays under a detectable 10 seconds, and shows that even Sirius approached by the whole orb-diameter would grow imperceptibly. Then the first formal argument: were the Earth moving, a sensible parallax would appear in the fixed stars’ meridian altitudes after three or six months, yet neither Tycho nor Riccioli ever found any. Kepler’s response is stated and Riccioli begins listing its four faults, continuing on p. 453.)

[Header: BOOK IX. SECTION IV.]

But it must be Observed that Lansberg formerly, together with Hortensius, admitted a Parallax of the annual orb in the Fixed [stars] of 30″ (that is, really a semi-parallax), as though detectable by no observations; but afterward, in the Uranometria (bk. 3, element 7), he reduced it to 7″ 22‴; wherefore, doubling it (as is fitting), the whole parallax becomes 14″ 44‴, or nearly 15″.

[Margin: The distance of the Fixed stars to be asserted, lest the Parallax of the annual orb become greater than 10 Seconds.]

[IV.] Besides the knowledge of the aforesaid parallax — because we judge that, by our Instruments rightly applied, a difference in the altitude of the stars can be detected which is not less than ten Seconds — it pleases [us] to investigate how great a distance of the Fixed [stars] from the World’s center the Authors must establish, if they wish so great a parallax of the Annual Orb [as is] possible not to exceed 10″ — that is, [if they wish] the said angle CFB to be 10″, and AFB 5″, the side AB (the distance between the Sun and the Earth) being posited as great as each of the below-written Authors supposes. Which we shall easily accomplish by the analysis of the said triangle ABF (right-angled at A), in which AB is given, and the opposite angle AFB is assumed [to be] 5″. Therefore, by [proposition] 7 of Plane Right-angled Triangles, we shall find how great the distance of the Fixed [stars] AF must be. Let, then, the below-written table [be set down]:

[Given the angle AFB = 5″ and CFB = 10″; AB and AF in Earth-semidiameters]

[Margin: There is no sensible diversity in the apparent magnitude of the Fixed stars on account of the Annual-orb Parallax.]

[V.] It remains to inquire how much larger a Fixed star of the first magnitude — such as is Sirius — would appear, if the Earth approached it by the whole diameter of the annual orb, and such a star were imagined to be in the Ecliptic, the minimum distance of the Fixed [stars] from the World’s center being posited which some Copernicans posit — namely 1,440,000 terrestrial semidiameters, as great as Herigone posits. But the apparent diameter of Sirius is only 18″, as we taught (bk. 7, sect. 6, ch. 11, in the first table). Let, then, in the following figure, AC be the distance of Sirius from the world’s center C, and the apparent semidiameter of Sirius [be seen] under the angle ACB of 9″. For, the tangent CB and the semidiameter AB being drawn, there is given, in the triangle ABC (right-angled at the contact B), the Base AC, 1,440,000, and the adjacent angle ACB, 9″; from which, by the first [proposition] of these Triangles, the true semidiameter of Sirius is gathered [to be] 62 738/1000 terrestrial semidiameters. Now let the eye approach the star, so that Sirius is distant from the eye C by the interval CP — namely 1,437,600 semidiameters (2400 being subtracted, as many as Herigone gives to the diameter of the annual orb, from the 1,440,000 semidiameters — although, on the Copernican hypothesis, only the semidiameter should be subtracted, unless, instead of distances from the World’s center, distances from the earth’s center be taken); and let DP be 62 738/1000. For in the triangle PDC (right-angled at D), there will be found, by the Second [proposition] of plane right-angled triangles, the angle PCD, 9″ 2‴; wherefore the whole apparent diameter of Sirius will be 18″ 4‴, and so, on account of the said approach, not greater than [by] 4‴ — which difference is utterly unobservable. How much less observable, then, will it be in other smaller stars, both outside the Ecliptic and more distant from the world’s center? These [things] being prepared, let us come to the Arguments.

