[I.] Various opinions on this matter are related—chiefly by Simplicius (On the Heavens I, text 88, comment 86), Francesco Bonamico (On motion, bk. 4, ch. 37), the Conimbricenses (On the Heavens bk. 2, ch. 6, q. 1), Zabarella (On the motion of heavy and light bodies, bk. 1, ch. 16), Chiaramonti (On the Universe, bk. 12, ch. 30), Contarini (On the Elements, bk. 1), and Pererius (Physics 4, ch. 3). These we must here touch upon, that we may show there is not lacking, in the hypothesis of an immobile Earth, a way of adducing an a-priori cause for which Heavy and Light bodies move faster and faster toward the end—in the rendering of which there is now greater difficulty than there was before our experiments and Galileo’s. For previously it was unknown what the proportion was which they continually keep in such an increment, and so it sufficed to bring [forward any] cause of the greater and greater velocity; but now it is known—at least by those who have made experiments, or have wished to experiment accurately—that this increment (at least speaking of heavy bodies) is according to the progression of the odd numbers counted from unity; wherefore the reason for this proportion too must be rendered.
[Margin: 1st Opinion, of Hipparchus, on the cause of the increment of velocity.]
[II.] The First Opinion was that of Hipparchus, saying that, if heavy bodies are thrown upward by force, they begin their downward motion when gravity begins to prevail over the force impressed by the thrower; but, because something of that impressed impetus still endures (yet is more and more overcome by the innate gravity), therefore the descent is slower at the beginning and faster at the end. But if heavy bodies have not been thrown upward, but let down, still something of the force which detained them remains in them, which is consumed only gradually; and therefore the motion of heavy bodies is slower at the beginning, but faster toward the end. Which cause Alexander, in Simplicius, disproves thence: because heavy bodies are not rarely generated outside their place, where they were held back by nothing, and yet they are still faster toward the end of the motion. I add, moreover, that if Hipparchus’s reason held, it would come about that heavy bodies thrown downward would always move faster at the beginning than at the end, on account of the conjoined double downward-moving force—one from the intrinsic force, the other from the external thrower—of which the second force gradually grows faint: but this is not universally true, for sometimes the impetus by which they are sent down is so slack that the increment of velocity toward the end of the motion is far greater than at the start. Finally, he who does not simply lower [a body], but releases a heavy body downward, ceases altogether to hold it, and leaves no retentive quality in that heavy body.
[Margin: 2nd Opinion, attributed to Alexander.]
[III.] The Second Opinion, which some attribute to Alexander, supposed the cause to be that heavy bodies are less altered by contrary qualities the nearer they approach their place. But surely an iron ball descending through air is in no way altered by the air; and if it is altered, [it acquires] rather some
[…continues on p. 407 (PDF 442) with the catchword “ali-” (aliquam): “…some [moisture]“—the refutation of Alexander’s opinion continues.]
(printed p. 407 — within Chapter XVIII, the survey of proposed a-priori causes of free-fall acceleration on a stationary Earth, each refuted in turn: Alexander’s diminished contrary alteration, Themistius’s and Kepler’s attraction, the less-medium theory, air rushing in behind, air pushed in front, and finally the Aristotelian accumulating-impetus account — which would suffice for mere acceleration but naively yields the arithmetic (triangular) progression, whereas experiment proves the odd-number (squares) law, so the true cause is still sought.)
[Header: ON THE SYSTEM OF THE MOVED EARTH — 407]
…some [moisture], which grows toward the end; and much less is a glass ball altered in descending through water.
[Margin: 3rd Opinion, of Themistius and Kepler.]
[IV.] The Third Opinion was that of Themistius (Physics 8, text 76), saying [bodies] move faster at the end because they are allured by the kindred places to which they are borne; or [the opinion] of Kepler (in the Introduction to the Commentaries on Mars), saying they are attracted by the globe of earth and water by a certain magnetic force. But whether the place allures only objectively or effectively, certainly it would allure and attract a body kindred to it more vehemently when it is nearer than when it is farther—even if it had not been made nearer by motion from afar. And so, if one ball were dropped from a high window of 50 feet, then, when another ball of the same species, figure, and weight, dropped from a height of 100 feet, had reached that window, they would be attracted equally on account of the now-equal distance; and yet the one dropped from a height of 100 feet reaches the earth faster, and strikes it with a greater blow.
