[I.] Most advisedly and on purpose we premised, in chapter 16, so many experiments concerning the natural descent of heavy bodies and the ascent of light ones, that we might lay more solid foundations for these arguments which we shall soon bring forward against the motion of the Earth—which will indeed have as much Physical evidence among those willing to make the same experiments, or as much credit among those believing our experiments, as the experiments themselves have evidence. But first one Axiom must be premised—or quasi-Axiom, to be denied by no Physicist—which is of this kind:

Axiom. Such and so great is the Increment of Velocity of the same Heavy body naturally descending downward from the same altitude through the same medium—or of a Light body of the same [kind] naturally ascending to the same altitude through the same medium—in any place of the Terrestrial globe, as it is in another place of the terrestrial globe, if all its intrinsic conditions be the same in both [places].

[Margin: Exposition of the Axiom.]

For the sake of explaining the Axiom, let there be an example in the clay globe of 8 ounces, which we have often used (from what was said at chapter 16, number 12), and which we found, by repeated and most certain experiments, to have descended from a height of 240 feet in the time of 4 Seconds of an hour—but in such a way that at the end of the first Second it had already traversed 15 feet, at the end of the second 60 feet, at the end of the third 135 feet, and at the end of the fourth 240 feet; and this in our Bolognese parallel [latitude]—experiences similar to which Galileo attests that he himself made, in Dialogue 2.

[…continues on p. 409 (PDF 444) with the catchword “logo 2” (dialogo 2): the exposition of the Axiom continues with the engraved figure.]

(printed p. 409 — within Chapter XIX. The Axiom is completed and justified: a given body’s velocity-increment in fall is the same in any place of the Earth. The First Argument then follows in syllogistic form: many bodies dropped in the equatorial plane fall with a real increment (proved by harder blows and other physical effects), but on a diurnally-rotating Earth the increment would be merely apparent — the fall tracing a circle whose segments come out in the squares-proportion — therefore the Earth does not move diurnally alone.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 409]

[Dialogue 2,] On the System of the World, at some other Italian latitude of the Equator. I say, therefore, that if this same globe—retaining all its intrinsic conditions (namely gravity, figure, bulk, etc.)—were dropped from the top of a tower existing in any other place of the terrestrial surface (whether on the Equator, or at the poles of the terrestrial globe, or in any other parallel), and the height from which it were dropped were 240 Roman feet (such as was the height we used), and the medium were the same (namely air of the same rarity and tranquillity)—it would come about that the increment of velocity would be such and so great as we found in the Bolognese parallel. Such, I say—that is, really unequal, or with a real and sensible increment of impetus and velocity—if in our parallel it was with a real inequality. And so great, besides—that is, according to the order of the square numbers counted from unity, in the spaces completed after equal times—if so much was found in our parallel by certain experiments.

[Margin: The reason for the Axiom.]

The reason for the Axiom is that, since all things which can concur to determine the increment of velocity are the same in both [places], and no miracle is supposed to occur (for we speak of natural descent), no cause can be solidly assigned for which that increment should be real in one place and only apparent—not real—in another, or in one place according to the aforesaid proportion and in another not. Otherwise, no one could, from an induction made in some places concerning the motion of natural bodies or its other sensible effects, gather for himself universal first principles, true in every place—such as this: “Every natural body is movable”; “Every fire is naturally combustive”; and so of very many [propositions] which are commonly held in place of Axioms in Physics. This being posited, [let us proceed] now to the arguments.

First Argument, from the Real Increment of Velocity of Heavy bodies, against the Diurnal Motion of the Earth

[II.] Many heavy bodies, dropped through the air existing in the plane of the Equator, would descend to the earth with a real and notable increment of velocity—and not merely an apparent one. But if the Earth were moved by the diurnal motion only about its center, no heavy bodies dropped through the air in the plane of the Equator would descend to the earth with a real and notable increment of velocity, but only with an apparent one. Therefore the Earth either is not moved, or is not moved by the diurnal motion only.

[Margin: Proof of the Major.]

The Major is proved: for any globe dropped by us through the air in the Bolognese parallel descends to the earth with an increment of velocity not only apparent (and growing according to the order of the square numbers, as we showed at ch. 16, no. 12), but also real—as was established from the more vehement percussion, the greater sound, the deeper perforation of water, the greater rebound and recoil, the elevation of a greater weight, and other similar effects, the higher the place from which the globes were dropped (per the same ch. 16, from no. 6; and, for the elevation of weight and the perforation of water, no. 23, experiments 12 and 13)—and as anyone can easily test for himself. Which effects, unless one wishes to be obstinate, are a natural indication of a greater and greater impetus acquired in descent, and accordingly of a real increment of velocity naturally following from the greater impetus (so long as there is no impediment retarding its second act)—an indication, I say, of this as of its cause; just as the same projectile, [thrown] by an arm impelling more strongly, and therefore making a greater wound or blow in the same subject, is a physically evident sign of a greater impressed impetus, or of a greater commotion of air (as certain haters of impetus and of impressed quality say). Therefore, by the premised Axiom, the same would happen to these globes in the Equator, dropped in like manner through air of the same character; and so the increment of velocity there too would be real, and not merely apparent.

[III.] The Minor—in which the whole difficulty lies, and which requires some Mathematics—is proved by the figure premised, which we already set out at chapter 17, number 7.

[Margin: Exposition of the Figure.]

Let there be, then, with the Earth’s center A, the arc CD described, and [the arc] BM—the former passing through the tower BC and its vertex C, the latter through the tower-foot B—[the tower] existing in the plane of the Equator. And the Earth’s semidiameter AC being bisected at E, let another arc be described from C, toward the same eastern quarter M, which shall be either the semicircle CIA, or at least so great a portion of it that the straight lines to be drawn from A to the arc CD fall upon it, or even cut it. And since the tower BC is supposed to be moved—by the motion of the diurnal rotation alone—toward DM uniformly (that is, by describing, with its vertex and with its foot, equal arcs in the same circumference in equal times), let the arc CD be divided into however many equal arcs answering to the aforesaid equal times—say into the arcs CF, FG, GH, HL, LD—and to the points of division let there be drawn, from the Earth’s center A, the straight lines AF, AG, AH, AL, AD. And at last let there be drawn, from the center E of the lesser periphery, a straight line to some point of the common section of the periphery CIA with one of the aforesaid straight lines—say to the point I.

[Margin: First Proposition, for the proof of the Minor.]

These things being designated: Firstly I say that, if the Earth were moved by the diurnal motion alone, some heavy body dropped from the tower-vertex C (in the plane of the Equator) would describe by its natural motion a portion of the line CTI, which would be, to all sense, circular—or a portion of a circular periphery. Which I show thus. The clay globe dropped by us from the tower-vertex C descended to the earth in 4 Seconds of an hour, and in such a way that at the end of the first Second it had traversed 15 old Roman feet, at the end of the second 60 feet, at the end of the third 135 feet, at the end of the fourth 240 feet. Therefore, by the premised Axiom, it would do the same in the Equator; and, the same proportion of the square numbers being preserved, it would describe in that time a line circular to all subtlety of sense—for the spaces completed in the four Seconds, namely FS, GT, HV, LX, would be just as great as they ought to be if the line CSTVX, drawn through their ends, must be circular; and conversely, if the circular line CSTVX be drawn from center E, and the quantity of the spaces FS, GT, HV, LX be investigated, FS is found as 1, GT as 4, HV as 9, and LX as 16—which is the proportion of the squares counted from unity; or, if the first space FS be taken as 15 feet, GT will be found 60 feet, HV 135 feet, and LX 240 feet.

