[I.] Up to now, the arguments against the Earth’s motion have been sought out from those [things] which appear in elementary bodies; now they are to be deduced from the celestial Phenomena. But these too we shall at length recognize [to be] weak — which, nevertheless, are not to be despised, lest they should seem to be rejected unheard, for nothing.
I. Argument, from the apparent Setting of the Stars
[Margin: 1st Argument. Form.]
If the Earth were moved by the diurnal motion, the Western mountains would seem to us to ascend, rather than the star to descend below the horizon; and the Eastern mountains [would seem] rather to descend, than the star to ascend above the horizon. But observation holds otherwise: for the mountains seem immovable in both quarters, and the star [seems] to ascend from the East and to descend into the West — indeed, it is from this that that part of the heaven is called the East, and this [other] the West. Therefore the Earth is not moved by the diurnal motion.
[Margin: Kepler’s Response to Argument 1.]
Thus Kepler objects to himself (bk. 1 of the Epitome of Copernican Astronomy, p. 129). But he responds at once by denying the Major’s consequence: for ascent, says he, is judged by sight from the desertion of the plane on which we stand, and from the approach of a thing toward the zenith, toward which a man’s stature is raised erect. But the mountains themselves are the very [things] which form the visible plane for us, and they do not approach the zenith — because, by as much as they themselves advance, by just so much does the line in which the spectator’s stature stands erect advance at the same time; and in that line [is] the zenith too: whence it comes about that the zenith is visibly always equally distant from the ends of the ground on which the spectator stands, that is, from the extreme [end-]points [of the motion]. The mountains, therefore, cannot be seen to ascend — granted that, on the hypothesis of a moving Earth, they truly do ascend.
[Margin: Corroboration of the argument.]
But if you press the argument, and say that it is necessary that they at least seem to be moved, even if not to ascend or descend — it is responded by denying this consequence too: for if they seemed to be moved, [it would be] by no other [motion] indeed than the motion of ascent or descent; for whatever cause, then, they are not seen to ascend or descend, for that very [cause] they will not even seem to be moved. And here that Virgilian [verse] holds good:
We are carried forth from the harbor, and the land and the cities recede. [Aeneid III.72]
— granted it is the ship which really recedes. So those who sail downstream, if they were not beforehand aware of the motion of themselves and of the ship, will think that the neighboring shores and the mountains come to meet them against the stream: and so of like cases.
II. Argument, from the Fixed Stars seen from a Well
[Margin: 2nd Argument. Form.]
[II.] If the Earth, rather than the Firmament, were moved by the diurnal whirling, we — looking from the bottom of a well at some Fixed star overhanging our zenith — could not see it as long as we in fact do see it; but in one beat of the artery, snatched together with the well toward the East, we would lose sight of it. The consequent is false; therefore the antecedent [is false] too, whence that [conclusion] follows.
Thus indeed argue, with Alessandro Tassoni, [also] Bartolomeo Mastri and Bonaventura Belluto (Disputation 4, On the Heaven, q. 4, art. 3, no. 117). They try to prove the Major’s consequence from this: that the well’s mouth is not wider than one ell — as great a little space as the Earth, by its diurnal whirling, traverses in the blink of an eye; while we cannot see more of the heaven from the well’s depth than what is comprehended under the angle made at the eye’s center by two lines drawn from the eye through the well’s outermost edges into the heaven. Since, therefore, the space of the heaven comprehended by the said angle is far greater than the well’s mouth, from this they deduce that the same star can be beheld from the well’s depth for so long a time, and that the Fixed [stars], rather than the Earth, are moved by the diurnal motion. But these authors are to be pardoned, as being little devoted to Geometry — since even some friend of ours (not lightly imbued with Geometry, I know not who) thought that this argument had a demonstrative force against the Earth’s motion, and tried by his calculations to add weight to it. For they say that in one Second of an hour (or in about one beat of the human artery) the Earth, by the force of the diurnal whirling, traverses under the Equator very many paces — much more than one ell, as great as the well’s width is — and therefore that immediately the whole width of the well would withdraw itself from the space which lies beneath the celestial space embracing the Fixed star, etc. Nor, however, is it necessary to descend into a well; but it suffices to look at the passing stars through a window, from a very long tube, or from the bottom of some portico closed on all sides.
[Margin: Response to argument 2.]