[Engraved figure (#42) — the Sirius-approach triangle. A small circle (Sirius) stands at the top, marked B and A (A its center, B the point where the tangent from C meets it); inside the circle, lower, are the points D and P. From the far lower vertex C, two long lines rise to the star-circle — the tangent C–B and the line C–P (through D) — forming a tall, extremely narrow isosceles triangle whose apex-angle at C is the few-arcsecond angle subtended by the star, illustrating that even the whole annual-orb approach changes Sirius’s apparent size by only 4‴.]

I. Argument, from the Parallax of the Annual Orb in the Altitude of the Fixed stars

[VI.] If the Earth were moved in the Annual Orb, some sensible parallax would be observed in the meridian altitude of the Fixed [stars], after three or six months, in the same Horizon, taken by great instruments exactly made and applied. But no such parallax has hitherto appeared. Therefore the Earth is not moved in the Annual Orb.

[Margin: Proof of the Major.]

The Major seems proved from the Trigonometric calculation, and the table at the end of number 3, at least for stars at the pole of the Ecliptic, or placed not far [from it] — of which kind is the star in the bend of Draco, while in the other hemisphere there is a star in the breast of the fish Dorado. It can moreover be proved from Sirius, which we have observed in the Meridian more than once — a little after the autumnal Equinox, in the morning at sunrise; and a little after the vernal Equinox, in the evening at sunset; for so great is the brightness of this star that even for about one quarter-hour before sunset, or after sunrise, it can be seen in a clear sky, and its distance from the Sun observed, as well as its meridian altitude. Now let Sirius be at L, in the figure set out at number 1 [fig. #41], and its Southern latitude GL — that is, the angle GAL — [be] 39° 32′ 58″ (from what was said in bk. 6, ch. 24). Let, then, according to Herigone, AB be 1200 terrestrial semidiameters, and AL 1,440,000. For from these and the comprehended angle BAL of 39° 32′ 58″, there comes out, by [proposition] 3 of Plane Oblique-angled Triangles, the angle ALB, 1′ 50″; wherefore the whole parallactic [angle] CLB is, very nearly, 3′ 40″. Now let the Pole star be at V, whose latitude EV — that is, the angle EAV — is 66° 2′; and in the triangle CAV, besides the said angle, let there be given (as above) AC, 1200, and AV, 1,440,000 terrestrial semidiameters; for there will be found the angle AVC, 2′ 38″, and BVC, nearly 5′ 16″. So great, then, a parallax in the altitude of these stars would appear, if they could be observed in the Meridian from the earth transferred from B to C, through the whole diameter BAC. But if the Keplerian distances be taken — AB 3469, and AL (or AV) 60,000,000 — there comes out ALB 8″, and CLB 16″ nearly; but AVC 12″, and CVB 24″ nearly.

[Margin: Proof of the Minor.]

The Minor is proved, because neither we, in the altitude of Sirius, nor we, or Tycho, in the altitude of the Pole star, detect so great a parallax. Indeed Tycho, in the year 1586, observed at midnight the maximum meridian altitude of that star after the autumnal Equinox; but in the years 1577, 1581, 1586, and 1589, after the winter Solstice, he observed toward morning the minimum, and after evening the maximum, altitude of the same; and from all these observations he found always the same distance of it from the world’s pole — indeed, in the year 1586, from the Equinoctial and the winter observation, he found it [to be] 2° 56′ 10″ — when he ought to have found [it] different, on account of the earth’s translation made in a three-month interval; nor did he detect any parallax, not even of ten Seconds, as Longomontanus also relates (bk. 1 of the Theorica, ch. 1, p. 159). Which Kepler too (bk. 4, Epitome of Copernican Astronomy, p. 493) confirms, saying of Tycho: He observed the maximum altitude of the pole star (which in this age is in the 7th [degree] of Aries) in the year 1586, at midnight after the autumnal equinox, and it was 58° 51′; he observed the same also around the winter Solstice, on 26 December, in the evening about the 6th hour, and found again 58° 51′; and so there was no difference — although in September the horizon would cut the sphere of the Fixed [stars] lower by almost the whole semidiameter of the orb in which the earth is carried, than on 26 December (since there the Sun appeared in Libra, here in Capricorn). The same happened also when the minimum altitude was observed at midnight, after the vernal Equinox, and after the winter Solstice in the morning at the 6th hour; for in both there were found 52° 59′ 30″ — although in March the horizon would cut the Fixed [stars] higher by almost the whole semidiameter of the orb in which the Earth [is carried], than in December. Therefore that diameter of the orb in which the earth is carried is not sensible through the Brahean instruments.

