[I.] For now twenty and more centuries it has flourished in the Academies of the Physicists—especially the Peripatetics, with Aristotle (bk. 1 On the heaven, ch. 88)—that principle and axiom, collected from the induction of experiments and of itself sufficiently known to sense: “Heavy bodies, descending by natural motion through a lighter medium, are continually moved more and more swiftly, the nearer they approach to the terminus toward which they tend”—or, the nearer they approach to the center of the Earth, whether this happens because this center is the center of gravity, or because it is the center of the Whole terrestrial element, to which (as a whole) all terrestrial bodies desire to be joined. Which increment of velocity, manifest to ocular attention itself, appeared much more evidently from the far more vehement sound, and from the much stronger percussion, which follow from some heavy and hard body if it be let fall from a higher than if from a lower place. And Aeschylus could have attested this, if he had survived—on whose bald cranium (mistaken for a rock) an Eagle, on high and in lofty flight, poised perpendicularly, let fall a tortoise previously seized in [its] talons, and, looking down on the broken [shell], suddenly flew down, that it might devour the flesh of the animal (like a kernel freed of its native rind); for they say Aeschylus died from that percussion—which would by no means have happened to him thence,

[…continues on p. 382 (PDF 417) with the catchword “si ex”: “…if [the tortoise had been let fall] from [a lower place]“—the rest of the discussion of falling bodies and the increment of their velocity.]

(printed p. 382 — within Chapter XVI (falling bodies). ¶II surveys five errors about descending bodies: that they fall uniformly, that the apparent acceleration is circular motion, that bodies of different weight fall in equal times (against which Riccioli inserts his own 1634 Ferrara tower experiment with Cabeo, where the heavier struck a little sooner), and that velocity is proportional to weight (Aristotle).)

[Header: BOOK IX. SECTION IV. — 382]

[which would by no means have happened to him] if that body had been let fall from near at hand. And so the tortoise of the brute animal, when it was struck back by the percussion, so broke the tortoise-shell of the rational soul (containing a not-ignoble brain), that, broken, it perished in the wound of another about to perish soon; and while it made the corpse of Aeschylus, in that field, as far as in it lay, exposed to the Crows, it [the tortoise]—the more ignoble food, and the slowest—yielded as nourishment to the nobler and swiftest bird. This increment of impetus, therefore, inseparably connected with the increment of velocity, and manifest to the three chief senses (sight, hearing, and touch)—nay, even to the birds themselves—there were not lacking those who dared to call it into doubt, or even utterly to deny it.

[Margin: Five errors in the descent of heavy bodies. — 1st Error. — Of Simplicius. — Of Arriaga. — Of Cabeo.]

[II.] But I have found five diverse errors in this matter. For, first, some deny that heavy bodies descending naturally are moved uniformly-difformly [uniformly accelerated], or with an unequal but greater and greater velocity toward the end, and dare to affirm that they are moved equally [uniformly]—not only [as to] the real motion considered, but also [as to] that [motion] which appears to us to happen along a straight perpendicular line: among these is Simplicius (on bk. 1 On the heaven, text 88, comment 86, toward the end), where he calls this into doubt; but manifestly in this class is our [Father] Rodrigo Arriaga (disput. 4 On Generation, sect. 5, subsection 3), affirming that the same movable—whether it be heavy and [move] downward, or be light and move upward—is always moved with an equal velocity. But Nicolaus Cabeo also (bk. 1 of the Meteors, text 17, at the end of question 3) professes himself doubtful concerning this matter, when he said of a heavy [body] descending: “Neither is this very thing established to me, whether the velocity increases uniformly-difformly in equal parts of time and of space, and consequently whether it can make the triangle of velocity: therefore I can establish nothing certain”—but he alludes to the triangle of Galileo, of which below.

[Margin: 2nd Error. — Of Galileo. — And of the revived Philolaus, or Bullialdus.]

Secondly, Galileo concedes the apparent inequality in the descent of heavy bodies along the perpendicular line (dialogue 2 On the twofold System of the World, Latin p. 120, Italian p. 159), but says that this motion, on account of the annual motion of the Earth, really happens along a circular line, and so that in equal times equal arcs are traversed. The author of [the revived] Philolaus, namely Bullialdus (for he, bk. 1, ch. 7, of the Philolaic Astronomy, professes himself the author of that book), in book 1 On the true System of the World, ch. 4, teaches that this motion—which appears to us [to be] in a straight line and irregular—really happens, or is composed, from two circular motions, of which each is equal and uniform (granted that one of them is faster than the other); all which [things] we shall declare in the following chapter, with figures set alongside.

