(printed p. 398 — Chapter XVII opens: whether the Earth’s diurnal-plus-annual motion better explains the acceleration of falling bodies, and what figure such bodies would describe on a moving Earth. The page surveys Copernican views of the fall-line (Kepler, Gassendi, Galileo, Bullialdus), expounds Bullialdus’s two-circle (Tusi-couple) derivation of the apparent acceleration from uniform circular motion, and reports Chiaramonti’s three objections against it, the third of which Riccioli begins to dismiss as frivolous.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 398]

[I.] When the Copernicans saw that, the Earth being moved daily and annually, the line which Heavy bodies describe in descending, and Light bodies in ascending, naturally, cannot really be a straight line perpendicular to the Horizon (such as appears to us), not a few of that sect labored greatly to determine the figure of this line. And Kepler indeed, with no proof added; but Gassendi, by a similitude drawn from the case of a ball dropped from a mover that is meanwhile carried along, judged it to be parabolic, or very nearly akin to a parabola—as we have already set out in chapter 4 of this section, in scholium 2. [3.] But Galileo and Bullialdus taught it [to be] circular, or composed of circulars; and not stopping here, from the motion made through such a line they seemed to themselves to adduce a solid and true cause for which the motion of Heavy bodies appears faster and faster toward the end, and accordingly took thence an argument—at first sight most strong—to confirm the Earth’s motion. But we shall sharpen their argument more than they themselves [do], and yet thence we shall twist the wound back into the Copernican hypothesis. It pleases [us], however, to deal first with what Bullialdus devises.

[Margin: Bullialdus’s Opinion.]

[II.] Firstly, therefore, Bullialdus, in chapter 4 of his Philolaus, censures the various opinions of those who have tried to bring an a priori cause of the greater and greater velocity of heavy bodies toward the end—but especially [the opinions] of those who take refuge in a greater gravitation, or a second act of gravity—and he says: “Some will have it that the descending body acquires a greater gravity”; which he at once carps at, saying: “A flat and insipid reason, laboring under manifest falsity: for gravity, whatever it ultimately be, follows the mass of matter; and the same [matter] remaining, the same gravity remains.” From their opinion, therefore, it would follow that some [bit] of substance could be produced by local motion—which is false. But, as Scipio Chiaramonti rightly answers him (part 1 of the Antiphilolaus, ch. 4), those authors do not understand a new gravity in the first act, acquired through the motion itself, but in the second act—which accordingly not a few call a gravitation, or a downward-translative impetus. Having spurned, therefore, this and the remaining opinions, Bullialdus concludes: “The reason for the acceleration of the motion of falling bodies, therefore, depends wholly on the circular motion which the part must have in common with the whole, and which it would without doubt observe in [its] fall, unless the violence of its [own] state moved it by a violent motion.” He thinks, moreover, that this motion comes about through two opposite circular motions, by which the movable is kept—to appearance—in the same straight line, and seems unequal in it, although in reality the individual circular motions are equal. To show which, he employs the demonstration of Copernicus (bk. 3, ch. 4) made for the motion of libration and of the Anomaly of the Equinoxes; which we too set out in bk. 2, ch. 29, no. 10. But here it must be repeated summarily, and applied to this case.

[Margin: Rectilinear motion from two circular [motions].]

[III.] In the following diagram, from center A, describe the circle BKE, and the smaller [circle] CIDL, whose semidiameter AC is subduple [half] of the semidiameter AB; and to the smaller circle let there be equal another circle HGA, whose center F is carried in consequentia [eastward, with the order of the signs] by the periphery of the circle CIDL, from C to I. But meanwhile let some movable on the periphery of the circle AGH be moved with a double-faster motion in praecedentia [westward]—the motion of the center F having begun from C, but [the motion] of the other movable H from B. These things being posited, Copernicus shows that it comes about that, these motions meeting one another in contrary directions, the movable is detained in the straight line AB; and when the center F has completed the semi-quadrant CF, the movable H, which was at B, having completed the quadrant HG, is at G. But when the center F is at I—the quadrant CFI of its circle being now completed—it follows that that movable, the semicircle HGA being completed, is at A. Wherefore, if these motions proceed thus, the movable will be kept in the straight line BAE; and although the motion in it will appear unequal, yet in reality it will be equal in the aforesaid circles. Thus, therefore, the motion of Heavy bodies, which appears to us to be made in a straight line, and faster and faster, is in reality nevertheless composed of two circular—and equal—motions.

[Translator’s note — the engraved figure (upper right): center A (the Earth’s center); the large circle BKE with B at the top (tower-summit), E at the bottom (antipode), K at left and L at right, the vertical diameter B·A·D·E and horizontal diameter K·A·L; the smaller circle CIDL (radius AC = ½ AB), with C on the vertical above A, D below, I and L on the sides; and the equal circle HGA (center F, on the vertical between A and B), with H and G in the upper region. As F runs eastward along CIDL while the point starting at B runs westward (twice as fast) along HGA, the two equal circular motions compound to drive the falling point straight down the diameter BAE with an apparently accelerating, really uniform-circular, motion — the Ṭūsī-couple device Copernicus used for the libration of the equinoxes.]

[IV.] Chiaramonti, in that chapter 4, disproves this doctrine as ill-applied to this increment of velocity of which we treat.

[Margin: Chiaramonti’s arguments against Bullialdus.]

Firstly, that Bullialdus wrongly calls the motion of Heavy bodies downward, toward the center of the Earth, violent—since it is most natural, on the supposition that they are outside their [own] place. Secondly, because he supposes KAL to be the surface of the Earth, and a Heavy body dropped from the tower-top B to have an inclination to the circular motion BHK; but [supposes] the points of the arc HG to resist it, and to be carried toward B with double the velocity with which the Heavy body from B would be carried toward K; from this contrariety, therefore, [he says] it comes about that the Heavy body falls through a straight line perpendicular—to sense—by [its] motion into the Earth A. But the diurnal and annual motions of the Earth are not contrary, since they are toward the same quarter; nor can a motion be feigned resisting the motion of the heavy body, for the air near the earth is not moved in a different direction, but the same; nor does gravity, resisting the circular motion, move through a circle striving against the prior circle.

[Margin: Chiaramonti’s double error concerning the velocity of Heavy bodies.]

Thirdly, he objects to him that an excessive speed would follow from such motion. For let the Heavy body be at B, and the tower BA [be] of 10 perches, of which BG is supposed to be a tenth part, that is, of one perch—that is, of 10,000 parts, of which AB (the whole sine) is 100,000; for to the versed sine BG of 10,000 parts there answers the arc BH of 25° 51′. Now, by Archimedes, On the dimension of the circle, the periphery is to the diameter as 22 to 7; therefore, if the periphery BKE, etc., is 360° and the diameter BE is 200,000 parts, the arc BH (which is 25° 51′) will be 45,134 [parts]; and its proportion to BG, set at 10,000 parts, will be greater than quadruple-and-a-half. But now a stone dropped from B traverses the perch BG in a shorter time than two beats of the human artery [pulse]; yet let there be two beats in which the arc BH is revolved by an equal motion (as Philolaus [Bullialdus] will have it)—namely the arc of 25° 51′. The whole circle, therefore, of 360° will be revolved in the time of about 27¾ beats, or nearly 28. But the circumference, whose diameter BE is 20 perches, is most nearly 63 perches; and in a mile there are 333⅓ Cesena perches. Wherefore, if 63 perches are run through in 28 beats, a mile will at any rate be run through by such a movable in 148 beats—that is, in about two minutes of one hour. Wherefore in one hour, by the force of circular motion, it would complete 30 miles, and in one day 820 miles—which monstrous velocity Chiaramonti calls [it], and denies that it suits any flying thing.

But this third impugnation is frivolous, and errs doubly. Firstly, [it errs] in the Copernican hypothesis: for from it, in the motion of birds, besides the motion which appears to us as proper to them and theirs alone, there must also be included the common, circular motion by which they accompany the earth—if they fly toward the East; and so in such a hypothesis they fly past far greater space

[…continues on p. 399 (PDF 434) — the sentence completing “praeteruolant” (“they fly past”)—Riccioli’s rebuttal of Chiaramonti’s speed-objection continues.]