[Margin: Kepler’s 1st response, faulty.]

[VII.] Kepler would respond, first (same p. 493), by denying the Major, because from his distances there does not follow a parallax of the annual orb greater than 12″ — which, he denies, can be perceived by any diligence of the craftsmen, especially since the diameter of the pole star seems to equal at least one minute [60″], and one ought not to trust the craftsmen’s diligence concerning the fifth part of one minute [12″]. But this response labors under four faults: First, he estimates the parallax [as] 12″ from the semidiameter alone of the annual orb, whereas the whole parallax, to be estimated from the diameter, is 24″ (as is clear from the table at number 3, premised). Secondly, the apparent diameter of the Pole star is only…

[…continues on p. 453 (PDF 488) with the catchword “tum-” (tan-tum-modo) — “…only 7″ 54‴, or at most 8″,” completing the second of Kepler’s four faults.]

(printed p. 453 — within Chapter XXVIII: the responses to Argument I conclude — Kepler’s remaining faults, Hortensius’s flawed 30-second concession, and Mästlin’s and Herigone’s per-accidens excuses, all judged insufficient. Riccioli then gives his own response denying the Major: on truly Copernican distances all parallax vanishes, and the geometry of the Earth’s position defeats the three-month and Sirius observations. Argument II, from annual-orb parallax in Sirius’s distance from the Sun (with Grimaldi’s 180-degree variant), is stated and answered — the acute/obtuse difference arises from Sirius’s place relative to the colure, not the Earth’s motion.)

[Header: DE SYSTEMATE TERRÆ MOTÆ — 453]

…only 7″ 54‴, or at most 8″, by our corrected observations (handed down in bk. 7, sect. 6, ch. 11, in the first table). Thirdly, even if this diameter were of one minute, nevertheless threads can be erected at the center of an Astronomical quadrant so subtle that through them one may collimate [sight] upon the very center of the star, and its disk be bisected by them. Fourthly, he does injury to his own Tycho, who in his Mechanica often asserts that he observed the altitudes of the Sun and the Fixed [stars] up to a difference of ten Seconds, or the sixth part of one minute.

[Margin: Hortensius’s 2nd response, faulty.]

[VIII.] Hortensius responds, secondly (in the dissertation with Gassendi On Mercury seen under the Sun), by denying the Minor; for he admits a parallax of 30″, but denies that this can be evidently perceived by any instruments. But his response too errs doubly. First, that parallax was deduced by him from the semidiameter alone of the annual orb; but from the whole diameter it comes out [to be] 60″, that is, of a whole minute. Secondly, he too does injury to Tycho and the other more diligent Astronomers of this age, who descend, in observing the stars’ altitude, at least to tens of Seconds.

[Margin: Mästlin’s and Herigone’s 3rd response, insufficient.]

[IX.] Mästlin responds, thirdly (in the addition to the First Narration of Rheticus, p. 114), and Pierre Herigone (vol. 5 of the Mathematical Course, p. 615), by denying the Major. For although the parallax per se from the annual orb is so great that, if the same star could be beheld in the Meridian or the same vertical by the same observer (remaining in the same Horizon) by two observations after a six-month interval, it would be 3′ near the pole of the Ecliptic (as Herigone concedes), or 2′ or 3′ in stars rising or setting around midnight (as Mästlin admits) — yet per accidens it is unobservable: either because of the horizontal refractions of rising or setting stars, or because, if a star once appears at night in the Meridian, after six months it cannot appear in it (the daily light of the Sun obstructing); and after three months that parallax is much smaller — 2′ or 3′, and so, for [its] smallness, not evidently sensible. But this response too, unless something else be added, is insufficient, at least as regards stars observable in the Meridian. First, Sirius, the Sun not obstructing, is observable after six months in the same Meridian in which it was previously observed, as I said in the proof of the Major. Secondly, that very parallax which Herigone admits would occur after 6 months is — by his distances — of nearly three Minutes, since [it is] estimated from the semidiameter; for that which [is reckoned] from the whole diameter would be 6 minutes, as is clear from the table of number 3. But who would deny that a parallax of 3 Minutes can be perceived, on account of its smallness — unless [he be] most unskilled in Astronomical observations?