But the aforesaid spaces are investigated by the method already delivered at ch. 17, no. 14, which is this. Since, in one Second of an hour, a point of the terrestrial Equator, by the force of the diurnal rotation, runs through the arc CF of 15″ [arc-seconds]—and equal to this, severally, are the arcs FG, GH, HL—but the arc CS (being described from E, which bisects AC, and intercepted by the same lines AC and AF) is of double the [number of] seconds (as we showed at ch. 17, no. 8): therefore the arc CS will be of 30″. And since the arc CG is of 30″, the arc CT will, for the same reason, be of 60″, or one minute; and since the arc CH is of 45″, the arc CV will be of 90″, that is, of one minute and 30″; and at last, since the arc CL is of one minute, CX will be of 2 minutes. The Complements, therefore [of these arcs to the semicircle], will be: AIS = 179° 59′ 30″; AIT = 179° 59′ 0″; AIV = 179° 58′ 30″; AIX = 179° 58′ 0″. And the Chords of these, taken in order from the doubled sines, will be: AS = 19,999,999,947 parts (of which the diameter, or greatest chord, is 20,000,000,000); AT = 19,999,999,788; AV = 19,999,999,524; AX = 19,999,999,154. Which chords, if subtracted from the greatest chord AC (to which the radii AF, AG, AH, AL are equal), leave the spaces: FS = 53 parts, GT = 212, HV = 476, LX = 846. But the proportion between

[Translator’s note — the engraved figure (upper right) is the same semicircular-fall construction used in chapter 17 (there, fig. for ¶VII, p. 399): center A (Earth’s center, lower right), the outer arc C·F·G·H·L·D (the tower-top’s path, divided into equal time-arcs), the tower-foot B and point M, the radial lines from A, and the fall-semicircle C·S·T·V·X·…·I·A (center E, on diameter AC). Here it serves to compute the fall-segments FS, GT, HV, LX on a rotating Earth.]

[…continues on p. 410 (PDF 445) with the catchword “inter”: “But the proportion between [these segments]…”—showing the segments fall in the squares-ratio, so the rotating-Earth acceleration would be merely apparent.]

(printed p. 410 — within Chapter XIX, finishing the First Argument. The demonstration closes with a synopsis table and a second proposition showing the fall-arcs equal in true length, so on a rotating Earth the acceleration would not be real — proving the Minor. Two Copernican responses are then taken up: the first (uniform forward motion explains the harder blow) is rebutted as frivolous, and the second (the annual motion makes the forward-arcs unequal) begins to be answered.)

[Header: BOOK IX. SECTION IV. — 410]

…between these is altogether such as between the squares 1, 4, 9, 16. For if it be made: as the first space FS, 1, to 4, so 53 to another, GT will come out of 212 parts; again, if as FS, 1, to 9, so 53 to another, HV will come out of 476 parts; and at last, if as FS, 1, to 16, so 53 to another, LX will come out of 846 parts. Therefore, if a curved line be continuously described through the ends of the lines completed by that globe—the proportion of the squares being preserved in the spaces—that line will be circular; and conversely, if it be circular, those spaces, or segments, intercepted between that circular line and the arc CD, will have [the following]:

[Margin: Second Proposition, for the proof of the Minor.]

[IV.] Secondly I say that the arcs CS, ST, TV, VX are each equal to the arcs CF, FG, GH, HL, and hence are equal among themselves (just as these are equal among themselves), and correspond to equal times. For the periphery CD has, for its semidiameter, the straight line AC, which is double AE—the semidiameter of the lesser periphery CI—since its center E, by construction, bisects AC. Now, by theorem 5 of book 11 of the Mathematical Collections of Pappus of Alexandria, the circumferences of circles are to one another as their diameters (or as their semidiameters); therefore, just as a quadrant whose semidiameter is AC equals two quadrants of a periphery having for semidiameter AE (double-smaller than AC), so the arc CF (which is of 15″) is equal to the arc CS (which is of 30″); and, for the same reason, the arc FG to the arc ST, the arc GH to the arc TV, and the arc HL to the arc VX, are equal as to true and absolute length—because the former are each of 15″, the latter of 30″. That these arcs are of 30″, and of double the [number of] scruples [arc-seconds] of the arcs CF, FG, GH, HL, was already shown at ch. 17, no. 8; and it is easily proved by drawing straight lines from E to the points S, T, V, X (as EI was then drawn to the point I): for the angles at the center E (by Euclid III.20) come out double the angles constituted at the circumference at A and made by the same lines; but the angles at A are at the center of the periphery CD and measure it, just as the angles at E measure the arcs of the circumference CI. Wherefore, if the angles at E are double the angles at A made by the same straight lines—say the angle BEI is double the angle BAI (that is, CAD)—the arc CI too is of double the [number of] scruples than the arc CD, and so of the rest. These things being shown beforehand, the Minor is now to be proved.

[Margin: Proof of the Minor.]

The MINOR, therefore, is proved: for the Globe, of which it was spoken in the proof of the Major, would describe in the Equator, by its descent, a line circular to all subtlety of sense (as we showed at number 3); this being posited, it would traverse equal arcs in equal times (as was shown here at number 4). Therefore it would not descend with a real inequality—that is, with a real increment of velocity—if, namely, the Earth be moved by the diurnal motion only. Which was to be demonstrated in the Minor. The Major therefore being demonstrated physically, and the Minor physico-mathematically, it follows that either the Earth is in no way moved, or is not moved by the diurnal motion only.

[Margin: 1st Response of the Copernicans, but frivolous.]

[V.] The Copernicans would perhaps respond that the impetus—by whose force a heavy body, descending from a higher place, makes a greater percussion—does indeed grow continually as [it comes] from the principle moving downward (namely, from gravity), but does not grow as [it comes] from the principle moving forward toward the East, but is equal (or produced equally and uniformly)—the equality of the diurnal motion so requiring, to which the heavy body, in its forward motion, must be accommodated; and so that a real increment of impetus cannot be argued from the greater percussion in relation to that other effect, namely the greater velocity in the real descent through the aforesaid circular line. But the Response—although it must be defended on the strength of the Copernican hypothesis—is necessarily feigned; yet absolutely [it does not escape] the aforesaid proportion. Since, therefore, that heavy body proceeds through a single line and way—nor any other than that which the ends of the completed spaces describe—that [line] will be circular; and the same will happen if the time of descent does not exceed 4 minutes of time, and so if the whole arc CL does not exceed one degree, as will be clear to one examining the chords and the complements of the chords to the greatest chord. It is pleasing, moreover, to set the aforesaid numbers before the eyes in the synopsis of one little table—for which, look at the preceding figure, which is also placed at p. 399.

[This response] is frivolous. For by induction it is well enough established that a corporeal movable, by the force of an impetus flowing whether from an intrinsic or from an extrinsic [source], is by nature moved prior to striking another body toward which it is moved, and that the percussion necessarily presupposes the local motion of the striker; and that if, the impetus increasing, the percussion becomes greater, then a greater—or faster—motion also precedes by some priority, so that the same impetus, of itself and naturally (impediments removed), produces these two effects, namely motion and percussion; and just as percussion cannot be without prior motion, so neither [can there be] a greater percussion without a greater motion arising from the increment of impetus. Nay, per se and primarily the greater impetus is ordered to a faster motion, and secondarily (or quasi per accidens) to the percussion of the body impeding its further motion. Granted, therefore, that the impetus of a heavy body descending through a circular line did not grow by the force of the common principle moving [it] forward toward the East—yet by the force of gravity it ought to grow, and by its increment first to cause a faster motion through the way by which it really descends, [rather] than a greater and greater percussion. Nor can any reasonable physical cause be assigned which would permit this impetus to grow insofar as it is percussive while it tends downward, and yet impede its increment insofar as it is really motive downward (and so [impede] the increment of velocity). And conversely, if anything could physically impede the increment of the impetus as motive—by inhibiting it, or by tempering and holding it in perpetual equality—by that very thing it would also impede the secondary effect (which depends, by physical necessity, on the primary): namely, the greater percussion. Since, therefore, the greater percussion is manifest—the higher the terminus from which the heavy body falls—it is also manifest, and physically evident, that a greater and greater velocity preceded, not merely apparently but really; but this could not have preceded on the supposition of the equality of the Earth’s diurnal motion; therefore the supposition of this motion is false and absurd.

[Margin: 2nd Response for the Copernicans. — But useless.]

Someone would perhaps respond, secondly, that the motion of descending heavy bodies—insofar as it is forward, in imitation of the terrestrial diurnal and annual motion—is not equal, because it does not occur at the same distance from the center of the earth and of the annual orb, but at a less and less [distance]; and so [the body] completes arcs of circles indeed similar among themselves in equal times, but smaller (of fewer paces or feet); wherefore, by the force of the principle moving [it] circularly of itself, the motion of descending heavy bodies is retarded as motion, though it is not diminished as percussive. But neither does this response avail anything, for the aforesaid dilemma always holds: For either it so retards the real motion as to reduce it to perpetual equality—and then, by the necessity by which percussion follows the vehemence of the motion from which it [arises], the percussion would be rendered equal, from whatever height it occurred; or it does not [so] reduce it, but leaves it still—if not in the original [inequality], at least in a very notable inequality—and then the motion too, though not as great as it otherwise [would be], would in reality have a notable inequality. But in reality it would not notably diminish the inequality of Heavy bodies, because—in the figure to be set out at number 9—the semidiameter of the Earth’s Orbit CF is 25,870,000 Roman feet, nor much greater or less on the Coperni-

[…continues on p. 411 (PDF 446) with the catchword “Coperni-” (Copernici): “…on the Copernican [reckoning]“—the vast orbit-radius makes the annual motion change nothing.]