(printed p. 399 — within Chapter XVII. Riccioli finishes rebutting Chiaramonti’s “monstrous speed” objection, then gives his own three arguments against Bullialdus: uniform circular fall would strike no harder from a height, the two-circle construction fails to yield the odd-number increment (shown by a comparison table), and it cannot explain rising light bodies. The page then begins Galileo’s demonstrations from the Dialogo that the fall-line on a rotating Earth is circular — a semicircle terminating at the Earth’s center.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 399]

…than what appears to us. Let it suffice for Bullialdus if, as to the apparent motion, only 10 perches seem to be traversed. Next, he [Chiaramonti] errs absolutely when he says that a passage of 820 miles made in one day is incredible and monstrous; for it is far less than the velocity which would really suit a stone, if it were dropped from so great a height that one day were required to traverse it: for from what was shown in chapter 16, number 24, Theorem 3, and number 26, Problem 2, a clay globe of about one pound would traverse 30,662 Italian miles in one hour, and in 24 hours 17,661,212 miles—which is far more wonderful, and yet truer.

[Margin: Our arguments against Bullialdus.]

[V.] Bullialdus, therefore, must be assailed otherwise than by that third argument. And firstly indeed: if in equal times any heavy body really completed equal spaces in its circle, it would strike the earth with no greater impetus when dropped from a high place than from a low one—which is against the most manifest and very many experiments, indicated in chapter 16 from number 6. Secondly, from the motion ordered through the two aforesaid circles, there does not follow the increment of velocity which manifestly appears in the natural motion of heavy bodies—which we showed (chapter 16, nos. 11 and 12, and drove home at no. 24, Theorem 3) to be according to the odd numbers counted from unity. That this does not follow is easily demonstrated: for, by Bullialdus, the Heavy body B reaches A when the circle CFD, proceeding by equal motion in consequentia together with the circle BKE, has completed a quadrant—for then the movable on the circle HGA has completed a semicircle in praecedentia. Now let the whole time of descent be distributed into four equal parts; for in the first time the outermost point H of the diameter AFH will complete a fourth part of the quadrant, namely 22° 30′—which suppose to be contained in the arc BH; its versed sine BG, therefore, will be 76 parts, of which the whole sine AB is 1000. But at the end of the second time it will complete 45°, whose versed sine is 293; and at the end of the third time it will complete 67° 30′, whose versed sine is 617; and at last, at the end of the fourth time, it will complete the quadrant BHK, whose versed sine AB is the same as the whole sine, of 1000 parts. Now let the first versed sine be subtracted from the second, the second from the third, and the third from the fourth; there will remain the spaces completed in each of the four times, as you see in the first column of the following table.

[Translator’s note: in the middle column the second value is printed “218,” but by the odd-number rule (first space = 76, the rest as 1 : 3 : 5 : 7) it should be 228 (= 76 × 3); the 380 (= 76 × 5) and 532 (= 76 × 7) confirm the pattern, so “218” is a misprint. Riccioli’s argument is unaffected.]

But according to the increment of the odd numbers, which is due to the velocity of heavy bodies—if the space of the first time is 76—the spaces of the other times ought, separately, to be as you see in the second part of the table. Although, therefore, the space of the second time completed by the force of Bullialdus’s circles is most nearly equal to the space that ought to be completed, yet the third, and much more the fourth, falls short of the space due to the increment of velocity. Wherefore, since a clay globe—by our experience, as I said in chapter 16, no. 12—completed 240 Roman feet in four Seconds of time, and 15 feet in the first Second, if Bullialdus’s reasoning held, it would not have completed, at the end of the fourth Second, except 76 feet. Which is absurd, and contradicts the most evident experiments. Thirdly, Bullialdus cannot, by the aforesaid circles, render the reason for the increment of velocity of Light bodies upward—such as purer air and fire—because these are not carried toward the center of their whole, but toward the circumference.

Let us therefore hear Galileo, who, by a single circular motion, is believed to have come nearer to the truth.

Galileo’s Demonstrations, by which he tried, in the hypothesis of a moving Earth, to bring a Reason for the Increment of velocity of Heavy bodies

[Margin: Galileo’s doctrine of the line described by the motion of heavy bodies.]

[VI.] Galileo, in the Dialogue on the two Chief World Systems (from Italian page 157, but page 119 of the Latin version), pronounces the following propositions under the person of Filippo Salviati.

[Margin: 1st Proposition of Galileo.]

The First is: If the straight motion of heavy bodies toward the center of the earth appeared uniform, since the circular motion of the earth toward the East is posited (and itself uniform), then from both motions composed there would result a spiral line—one, namely, of those which Archimedes defined in his book On Spirals; and he said it comes about when a point is moved uniformly along a straight line, [that line being] uniformly carried around about one of its ends [as] fixed, as the center of the revolution. But since the apparent motion of heavy bodies is uniformly difform and is continually accelerated, it is necessary that the line described from this and the diurnal motion of the earth composed should recede successively, by an ever-greater proportion, from the circumference of that circle which the center of gravity would have described if the stone had remained at the top of the tower; and moreover that this recession at the beginning be slight and least, because the heavy body passes from rest—that is, from privation of downward motion—and enters into downward motion, and so necessarily passes through all the degrees of slowness lying between rest and any velocity whatsoever, which he had earlier concluded to be infinite.

[Margin: 2nd Proposition of Galileo.]

Secondly, since the natural descent of heavy bodies from the top of the Tower to the center of the earth is borne of itself, as to its terminus, he says it necessarily follows that the line of the composite motion recedes thus from the periphery of the circle described from the top of the tower, by an ever-greater proportion, yet so that it terminates at the center of the earth.

[Margin: 3rd Proposition of Galileo.]

[VII.] Thirdly, he affirms (Italian page 160, Latin 121) that the aforesaid line is circular, or supremely near to circular—because in it the conditions prerequired in the 1st and 2nd proposition are preserved. Which circular line he teaches to be designated thus, on the preceding page. Let A be the center of the earth, from which, with the interval of the semidiameter BA, let the quadrant BIM, etc., of the terrestrial periphery be described. Let there be, besides, the tower BC, whose vertex C, about the center A, describes a quadrant of a circle toward the same quarter—namely the East, toward which the earth is moved by its rotation. Now let the whole interval AC be bisected at E, and from center E, with interval AE, let the semicircle AIVC be described. “Through this,” says Galileo, “I now affirm that it can be believed, with sufficient probability, that a stone falling from the top of the tower C is moved by a motion composed of the common circular and its own straight [motion].” For if on the circumference CD some equal arcs be marked—namely CF, FG, GH, HL, LD—and from center A straight lines AF, AG, AH, AL, AD be drawn to the ends of the aforesaid arcs, the portions of these lines intercepted between the two circumferences CD and BI—namely OF, PG, QH, RL, ID—will always represent to us the same tower CB, carried in a circle by the rotation of the terrestrial globe toward D; through which [positions] the stone, by its fall, will seem to descend by a rectilinear motion to us—who, stationed at the foot of the tower B, are carried through the arc BI into the same quarter by a common and equal motion. Moreover, the portions of the aforesaid lines, intercepted between the periphery CD and CI [the semicircle]—namely FS, GT, HV, LX, DI—will be the spaces apparently completed downward by the stone at the end of each [equal] time; and the places in which the stone will be found at the end of those

[Translator’s note — the engraved figure (lower right): center A (the Earth’s center, lower right); the inner terrestrial quadrant B·I·M (B at right, M at bottom) of radius AB; the tower B·C standing on it; the outer arc C·F·G·H·L·D (the circular path of the tower-top C, swept eastward by the Earth’s rotation) divided into equal arcs CF, FG, GH, HL, LD; radial lines A·F, A·G, A·H, A·L, A·D cutting the inner periphery at O, P, Q, R, I (so OF, PG, QH, RL, ID are successive positions of the tower); and the fall-semicircle on diameter AC (center E, the midpoint of AC) passing through C, S, T, V, X, … , I, A — the curve the falling stone traces, with FS, GT, HV, LX, DI the apparent fall-distances at the ends of the successive equal times. This is Galileo’s celebrated “fall along a semicircle” figure.]

[…continues on p. 400 (PDF 435) with the catchword “illo-” (illorum): “…of those [times]“—the rest of Galileo’s semicircular-fall construction.]

(printed p. 400 — within Chapter XVII. Galileo’s semicircular-fall demonstration concludes with his “admirable” fourth proposition: the stone’s real motion is uniform and circular, so the acceleration is mere appearance. The objections then begin: Chiaramonti’s is dismissed as fallacious, but Riccioli and Grimaldi open more solid ones — the harder blow from a height proves the acceleration real, and unequal fall of unequal bodies is incompatible with a single semicircular path.)

[Header: BOOK IX. SECTION IV. — 400]

…of those times, will be S, T, V, X, I—which points always recede from the periphery CD by an ever-growing proportion; and so the motion of the stone appears to us to be more and more accelerated; and at last this motion is understood, if nothing resisted, to be terminated at the Earth’s center A, the semicircle CIA being completed.