[Margin: Our 4th response.]

[X.] It is responded, therefore, fourthly, by denying the Major. First, indeed, because Copernicus wishes so great a distance of the Fixed [stars] to be assumed that all parallax of the annual orb vanishes — even if this interval had to be increased [from] finite toward infinite. And although the Major militates against Herigone, and nearly against Mästlin and Hortensius (especially in [the case of] Sirius, which some of them thought could not be observed in the Meridian once and again after six months) — yet against the other Copernicans in the table of number 3, and much less against Kepler, it does not so militate; because that parallax, [reckoned] so great from the whole diameter of the annual orb according to their distances, cannot be accommodated except to stars established at the pole of the Ecliptic — none of which can be seen once, and again after six months, in the Meridian, the Sun’s brightness obstructing one of these observations. But Sirius, which can be seen twice at the extremes of about six months, is so far distant from the pole of the Ecliptic that the parallax of the annual orb, from Kepler’s distances, does not come out greater than 16″. And what, in the pole star, would reach 24″, is not observable, the Sun’s light obstructing one of the observations. But the distance of these and of all Fixed [stars] from the Ecliptic pole, like [their] latitude, is invariable. Secondly, because that Parallax which, per se, after a three-month interval would be observable in other Fixed [stars] visible at night in the Meridian — if the Earth’s center ran from N to A through the very diameter of the Annual Orb — nevertheless cannot be observed, because the Earth’s center is transferred through the periphery of the annual orb from C to D; in which situation the diameter or semidiameter of the Annual Orb cannot fall into the plane of the Meridian, and so a triangle cannot be formed in which the parallactic angle is made; nothing, therefore, [is] wonderful if, after three months, no parallax of the annual orb or semi-orb can be observed — on this hypothesis of a moving Earth equally. Thirdly, because not even the said Parallax of Sirius itself is observable, for another cause: namely, that when it appears in the Meridian at sunrise or sunset, the Sun appears to the Earth under the Equinoctial points (or near), not under the Solstitial points; and on the Copernican hypothesis the Earth is at, or near, the Equinoctial points — in which situation the diameter of the annual orb through the earth’s center is in a plane most diverse from (and such as is) the plane of the Meridian; for it is in the plane of the Horizon, and cuts the Meridian plane nearly at right angles. Wherefore a triangle cannot be formed upon it, in which the Meridian altitude and its parallax is made.

II. Argument, from the Parallax of the Annual Orb in the Distance of Sirius from the Sun

[XI.] If the Earth were moved through the Annual Orb, some Parallax of the Annual Orb would appear in the distance of Sirius [measured] from the Sun — taken once at sunrise, and again after 6 months at sunset. But none appears. Therefore, etc.

The Major is proved, the figure premised at number 1 [fig. #41] being resumed — in which, however, let EFG be a great circle drawn through the Equinoctial points (which here let be imagined G and E) and through the center L of Sirius. For since [Sirius] is seen not only at sunrise or sunset (the Sun being at, or near, the equinoctial points), but also somewhat after sunrise and before sunset, its distance from the Sun can be taken. Let, then, AF be the colure of the Solstices, and Sirius at L: for when the Sun A rises to the eye O, near the autumnal Equinox, [Sirius] will appear distant from the Sun by the acute angle AOL; but when, near the vernal Equinox, the Sun sets to the eye I, Sirius L will appear distant from the Sun by the obtuse angle AIL (supply what is lacking in the figure). Therefore the difference between the said angles will be the parallax arising from the annual orb. The Minor is proved by experiment, because no such parallax in fact appears in the distance of Sirius from the Sun.