(printed p. 411 — within Chapter XIX, finishing the First Argument’s defense. The second Copernican response is rebutted (the annual orb is so vast the composite motion remains sensibly uniform), and the third — the ship-mast analogy — is dismissed as unverifiable, Riccioli proposing his own rotating floating-beam apparatus as a real test. A sub-head then extends the argument to many heavy bodies beyond the 8-oz clay globe, since even a full minute’s fall keeps the line sensibly circular.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 411]

…on the Copernican [reckoning]; and the semidiameter of the annual orb is, on the Copernican [view], 29,543,740,000 Roman feet, as will appear from what is to be said in the table at the end of number 13. Wherefore the whole AF is 29,569,610,000 feet.

[Margin: Look at the figure to be set below at number 9.]

But the Asinelli tower, from whose height we dropped the Heavy bodies—namely FH—is 240 feet; therefore the whole AH is 29,569,610,240. Now, in one Second of an hour, by the force of the diurnal circular motion, 15″ are run through at the Equator, to which there answer 1875¼ feet; but by the force of the annual motion there occur 2″ 28‴ (as many, namely, as the mean motion of the Sun would make)—that is, 5965 feet on the Copernican [view], as will appear shortly from the same table. Therefore the motion composed of the diurnal and the annual, answering to one Second of an hour, is 7840¼ feet—by which, namely, a point on the surface of the terrestrial Equator (say the point F) is carried in one Second of an hour to X. Now let it be made: as the semidiameter AF, of 29,569,610,000 feet, to FX of 7840¼ feet, so the semidiameter AH, of 29,569,610,240 feet, to another; and there will be found the arc HE of 7840 feet [and a small fraction]—which is insensibly greater than the arc FX; much less sensibly, therefore, do the arcs HE and FX differ from each other; and yet the inequality of the motion of Heavy bodies in descent from the aforesaid tower is as great as [the inequality] among these numbers 15, 60, 135, 240 (from what was said at ch. 16, no. 12). Therefore, since the diurnal motion together with the annual is retarded, on account of the variation of the circles, not even by a quarter of a foot, that whole inequality which is from the downward-moving principle is left entire as to the integrity of the feet, and is still according to the numbers 15, 60, 135, 240. Whence one might rather argue: if the way by which the heavy body in fact descends is rendered equal by the principle moving [it] forward circularly (and retarding the principle that moves downward unequally of itself), then this circular principle must be called very unequal of itself, and intending a motion very different from the diurnal and annual—which are feigned [to be] equal of themselves.

[Margin: 3rd Response for the Copernicans. — But insufficient.]

The Copernicans would respond, thirdly—taking refuge in the familiar example of the ship, as in the last little boat in the shipwreck of their hypothesis—and will say that it can happen, in the hypothesis of a stationary earth, that, [just as] on a ship borne eastward a ball dropped down from the top of the mast (and at the same time receiving the ship’s impetus from the mast) really falls along a curved line to the foot of the mast, so as to make a greater percussion the higher the mast is—and yet its motion, in reality, and such as would be observed by one standing on the shore, would always be equal. But this response can neither be confirmed by any a-posteriori experiment, nor is it persuasible by any a-priori reasoning. For if the ball’s motion were so blunted that, from being unequal of itself, it were rendered always equal, the percussion too, made from any interval whatever, would be equal. But because this experiment is not easy in practice, we ourselves will suggest to the Copernicans an easier way of testing the effect [of a motion] arising from two principles—of which the one should move [the body] along a straight line, unequally of itself, the other through a circle, and equally.

Let the figure set down at number 3 be repeated here, in which let AC be a beam hollowed lengthwise in the manner of a channel (or troughed), and so floating upon standing water that the end A is held immobile, as if at the center of motion, while the other [end] C can be carried around by an equal and uniform motion from C, through G, to D. Let there be, besides, some globe at C, connected by a cord to a great weight hanging under the water from A, so that it can be drawn through the trough of the beam; and let the depth of the standing water be so great that the weight does not reach the bottom of the water when the globe has come from C to A—that is, let the depth of the water be greater than the length of the beam AC. First, then, with the beam at rest, let the weight be dropped from A downward, and let the time in which the globe moves from C to A be measured by the pendulum (of which we have often made mention in measuring Seconds of an hour); and at the same time let the percussion made by it upon some body placed at A be carefully observed; and let the same be done while the globe again comes from E to A, which is a journey half-smaller. Then, the globe being brought back again to C, at the same instant in which the weight is dropped from A and the globe begins to move from C, let the end C of the beam begin to be carried around uniformly toward D; and let the time in which the globe C reaches A be measured, and the percussion noted. For it will appear that the percussion is greater when [the globe comes] from C than when from E (while the beam was at rest); and that the time from E to A exceeds half the time spent from C to A. But, the beam being carried around uniformly, an inequality will indeed be observed in the aforesaid percussions and times, but not so great—because the transverse motion blunts the motion that would be along the straight line CA of itself; and the more so, the faster the transverse motion of the beam shall have been. But if the velocity of the circular motion be precisely so great that the motion of the globe (dropped whether from C or from E) turns out really equal—at least to sense—whether its real way be made through the circle CSTI or not, then the percussion too will be, to sense, as great for the globe dropped from C as [for one dropped] from E; and the time spent from E to A will be half the time spent from C to A. All which will be still more easily achieved if the beam AC be so inclined to the plane of the horizon that the globe runs freely down to A, the immobile end, while the other end C, raised, is carried around through a circle equidistant from the horizon. And hence the Copernicans will learn that an impetus which of itself would grow unequally cannot be inhibited to equality without the inequality of percussion being impeded too. Finally, from other more certain experiments it is manifest that the same heavy body, if dropped straight down, is moved faster than if [it descends] obliquely—as in the experiments of the 11th Class, adduced above at ch. 16, no. 22, compared with others in which the same globe was dropped perpendicularly down; and in rivers, of which one branch flows down through a straight channel, the other through meanders and a less steep channel.

Application of the same Argument to very many Heavy Bodies

[VI.] Although we have used the example of the clay globe of 8 ounces—because we employed it, of its kind, most frequently, as being more easily procurable on account of the multitude of many such globes needed for repeating the experiments often (for each of them a fresh globe was required, to be substituted in place of the preceding one, now broken or bruised)—nevertheless the argument made holds for the greatest part of those heavy bodies which are heavier than our air to such a degree that they can overcome the fluctuations of the air made by an ordinary breeze, without disturbing the aforesaid proportion. For such bodies are either heavier in species than the clay globe, or not much lighter—so that they descend from a height of 240 feet either in a shorter time than 4 Seconds, or not much longer. Nor will you find any [such body] which, to traverse so small a height by natural descent, requires one whole minute of time. For a leather playing-ball, and an orange, are not apt to overcome the usual commotion of our air without fluctuating somewhat in descent (as we have experienced more than once); and I say the same of certain wooden globes of lighter grade—and yet these very globes reached the ground from a height of 280 feet (namely, from the cornice of the Asinelli tower) in a shorter time than 8 Seconds. Wherefore all bodies (which are indeed very many) apt to overcome the fluctuation of the air without disturbing the due proportion of the velocity-increment descend from a height of 240 feet in less than 8 Seconds. And yet, if they required for this a whole minute of time—in which, namely, a point of the terrestrial surface, by the force of the diurnal rotation, traverses 15 minutes of the Equator (or of a parallel of the Equator)—still the argument would hold, because within the first minute of the first degree of this motion the line described by the heavy body (on account of the diurnal motion alone) is, to all subtlety of sense, circular; from which it would follow that its motion is in reality equal, as we showed at number 4. And that it is circular will be clear to one investigating the chords, and thence the spaces which are the complements of the chords, in the manner we taught at ch. 17, no. 14, and in this chapter at number 3. Which, that it may be more evidently established, we shall investigate in this place. Let, then, the whole time of the globe’s descent from a height of 240 feet be 60 Seconds, which make one Minute; which, accordingly, divided into four equal times—namely, into fifteen Seconds of an hour [each]—would require, for each of them, 3 minutes of the Equator and 45″ [of arc, i.e. 3′45″]. But, for the sake of abundance and ease, let there be four minutes [of arc] in each of the arcs CF, FG, GH, HL of the premised figure; for they will be of double the number of

[…continues on p. 412 (PDF 447) with the catchword “Minu-” (Minuta): “…minutes”—the computation generalizing the First Argument to a one-minute descent.]