[Margin: 4th Opinion, of Iamblichus, Serenus, and Durandus.]
[V.] The Fourth Opinion, of Iamblichus and Serenus, was that this happens from the lesser resistance of the medium—because, namely, a smaller depth of air remains to be penetrated; or certainly, as Durandus says (on [the Sentences] bk. 2, dist. 14, q. 1), because the lower air, the nearer it is to the earth, the thicker it is, and therefore—being less buoyant upward—resists the downward motion less. But against these is the case just brought forward: of two heavy bodies equal in species, bulk, figure, and weight, of which one is dropped from a tower 100 feet high, and, when it reaches a window 50 feet high, the other is dropped; for then an equal depth of air remains to be overcome by each, and yet the one dropped from the hundred-foot height reaches the earth much faster. And against Durandus in particular the very thickness of the air militates—[as] less apt to yield, and to [permit] motion upward or sideways. And when a ball heavier than water is dropped through water of uniform thickness (as we have often done), it nevertheless descends faster at the end.
[Margin: 5th Opinion, of Contarini and Burley.]
[VI.] The Fifth Opinion is that of Contarini (On the Elements, bk. 1) and Burley (Physics 8, text 76), who place the cause in the air running up from behind to fill the void, and—more and more rarefied by the motion—therefore driving the heavy body downward with greater effort. But if half-burnt and still-smoking material be dropped downward, or a heavy body to which a piece of lighted candle has been fixed on top, the smoke or flame appears long erected upward; therefore the air does not press that heavy body from behind. So too, when light bodies tend upward through water, if you hang a thread below, it remains hanging downward, nor is it driven upward by the water. Besides, the lower air is not more rarefied by reason of the motion performed in the upper parts.
[Margin: 6th Opinion, of Zabarella, Pererius, Dandini.]
[VII.] The Sixth Opinion is that of Zabarella (On the motion of heavy and light bodies, bk. 1, ch. 16), Pererius (Physics bk. 4, ch. 3), and Dandini (On the Soul, bk. 2, in a digression on this motion): who, although they grant that the increment of velocity is helped by the air running up behind, nevertheless refer the chief cause to the air in front—already moved and dissipated toward the direction in which the heavy or light body is borne. Because, just as a movable is moved faster through air already moved in the same direction (and a ship advancing downstream faster than through unmoved air or wandering water), so it is moved much faster through air twice-moved, and faster still through air thrice-moved, etc. But against the air-behind, enough was said at number 6; while against the air-in-front there is the wind that impels that air crosswise, yet is unable to move from its perpendicular path any heavy body of great weight. Next, it is not numerically the same part of air that is moved again and again by the descending heavy body—since it runs to the sides and around the heavy body, to fill the place successively abandoned by it; wherefore the air is not rightly called twice-, thrice-, and four-times-moved. But universally, against those who recur to the motion of the air or water in front, [there] makes the priority of the heavy body’s own motion: for the underlying air or water is moved because it is impelled by the heavy body tending downward, and therefore the air or water is moved faster because that heavy body is moved faster; the increased velocity of the [body] is therefore presupposed prior to the air or water being moved faster—at least by a priority of nature and of cause. The commotion of the air or water cannot, therefore, be the cause of the greater velocity in the descent of heavy bodies. And the river-analogy does not hold here, because, antecedently to the motion of the ship, the motion of the river is presupposed. Finally, against this opinion is the slower descent of a lighter globe dropped together with a heavier.
[Margin: 7th Opinion, of Aristotle, Alexander, Aquinas, Albert of Saxony, the Conimbricenses, Cabeo, Chiaramonti, Bonamico.]