Otherwise Fr. Francesco Maria Grimaldi proves the same Major: namely, because if the earth is at A, the distance of Sirius from the Sun B — [measured] at the rising [Sun] at the point of the Autumnal Equinox, observed by the angle BAL — together with the same distance of Sirius from the Sun, but at the setting [Sun] in C, at the point of the Vernal Equinox, observed by the angle CAL, will complete two right angles, or 180° (by [proposition] 13 of the first [book] of Euclid). But if, the Sun being immovable at A, the earth — placed now at B — observe the distance of Sirius from the Sun by the angle LBA, and now at C observe the same by the angle LCA, these two distances will not complete 180°, but the quantity of the parallactic angle BLC will be lacking.

[Margin: Response to Argument 2.]

[XII.] I respond by denying both the Minor and the sequel of the Major: for that diversity of the obtuse and acute angle does not arise from the motion of the Earth through the annual orb, but from the distance of Sirius from the colure of the Solstices — which Sirius, in this age, is nearer to the autumnal Equinox than to the vernal; whence it comes about that, whether the Earth move through its annual orb, or the Sun through its own annual [orb], [Sirius] ought to appear distant from the Sun (rising at the autumnal Equinox) by an acute angle, and at the vernal Equinox (the Sun setting) by an obtuse angle. But if Sirius were at F, the point of the colure of the Solstices, it would appear — on the hypothesis of the Earth both moving and resting — distant from the Sun, at either Equinox, by the right angle AIF and AOF, formed by one line tending straight from the eye to the center of Sirius, and by the other, which would be the tangent to the earth at the point of the eye (and the same would be the physical Horizon, in which the Sun’s center would rise or set to the eye O and I). To say nothing meanwhile of the variety of Solar refractions at the horizon, which could disturb the evidence of this parallax, even if it were otherwise sensible per se — although Copernicus would avoid this sensibility by increasing the distance of the Fixed [stars] to infinity. And hence likewise comes the response to the other proof of the Major [Grimaldi’s]: because, namely, on account of the immense and free distance of the Fixed [stars], the lines AL, BL, CL would all turn out parallel, and so the angles at A, B, C would appear equal; besides that, that observation is too slippery, and it can scarcely happen that at the Sun’s rising and setting an Equinox occurs.

[…continues on p. 454 (PDF 489) with the catchword “III. Ar-” (III. Argumentum) — the third parallax-argument.]

(printed p. 454 — the third parallax-argument, from variation in the fixed stars’ apparent magnitude, closes Chapter XXVIII: even Sirius at the least Copernican distance would vary by an unobservable four thirds of a second. Chapter XXIX opens with three arguments against the annual motion from the excessive distance of the fixed stars and the bulk of the eighth sphere, beginning a review of opinions on that distance. The first two Copernican opinions are given: Aristarchus (via Archimedes’ Sand-Reckoner) and Mästlin, whose figures Riccioli calls inconstant.)

[Header: BOOK IX. SECTION IV. — 454]

III. Argument, from the Parallax in the apparent Magnitude of the Fixed stars

[XIII.] If the Earth were moved through the annual orb, a great diversity would appear in the apparent diameter of the Fixed [stars]. But none appears. Therefore, etc.

[Margin: Response to Argument 3.]

It is responded by denying the Major: because not even in Sirius would the diversity be greater than 4‴ — even if Sirius were supposed to be in the Ecliptic (as was shown at number 5), and the stars were distant from the World’s center by not less than 1,440,000 terrestrial semidiameters (as Herigone posits); but a little difference of 4‴ is evidently observable by no instruments. How much less [observable], if the star were far from the Ecliptic, or the Keplerian distance of the Fixed [stars] were assumed, or as great [a distance] as Copernicus allows — who allows so great [a one] in his hypothesis that every difference of this kind comes out utterly insensible. But that in the Planets — especially in Mars and Venus — a notable diversity of apparent magnitude is seen, is no more an argument of these Planets approaching the unmoved Earth than of the Earth moved and approaching them.

✠ [Chapter XXVIII ends here.]