(printed p. 412 — within Chapter XIX. The extension to many bodies concludes with a synopsis table showing that even for a one-minute descent the fall-line on a rotating Earth deviates insensibly from the squares-proportion circle, so the First Argument holds for all bodies heavy enough not to fluctuate in air. A further sub-head extends the argument to bodies off the Equator, whose fall traces a curve on a right cone, sensibly planar over four seconds; a demonstration for 45 degrees latitude (Bologna) begins.)

[Header: BOOK IX. SECTION IV. — 412]

…minutes—namely of 8 minutes [each], [for] the arcs CS, ST, TV, VX; wherefore the arc CT will be of 16 minutes, CV of 24, and CX of 32. Their complements, therefore, will be: AIS = 179° 52′, AIT = 179° 44′, AIV = 179° 36′, AIX = 179° 28′; from the halves of which the right sines will be found, which, doubled, will give the chords of the aforesaid arcs; and the chords, subtracted from the greatest chord AC, will leave the spaces traversed at the end of the equal times, in parts of which AC is taken (as above) of 20,000,000,000—as may be seen in the following table:

[Translator’s note: the chains complement → half-complement → sine → chord → segment are internally consistent and imply the fall-arcs CS, CT, CV, CX = 8′, 16′, 24′, 32′ (and tower-arcs 4′, 8′, 12′, 16′), as the surrounding text states; the table’s printed “Tower-arc” and “Fall-arc” columns appear at double that scale (a source slip). The conclusion is unaffected.]

Let it now be made: as the first space FS, 1, to the second GT, 4, so 13,538 to another—and you will find 54,156; again, let it be made: as FS, 1, to HV, 9, so 13,538 to another—and you will find 121,842; lastly, let it be made: as FS, 1, to LX, 16, so 13,538 to another—and you will find 216,608. The spaces found from the chords subtracted from the greatest chord, [compared] with those found by the proportion due to the square numbers, are therefore deficient in GT by 2 parts, but in HV exceed by 6 parts, and in LX by 8 parts—of which parts the whole AC (that is, the semidiameter of the earth) is 20,000,000,000. Now the semidiameter of the Earth, as we showed in bk. 2, ch. 7, contains 4139 recent Italian and Bolognese miles—that is, 4,139,000 paces, or 20,695,000 feet, or 248,340,000 inches. Let it therefore be made: as 20,000,000,000 to 8 parts, so 248,340,000 inches to another—and you will find not one whole inch (or finger in breadth), but this fraction of one inch [≈ 1/10], that is, about a tenth part of an inch. Wherefore LX would differ, at the most, by one tenth-part of a finger taken in breadth, from the space due [for it] if the line CSTVX were geometrically circular; but HV and GT [would differ] much less. Therefore, if a Heavy body, for descent from a height of 240 feet, did not require a whole Minute of an hour, it would nevertheless, in descent, still describe a line sensibly circular at the Equator (according to what was said at number 3), and consequently would really be moved equally (according to what was said at number 4). Wherefore the aforesaid argument holds for all bodies so heavy that, for natural descent through a height of 240 feet, they require no time longer than one Minute of an hour—of which kind is the greatest part of heavy bodies, as being apt to descend through our air without sensible fluctuation, or [without] that [air] moving them from the line sensibly perpendicular.

Application of the same Argument to Heavy bodies placed outside the Equator

[VII.] We have hitherto used the argument as made, supposing the Heavy body to descend in the plane of the Equator—because the demonstration of the real equality of motion (which would necessarily follow from the circular motion) has greater evidence if the line of motion be in one and the same plane, as the nature of a perfectly circular line requires. Yet it has its Physical evidence, sufficient to infer the same equality, in places outside the Equator too—the more truly, the less they are distant from the Equator. For although outside the Equator heavy bodies, in descending (the diurnal motion of the earth being admitted), describe the line of their motion as a curve on the surface of a right cone having its vertex at the earth’s center and its axis coinciding with the Equator’s axis—and that motion begins from the periphery of the conic base, as we showed at ch. 17, no. 18—yet within the time of 4 Seconds of an hour (in which, in the premised argument, we supposed the globe to have descended from a height of 240 feet) the revolution of the aforesaid cone is so small with respect to the transverse magnitude (or thickness of the cone taken near the base) that, to sense, it is just as if the motion were made on a plane surface; nor could that slight deflection from one plane to another be so great as to effect a real, sensible inequality of motion, as great as the percussion and other similar effects evince—[effects] indicating a real increment of impetus.

But, lest anyone think this said from a certain supine carelessness, let us undertake the demonstration of the said [point]—and indeed one accommodated to a Parallel declining from the Equator by 45 degrees, such as is most nearly that in which we instituted our experiments, and Galileo his, so that from this, as from a mean, judgment may be made of the others. Let there be, in the following little diagram, a cone ABC, whose vertex is at the earth’s center B, and whose axis (coinciding with the Equator’s axis) is BM; and let the base of the cone be the circle AECD, turned toward the viewer of this page in a direct position, for ease of demonstration—although, by the laws of Perspective, it ought to be represented in an oblique aspect, as an ellipse. And let the Parallel of the Equator, on which is the tower-foot, be FHG; let the tower itself be GC; and let a globe, dropped from its vertex (or window) C, descend to the point H of the terrestrial surface, through the curved line LH, in the time of 4 Seconds of an hour—in which the point C, by the force of the diurnal rotation, completes the arc CK, which is of one Minute (since, in a circle parallel to the Equator, so great is the arc of the Equator answering to 4 Seconds of an hour); but its half, namely the arc CL, is of 30″. And the second sine PC of this arc—equal to the straight line MO—is 9,999,999,894 parts, of which the Radius ML is 10,000,000,000; from which, if you subtract the sine PC, there is left the versed sine, or sagitta, OL, of 106 such parts. But that the quantity of the arc CLK, and of the line OL by which the arc CLK recedes from the straight line COK, may be known in determinately taken parts, it must be recalled that in the semidiameter of the earth there are 20,695,000 Bolognese feet, and in the diameter 41,390,000 (as I said at number 6, toward the end)—whence it comes about that the circumference of the greatest circle is 129,964,600 feet; for, from what was said in bk. 1, ch. 4, the diameter is to the circumference as 100 to 314, and therefore in one Minute of the Equator are contained 6020 feet. But because we suppose the circle AECD to decline from the Equator by 45 degrees, the arc CK—by the problem of Giovanni Antonio Magini and Giuseppe Moletti (in bk. 1, ch. 20, of the Geography of Ptolemy), and [by] the table—

[Translator’s note — the engraved cone figure (lower right): a right cone A·B·C seen face-on, vertex B at the bottom (the Earth’s center), axis B·M rising to the top center M; the base circle A·E·C·D (A at left, D top, C right, E lower); the latitude-parallel F·H·G crossing it (F left, H center-low, G right), carrying the tower G·C; and near C the fall-geometry points P, O, K, L with the descent-curve L·H. Drawn upright for ease, though perspectively it would be an ellipse.]

[…continues on p. 413 (PDF 448) with the catchword “lam” (tabulam): “…and the table”—the 45°-latitude computation of the fall-curve’s deviation continues.]