[Margin: 4th Proposition of Galileo.]

[VIII.] Fourthly, Galileo affirms (Ital. p. 159, Latin 120) that the natural motion of the stone is in reality circular—although it appears to us straight, and indeed [moves] through the simple circumference of a single circle. Add to this that this movable completes, in falling, no greater space than if it had remained at the top of the tower C: since the whole arc CD, which it would have described by remaining at the tower’s summit, is precisely equal to the arc CI which it completes in falling—just as, severally, the arc CF is equated to the arc CS; the arc FG to the arc ST; the arc GH to the arc TV; the arc HL to the arc VX; and the arc DL to the arc XI. From which follows the third member of this proposition, which he himself calls admirable: namely, that the true and real motion of this stone is not accelerated, but is in reality equal and uniform—though it appears to us unequal—since to the equal times in which the tower-top C is moved (marked by the equal arcs CF, FG, GH, HL) there answer the real equal spaces CS, ST, TV, VX, XI.

“Which fact,” Galileo adds, “frees us from the labor of investigating the causes of the acceleration of this motion: since the movable, both in standing above the tower and in falling, is always moved in the same way—that is, circularly, with the same velocity and uniform tenor.” That the aforesaid arcs are equal among themselves is most readily demonstrated by drawing the line EI from center E. For the semidiameter AC of the circle CD is, by construction, double the semidiameter EC of the circle CIA; therefore both the whole circumference of the former to the whole of the latter, and quadrant to quadrant, and any arc of the greater circumference to any similar arc of the lesser, have a double proportion [ratio 2 : 1]—for the circumferences of circles are to one another as their diameters, by theorem 5, book 11 of the Collections of Pappus of Alexandria. Wherefore any half-arc of the circumference CD is equal to a similar arc of the lesser circumference CI. Now the angle CEI, standing at the center E, is double the angle CAD—which stands at the circumference of the lesser circle, but at the center of the greater—by Euclid III.20; therefore the arc CI, on which the angle CEI stands, is double the arc CD, on which the angle CAD stands; therefore CD is the half of the greater circle’s arc similar to the arc CI, and accordingly the arc CD is equal to the arc CI. And by the same reasoning, drawing straight lines from E to the points S, T, V, it will be proved that the arc CF is equal to the arc CS, the arc CG to the arc CT, and the arc CH to the arc CV; wherefore, since the arc CD is divided into the aforesaid equal arcs, FG will be equal to the arc ST, the arc GH to the arc TV, the arc HL to the arc VX, and the arc LD to the arc XI—which was to be demonstrated.

These things accomplished, Galileo concludes (p. 160, but Latin 121) in these words: “Although I would not now say that, as to the motion of descending heavy bodies, the matter stands precisely thus; yet this certainly I affirm: if the line described by the falling body is not exactly this very one, it is nevertheless supremely near to it.” Now let us see the objections—one indeed of Chiaramonti, the rest our own—against Galileo.

[Margin: Chiaramonti’s objection against Galileo. — But fallacious.]

[IX.] For Scipio Chiaramonti, in the Italian defense of his Antitycho (part 4, ch. 10), objects to Galileo that it is impossible for a stone in falling to move through the circumference CI and yet represent to us a motion made through a straight perpendicular line; and [he says] whoever should say otherwise commits a very great error in Mathematics. For, assuming the arc CI [to be] of one Italian mile, the height of the tower BC [being] 50 paces, he shows that that arc would have a declination from the horizontal line six times smaller than channels suitable for turning mill-wheels commonly have. But Chiaramonti is surely deceived: for Galileo never said that the arc CI itself, or its parts, represents to us—by itself or through its little portions, however small—a straight perpendicular line; which, had he said it, Chiaramonti’s objection would be valid. But he said that, in the hypothesis of diurnal rotation, a stone falling from a tower is, in every instant of its motion, at such a point of the circumference CI that a straight line drawn from the tower-top (or the terminus from which it fell) through the stone falls upon the same point of the earth on which, at the beginning of the fall, it stood perpendicularly—because the tower-top and that point of the earth are moved uniformly toward the East by circular motion; and so, our eye remaining in that place of the earth (and consequently carried along with the earth), the motion appears to be made through a straight perpendicular line.

Chiaramonti, therefore, being dismissed, let us come to more solid objections, devised partly by us, partly by Fr. Francesco Maria Grimaldi, with whom I have often been wont to discuss this hypothesis.

[Margin: The objections of ourselves and Fr. Grimaldi against Galileo.]

[Margin: 1st Objection. — The inequality of impetus argues an inequality of the motion really made.]

[X.] Firstly, then, I say what I adduced against Bullialdus: if the motion of naturally falling heavy bodies were in reality equal, and only seemed unequal through deception of the eye, [then it would] fall with no greater impetus from a higher place than from a lower; and accordingly those innumerable effects would not follow which follow from the more vehement percussion of bodies falling from a higher place—some of which we reviewed in chapter 16 from number 6. Since, therefore, touch itself and hearing evidently discern that greater percussion, it cannot be said to be a mere appearance and a mockery of the senses; but that greater percussion necessarily, naturally, presupposes a greater impetus, and a greater impetus a greater real increment of velocity. They are not, therefore, moved equally, but with greater and greater real velocity.

[Margin: 2nd Objection. — Heavy bodies descending unequally through the same circle would not always appear in the same straight [line].]

[XI.] Secondly, by most evident experiments before many most trustworthy witnesses—set out in chapter 16, number 13, and deduced into various corollaries at number 14—it is established that, if two heavy bodies of different weight are let fall together from the same altitude, the heavier descends faster: whether it be heavier both in the individual and in species (as there at corollary 4); nay, even if it be heavier in the individual but equally heavy with the other in species (as at corollary 2); or if it be heavier in species but equally heavy in the individual. But, lest any place be given for evasion, I shall choose this last case, of which (in the same chapter, no. 13, experiment 12) I related that a wooden globe of 2½ ounces, let fall together from the cornice of the Asinelli tower at Bologna with a lead globe likewise of 2½ ounces, was 40 feet (or 8 paces) from the pavement of the same tower’s base when the lead [globe], now striking the pavement, had measured a height of 280 feet (or 56 paces); and so the wooden one had completed, in the same time, only 48 paces. From which, by the second example of Problem 5 (in the same chapter 16, no. 29), I deduced that if such globes were let fall from a height of 929,210 Italian miles, it would come about that at the end of 6 hours the lead would strike the earth, and then the wooden would still be 129,530 miles from the earth—that is, most nearly 31¼ terrestrial semidiameters. Experiments of this kind, and similar ones, cannot be saved by the motion of heavy bodies in the circle determined by Galileo, or [in one] insensibly differing from it. For, the figure of number 7 being repeated: if both globes are let fall together from the tower-top C, and each must reach the earth’s center A through the semicircular periphery CIA, it is necessary that each globe, at every same moment of the time during which the motion lasts, appear—nay, really be—at the same point of the same periphery; and so, when the lead is at S, let the wooden too be at S, and there it would appear to our eye remaining at O; and when the lead is at T, let the wooden too be there; and when the lead has struck the surface of the earth at I, there too let the wooden be and strike the earth. Otherwise, if the wooden were not at the same points of the same periphery, but of another, greater one, drawn however through the vertex C, this circular line would not terminate at the center of the earth, but at another point below that center—which is against the nature of heavy bodies, and against Galileo’s 2nd proposition, of which I spoke at no. 6. But if the wooden globe had been, during the whole time of the motion, at the same points of the same periphery in which the lead was, it would have descended with precisely equal velocity, and would not have been a whole 40 feet from the earth—which is against the most evident experiment. Hence Fr. Grimaldi suspected that Galileo, in that same 2nd Dialogue on the world-system, therefore denied that, of two heavy bodies (however different in weight) let fall together from the same terminus, the heavier reaches the earth sooner—lest, namely, this should harm this his motion of heavy bodies through the aforesaid circle. But perhaps the occasion for denying it was that he used two globes of the same species, but of different weight and different bulk, in which the difference of descent and of percussion is far

[…continues on p. 401 (PDF 436) with the catchword “nor” (minor): ”…[far] smaller”—the rest of Riccioli’s second objection.]

(printed p. 401 — within Chapter XVII, continuing the objections to Galileo’s circular fall. The second objection closes (unequal bodies cannot share one semicircle); the third shows the circle saves only equatorial, six-hour falls, Galileo himself conceding to Scheiner cones and cylinders off the Equator; the fourth shows the annual motion makes the path grossly non-circular; and the fifth begins demonstrating that even with diurnal motion alone the true fall-line lies inside Galileo’s semicircle, else the squares-of-times law would fail.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 401]

…[far] smaller than in other comparisons; nor does it betray itself evidently unless they are dropped from a notable height—but Galileo did not use a height greater than 100 cubits, as he testifies in the same dialogue.