It is responded nonetheless by distinguishing the Major’s consequence, and conceding it — if the eye and the well-bottom were at the Earth’s center, and the well’s mouth were not much wider than 300 of our paces; or [if] the eye and the well-bottom were near the earth’s surface, but the visual angle made by the straight lines drawn from the eye through the opposite edges of the well comprehended no more Seconds in the heaven than the fifteen [15″] of the circumference described from it. But otherwise the Major’s sequel is denied. As to its proof, it is conceded that there are more paces — which a point of the terrestrial surface traverses in one beat of the human artery (that is, in about one Second of an hour), especially in the parallels near the Equator — than the usual width of the well; but the consequence is denied, because the measure of the time during which some Star lasts under the gaze of an eye placed at the well-bottom is not the physical quantity of the well’s width (determined in paces or in other measures known to us), nor its proportion to the physical quantity of the whole terrestrial circumference; but it is the optical quantity — namely, of the visual angle, relative to the arc of that circumference which is described from the Earth’s center (for this circumference is the one whose whole revolution measures the diurnal time of 24 hours). Hence it comes about that, if the eye is at the earth’s center, the measure of that time (or duration of the star under the eye’s gaze, [the eye] placed at the bottom of so deep a well) is the arc of the circumference — both terrestrial and celestial — comprehended by the two straight lines drawn from the eye through the opposite edges of the well: and in this case I say that, if the star must remain under view for a longer time than one Second of an hour (or one beat of the human artery), the well’s mouth must be wider than 300 of our paces. But if the eye is near the earth’s surface, the measure of the said time is not the arc of the terrestrial globe’s circumference comprehended by the well’s edges (because that arc is not described from the eye as its center); but it is a celestial or terrestrial arc similar to the arc comprehended by the visual angle (or by the lines drawn from the eye through the opposite edges), if from it the doubled parallax of the star be subtracted. And in this case I say that, if the star is not to remain under the gaze of an eye placed at the well-bottom more than one Second of an hour, this arc and angle must comprehend no more than 15″.
[III.] These [things], though they can easily be understood by those skilled in Geometry, it is nevertheless worth while to assist — both them and others — by some diagram, and by some problems (the occasion taken from this argument) very opportune. From the Earth’s center A, describe the semi-periphery BCD of the celestial Equator, and under it the semicircumference ELF of the terrestrial Equator, in whose Equator’s plane let there be a well a–S–O–b, through whose bottom describe from A the semicircumference GIK, which the eye I, placed at the well-bottom, is supposed to describe while it is moved by the diurnal whirling with the Earth around the center A — toward B, the Eastern point; and let the width of the well be the straight line SO, subtended to the arc S–L–O. But since the walls of all wells and buildings constructed to the perpendicular, and the perpendicular lines of [those] walls, meet at A, the center of the Earth, if the well be imagined dug all the way to the earth’s center, the well will be A–S–O. Let there be, then, such a well, and the eye placed at A: for, the straight lines ASP, AOQ being produced from A through the edges of the well…
[…continues on p. 435 (PDF 470) with the catchword “tei” (pu-tei S, & O) — ”…[through the edges] of the well S and O, all the way up into the heaven,” carrying on the geometric construction and the three Problems that resolve the well-argument quantitatively.]
(printed p. 435 — within Chapter XXIII: the geometric resolution of the well argument continues through three worked Problems — finding the well-mouth needed for a given visibility-delay with the eye at the Earth’s center (300 paces for one second) and near the surface, and conversely finding the visibility-delay from a well’s given width and depth (a 4-ft-wide, 100-ft-deep well yields over a minute of visibility).)
[Header: DE SYSTEMATE TERRÆ MOTÆ — 435]
…[through the edges] of the well S and O, all the way up into the heaven, where, as soon as the Eastern edge of the well comes to S — if in the line ASP there be a star at P — [the eye] will begin to behold it, and will be able to see it until the well’s edge O comes to S. But I said that the star cannot be seen by an eye placed at the Earth’s center for longer than one Second of an hour, if the well’s mouth OS is not wider than 300 paces — which, by the help of the following Problem (that the benefit may be more universal), I shall demonstrate.
[Engraved figure (#39) — the well-and-zenith construction. A great semicircle B–C–D (the celestial Equator) stands on the diameter B…D, with center A on the baseline; C is the zenith (top). Along the baseline from west (left) to east (right): B, E, G, A, K, F, D. Nested over A are the smaller concentric arcs — the terrestrial Equator E–L–F and the well-bottom semicircle G–I–K — and at the very center the well-mouth chord S–O (with the well A–S–O–b and points a, b). Radial lines run from A up to the lettered points on the great arc: M, P, C, O[uter], N (M and N farthest left and right, P and the rest between, C at the apex). The visual lines from the surface-eye, I·S·M and I·O·N, frame the wide arc MCN; the inner letters R, S, L, X (with small a, b) mark the well-mouth and axis near A; V, T, H lie on the lower left. The figure serves all three Problems below.]