(printed p. 413 — within Chapter XIX. The 45-degree-latitude computation finishes: the fall-curve recedes at most about two inches from the plane, an insensible amount, so the First Argument holds even off the Equator. The First Argument ends and the Second Argument opens, aimed at the diurnal and annual motions together: bodies dropped in the equatorial plane would fall with a real, equably-growing increment at any hour, but on a doubly-moving Earth some would not; the figure-exposition and its computation begin.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 413]

…[the arc CK] there, in Moletti, will be 42″ 25‴ of the Equator. Now the periphery AECD is to the periphery of the terrestrial Equator as 42″ 25‴ to 60″; and because circumferences are to one another as their diameters (or semidiameters), and conversely, therefore [they are] as 60″ to 42″ 25‴; so [is] the semidiameter of the earth to the semidiameter ML of the parallel. Let it therefore be made: as 1 Minute, or 60″, to 42″ 25‴, so 6020 feet to another—and there will come out the arc CLK of 4219 feet. Again, let it be made: as 60″ to 42″ 25‴, so the 20,695,000 feet of the terrestrial semidiameter to the semidiameter ML, [namely] 14,441,500 feet. Finally, if it be made: as the Radius ML, of 10,000,000,000 parts, to the versed Sine OL, of 106 parts, so 14,441,500 feet to another—there will be found the sagitta OL of 0.153 feet, that is, nearly the 6½-th part of a foot. Wherefore the arc CLK, which is of 4219 feet, does not deflect from the straight and plane line COK—where most [it does]—by more than two inches of one foot, which, in an arc of 4219 feet (or of 844 paces, or in a circular periphery of nearly one Mile), is not a sensible portion. Therefore, if a heavy body descending from C were to proceed along the arc CLK in the time of 4 Seconds of an hour, it would not decline from a plane surface by [more than] 2 inches of a foot, or 3 fingers, in breadth. But it proceeds along LH, which recedes less from a plane surface, and approaches nearer to the straight line CLG existing in the same plane; therefore it deflects from a plane surface even less—although, in Geometrical rigor, it proceeds along a conic surface. Nor does this clash with what was said at ch. 17, no. 18; for there we disputed against Galileo, pronouncing universally that the line of motion of Heavy bodies is circular, [yet] not restricting his proposition to a heavy body which completes its descent in 4 Seconds, and [Galileo] supposing—besides the diurnal motion—the annual motion of the Earth, and speaking of the whole semicircle continued right to the center of the Earth.

II. Argument against the Diurnal AND Annual Motion of the Earth together, drawn from the Real Increment of the velocity of Heavy bodies

[VIII.] I see that the Argument made hitherto does not militate except against the Semicopernicans, who attribute to the Earth the diurnal whirling only—whom I reviewed at chapter 2. But the Copernicans could refer the real inequality of the descent of heavy bodies to the annual motion of the Earth mixed with the diurnal; granted that Galileo, an asserter of both motions, denies this inequality, as we said at chapter 17, no. 8. Therefore the weapons must be directed against both motions—namely, by the following argument:

Many heavy bodies dropped through the air existing in the plane of the Equator would descend to the earth with a real, notable, and equably-growing increment of velocity, at whatever time of day they were dropped. But if the Earth were moved by the Diurnal motion together with the Annual, some out of those many Heavy bodies dropped through the air existing in the plane of the Equator would not descend with a real, notable, and equably-growing increment of velocity, [at] whatever time of day dropped. Therefore the Earth is not moved by the Diurnal motion together with the Annual.

[Margin: Proof of the Major.]

The Major is proved at number 2; for here nothing else is added to the Major of that argument, except that the increment of velocity grows in the same way at whatever time of day the heavy body is dropped downward—which is sufficiently established from our experiments, especially in the clay globes, made both morning and evening and around midday; for we always obtained the same proportion of the square numbers, and the same percussion, and elevation of weight, and perforation of watery depth, in the same globe dropped from the same height; and the same will be established for anyone not neglecting these experiments.

[Margin: Part of the Minor is proved.]

[IX.] The Minor is proved as to the particulars real and notable Increment. For by the force of the precisely diurnal motion, the descent of the clay globe of 8 ounces through 240 Roman feet in the time of 4 Seconds of an hour would be really equal (granted apparently unequal), and the increment of velocity would turn out in reality none, or contemptible, as was shown from number 2 to 5—which demonstration was applied at number 6 to very many heavy bodies. By the force of the annual motion too, the motion of that globe would be equal without any difference worth caring about, because the annual motion of the earth’s center, within 4 Seconds of an hour, is entirely equal [uniform], and in each [Second produces just as much]—not otherwise than the Sun, in the hypothesis [that moves the Sun]; and the vertex of the tower, from which the Globe is dropped, in this time describes a line insensibly differing from a circular one; and the aforesaid Globe meanwhile describes arcs equal to the arcs described by the tower-vertex, without any notable difference, or any to be cared about, as we shall show presently.

In the following diagram, from A, the center of the annual orb, describe the arc BC, in which the Earth’s center, carried from C to B (the Eastern point), is at the beginning of the globe’s descent at C; but at the end of the first Second of an hour at P, at the end of the second at D, at the end of the third at Q, at the end of the fourth at B. And let the straight line AH, produced from A, have upon it the Tower (or height) HF, of 240 feet—so that its vertex H, with the interval AH, describes in the time of 4 Seconds the arc HG, similar to the arc CB, such as it would describe by the force of the Earth’s annual motion alone. And if this were done, the globe dropped from H would, at the end of the first Second of an hour, appear at R, the space ER (which is as 1) being completed, and the tower would be at EX; at the end of the second Second the globe would be at S, the space LS (as 4) being completed, and the tower at LY; at the end of the third the globe would be at T, the space MT (as 9) being completed, and the tower at MZ; and at last, at the end of the fourth Second, the globe would be at O, the space GO being completed—of 16 parts, of which ER is one—and the tower would be at GO, and the globe’s path would be HRSTO.

In reality, however, in Geometrical and Astronomical rigor, according to the hypothesis of Copernicus, the Tower FH is also moved in the Equator toward P by the diurnal motion, completing in the time of 4 Seconds of an hour one whole Minute of the Equator; wherefore at the end of the fourth Second it is not at GO, but a little more eastern and inclined, say at KI—the arc IO of one Minute being completed, namely; and therefore the vertex H described not the arc HG, but the arc HK. Yet I say that, to sense (or without any difference to be cared about), it described an arc similar—nay, even equal—to the arc HG, and coinciding with it; and consequently that the Tower KI can be assumed in the position GO, and in the preceding times in MZ, LY, EX, just as if it had not been inclined at all by the force of the diurnal whirling. To show which, I produce the straight line KI to B, and connect K with the center A by the straight line AK. For in the triangle KAB, besides the angle ABK of 179° 59′ (as being the complement of the angle KBG, which measures the arc OI of one Minute, traversed by the tower in the time of 4 Seconds), the side BK is given—the aggregate of the height of 240 Roman feet and the Earth’s semidiameter BI, which (from what was said in bk. 2, ch. 7) is 5174 old Roman miles, that is 5,174,000 Roman paces, namely 25,870,000 Geometric feet. Wherefore the whole BK is 25,870,240 feet. Besides, from what was said in bk. 3, ch. 7, the mean distance of the Sun from the Earth is 7300 terrestrial semidiameters, that is [so many] Roman feet—

[Margin (beside the figure-exposition): “The Earth’s Annual motion, in the time of 4 Seconds, would be entirely sensibly Equal.” — “Exposition of the Figure.” — “1. Progress of the demonstration.”]

[Translator’s note — the engraved figure (fig. #36, top right) is the compound-motion radial fan: a sheaf of straight lines converging downward to the annual-orb center A; across the top runs the short arc of the Earth’s path (carrying the Earth-globe through C, P, D, Q, B from west to east), and just above it the tower-tops and successive fall-positions are marked — the tower positions E·X, L·Y, M·Z, G·O / K·I and the globe’s curved path H·R·S·T·O, with the point V lettered on one of the radii. It is the same construction used for the diurnal case (fig. #32, the equatorial fan, p. 404), here re-purposed for the combined diurnal-plus-annual motion.]

[…continues on p. 414 (PDF 449) with the catchword “Roma-” (Romanorum): “…of Roman feet”—the Sun-distance reduced to feet, continuing the triangle KAB computation of the Second Argument.]

(printed p. 414 — within Chapter XIX, Second Argument. The triangle computation is completed, showing the tower’s arc on the annual orb is sensibly circular, so a globe dropped from the tower would move sensibly equally, with no notable real velocity-increment, even on the annual motion. To remove any scruple about the diurnal motion’s addition, Riccioli then treats both motions together with an epicycle-on-eccentric figure, distinguishing three cases in which the diurnal motion adds to, subtracts from, or leaves unchanged the annual.)

[Header: BOOK IX. SECTION IV. — 414]

…of Roman feet, 188,851,000,000. With BG therefore joined (which is as great as BK), together with AB, the whole AG comes out 188,876,870,240 Roman feet—as great as are the several lines AH, AE, AL, AM. Now if, by the laws of Triangles, you use a Radius (or whole Sine) of 12 ciphers [10¹²], as it is in Pitiscus’s great Canon, and you make: as the sum of the sides AB, BK, to the difference of the same, so the Tangent of the half-sum of the angles adjacent to the base AK (which half-sum is of 30″) to another number—you will find the Tangent of the angle of 0° 0′ 29″ 56‴ 40⁗; which, added to the aforesaid half-sum (that is, to 30″), makes the angle AKB of 59″ 56‴ 40⁗; but subtracted from the same half-sum, leaves the angle BAK of 3‴ 20⁗.