[Margin: An escape is shut off to Galileo.]

Nor may you say that that inequality can be defended [explained away] if the wooden globe is carried around through another periphery insensibly greater than the periphery CI; for if it were so, then both globes would have struck the point I at sensibly the same moment, and the difference of interval would not have been so sensible as to amount to 40 feet. Moreover, if each globe were dropped from a height of 929,210 miles, and the medium were of the same density as the air near us (and were not consumed by heat conceived from the motion, or dissipated before its arrival at the earth), then, when the lead, 6 hours having elapsed, had reached the earth’s center A, the wooden one would be distant from the center A by 31¼ terrestrial semidiameters, as I showed at chapter 16, number 29, at the end. Which difference is very notable.

[Margin: 3rd Objection. — The descent of heavy bodies is not everywhere of the same character.]

[XII.] Thirdly, the motion of heavy bodies through the circle assigned by Galileo is unfit to save the greatest part of the motions by which heavy bodies naturally descend: both because it saves only the motion of heavy bodies descending along the plane of the Equator, and because it does not save even one hour’s motion of all [bodies], but only of those which—on account of their lightness, or defect of greater gravity—are so slow that they cannot reach the center of the earth except in the time of 6 hours; of which scarcely any can be found, if the descent must be made through air; but if through water, very rare are heavy bodies of this kind that descend so slowly—such, however, is a globe of rarer ebony let fall through water, as appears from the problem which I delivered to this end in chapter 16, number 30. But by far the most heavy bodies, in number and species, neither descend along the plane of the Equator, nor are so slow that for the passage of a terrestrial semidiameter (or 4139 miles) they require 6 hours, as appears from what was said in chapter 16, no. 25, example 3, and number 30.

I said, firstly, that the motion of heavy bodies is not saved unless they descend along the plane of the Equator—and so unless the tower from which the stone falls is on the Equator itself; because Galileo’s demonstration adduced at numbers 7 and 8 requires that the circle CIA, through whose periphery the stone is carried, be in the same plane in which are both the earth’s center A and the tower-top C, and so the circle CD described from that same tower-top by the force of the diurnal rotation (as is manifest from the construction of the figure and the hypothesis on which that demonstration rests); and this cannot happen unless that tower is on the Equator. For if it is outside it, under the poles, the descent of the stone will in reality be along a straight line, and so not through a circle; but if [it is] on some parallel to the Equator, one parallel will be described by the tower-foot B, another by the tower-top C; and moreover the plane of neither will be in the plane in which is the center of the earth, but in a far different plane. Moreover, Galileo himself, in that 2nd Dialogue (Ital. p. 237, Latin 179), when he answered the instances of our [Father] Scheiner, acknowledged that this line is not circular except on the Equator, but under other parallels describes a conical surface—being prompted, namely, by that question of Scheiner which is found in the Disquisitions, p. 31, and is of this kind: “Why does the center of a sphere dropped under the Equator describe a spiral in its plane? under other parallels, a spiral on a cone; under the pole it descends in the axis as a gyral line running on a cylindrical surface?” To whom he answers in these words: “Because, of the lines drawn from the center to the circumference of the sphere (along which, namely, heavy bodies descend), that one which terminates at the Equinoctial designates a circle; but those which terminate in other parallels describe conical surfaces; and the axis, finally, describes nothing else, but remains in its own being.” Wrongly, therefore (p. 121 Latin, but Italian 160), did he universally—or indefinitely—define this line to be circular; nor did he correct this his statement, as he ought, having forgotten (I believe) what he had said.

[Margin: Galileo does not correct his statements, where, from what he himself concedes, he ought.]

I said, secondly, that not even under the Equator is the motion of heavy bodies saved, unless they be so slow that their arrival at the center of the earth (if access there were given through a well full of air or water) requires precisely 6 hours. Because, for Galileo’s demonstration to hold (of which at nos. 7 and 8), it is necessary that the semicircumference CIA, through which the heavy body is carried, be equal to the quadrant CD, etc., which the tower-top C describes by the force of the diurnal rotation; and so that, when the heavy body from C, through I, reaches the earth’s center A (the semicircle CIA being completed), the tower-top C should meanwhile have completed a quadrant of the diurnal rotation, that is, in 6 hours. Otherwise, if the heavy body reaches A either faster or slower than in 6 hours through the arc CIA, the whole demonstration of Galileo collapses. Since, however, these two conditions can really be conjoined at once—namely, that the place from which the heavy body is dropped be on the Equator, and be of such slowness that it requires 6 hours to traverse the earth’s semidiameter—hence we shall draw a most strong argument against the diurnal motion of the earth, and so against the annual (which does not stand without the diurnal); and it is what follows.

[Margin: 4th Objection. — Nowhere does a heavy body descend through a circle, if the annual motion be admitted.]

[XIII.] Fourthly: not even on the Equator—even if a heavy body were such that in 6 hours it would reach the center of the earth—is it true that the line described by it is circular, or differing insensibly from a circular one, unless the motion of the Earth be diurnal simply, about its own immobile center. But if the center of the Earth be moved, as Galileo wishes with Copernicus, by the annual motion, the line of the said movable will not be circular, but in mundane space will describe a line very different from a circular one—as appears, because while in one hour, by the force of the diurnal motion, it ought to describe 15 degrees of the circular line, meanwhile the center of the Earth is moved forward in the Great Orb [by] 2′ and 3″ [≈ 2′ 30″], which, however, are far greater than 15 degrees of the terrestrial circumference—even if the great orb be not so great as the distance of the Sun from us, found in bk. 3, ch. 7, requires, but [only] as the Copernican distance: that is, if its diameter be only 2284 terrestrial semidiameters, or 1142 terrestrial diameters. For it follows that the circumference of the great orb to the circumference of the terrestrial globe is as 1142 to 1, since circumferences are to one another as their diameters, by theorem 5 of Pappus of Alexandria, Mathematical Collections book 11; therefore also one degree of the great orb to one degree of the terrestrial circumference is as 1142 to 1. But 2′ 30″ are the 24th part of one degree; wherefore, 1142 being divided by 24, there will result 47 7/12. The Great Orb [-motion], to one degree of terrestrial circumference, is as 47 7/12 to 1; but to 15 terrestrial degrees, as 47 7/12 to 15—that is, of a proportion greater than triple. Wherefore in one hour, by the force of the annual motion alone, the said heavy body completes, in mundane space, a space more than triple greater than it would complete by the force of the diurnal motion in the terrestrial circumference. But if the diameter of the Great Orb is, according to us, of 7300 terrestrial diameters, these divided by 24 make 304¼; and so the space completed in one hour by the force of the annual motion will be to the space completed by the force of the diurnal motion as 304¼ to 15—that is, twentyfold and more greater. Since, therefore, the circumference of the Great Orb is not parallel to the terrestrial circumference, but so passes through its center that, on account of the inclination of the Equator to the Ecliptic and the parallelism of the Equator with itself, a heavy body following the motion of the center and circumference of the earth is perpetually in different planes, and the portion of the great orb answering to one hour is triple—nay, twentyfold—greater than the portion of the terrestrial circumference answering to one hour: it is necessary that the line composed from both motions be remarkably different from a circular one; and much more if the motion lasts for 6 hours.

[Margin: 5th Objection. — No heavy body descends through a circle, even granting the diurnal motion alone.]

[XIV.] Fifthly, I say also: even if some Heavy body descended on the Equator from the surface to the center of the earth in precisely 6 hours, and the Earth were moved only by the diurnal motion, still the line described by it would not be circular, but would fall within the semicircle’s periphery. Otherwise the apparent increment of velocity would not be in progression according to the squares of the times—whereas it is so, not only by our experiments, but also by Galileo’s experiments and confession. That this may be made manifest, let the figure set out at number 7 be repeated here, and let the quantities of the spaces FS, GT, HV, etc.—completed by the heavy body in equal times, until the tower-top C has completed an entire quadrant in 6 hours, and the heavy body an entire semicircle CIA—be investigated: for the aforesaid spaces will be found to fall short, more and more, of the quantity necessary that they keep the proportion with the squares of the times.

[Margin: Method of finding the spaces completed by a heavy body, if it descended along a circular line.]