I. Problem. Given the delay of time during which a Star can be seen by an eye existing at the bottom of a well at the Earth’s center under the Equator; and given moreover the Earth’s semidiameter: to find the width of the well’s mouth necessary for such a delay.
[IV.] From A, the Earth’s center, let the perpendicular ALC be drawn, cutting in half both the arc SLO, and the chord SO, and the angle SAO (by [propositions] 9, 10, and 11 of the first [book] of Euclid); for there will arise the rectilinear triangle ALO, right-angled at L, in which is given the base AO (the earth’s semidiameter) and the angle OAL (which is half of the angle OAS subtending the arc OS — known from the given time-delay, converted into parts [degrees] of the Equator). Therefore, by the laws of Trigonometry, the side OL will not lie hidden, in the [same] parts of which AO is known; and so OL, doubled, will give the chord OS, that is the required width of the well.
[Margin: Example of a one-second delay.]
EXAMPLE. Let the said time-delay be of one Second of an hour; to this, from the tables of the Primum Mobile, corresponds the arc OS of 15″; whose half is the arc OL of 7″½, and just so great too is the angle OAL. But since, from what was said in book 2, ch. 7, the Earth’s semidiameter AO is 4139 Bolognese Miles, AO will be 4,139,000 Bolognese paces; from which data the side OL is gathered [to be] 150 paces. Therefore the required width of the well, OS, is 300 paces.
Now let the eye be imagined at the well’s bottom I, equally distant from the sides O–b and S–a, and let the straight lines I·S·M and I·O·N be produced from I through the well’s edges O and S all the way into the heaven; for they will intercept the arc MCN, much greater than the arc PCQ which was intercepted by the visual angle SAO. Wherefore, by so much longer a delay will the eye I see the star (rising for it from M) sooner than the eye A [sees] the star rising at P, as the Equator’s arc MCN is greater than the arc PCQ — both arcs converted into time. But if this time, in this position of the eye too, must be no greater than one beat of the human artery (or one Second of an hour), it will be necessary, as I said, that the depth of the well IL and the narrowness of the mouth OS be so [great] that the angle MIN comprehend no more than 15″ of the Equator — at least if the star is Fixed, or lacking sensible parallax. Let us suppose the well’s depth to be at least 100 paces (or 500 feet) — although scarcely any well is wont to be found which exceeds 100 cubits (that is, 150 feet); and let the delay of the star under the gaze of the eye I be of one Second of an hour; and let it be proposed to find the width of the well, beyond which it cannot last for the given delay — for this will be investigated by the following Problem, with an example also added, as you see.
II. Problem. Given the delay of a star under the gaze of an eye looking at it from a well-bottom, [the eye] existing at the Equator; given moreover the depth of the well: to find the width of the well’s mouth congruent to that delay.
[V.] In the preceding figure, since the arc MCN is given (from the time of the star’s delay under the gaze of the eye I, converted into parts of the Equator), the angle MAN is also given (from which that arc was described); and, AIC being produced cutting the angle MAN in half (just as it cuts SIO), the angle CAN will be known, because it is half of the angle MAN. Now let the straight lines AM and AN be produced from A: for in the triangle AIN, the angle ANI (which is the star’s Parallax) and the angle NAI, taken together, are equal to the external angle CIN (by [proposition] 32 of the first [book] of Euclid’s Elements); wherefore, if the parallax N be known and added to the angle NAI, the angle CIN — that is, LIO — will be known, in the small rectilinear triangle LIO, right-angled at L, in which the well’s depth IL is given. Therefore, by the canons of Triangles, the side LO will be made manifest, which, doubled, will give the well’s required width OS.
EXAMPLE. Let the given delay of the Star be of 1 Second of an hour; to which, at the Equator, corresponds the arc MCN, that is the angle MAN, of 15″; wherefore the half of this angle, CAN, is 7″½; and just so great — to all the subtlety of the senses — is the angle CIN, that is LIO, if the angle N (the parallax) be supposed insensible, as it is supposed [to be] in the Fixed [stars]. Let there now be given, in the triangle LIO, the well’s depth (or side LI) of 100 paces, or 500 feet, that is 6000 inches. For from this and from the angle LIO of 7″½, the side OL will come out [to be] 21⅞ inches. Wherefore the whole width of the well, OS, will be 43¾ inches — that is, 3 feet and 7¾ inches. But if it were greater, the star would last under the gaze of the eye I for a longer time than one Second of an hour.