[Margin: 2. Progress.]

These angles being acquired, if again, by the rules of Trigonometry, it be made: as the Sine of the angle AKB to the given opposite side AB, so the Sine of the angle ABK to the opposite side AK—the side AK will be found [to be] 188,876,870,238 7/10 Roman feet. Wherefore the line AK—by which, at the end of the globe’s descent, the tower-vertex is distant from A, the center of the Great Orb—is very nearly equal to the line AG, and to the rest, AM, AL, AE, AH, which are severally of 188,876,870,240 Roman or Geometric feet, nor differs from them save by 1 foot and 3/10. Which little difference is utterly contemptible with respect to the Radius AG, which is of more than 180,030,000,000 feet. Nay, even if AK differed from AG by the whole tower-height IK, that is, by 240 feet (which, however, is impossible, since the tower has described only the arc OI, which is of one Minute), still that difference could be despised. Therefore, if AK is sensibly equal to AG itself, much more are the rest of the straight lines drawn from A to the arc HK equal to the same lines produced to the arc HG; wherefore the arc HG may be taken, without any estimable difference, for the arc HK; and the arc HK [may be taken] as an arc perfectly circular, so far as concerns our purpose; and the tower’s situation at the end of the descent may be taken just as if it were at OG, or on the line AG—and much more, in the preceding time, as if it were on the lines AE, AL, AM. Moreover, in the time of 4 Seconds of an hour the Earth’s center (whose motion is as great as the mean motion of the Sun) traverses the arc CB of 9‴ 51⁗ 24⁗ᵛ; and the tower-vertex, from H to G, traverses very nearly as much; and of just as many scruples is HK, described from its own center.

[Margin: 3. Progress.]

Let AH now be bisected at V; and from the center V, with the interval VH, let there be described through the tower-vertex H the circular arc HO; for it will be of double the scruples that the arc HG is—namely, 19‴ 42⁗ 48⁗ᵛ; and any arc of it will be of double the [number of] minute-parts than any arc of the periphery HG, intercepted by the same straight lines drawn from the center A. And accordingly, as to absolute length, the whole arc HG will be equal to the whole arc HO; and the arc HE to the arc HR; and EL to RS; and LM to ST; and MG to TO. All which proportions are evident from the things demonstrated in the like figure at number 4 of this chapter. And I chose the tower’s situation opposite to the Sun in the nocturnal hemisphere, because in such a situation there is danger of a greater inequality in the increment of velocity, as I shall say at number 12.

[Margin: 4. Progress.]

[XI.] Lastly, I say that the globe spoken of above, dropped from H in the time of 4 Seconds, describes—to sense—the circular line HRSTO, equal to HG (or HK) itself; and accordingly, from what was just said, completes in equal times arcs as great as the point H traverses on the arc HG, that is, [arcs] sensibly equal. For since at the end of the four Seconds the globe appears severally on the perpendicular of the tower—that is, at EX, LY, MZ, GO—and (as to appearance, from what was proved at ch. 16, no. 12) the first space completed by the globe (namely ER) is as 1, the second LS as 4, the third MT as 9, and the fourth GO as 16; and since these very spaces, this proportion being preserved, lie between the circular line HG and HO (if that be designated with a semidiameter double-greater than this, as we supposed by construction), so long as the first arc HK does not exceed one Minute (as is clear from the things demonstrated a posteriori by the tables of Sines and Chords at number 3)—since, I say, these things are so, and the aforesaid globe describes only one line by its descent, it follows that it describes the circular line HRSTO, and therefore moves sensibly equally, without any notable real increment of velocity.

[XII.] But because some scruple could still remain in someone’s mind, on account of the appendix added to the annual motion by the Earth’s diurnal motion—and [because someone might] thence suspect some notable, and great, inequality in the motion of heavy bodies, which, though it would be imperceptible to Astronomical instruments, would yet absolutely be of many Miles or paces or feet, and that from it would arise that more vehement percussion and real increment of impetus, manifest from that very percussion—we shall show that not even in this way is that inequality notable, or so great that the more vehement percussion could be ascribed to it; nay, that sometimes it will be utterly none, sometimes even with a decrement, but for the most part contemptible and insufficient to uphold the experiments of the real increment of impetus of heavy bodies (which is [argued] from the more vehement percussion, the higher the place from which they descend).

[Margin: The second part of the Minor is proved.]

Now we shall use the figure we used in book 7, section 5, chapter 3, for the Stations and Retrogradations of the Planets. Since, from Galileo himself (Dialogue 4, On the System of the World, according to Copernicus), the Orb of the terrestrial globe is to the Great Orb as the Epicycle to the Eccentric; and the points of the terrestrial surface, and the bodies completing the diurnal conversion together with it, become daily quasi-twice stationary, and through almost the whole day are in a manner retrograde, but through the whole night quasi-direct, as to that addition of motion which the diurnal adds to the annual. Therefore, from E, as center, let there be understood described the arc LDK of the Great Orb, carrying the terrestrial globe ALCK, [whose] center [is] D, toward the Eastern region L; and let the point A, on the periphery of the terrestrial Equator (or a terrestrial body placed there), be moved toward L, completing the whole conversion ALCKA in 24 hours.

[Margin: 1st Case.]

For, firstly, it is manifest that around the point L (where the Sun, [seen] from E, rises, [the observer] being placed [there]) and K (where it sets), the diurnal motion adds or subtracts nothing to the annual motion of the center D; wherefore, if a globe descending—and, together with the earth, completing a particle of the diurnal conversion within 4 Seconds of an hour (that is, traversing one Minute of the Equator, ALCK)—be nearest to the morning place L, or the evening place K, its motion is as equal as the annual motion itself is, nor is it at all varied by the diurnal.

[Margin: 2nd Case.]

Secondly, since in the whole arc KAL the motion is composed of the annual [motion] of the center D and the diurnal (as promoting [the body] toward the same region as the center), the diurnal so adds to the mean annual motion that it adds more and more from K, through H, to A—and most of all at A—but less and less [as it adds] from A to L. On the contrary, because the diurnal motion in the whole arc LCK tends toward the contrary region, [it serves] to subtract from the annual motion—yet so that it subtracts more and more from L to C, but less from C to K. Hence it comes about that, although any terrestrial body always progresses in world-space, yet it really moves unequally (if we wish to keep Astronomical rigor): and it becomes faster than the mean motion of the center in the arc KAL, and fastest at A, the midnight point, but slower in the arc LCK, and slowest around C, the midday point; and by that reasoning [it serves] to imitate the Direction and Retrogradation of the Planets, and the inequality of the prosthaphaereses which the Epicycle introduces upon the Eccentric.

[Translator’s note: the print reads “the midnight point C,” repeating the phrase used of A; but C is the point of the epicycle nearest the orb-center E (the Sun), hence the midday point, diametrically opposite the midnight point A — a compositor’s slip. The physics is unambiguous: maximum addition (fastest) at the antisolar point A, maximum subtraction (slowest) at the subsolar point C.]

[Margin: 3rd Case.]

Thirdly, from what has just been said it follows that in the whole left arc ALC the inequality of the annual motion made by the diurnal is with a decrement, because the diurnal in the arc AL adds less and less continually to the annual, but from L to C subtracts more and more. On the contrary, in the right arc CKA, the inequality is with a continual increment of mo[tion]—

[Translator’s note — the engraved figure (fig. #37, lower right) is the epicycle-on-eccentric diagram borrowed from bk. 7, sect. 5, ch. 3 (planetary stations & retrogradations): a sheaf of straight lines converging downward to E (center of the Great Orb / the Sun); across the top the eccentric arc L·D·K (Great Orb, carrying the epicycle-center D eastward); and the small circle of the terrestrial globe A·L·C·K (epicycle) with A at top (antisolar = midnight, farthest from E), C at bottom (subsolar = midday, nearest E), L at left (sunrise) and K at right (sunset), plus the prosthaphaeresis lines G·V·T and the auxiliary points carried over from the planetary figure.]

[…continues on p. 415 (PDF 450) with the catchword “tûs” (motûs): “…of motion, in that from C to K the mean annual motion is diminished less and less, and from K to A increased more and more.”]