Moreover, the method of inquiring into the quantity of the aforesaid spaces will be most briefly declared by an example. Let the first assumed arc FC be of one degree, which the earth by the diurnal motion completes in 4 minutes of an hour; for the arc CS, intercepted between the same lines AC, AF, will be double as to the number of degrees (by what was demonstrated at number 8)—that is, of 2 degrees. Therefore the complement [to

[…continues on p. 402 (PDF 437) with the catchword “ad”: “…to [a quadrant]“—the rest of the worked construction showing the semicircular fall-line fails the squares-of-times law.]

(printed p. 402 — within Chapter XVII, the heart of the fifth objection. Riccioli completes a trigonometric demonstration that Galileo’s semicircular fall-line cannot obey the squares-of-times law: the circle’s successive segments fall short of the required proportion, so the true fall-line must recede inward from the semicircle toward the diameter. A large table compares, for each step of arc, what the circle yields against what the squares-law demands, the divergence growing to impossibility at the center.)

[Header: BOOK IX. SECTION IV. — 402]

…to a semicircle—namely the arc SIA—it will be of 178 degrees, and its half of 89 degrees, whose right sine is 99,984½ (of such parts as the radius EC is 100,000; but the diameter, or greatest of the chords, AC, is 200,000). That sine doubled will give the chord AS of 199,969 parts, which, subtracted from the radius of the quadrant AC—that is, from AF—of 200,000 parts, will leave the segment FS of 31 parts. In the same way, if the arc CG be of 2 degrees, CT will be of 4 degrees, and TIA of 176 degrees, and its half of 88 degrees, whose right sine is 99,939; doubled, it will give the chord AT of 199,878; and this, subtracted from the whole radius AG, will leave the space or segment GT of 122 parts. And so of the rest.

Now, if the space FS of 31 such parts be taken as the first—and as 1—and as the space traversed by the heavy body in the first time (or first 4 minutes of an hour), then the space GT, traversed at the end of the second of the equal times, ought to be to the space FS as 4 to 1: for the square of the unity ascribed to the first time is 1, but the square of the binary ascribed to the second time is 4. Let it therefore be made: as 1 to 4, so 31 to another [number], and there will be found not the 122 parts which we said are in the space GT, but 124 parts. Wherefore, since GT—if the proportion due to the squares of the times be preserved—is greater than that which arises from the circular line, it is necessary that the line which the descending heavy body describes (while preserving the said proportion), if the earth be moved by the diurnal motion alone, recede inward from the line CIA, ever more toward the diameter AC—although it will at last pass through the center A.

But lest anyone think that this difference, which at the beginning is slight, is also slight and to be despised in the progress—and accordingly that that line is insensibly different from a circular one—let him inspect the following table, whose explanation (to be sought from what was just said) will be subjoined after it.

Table — for each arc of the semicircle: the chord (of which the greatest is 200,000); its complement to the greatest chord (the segment the circle yields); the square number (to be multiplied by 31); and the space due by the proportion of the square numbers (= 31 × the square):

[Translator’s note — the table’s internal relations: column 3 (the circle’s segment) = 200,000 − column 2 (the chord); column 5 (the space the squares-law demands) = 31 × column 4 (the square number, running 1², 2², … 90²). A few of the source’s printed values carry evident typesetting slips against these relations — e.g. at arc 160°, 3,010 should be 3,100 (= 31 × 100); at 90°, 65,775 should be 62,775 (= 31 × 2,025); at 20°, 194,000 should be 198,400 (= 31 × 6,400); at 64°, the complement 95,016 should be 94,016 (= 200,000 − 105,984). The argument is unaffected: at the center (arc 0°) the circle has the body at A (segment 200,000), whereas the squares-law would demand 251,100 — beyond the diameter, hence impossible.]

[Margin: Look at the preceding figure.]

Now, for the explanation of the preceding table, note in it how—the quadrant CD, etc., being divided into 90 degrees, and the semicircle CIA likewise into 90 pairs of degrees—I traced the chords of the complements to the semicircle, which are contained in the second column, and the segments intercepted between the semicircle CIA and the quadrant CD, etc., which are had in the 3rd

[…continues on p. 403 (PDF 438) with the catchword “co-” (columna): “…column”—the rest of the table’s explanation and Riccioli’s conclusion.]

(printed p. 403 — within Chapter XVII. The table’s explanation concludes: the semicircle increasingly falls short of the squares-law. A sixth objection follows: Galileo’s uniform circular fall requires an immobile Earth-center, yet his own tide theory makes the Earth’s composite motion unequal — he contradicts himself. A seventh objection then opens a “triple figure”: the first (a polar drop) shows that with the annual motion added the real fall-line is a curved, non-planar path, neither circle nor ellipse.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 403]

…column. In the Fourth column is a continuous series of square numbers, begun from unity, by which—multiplying the first space FS of 31 parts—there come forth the spaces, or segments, due, if the proportion of the squares be preserved. For you will see that, at the end of the quadrant from the vertex C (completed in 6 hours), the space completed by the heavy body ought to be of 251,100 parts; and yet, if the heavy body must descend through the semicircle CIA, it cannot be except 200,000—as great, namely, as AC; and at the end of the other times too, that proportion falls short more and more—whereas it ought either to be preserved, or rather to grow.

[Margin: Objection — the motion of heavy bodies is not really equal, therefore not circular either.]

[XV.] Sixthly, the demonstration of the equal motion of the aforesaid heavy body—namely [one] descending on the Equator so slowly that it requires 6 hours to reach the center of the earth—which we adduced from Galileo at number 8, does not hold, unless the tower-foot B and the vertex C (that is, the circumference of the terrestrial globe) be supposed to be moved equally, and A, the center of the Earth, to be immobile. But if, besides the diurnal motion, the annual motion of the center be also added to the Earth, Galileo judges that the composite motion of any heavy body following the motion of the earth is really unequal—now slow, now fast—which he thinks so true in this hypothesis that, upon the inequality of the earth’s motion (composed of diurnal and annual), Galileo founded the cause of the sea’s tide, as we already saw in chapter 14, from number 4.

[Margin: Galileo’s self-contradictions.]

Galileo therefore contradicts himself, while he attributes a real equality to the motion of such heavy bodies, which he does not even concede to the earth. And granting that this inequality is not that which we experience in the apparent motion of heavy bodies, yet in its place another inequality, unknown to sense, is substituted, in order to take away that which is manifest to sense—which is a greater incongruity.

[Margin: 7th Objection.]

[XVI.] Seventhly, Galileo errs while he pronounces, universally or indefinitely, that the line which heavy bodies describe by natural descent is, in the Copernican hypothesis, circular, or very near to it. For this is not true even in the plane of the Equator, where it ought most of all to be verified, as will appear from what has been said hitherto and is to be said; but it is much less true outside the Equator. That we may show this to the Reader, we shall set forth a triple figure, by which we shall sketch this line as best we can: for we cannot represent, in the one plane of this page, the diversity of planes through which, in this hypothesis, heavy bodies are borne in descending, nor preserve the due proportions, on account of the narrowness of the leaves. Hence, however, you will better learn the things which Scheiner very lightly indicated, concerning the line of this motion on a cylindrical or conical surface, in the Mathematical Disquisitions, p. 31; and at his prompting Galileo conceded [it], as we have already said at number 12.

[Margin: First figure, for a Heavy body’s descent in a parallel sphere.]

The First figure is for a heavy body dropped perpendicularly above one pole of the terrestrial Equator, a spectator of this descent being imagined in a parallel sphere [i.e. at the pole]. Let ABCD be the Ecliptic, or Great Orb, carrying around—by the annual motion—the center of the Terrestrial globe from A, through B, toward the Eastern quarter C, and from C, through D, to A, the Western quarter. For as the axis of the terrestrial ecliptic, by its annual revolution, describes its own cylinder, so also the axis of the Equator describes its own, but inclined to the ecliptic’s cylinder. Let the Earth now be understood as ILRP, its center translated along the Great Orb from the point K to the points V, B, X, Y, at the interval of four equal times—to which times answer the equal arcs of the Great Orb, KV, VB, BX, XY. And let the terrestrial Equator be IR, its axis LP, and its North pole P; and let the axis be produced through the pole P, outward toward the North, by as much as the height of the tower or place from which the heavy body is dropped—namely, as much as KPM; for this too, carried around with the axis of the Equator, will describe the cylinder ACGE, of which one base is ABCD, the other EFGH. And at the end of the first of the equal times, the tower (which was in the position PM) will be at PN; at the end of the second, at PF; at the end of the third, at PS; at the end of the fourth, at Z·a.