III. Problem. Given the Width and Depth of a Well under the Equator: to find the time-delay during which a star can be seen by an eye placed at the well-bottom.
[VI.] In the triangle LIO of the preceding figure, right-angled at L, the well’s depth LI is given for one side about the right angle, and LO for the other (since it is half of the well’s width OS). Therefore, from Trigonometry, the angle LIO will be made known; from which, the star’s parallax (if there be any) — that is, the angle ANI — being subtracted, the angle IAN will be known (by [proposition] 32 of the first [book] of Euclid), that is the angle CAN, measuring the arc CN of the Equator; and this, doubled, will give the arc MCN, which finally, converted into time, will make manifest the delay of the star under the gaze of the eye I.
EXAMPLE. Let the well’s depth LI be given [as] 100 feet, and the width SO [as] 4 feet, and therefore the half-width LO [as] 2 feet. For, by the laws of right-angled Triangles, the angle LIO will be 8′ 50″; and just so great will be CAN, if N, the parallactic angle, is insensible (as it is in the Fixed stars). Wherefore the whole angle MAN — that is, the arc MCN — will be 17′ 40″; to which corresponds a time of one Minute and 11″ nearly, that is about 71 beats of the human artery; which having elapsed, the star M — seen as caught beneath the eye I, under the edge S — will set for it beneath the edge O, [the star] being now translated to N by the force of the celestial diurnal conversion. But if it be the Earth that turns, the well’s edge O will, in so great a time, be translated together with the well toward T, the Eastern point, by so great an arc of the terrestrial circumference that through it a straight line — namely IM — can be drawn from the eye to the Fixed star at M (now supposed immovable): concerning the quantity of which arc, consult the following Problem.
[…continues on p. 436 (PDF 471) with the catchword “IV. Pro-” (IV. Problema) — the fourth Problem of this well-geometry, finding the terrestrial arc through which the well’s edge is carried while the star remains in view.]
(printed p. 436 — the well-geometry of Chapter XXIII finishes with Problem IV, exposing how greatly the argument’s proposers sinned against Geometry. The Third Argument (solstitial gnomon shadows at Syene) and Fourth (the three-hour eclipse at Christ’s death, answered as miraculous by the Moon’s supernaturally varied motion) are given and answered, and the chapter closes. Then Chapter XXIV opens, on four arguments from the principle and simplicity of the motion, beginning with Aristotle against Anaximander.)
[Header: BOOK IX. SECTION IV.]
IV. Problem. Given the Depth and Width of a Well, to find the arc of the terrestrial Equator’s circumference which, by the force of the diurnal whirling, must pass from the rising to the setting of the star, with respect to an eye [looking] from the well-bottom. And if the quantity of one Degree of the terrestrial Equator be known, in feet or paces, etc.: to determine, in these measures, the quantity of the said arc.
[VII.] Let the time of the star’s delay under the gaze of an eye at the bottom of the given well be investigated, by Problem 3; and thence the arc of the Equator corresponding to that time — for the sought arc of the terrestrial circumference will be similar to it. For example, if of the well S–a–b–O there be given LI (the depth) and OL (the half-width), and thence the angle LIO; the angle CAN is given, provided the parallax N (if there be any) has been subtracted from the angle LIO; this [CAN], doubled, will give the angle MAN, which measures both the arc MCN and the arc ZLX — because each is described from the common center A, and intercepted by the same two straight lines AZM and AXN drawn from that same center A. The arc ZLX being now known, if an arc equal to it be taken from the edge S toward T, you will have the arc ST; and the point T will be that to which the eastern edge S will have been translated after the said time. Therefore, taking toward the right the arc TR — determined by the chord TR equal to the well-width OS — R will be the western edge; and a line drawn from the eye H (now translated hither) through R will fall upon the star M, setting for the eye H. Finally, if it be made that, as 60′ to the paces or feet contained in one degree of the terrestrial Equator, so the minutes of the found arc ZLX (or TRS) to another [number], the said arc will be known in these measures too.