[VIII.] The Seventh Opinion was that of Aristotle and Alexander (in Simplicius, On the Heavens I, text 88—which Simplicius himself prefers to the others, on the supposition that such an increment be conceded, for he himself doubts it there); likewise of St. Thomas and Albert of Saxony (in Bonamico); of Bonamico himself (On motion, ch. 37); of the Conimbricenses (On the Heavens 2, ch. 6, q. 1); of Cabeo (Meteors 1, text 17, q. 3); and of Chiaramonti (On the Universe, bk. 12, ch. 30). These place the cause in a greater and greater gravitation—that is, in a greater accidental gravity, or in a second act of gravity, which by the motion itself becomes in second act more and more perfect, [and] which Aristotle called the “apposition of gravity, or of levity” (understand: in second act). For in On the Heavens I, text 88, he says that if the velocity of the Elements were increased to infinity, gravity or levity would be increased to infinity; and in the Mechanics, q. 19, he teaches that, if a heavy weight be placed upon an axe, the axe nevertheless often does not split that body which it does split if the axe alone (without other weight) is dropped from on high—because it takes to itself something of gravity. From which it appears that by the name “gravity” he does not understand gravity in first act (for, another weight being superimposed, that aggregate already had a greater gravity, and yet did not split [the body]), but understands an impetus, and gravity in second act, or a translative quality. Although Chiaramonti calls the second act of gravity itself “motion,” which—by means of gravity—he admits can produce another motion; but others understand an impetus distinct from motion formally taken. Since, therefore, in every moment in which a heavy or light body is outside its place and is hindered by no impediment or contrary mover, it produces some determinate degree of impetus suited to its gravity or levity, and there is no reason why the prior impetus should cease, it comes about that—new and new degrees of the same kind of impetus being superadded in the same part of the subject—the impetus becomes more and more intense, and therefore the motion faster and faster. But Bonamico, besides this chief cause, adds also a less and less resistance of the medium.
Which opinion would doubtless satisfy us, if the cause of the continual increment of velocity alone were to be assigned. For what is more agreeable to the nature of heavy or light bodies than that, in every instant in which they are away from their natural place, gravity or levity should produce such and so great an impetus as it produced in the first instant of motion, and that this [impetus] should remain so long as it is not destroyed by a contrary impetus, or the body does not reach its natural place? But from this it would follow that the proportion of this increment is according to a simple and natural Arithmetic progression. For let us suppose, for easier explanation, that the impetus produced in the first instant of motion is so much, in such a determinate heavy body, that by its force, in the first least [unit] of the equal times, that heavy body traverses a single palm; then, a second degree of impetus equal to the first being added, it ought, in the second of the equal times, to traverse only two palms, which composed with the first would make, at the end of the second time, 3 palms; and, a third [degree of] impetus being added, it ought, in the third of the equal times, to traverse separately 3 palms, which composed with the 3 preceding would make six palms, and so on. Wherefore the spaces completed in the individual equal times, taken separately, would stand according to the increment of the arithmetic progression 1, 2, 3, 4, 5, 6; from which would arise the spaces traversed at the end of the aggregated times, with this proportion: 1, 3, 6, 10, 15, 21. And that this increment is of this kind, Cabeo (in the place just reviewed) and Baliani (On the motion of heavy bodies, bk. 4, p. 110) judged more probable. But now, to Galileo, to us, and to anyone willing to experiment, it is certain that this increment is much greater—and indeed according to the progression of the odd numbers counted from unity—so that if, in the first of the equal times, a heavy body traverses 1 palm, in the second it traverses separately 3 palms, in the third 5, in the fourth 7, in the fifth 9, etc. From which the spaces aggregated at the end of the times arise according to the order of the square numbers: 1, 4, 9, 16, 25. The reason, therefore, for this increment is sought.
[…continues on p. 408 (PDF 443) with the catchword “Octa-” (Octava): “[IX.] The Eighth Opinion…”—the next opinion on the cause of the increment.]
(printed p. 408 — the end of Chapter XVIII and start of Chapter XIX, a turning point of the Section. An eighth opinion (self-replicating, decaying impetus) is rejected; Riccioli then gives his own ninth opinion in three formulations, whereby gravity and self-begetting impetus jointly yield the odd-number law, the ultimate a-priori cause being the will of God. The transition to the arguments FOR the Earth’s immobility follows, and Chapter XIX opens its five arguments from the velocity-increment, stating the Axiom that the increment is the same in every place of the Earth.)
[Header: BOOK IX. SECTION IV. — 408]
[Margin: 8th Opinion, of a recent Philosopher.]