Now, since heavy bodies, descending naturally, so descend that—if the whole time of descent be divided into 4 equal parts (as we have here assumed) and the first space be taken as 1—at the end of the second time the space completed is as 4, at the end of the third as 9, and at the end of the fourth as 16 (as is established by our experiments, chapter 16, number 12, and by Galileo’s confession, Dialogue 2, On the System of the World): let the height PM be divided into 16 equal parts, and let there be marked in it the part M1 (of one such part), the part M4 (of four), and the part M9 (of nine). Now if a globe heavier than the medium through which it descends were dropped from the vertex M, above the pole P, and the Earth were moved by no other than the diurnal motion about its center K (from I to R), and its center always remained at K—then assuredly this globe would descend, not only apparently but also really, through the same straight line MP, falling perpendicularly upon the pole P; and at the end of the first time it would have completed the space M1, at the end of the second the space M4, at the end of the third the space M9, and at the end of the fourth the whole space MP, which is of 16 [parts], of which M1 is one.

But because the center is translated from K to V, etc., by the force of the annual motion, and the tower (or height MP) is carried around above the surface of the cylinder—to be conceived in mundane space, on Copernicus’s hypothesis—therefore, when the tower at the end of the first time is at PN, the space NO will appear to have been completed by that globe, which is equal to M1, or as 1; and at the end of the second, the space FQ, or as 4; and at the end of the third, the space ST, as 9; and at the end of the fourth, the space a·Z, which is as 16. Moreover, because the eye of the supposed spectator at the pole P is likewise translated with that tower, the globe will appear to him to be always in the same straight line—even though in reality, on this hypothesis, the globe has described, on the cylindrical surface, a curved line MOQTZ, far longer than the straight line MP, and not lying in the same plane.

Wherefore this line will not be a cylindrical section that is either a circle (such as that whose plane, equidistant from the base, cuts the cylinder) or an ellipse (such as that whose plane cutting the cylinder is not equidistant from the cylinder’s base)—according to the demonstrations of Serenus of Antinoöpolis (usually published with Apollonius), bk. 1 of the Cylindrical Sections, Propositions 5, 6, and 17. The reason the points O, Q, T, Z are not in the same plane is the proportion of the square numbers; otherwise, if they were in the same plane, the section would be elliptical, and would be bisected by the straight line FP (which is the middle among the five [points] distant from one another by equal interval) at Q, and the parts FQ would be 8 of the 16 parts. But the second [line] NP and the fourth SP would be cut so that NO would be equal to PT—namely, of 7 parts each—as could easily be demonstrated, were there need, and were we not hastening to other things.

[Margin: The rotation due to a globe descending above the poles.]

It is to be noted, however, in this place, that a globe dropped perpendicularly above the pole of the terrestrial Equator, while it descended through the line MOQTZ, would at the same time be carried around about its own center, imitating the diurnal vertigo of the earth—which would that it were permitted to observe, by an inhabitant under the poles of the World! For from it the truth of the earth’s diurnal vertigo would become manifest, if such a rotation were discerned on the surface of that globe; or its falsity, if it were not discerned.

[Margin: 2nd Figure, for a Heavy body descending in the Plane of the Equator.]

[XVII.] Now let there be the second figure, for a globe dropped from the tower-top in the plane of the Equator, or [for one] existing in a Right Sphere. Let A be the center of the Great Orb, from which describe the arc BDC, divided into four equal parts Bρ, ρD, DP, PC, answering to the four equal times in which the globe, dropped from the tower-top, is supposed to fall to the earth’s surface; and let it be understood that, at the beginning of the mo-

[Translator’s note — the engraved 3-D figure (first of the triple figure): a slanted cylinder ACGE drawn in perspective, its lower base the ellipse ABCD (the Ecliptic / Great Orb), its upper base EFGH. Along the lower base sit five successive positions of the Earth-globe (center translated K→V→B→X→Y eastward), each a small sphere with North pole P up, South pole L down, equator IR. From each pole rises the tower-axis; the leftmost (PM) carries the squares-law fall-marks M, 1, 4, 9 down to P. The tower-tops at the four successive instants are M·N·F·S·a (on the upper region); the falling globe’s true positions are M·O·Q·T·Z — a curved, non-planar line bowing across the cylinder. Straight-line fall MP (diurnal-only) vs. the curve MOQTZ (diurnal + annual).]

[…continues on p. 404 (PDF 439) with the catchword “tûs” (motûs): “…of the motion”—the second figure (the equatorial drop) is worked out.]

(printed p. 404 — within Chapter XVII, continuing the “triple figure.” The second figure (an equatorial drop) is completed: with the annual motion added, both globe and tower-vertex trace curved, non-planar lines that are neither circular nor any conic, since the Equator’s plane is carried into ever-different planes, though the co-moving observer still sees a straight drop. The third figure (an oblique-sphere drop, a tower on an off-Equator parallel sweeping cones about the Earth’s center) then begins.)

[Header: BOOK IX. SECTION IV. — 404]

…of the motion, the center of the Earth [is] at C, and at the end of the first time at P, at the end of the second at D, at the end of the third at ρ, at the end of the fourth at B. Let the tower-height be FH, and the terrestrial Equator EFG, which—about the center C, in the four aforesaid times—is revolved toward the East, from F to E, so that the tower-foot F completes the arc FE, and the vertex the arc HK. Here, if there were no other motion than the diurnal, a globe dropped from the vertex H, at the end of the first time, would have completed the space LM, as 1 (of which FH is 16), and the tower would be on the line CL; at the end of the second, the tower would be on the line CN, and the space completed by the globe would be NO, namely of 4 such parts; at the end of the third, the tower would be on the line C·m, and the space completed m·I, or of 9 parts; and at last, at the end of the fourth time, the tower being translated to the line CK, the whole space KE, of 16 parts, would be completed.

But because, at the end of the first time, the center of the Earth is supposed translated from C to P by the force of the annual motion, and meanwhile the tower, by the force of the diurnal motion, completed toward the same part of the Equator the arc XQ—therefore the tower is on the line QR, and the space apparently completed by the globe is RS, of 1 part. But at the end of the second time, the earth’s center being carried to D, and the tower (by the diurnal vertigo) completing the arc TV (double the arc XQ), the tower will be on the line XV, and the space completed by the globe will appear [to be carried farther]; …[and at the third, on the line a·β, the arc being triple the arc XQ, the space β·γ]; …and [the center] transported to B, and the tower at the line ι·θ—on account of the arc δθ completed by the diurnal vertigo (and indeed quadruple the arc XQ)—the globe will appear to have completed the whole space ι·θ, of 16 parts.

Wherefore the line described by it in mundane space will be the curved line HSZγθ; although, to the eye translated equally with the tower-foot from F to the points Q, V, a, θ, the globe will appear always to descend through the same straight line, parallel to the tower. Meanwhile the tower-vertex H will describe the curved line HLRXβε; and neither will be Circular, because [the body] always approaches more nearly to the center A—from which both are most distant when the globe is at H (since it is then distant by the whole aggregate of the great orb’s semidiameter AC, the earth’s semidiameter CF, and the tower-height FH) than in any of the other aforesaid positions. Nay, since the plane of the Equator—on account of the parallelism kept with itself, it being always inclined to the Great Orb—is translated to one plane after another, the line HSZγθ cannot be in one plane; and accordingly it cannot be Parabolic, Hyperbolic, or Elliptic, since for these lines it is required that they lie in one and the same plane cutting (actually or potentially) a Cone or a Cylinder. The same happens, proportionally, in the other positions of the tower, although we have chosen a single case—namely, when it is midnight for that tower.

[Translator’s note — the Second engraved figure (lower panel, left): a fan of five Earth-globes whose centers lie along the Great Orb’s arc, the apex being A (the orb’s center) at the bottom. The successive Earth-centers are C, P, D, ρ, B (the body’s position at the start and at the ends of the four times); on the rightmost globe the terrestrial Equator EFG carries the tower (foot F, top H), with the diurnal-only fall-marks toward the marks L·M·N·m·K and the surface point E. As the annual motion shifts each globe leftward, the falling body’s true positions trace the bowed curve H·S·Z·γ·θ, and the tower-top the curve H·L·R·X·β·ε — both swept over the radii to A.]

[Margin: Figure for the oblique sphere.]