EXAMPLE. Let it be as in the example of Problem 3: the well’s width of 4 feet, and the depth of 100 feet. For the arc MCN, from what was said there, will be 17′ 40″. Therefore both ZLX and TRS, each singly, will be 17′ 40″. But, from what was said in book 2, ch. 7, in one degree of the terrestrial Equator there are 72,500 Bolognese Paces, that is 362,500 Feet. Therefore, as 60′ to 362,500 feet, so 17′ 40″ to 46,319 feet. Wherefore, although the said well is 4 feet wide, the same star will nevertheless last under the gaze of an eye placed at its bottom until an arc of 46,319 feet of the terrestrial circumference has passed by — if the well is 100 feet deep.
[Margin: The Error in the Geometry of the Authors of the 2nd Argument.]
From which is gathered how greatly they sinned against Geometry who thought that the star must vanish from the eyes as soon as only so much of the terrestrial circumference (as great as the well’s width is) had passed by.
III. Argument, from the Shadows of Gnomons under the Tropics on the Solstitial Days
[Margin: 3rd Argument’s form and Response.]
[VIII.] If the Earth were moved by the diurnal motion, gnomons erected in the Tropics on the solstitial days would — except at the single moment of midday — cast sensible shadows. But this is against the experiments and observations made at Syene and other places under the Tropic of Cancer, and reported by Strabo and Pliny and others. Therefore, etc.
It is responded, as above to argument 2, by denying the Major, for the same reason. For since gnomons perpendicularly erected above the earth’s surface meet at the earth’s center, those two extreme [gnomons] which, produced into the heaven, would touch the Sun’s edges, subtend a similar arc both on the earth’s surface and in the heaven [at the Sun] — namely of about 30′ — which require, of the diurnal revolution, two horary Minutes [of time]. For so great a time, then, throughout that whole space contained between the two said extreme gnomons, no sensible shadow of them appears — whether the Sun be moved by the diurnal motion while the Earth stands still with the gnomons, or the Earth [moves] while the Sun stands still; because, as I said, the arc is similar; and either arc is the measure of a time having the same proportion to the whole day as that arc has to its whole circumference. Furthermore, concerning Syene especially and other places under the Tropic of Cancer, and concerning the fact that no midday shadows can be seen on the day of the Solstice, [these may be consulted:] Cleomedes (bk. 1 of the Cyclic Theory, ch. 10), Strabo (bk. 17), Pliny (bk. 2, ch. 75), Ptolemy (bk. 4 of the Geography, ch. 5, in table 3 of Africa), and Macrobius (On the Dream of Scipio, bk. 2, ch. 8).
IV. Argument, from the Eclipse of the Sun seen at the death of CHRIST
[Margin: 4th Argument’s form.]
[IX.] The Eclipse of the Sun at the death of CHRIST was total, and lasted such [total] for three hours. But if the Earth were turned by the diurnal whirling, it would not have lasted total for three hours. Therefore the Earth is not turned by the diurnal whirling.
The Major is certain from the Gospel, where it is said that darkness was made over the whole earth from the sixth hour until the ninth hour. The Minor is proved, because the Earth’s most rapid whirling would, in those three hours, have completed 45 degrees, and would have drawn the whole of Palestine under another quarter of the heaven — [a quarter] between which and the Sun the Moon would not have been interposed. And it is confirmed from the 5th epistle of Dionysius the Areopagite to Polycarp (which we reported in bk. 5, ch. 18, no. 3), where he describes the motion of the Luminaries in that eclipse [as going] from East toward West; from which Tanner (On the Heaven, q. 9) and Inchofer (ch. 7 of the Sylleptic treatise) infer that the diurnal motion belongs to the Luminaries; but [they argue] that, if the Earth too had been endowed with the diurnal motion, it would have freed itself twice as quickly in transferring Palestine from a position suitable for seeing the Eclipse — the Luminaries meeting it by their own motion and going still further to meet the terrestrial conversion.
[Margin: Response to Argument 4.]
It is responded by denying the Minor, because that miraculous Eclipse was not total and of three hours by the force of the diurnal and common motion (although, had it been by its force, a miracle could have been worked in that motion too), but by the force of the proper motion of the Moon compared with the proper motion of the Sun — which motion was supernaturally varied in the Moon, the Moon being drawn back from the Full and the diametrical opposition with the Sun to Conjunction with the Sun, and kept under it [the Sun], between [it and] Palestine, for three hours; which could equally have come to pass whether the Earth moved or not. But the rest of the things pertaining to this miracle we have diligently pursued in that chapter 18 of book 5.