(printed p. 415 — within Chapter XIX, Second Argument. The third case of the diurnal modulation of the annual motion closes, leaving only the task of reckoning the tiny inequality in feet. The page lays the numerical groundwork — how far an equatorial point and the Earth’s center advance per second — with a full-width reference table of the Earth’s and Great Orb’s dimensions on three sizings, and begins computing the angles subtended by the four fall-arcs of the 240-foot, four-second drop.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 415]

…of motion, in that from C to K the mean annual motion is diminished less and less, and from K to A is increased more and more. And thus the two earlier parts of the proposition asserted at the beginning of number 12 are established—namely, that sometimes the diurnal [motion] brings no inequality to the annual motion, as at the points L and K; sometimes brings it in such a way that it is rather with a decrement of velocity, as in the arc ALC. It remains to prove the third part, concerning the exceedingly small inequality which the diurnal motion adds to the annual, even if it be regarded in paces or feet.

[XIII.] But before we show that, we must first know how many paces or feet a point placed on the terrestrial Equator is advanced, in each Second of an hour, by the force of the diurnal motion, and the Earth’s center by the force of the annual motion. Both of which depend on the paces and feet contained in the semidiameter of the earth, and in the semidiameter of the Great Orb; for, these being doubled, the diameter is had; and if it be made: as 100 to 314, so the diameter to another, the circumference is had; which, divided by 360, gives the paces or feet contained in one degree; and these, divided by 60, give the paces or feet of one Minute; and so, by continued division by sixties, the number of feet contained in one Second, Third, or Fourth becomes known. Wherefore, if it be known how many scruples of degrees the Earth’s center is advanced by the force of the annual motion in each Second of an hour, and how many [feet] likewise a point of the terrestrial Equator [is advanced] by the force of the diurnal motion, the number of feet that are completed by the force of each motion will at once be known.

It must therefore be supposed, from what was said in bk. 2, ch. 7, that the Earth’s semidiameter contains 5,174,000 old Roman paces, that is 25,870,000 feet; but the semidiameter of the Great Orb, from what was said about the mean distance of the Sun in bk. 3, ch. 7, [contains], on the Copernican [view] 1142, on Kepler’s 3381, on ours 7300 terrestrial semidiameters. Besides, from the tables of the Primum Mobile, any point of the Equator traverses, in one Second of an hour, fifteen Seconds [of arc] of the Equator; and from the tables of the mean motion of the Sun (according to what was said in bk. 3, ch. 17) the Earth’s center traverses, in each Second of an hour, 2‴ 27⁗ 51⁗ᵛ. These things being supposed, you will learn the rest from the following little tables, in which, as to terrestrial measures, we follow only our own opinion.

[Margin: the second part of the Minor is proved.]

[Translator’s note: in the miles column the dagger-cross ✠ of the source marks a value under one mile (here rendered “(under 1 mi)”). The last row is the Earth-center’s actual mean advance per Second of time — its 5965 Roman ft on the Copernican value matches the figure used at ch. 19, no. 5 (printed p. 411). Each paces-figure = miles × 1000, each feet-figure = paces × 5.]

We shall, however, apply these measures to recent Italian miles in another way, at chapter 22, from number 1; but we chose these here so as to use the Geometric foot (which is the old Roman one), just as we used it in observing the descent of heavy bodies.

[XIV.] Now, the figure of number 12 being resumed, let us suppose the plane of the terrestrial Equator (even though in reality it is inclined to the plane of the Great Orb) to be nevertheless in the same plane; for it will not be hard to judge of what befalls it when inclined to that [plane]. And since the clay globe spoken of above descends from a height of 240 feet in 4 Seconds of an hour, this time being divided into four, and the globe being supposed at G at the beginning of the motion, let there be, for the several Seconds, four equal arcs GV, VT, TI, IC—each of which, from what was said a little before, is of 15 Seconds [of arc], but of 1875⅓ feet, from the first part of the preceding table. And let the points of these divisions be connected with the centers D and E by the straight lines DG, GE; DV, VE; DT, TE; DI, IE. Now it is proposed to investigate the four angles at E. Which will not be hard: for in the triangle EDI the angle EDI is given, measuring the arc CI of 15 Seconds; and the Earth’s semidiameter DI [is] as 1; and the Great Orb’s semidiameter ED is, to me indeed, 7300 such [parts] as DI is 1, but on the Copernican [view] 1142; therefore the angle DEI will not lie hidden. Similarly, in the triangle EDT, there are given DT as 1, and DE as above, and the angle EDT of 30 Seconds; wherefore the angle DET will become known, from which, the [angle] DEI being subtracted, IET will be left; and in like manner the remaining angles at E will be found, since the angles EDV (45″) and EDG (60″, or of one Minute) are known.

[Margin: Sum of the Precept.]

Let it therefore be made: as the aggregate of the two given sides to the difference of the same, so the Tangent of the half-sum of the angles at the base [to another]—and there will come out the Tangent of the angle to be subtracted from the aforesaid half-sum, so that the sought angle at E be left. Or, if you are accustomed to the compendia of Logarithms, join into one sum the Residue of the Logarithm of the aggregate of the given sides, the Logarithm of the difference of the same sides, and the Mesologarithm of the half-sum of the angles at the base; for there will result the Mesologarithm of the angle to be subtracted from the aforesaid half-sum, so that the angle at E be left known. By which method we have detected the angles written below.

[…continues on p. 416 (PDF 451) with the catchword “Angu-” (Anguli): a table of the four angles at E, continuing the proof that the diurnal motion adds only a negligible inequality (in feet) to the annual motion.]

(printed p. 416 — within Chapter XIX. Three computation tables finish the Second Argument’s proof: in almost the whole diurnal circle the diurnal motion adds no sensible inequality to the annual, yet the real percussion-differences are large — so the increment cannot be saved as mere appearance. A sub-head extends the Second Argument to most heavy bodies even off the Equator; then the Third Argument opens against the diurnal motion alone, arguing from the Axiom that the real increment must be equal everywhere, which a rotating Earth would violate, beginning with the polar-drop figure.)

[Header: BOOK IX. SECTION IV. — 416]

[Margin: 1st Table.]

Now the arc CI, which is seen from E in a direct position, is of 1875¼ feet (as I said in the first part of the preceding table, no. 13); which [feet] the diurnal motion takes from the annual—for me, [the annual motion being] of feet [≈ 35,690] (from the last part of the table, no. 13)—when the point I traverses the arc IC. But the rest, IT, TV, VG, are seen more and more obliquely, and take away so many fewer feet from the annual motion as the angles they subtend are smaller. Wherefore, if it be made: as the angle 59 to 55, so 1875¼ feet to another, we shall have the very nearest feet which the diurnal subtracts from the annual in the arc TI; and so of the rest. Let there be, then, for these four arcs, another table, both in Copernican and in our measures—between which the Keplerian are intermediate.

[Margin: 2nd Table.]

Wherefore, if we subtract these arcs from the annual motion answering to one Second of an hour (which is, as I said, of 28,555¼ feet), we shall have the equated annual motion of terrestrial points, when they are not distant from the point C by more than one Minute—in which situation the differences, and the inequality, are greatest, just as also at the point A. But, the trigonometric calculation being extended from the point C up to an arc distant from the point C by six Minutes, I find the differences now to turn out so equal that they do not differ by one whole foot, as is clear from the following table, gathered from our measures.

[Translator’s note: the source prints the per-Second annual motion here as “28,555¼ feet,” but that conflicts with the value established just below (¶XV) and in the p. 415 table — namely 7 1/10 Roman miles ≈ 35,690 feet per Second (28 2/7 miles in 4″). The “28,555¼” is a compositor’s slip; the demonstration (the differences vanishing to under one foot) is unaffected.]

[Margin: 3rd Table.]

It is therefore manifest—as I said—that in almost the whole diurnal circle, the diurnal motion of 4 Seconds of an hour does not bring to the annual any sensible or notable inequality; since (except for the arcs of 6 Minutes before or after the points A and C) the difference of the additions or subtractions does not reach one whole foot, and yet the difference of the percussions is proportioned to the difference and inequality of the apparent increment, which is of many feet (for in the first Second the aforesaid globe completes 15 feet, in the second 60 feet, in the third 135 feet, in the fourth 240, as has often been said)—proportioned, I say, in its kind: so that the percussion is much greater from a height of 60 feet than from 15, and from 135 than from 60, and so on. Of which difference we gave a sufficient specimen at ch. 16, no. 23, in the 12th class of Experiments.