[XVIII.] Now let there be the third figure, for a heavy body dropped from the top of a tower existing in an oblique sphere (or in the periphery of some terrestrial parallel declining from the Equator), but standing perpendicularly upon the terrestrial surface, and so positioned that, if it were produced, it would meet the center of the earth. Let there be designated, therefore, the arc ACB, which is a portion of the Great Orb, in which let the Earth FKEG be first, [its] center at B; and let the circle of the terrestrial Equator (inclined to the great orb) be DBE, whose axis KBQ etc. is understood produced as far as H. Let there be the Parallel FQG, on whose periphery the tower GL is erected perpendicular to the globe of the earth; and let the terrestrial Equator be turned, by the diurnal vertigo, from E, through B, to D (the Eastern point), until—the revolution of 24 hours being completed—the point E returns to its former beginning of revolution. For it is certain, in the Copernican hypothesis, that the whole globe of the earth, with all the parallels of the Equator, is turned toward the same quarter; and just as the earth’s semidiameter BG describes the cone BGF (whose base is the parallel circle indicated by GQF, described by the point G, which is at the tower-foot), so the tower GL, joined with the earth’s semidiameter BG, will describe a greater Cone BLM, whose vertex (common with the prior cone) will be the earth’s center B, and whose base [is] the circle ML. For the semidiameter of this [circle] makes a right angle with the axis BH; and accordingly, if—this axis BH remaining as a side—the other sides be carried around about it until they complete one entire revolution, a cone will be described by them, by the 18th definition of book 11 of Euclid’s Elements.

Let us now imagine a globe dropped from the vertex L to reach the earth’s surface in four equal times, the earth meanwhile being moved by the diurnal motion only; and accordingly, the arc FG of the parallel being divided into four equal arcs answering to the aforesaid times—namely into GP, PQ, QR, RF—the tower [is] likewise transferred, first to the line PO, then to the line QH, thirdly to the line RN, and fourthly to the line FM.

Therefore, since heavy bodies descending naturally traverse spaces which stand to one another, in order, as the squares of the times: if the height GL be divided into 16 equal parts—of which the one traversed at the end of the first time be OS, that is 1; the second, at the end of the second time, HT, of 4 parts; the third, at the end of the third time, NV, of 9 parts; and the fourth, at the end of the fourth, MF,

[Translator’s note — the Third engraved figure (lower panel, right): the oblique-sphere construction. The Earth-globe (center B) sits on the Great-Orb arc ACB; its tilted Equator is DBE with axis KBQ produced to H; the off-Equator parallel FQG carries the perpendicular tower GL (foot G, top L). The earth-radius BG sweeps the cone BGF (base GQF), and the tower the larger cone BLM (vertex B, base ML). The tower-foot’s parallel-arc FG is divided into the four equal time-arcs GP, PQ, QR, RF, and the successive tower-positions are the lines PO, QH, RN, FM; the tower’s squares-law fall-marks are OS (1), HT (4), NV (9), MF (16).]

[…continues on p. 405 (PDF 440) with the catchword “par-” (partium): “…of 16 parts”—the rest of the oblique-sphere construction and Riccioli’s conclusion.]

(printed p. 405 — within Chapter XVII. The third figure (oblique-sphere drop) is completed, with a corollary: in no case is the fall-line a conic, so Kepler, Gassendi, Bullialdus, and above all Galileo erred about its shape. An eighth objection shows Galileo’s pendulum doctrine and tunnel-through-the-Earth thought-experiment, with their real growing impetus, contradict his uniform-circular-fall claim. The page then states, in syllogistic form, the pro-Earth-motion argument from apparent acceleration that Riccioli will dissolve.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 405]

…of 16 parts—then indeed the tower will so describe a part of the conical surface that the globe (which always appears in the same straight perpendicular line) will describe, about the surface of the cone MBL, the curved line LSTVF, which the proportion of the square numbers does not allow to be in one and the same plane; and accordingly, not even thus will that line be either Circular, or Elliptic, or Parabolic, or Hyperbolic—even if, meanwhile, B, the center of the earth, were not moved by the annual motion toward A.

Let the center B now be moved toward A (the Eastern point being at X) by the annual motion, and accordingly let it carry to another position the Cone described by the tower and the earth’s semidiameter. Meanwhile the tower, on account of the diurnal vertigo, will have completed the arc a·β, and will be on the line β·γ; for the globe will appear, at the end of the first of the four equal times, to have completed the space γ·δ, which is taken as 1. Then, at the end of the second time, let the earth’s center be at C, and the whole cone translated with it, so that the tower (by the force of the diurnal vertigo) has completed the arc ξ·ζ (double the arc a·β), and is on the line ε·ζ; for the globe will appear to have accomplished the space ε·ν, which will be as 4. Thirdly, by the force of the annual motion, let the Earth’s center be carried to Y, with the whole cone standing upon it, and the tower (by the diurnal rotation) have completed the arc ω·θ (triple the arc a·β), and be on the line x·θ; for the globe will appear at λ, the space x·λ being now accomplished, which will be as 9. Lastly, at the end of the fourth time, the earth’s center will have passed, by the force of the annual motion, to A, carrying the earth with it (and the aforesaid cone); and the tower will now have run through, on account of the diurnal motion, the arc π·μ (quadruple the arc a·β), so that the tower is on the line μ·Φ; for then the globe will be on the earth’s surface, or at the tower-foot μ. Although, therefore, the globe will appear always in the same straight line, in which is the eye stationed at the tower-foot (because the eye, by the force of both motions, is carried from G to the points β, ζ, θ, μ), yet in reality it will trace the curved line L·S·η·χ·μ, which will be able to be neither Circular (since it always approaches more and more to its center, which is the center of the great orb), nor Elliptic, nor Parabolic, nor Hyperbolic (since it is not in the same plane); for the reason that it is so described about a cone that it is nevertheless carried to one plane after another—both on account of the proportion of the square numbers which it keeps in approaching the great orb, and on account of the inclination of the terrestrial Equator to the great orb, which Equator remains always parallel to itself.

[Margin: A warning, that the figures are to be corrected.]

But although, in the aforesaid figures, we have drawn the terrestrial globe’s center at diverse positions—with arcs of the great orb interposed equal to the arc which the earth’s diameter subtends—lest the figures should introduce confusion to the viewer, yet those intervals can be greater or smaller, according to the time of the whole descent of heavy bodies. And what we said of the position of the tower for which it is midnight holds similarly for any other position of the tower.

[Margin: Corollaries for the figure of the motion of heavy bodies. — The errors of Kepler, Gassendi, Bullialdus, and others concerning the figure of the motion of Heavy bodies, but especially Galileo’s.]

From what has been said, it appears that in no case is the aforesaid line Circular, or Elliptic, or Hyperbolic, or Parabolic—because it is not drawn through one [same] plane, which is nevertheless required, as appears from the sections of Apollonius, bk. 1 (proposition 4, on the circle cutting the cone; prop. 11, on the parabolic section; prop. 12, on the hyperbolic section; prop. 13, on the elliptic section); and also from the cylindrical sections of Serenus, bk. 1, props. 5, 6, and 17 (on the circle and ellipse arising from the section of a cylinder). Wherefore Kepler erred—who (from what was said at chapter 4, scholium 2) described this line as if the earth’s motion were diurnal only, and [the line] approaching too closely, at the beginning of the motion, to a straight perpendicular line; and Gassendi, who (from what was said in the same place, scholium 4—namely, in the 2nd epistle On impressed motion) judged it to be universally parabolic; and Bullialdus (bk. 1 of his Philolaus, ch. 4), who asserted it to be composed, in the same plane, of two circular [lines]; and those who (in Chiaramonti, bk. 12 On the Universe, ch. 21) said it is similar to the quadratrix of Nicomedes; but most of all Galileo (Dialogue 2, On the System of the World) affirmed it to be circular, or very near to circular, and yet terminated at the center of the earth. Against whom, however—besides these errors—there is also the following objection. Granted that (as I said at no. 12) he acknowledged, as if through a lattice [in passing], that this line is, in the parallels of the Equator, a spiral about a cone, and, in the case above the poles, a spiral about a cylinder—but [only] at the prompting and warning of Scheiner, who of all most nearly approached the truth in this. The aforesaid figures, however, we had already drawn by our own ingenuity, before we had read that passage of Scheiner.

[Margin: 8th Objection. — The inequality of impetus being conceded, the inequality of motion is wrongly denied.]

[XIX.] In the Dialogue 2, On the twin System of the world, p. Italian 223 (but 168 of the Latin version), Galileo teaches these things about the pendulum, or plumb-line:

[Margin: Galileo’s doctrine on the velocity of descent and ascent of a pendulum through an arc.]

“That observation about the pendulum I proposed to you for this end: that you may understand the impetus acquired in the descending arc—where the motion is natural—to be of itself sufficient to drive the same pendulum-bob, through just as much space, by a violent motion, up a similar ascending arc; of itself, I say, all external impediments removed. I believe, too, that it is understood without doubt that, just as in the descending arc the velocity continually grows up to the lowest point of the perpendicular, so also from that same point, through the other ascending arc, the velocity is continually diminished up to the extreme and highest point—and diminished in the same proportion by which it grew at the start—so that the degrees of velocity at points equally distant from the lowest point are equal among themselves.” On which occasion I say the cause why this motion is not perpetual is that, in the ascent, gravity somewhat diminishes the upward-impulsive impetus, and therefore the arc of ascent is not precisely as great as that of descent; and the contrary impetus prevails more and more until it destroys it all, and the pendulum rests.