[IX.] The Eighth Opinion, then, of a certain recent professor of Philosophy—who had been present at some of our experiments, and was by no means doubtful of the aforesaid proportion—was that gravity, in the first least [unit] of the duration of this motion, produces 1 degree of impetus, by whose force the heavy body crosses one determinate part of space; but in the second instant (or rather least [unit] of duration), gravity produces another degree, and the prior impetus produces another impetus like itself, and so there are three degrees of impetus—and therefore in the second of the equal times a space triple-greater is traversed than in the first time. But meanwhile the older impetus perishes, and so two degrees remain, which produce another two degrees; and gravity produces its always-customary degree; and so there are five degrees, from which follows a passage through a space quintuple-greater than the first space. But in the fourth moment, of these 5 the two oldest degrees of impetus perish, and the three remaining produce another three, to which gravity, adding its new degree of impetus, completes 7 degrees—whence the motion follows through a space septuple-greater than the first, and so on. But this opinion is little consonant with the natures of things: for why should the impetus perish, if it neither has a contrary nor [has] the movable reached its place? Or, since all the degrees of this impetus are of the same species, why should so many degrees perish in that instant in which, nevertheless, others—and indeed more—are to be produced in their place? For example, why, in the fourth instant, should two of the 5 degrees perish, and three remain, if in the same [instant] 4 degrees ought to be produced, which with the three remaining make 7? Is it not of less expense that only 2 be newly produced, which, added to the five prior, make 7? Surely yes.
[Margin: 9th Opinion, our own, threefold.]
[X.] The Ninth, therefore, and our own opinion—to which we adhere until something better occurs—is [this]. In every instant in which a heavy body is outside its place, and the hindrance is removed, gravity produces one degree of impetus; and that very impetus, which was produced by gravity, produces in the next instant another like itself—so that all the [degrees] produced remain and render the prior impetus more intense; but not all the degrees produce [degrees] like themselves at each and every moment, but once only, and immediately as they were produced. For if this happens, it follows that in the first instant of descent one degree is produced by gravity alone; but in the second instant (to speak thus for ease) both gravity produces its own and the impetus immediately before produced by gravity produces its own like degree of impetus, and so there are three; in the third instant, gravity produces its own, and the more recent impetus its own, which added to the prior make five degrees of impetus; in the fourth instant, gravity produces its own, and the newest impetus begets its own degree, which joined to the prior make seven degrees; and so on, two degrees always being produced—one by gravity, the other by the most recently produced impetus. Otherwise, if all the existing degrees each produced their own degrees in every instant, they would exceed their proximate radical principle, which is gravity.
[Margin: 2nd Part of our Opinion.]
If this does not please, let us say that gravity, in the first instant (or least [unit] of time), produces only half of that impetus which it can produce whole in the remaining instants—on account of the difficulty of beginning the motion, whatever that [difficulty] ultimately be. Or rather, let it be said that the heavy body, in the first instant, descends by the force of an impetus which, before the descent, it always conserved produced in itself—but hindered (by the sustaining body, say) from effecting a descent; and that, in descent, that heavy body produces for itself, in each instant, a degree of impetus double that first impetus which it produced while not descending. And accordingly, if in the first instant it descends, e.g., a palm’s space by the force of the first impetus, in the second instant it will descend three palms—one by the force of that same first, permanent impetus, and two by the force of the impetus produced in the first instant of descent; and in the third instant it will descend five palms, because now another impetus has been added effecting a descent through two palms. So, proceeding thus, the progression of the odd numbers counted from unity will be preserved.
[Margin: 3rd Part of our Opinion.]
But if anyone is unwilling that impetus stand to impetus as velocity to velocity, but [holds] that the merely-doubled impetus, by reason of its union with another impetus equal to itself, can have as much force as if it were itself doubled—and therefore that the two first impetuses effect a motion as 4, when the first made [a motion] as 1—then certainly three impetuses at the end of the third equal time, each behaving as if it were tripled, would together effect a motion as 9; four [impetuses] as 16; five as 25; and so on. If, however, someone should say that the a-priori cause was the will of the Author of Nature—who could have established another proportion of this increment for heavy and light bodies, but determined this one as both per se beautiful (on account of the order of the square numbers, which the increments of circles also keep, the diameters increasing equally) and apt to the intended end (namely, that these bodies should reach their place as quickly as possible, yet within certain limits of velocity); and that GOD willed to leave the traces of His providence in these effects, so that—not finding in nature itself a sufficient cause of them—we should necessarily acknowledge the First Cause—[such a one] would neither introduce a new method into Philosophy, nor without any necessity summon God from the machine. Surely he would philosophize far less absurdly than one who would have recourse to the motion of the Earth—as [being] unfit to render the cause of this increment, and involving with itself very many other entanglements of spiral motions, as we showed in chapter 17.
So far concerning the Arguments devised FOR the motion of the Earth. There follow the Arguments FOR the Immobility of the Earth.