Application of the same Second Argument to very many Heavy bodies, even outside the Equator

[XV.] The Second Argument, resting on the clay globe of 8 ounces, holds for most heavy bodies of the same bulk which are heavier than it, and so descend from a height of 240 feet in a time no greater than 4 Seconds of an hour (per what was said at no. 6). It holds, besides, in very many—and nearly in all—places outside the Equator; because neither does the diurnal motion in so tiny a time carry the globe sensibly outside the same plane (as I showed at number 7), nor does the annual [motion], in that same tiny time of 4 Seconds, transfer the same globe into planes so diverse that, by reason of them, the aforesaid globe could acquire a real increment of velocity as notable as the percussion made by it evinces, when it is dropped from a higher place. For although the Earth’s center in so much time traverses (by our measures, set at the end of number 13, in the last table) 28 2/7 old Roman miles—completing in each Second 7 1/10 miles—yet the plane through which the globe began to descend, following the parallelism of the Equator, uniformly declines from its prior position.

III. Argument against the Diurnal Motion of the Earth

[XVI.] A heavy body, the same in species and in number [individual]—retaining the same gravity, bulk, density, figure, and other intrinsic conditions, and dropped from the same height through the same medium—descends naturally with an equal increment of real velocity, over whatever part of the Earth it be dropped. But if the Earth were moved by the Diurnal motion only, such a heavy body would not descend with an equal increment of real velocity, over whatever part of the Earth it be dropped. Therefore the Earth either does not move, or [does] not [move] by the Diurnal motion only.

[Margin: Proof of the Major.]

The Major is clear from the Axiom premised at the beginning of the chapter, and cannot be denied except by one who would now wish to defend the presumed hypothesis of the earth’s motion controverted in this place; for he has no other reason for denying this axiom, nor has he proved from more known [principles] that his hypothesis is really such, but only that it is possible, the Astronomical experiments being saved.

[Margin: Proof of the Minor.]

The Minor is proved by repeating figure 1, which we set out at ch. 17, no. 16. For in it, a globe dropped from the tower-vertex M, perpendicularly above either Pole of the terrestrial Equator, would in reality descend along the perpendicular line MP; nor by any other motion would it imitate the [diurnal] vertigo [of the earth, except by the vertigo of its own parts about the center of its own gravity, while the center itself, with the heavy body, would descend]—

[Translator’s note — the engraved figure (lower right) is the polar-drop “First figure” re-used from ch. 17 (originally fig. #31, p. 403): the Earth’s center carried along the Great-Orb arc A·…·C, with successive globe-circles (the polar region at successive instants), the vertical polar-axis drop-lines marked with the squares (M·1, M·4, M·9, M·P) and the points E, H, D, N·O, F, S, Z, G — showing that, on a stationary-center polar drop, the fall MP is straight and follows the squares-law 1 : 4 : 9 : 16.]

[…continues on p. 417 (PDF 452) with the catchword “deret” (descenderet): “…would descend straight down; for it would not be carried elsewhere by the force of the annual motion, which in this argument we do not suppose.”]

(printed p. 417 — Chapter XIX closes and Chapter XX opens. The Third Argument’s Minor finishes (a polar drop would be really unequal in appearance from an equatorial one on a rotating Earth), and a threefold Copernican response is rebutted. The Fourth Argument extends the same reasoning against the diurnal and annual motions together, and the Fifth argues from the perpendicular ascent of light bodies. Chapter XX then opens its six arguments from the rectilinear, perpendicular motion of heavy and light bodies against the Earth’s motion.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 417]

…[the heavy body] would descend straight down; for it would not be carried elsewhere by the force of the annual motion, which in this argument we do not suppose. Wherefore the increment of velocity and of the spaces completed in equal times would really be such as appears to those experimenting and observing—namely, according to the series of the square numbers beginning from unity: and in the first Second the space completed would be M1, as one; and at the end of the second M4, as four; and at the end of the third M9, as nine; and at the end of the fourth MP, as 16.

On the contrary, the same heavy body dropped in the Equator, by the force of the diurnal motion only, would move unequally indeed as to appearance, but equally as to the real space traversed in four Seconds of an hour, as we showed in Argument 1, no. 3 (which we also applied to other parallels of the Equator at number 7). Therefore, if the earth were moved by the diurnal motion only, the same heavy body—retaining the same intrinsic conditions, and dropped from the same height (say, of 240 feet) through air of the same rarity and tranquillity—would not descend with an equal increment of real velocity, over whatever place of the earth.

[Margin: Response, twofold, to the argument.]

[XVII.] The adversaries might perhaps respond: first, that air as rare as far from that region cannot be found under the world’s poles; then, that the earth attracts heavy bodies along a straight line, because the magnetic power pours itself out along the length of the terrestrial axis; lastly, that in heavy bodies there is a double power—one of moving themselves downward through the lighter medium (and this is exercised wherever they are dropped), the other of following the Earth’s diurnal motion (on the hypothesis that they are dropped above a place of the earth which is turned by the diurnal motion), and this latter is not exercised in act by the bodies’ center of gravity if they are dropped above the terrestrial poles, because the poles do not share in the diurnal vertigo; and therefore the rest is not equal, and on account of the first of these disparities the Minor can be denied, but on account of the other two the Major can be denied or distinguished—conceded, indeed, if everywhere [the bodies] are either drawn extrinsically by the earth or exert the same intrinsic motive power into second act, but otherwise denied.

[Margin: The response is rejected.]

But as to the first: the air under the world’s poles can never naturally be so thick as notably to retard the motion of descending heavy bodies, and to impede that real increment of the spaces traversed in equal time according to the proportion which the squares of the times have. Or, if you prefer, transfer this argument to a heavy body descending through water, which everywhere is of the same density, and so on. As to the magnetic power: already at chapter 18, no. 4, we rejected it from the fellowship of the causes of motion—unless it agree from experiments and a-posteriori reasons; nor is that reason to be obtruded which is not more known than the reason for the contradictory hypothesis: which is much to be noted. Here, however, we suppose that the other arguments for the Earth’s motion have already been solidly dissolved, as in truth they were dissolved.

IV. Argument against the Diurnal AND Annual motion of the Earth together

[XVIII.] This argument is exactly the same as Argument 3, if to the Minor you add the hypothesis of the annual motion too, and in the conclusion exclude the annual motion as well. For outside the world’s Poles, heavy bodies, by the force of both motions, would move without a real and notable increment of velocity, along a curved line whose nearly-equal parts would be traversed in equal times—or [a line] not so unequal as are the parts of the straight line by which they would move above the Earth’s poles (as is established from what was said for Argument 2). For here that portion of the way would cease which, outside the poles, would have to be traversed by the force of the diurnal motion, and only that would have to be traversed which is feigned to be run through by the force of the annual motion; for at the poles the diurnal [motion] is none.

V. Argument against the Diurnal AND Annual motion of the Earth, drawn from the motion of Light bodies

[XIX.] All the Arguments made hitherto hold, in their own way, for Light Bodies ascending upward by natural motion perpendicularly, as to appearance; for these too acquire a real increment of impetus, as has already been proved by proper experiments at ch. 16. But the argument would be more evident if we had some body so light that it did not share the nature of earth and water—as do glass spheres full of air, and artificial fires, and fiery meteors thrown upward, and smoke, vapors, little clouds, and the like, to which something of terrestrial or watery matter is always admixed. But if we had a merely aëreal or igneous body that ascended perpendicularly upward, in such a way that such ascent could be observed with the eyes, we would now be certain that the earth is not moved by the diurnal motion—at least, if beyond our [region of] air it still persevered in a perpendicular ascent. For a body of this kind ought not to follow the earth’s diurnal motion, or the annual, as not being akin to it; wherefore, if the earth were moved, it would surely be deserted by it in the aëreal space, nor could its ascent appear perpendicular to us, but would appear oblique toward the West. And, on the contrary, if it appeared perpendicular, it would be a sign that the whole Earth—and that part in which the observer was—remains unmoved. Yet by however much such an ascent has the greater probability in merely aëreal corpuscles, and in exhalations, by an a-priori reason (even if it cannot be noted by the eyes, except perhaps in some Comets), by so much is the immobility of the earth more probable than its mobility. But the rest of the arguments adduced above make, for us—to whom the experiments of the real increment of impetus are established—a Physical evidence of the Earth’s immobility.