[Margin: And on the motion of heavy bodies through a well as long as the diameter of the earth.]

But Salviati, sustaining Galileo’s person, goes on and says: “And hence (reasoning probably) I seem able to believe that, if the terrestrial globe were perforated to the center—nay, and beyond, to the nadir—a cannonball would acquire, by its descent, so much impetus of velocity that, the center being passed, it would drive itself upward through just as much space as was the space of descent; so that beyond the center the velocity would continually diminish by those degrees of decrement which would correspond equally to the degrees of increment acquired in the descent.” So far he, as to our matter.

[Margin: Galileo’s discord with himself.]

Galileo, therefore, confesses that a heavy body acquires a greater and greater real impetus, and so much that it is fit to drive the same globe upward. But if it really moves uniformly through the circle (as follows from his doctrine delivered at number 8, and from his most open opinion), and the inequality of velocity is a mere appearance of the eyes—or [appears] when the intervals are regarded, which it seems to traverse in a straight perpendicular line—then it cannot continually acquire an increment of greater and upward-impulsive impetus; or, if it does acquire it, the inequality of velocity following from it by natural necessity is real, not apparent. Galileo therefore fights with himself; nor can his propositions—page 120 [Italian], on the real equality, and page 168 Latin, on the increment of impetus—be reconciled with each other.

[Margin: The argument in form, for the Earth’s motion. — Proof of the Major and the Minor.]

These things being not inopportunely premised about this motion, it is now time to propose the argument which Galileo and Bullialdus contrived, from the motion of heavy bodies, for the motion of the earth—and to dissolve it. The argument, then, forced into syllogistic form, is of this kind:

[XX.] Heavy bodies, in descending naturally, acquire by their apparent motion a continual increment of velocity. But the reason for this increment is most readily rendered if the Earth be moved at least by the diurnal motion; whereas if it be immobile, no reason can be rendered in which the intellect may rest. Therefore the Earth is moved at least by the diurnal motion.

The Major appears from many experiments, but especially those delivered at length in chapter 16, from number 6. The prior, affirmative part of the Minor is proved [thus]: because, if the Earth be moved at least by the diurnal vertigo, the real descent of heavy bodies really turns out to be circular and equal, and the whole inequality of their velocity is thrown back into mere appearance—the eyes falsely judging that they are moved through the same straight and perpendicular line, and that in it they traverse greater and greater spaces successively in equal times—whether they then be said to move through two circles (according to Bullialdus’s opinion, set out at numbers 2 and 3) or through a simple circle (according to Galileo’s opinion, explained at numbers 7 and 8). But the latter, negative part of the Minor is proved from the insufficiency of the reasons which either the Peripatetics or others have hitherto adduced for such an increment—which Simplicius (On the Heavens I, t. 88) and others chiefly relate. For the strongest [reason] usually brought is the acquisition of a greater and greater gravitation, or impetus intensifying the prior impetus; but if this were the cause, then heavier bodies too would have to descend faster, and as much faster as they are heavier than others—and so the [cannon]ball would be carried

[…continues on p. 406 (PDF 441) with the catchword “fer-” (fertur): “…would be carried [faster]“—Riccioli’s resolution of the argument and the chapter’s close.]

(printed p. 406 — the end of Chapter XVII and start of Chapter XVIII. Riccioli dissolves the pro-Earth-motion syllogism: the moving Earth saves only the appearance, not the real effects, of the velocity-increment, and a sufficient a-priori cause can be given on a stationary Earth; a corollary rebukes Galileo’s errors. Chapter XVIII then opens the search for that a-priori cause on a resting Earth, beginning the survey of opinions with Hipparchus (refuted) and Alexander.)

[Header: BOOK IX. SECTION IV. — 406]

…an iron [ball] of two pounds would descend from the same altitude to the earth in a double-shorter time than an iron ball of one pound—which is established to be utterly false, as is clear from the premised experiments (ch. 16, no. 13) and from the 13th theorem of the same chapter.

[Margin: 1st Response.]

I respond, firstly: the Major being conceded, [I respond] by denying the prior part of the Minor and its proof. For, the diurnal motion of the earth being posited, by a really-equal motion some appearance of the velocity-increment is indeed saved, but the other effects are not saved—which presuppose a real increment of greater continual impetus, and so a really greater and greater velocity of motion: such as the stronger percussion, the more vehement sound, and the elevation of a greater weight, the higher the place from which the heavy bodies fall (according to what was said against Bullialdus at number 5 and against Galileo at number 10). Next, by Bullialdus’s two circles, the increment of velocity according to the odd numbers counted from unity is not saved, as we showed at number 5.

[Margin: Many errors of Galileo collected into one.]

But by Galileo’s single circle, the real increment of the impetus of Heavy bodies is not saved—which nevertheless ought to have been saved, even by what Galileo conceded elsewhere (according to what was said at numbers 10 and 19). Nor, again, is the unequal descent of two heavy bodies of different weight, dropped together from the same altitude, saved, as I showed at number 11. Besides, that descent through the semicircle is not saved, unless the Heavy body be on the Equator and so slow that it requires precisely 6 hours to reach the center of the earth; for these two conditions removed (which are verified in very few cases), the line of the said motion is not circular, as I showed at numbers 12, 14, 16, 17. Nay, not even on the Equator does that semicircle arise from the motion of a heavy body (however slow—such that it would take 6 hours to reach the earth’s center) unless the Earth’s motion be diurnal only, the annual motion of the center excluded; which, since it is not excluded by Galileo, makes that motion really unequal, and through a line different from circular—wherefore Galileo again contradicts himself, as I showed at numbers 13 and 15. Nay, even if the Heavy body be on the Equator and descend to the earth’s center in 6 hours, and the earth be moved by the diurnal motion only, the line of descent is not circular, but notably different from circular toward the end, as I showed at number 14. Finally, Galileo cannot, by this hypothesis, save the increment of velocity of light bodies naturally ascending, even above the air bordering the earth: for these are neither borne to one determinate point as to a center, nor do they follow the motion of the Earth—for which cause, perhaps, he preferred to deny positive levity, against the experiments adduced at chapter 16, number 4.

[Margin: An objection rejected.]

Nor may you say: granted that Bullialdus and Galileo did not save all the properties of heavy bodies naturally descending, they can nevertheless be saved in the hypothesis of a moving earth, since the line which they describe can be assigned, and in it the inequality of the parts traversed in equal time, together with the inequality of percussion, of sound, etc., and the inequality of velocity of heavy bodies of different weight. For it is answered that that real inequality cannot be saved on the Equator, if the heavy body be such that it requires 6 hours for its arrival at the earth’s center (as I shall demonstrate below); but that outside the Equator it can indeed be saved by such a line as we described at number 17—yet by it the difficulty of the real increment of the velocity of heavy bodies is not removed, but rather increased; because that motion turns out unequal through such a line in such a way that it is nevertheless made through curved lines, and of different kinds about a cone, by a longer way than if, the earth standing still, heavy bodies were moved through the simplest straight line (as happens in heavy bodies falling above the poles of the terrestrial axis). And—what is absurd—without any necessity the same heavy body, its gravity unchanged and dropped from the same altitude, is forced really to traverse far more of a journey than if it fell above the poles of the terrestrial axis, and more in one parallel than in another, and finally more by night than by day.

[Margin: 2nd Response.]

I respond, secondly, by denying the second part of the Minor, which was negative; for a sufficient reason for the said increment can be rendered in the hypothesis of an immobile earth, as will appear in the following chapter, number 10. Nor is it necessary, if the reason is to be sufficient, that the downward motion be as much faster as the body is heavier—just as it is not necessary that, if the cause of refraction is the inclination of a ray upon a medium of different density, and the inclination of one ray be double that of another (or the density on which it falls be double), the refraction of the former be also double the refraction of the other. Or, [just as it is not necessary that,] if the greater distance of a star from the earth is the cause of less refraction and parallax (other things being equal), the refraction or parallax therefore turn out double-less because its distance has turned out double-greater.

[Margin: A corollary against Galileo’s boasting.]

From what has been said thus far, then, it appears into how shameful errors—in Geometry and in Physico-mathematics—Galileo fell, who with such great but empty glory boasted that he had found the way by which heavy bodies, descending naturally, seem to acquire an increment of velocity.