[Margin: 3rd Error. — Of Galileo. — Of Baliani.]

Thirdly, some concede indeed that the aforesaid motion is uniformly-difform—at least as it appears to happen in a straight line, if any heavy [body] be considered separately; but if two heavy bodies of diverse weight be let fall from the same altitude, equally distant from the ground or pavement, they deny that the heavier is moved faster and reaches the pavement sooner than the lighter. So Galileo (whose person Salviati bears, dialogue 2 On the two Systems of the World, Latin p. 164, Italian p. 218): when he wished to investigate the time in which an iron globe would reach the earth from the concave of the Moon, and Sagredo had said, “Let us take, then, a globe of determinate weight—and indeed that very one whose time of descent from the Moon we wish to measure,” Salviati responds: “But that matters nothing; for globes which weigh one, which ten, which a hundred, nay which a thousand pounds, will measure those same hundred cubits in altogether the same time”; and he adds that he has certain and firm experiments of this thing, to be brought forth in his treatise or dialogues on motion in due time. Since, therefore, he wishes the calculation which he soon institutes from the experiment of the altitude of a hundred cubits to be valid for investigating the time in which an iron globe would descend from the Moon to the earth, he surely judges that, if from the Moon’s very concave two globes were let fall—one of one [pound], the other of a thousand or however many pounds—both would measure out that space in the same time, and would strike the Earth at the same moment, at least if they be of the same material or the same kind.

The same thing, at length, affirms Giovanni Battista Baliani, a Genoese Patrician, in his brief but most pith-full treatise on the natural motion of heavy bodies, solid and liquid, book 1—where he says that, when he was Prefect of the citadel of Savona, he detected, by repeated experiments, that “from two globes of war-engines [cannonballs]—whether both of iron, or the one of stone, the other of lead—let fall at altogether the same moment of time through a space of fifty feet (even if one were of only one pound, the other of fifty)—[both] strike in an indivisible moment of time, so that only one blow of both was perceived by sense.” He then adds that he used two bodies of nearly the same bulk, one of wax, the other of lead, and: “I experienced,” he says, “in the waxen [one] some longer delay in the descent—yet far below the proportion of the weights; for that waxen globe, in the given distance of fifty feet of descent, was distant about one foot from the ground when the leaden [one] was touching the plane beneath, the intervening air (if I am not mistaken) sensibly resisting and impeding the motion.” From these and other experiments, he confesses that he descended into this opinion: that perhaps gravity behaves as the agent, and the material body as the patient, and therefore that heavy bodies are moved according to the proportion of gravity to matter; and where they are borne naturally by a perpendicular motion without impediment, [they] are moved equally [among themselves], because where there is more gravity, there is equally more matter or material quantity; but if some resistance accede, the motion is regulated according to the excess of the agent’s power over the resistance of the patient, or the impediments of the motions.

[Margin: Of Cabeo.]

Nicolaus Cabeo agrees with Baliani—from [the time] when he knew him at Genoa, while he was preaching there—nor could he ever be torn by me from that opinion; and accordingly (in bk. 1 Meteor., text 17, q. 5 and 6) he most assertively affirms, from his own and oft-repeated experiments, that if two globes be let fall together from the same altitude—one of one ounce, the other of ten pounds, or of any greater weight, whether both be of lead, or one be of lead, the other either of stone or of wood—provided the air be tranquil, and that which is lighter be not of such small weight that, unable to overcome the resistance of the air or breeze, it fluctuates in the air (of which kind is a feather or paper)—both will reach the earth at the same moment, and no sensible difference can be noted in the fall; from which he infers that the velocity of all falling [bodies] is in itself equal.

But I do not know from how great an altitude he let fall those globes. This, however, I certainly remember: when we were together at Ferrara in the Year 1634, and we let fall, from the tower of our church of the Society of Jesus, stones of diverse weight, and also wooden globes—I having placed under one [spot] a brazen basin, under the other a wooden board, that from the diversity of the sound I might better distinguish which reached the ground sooner—I noticed that the [body] which was heavier reached [it] a little sooner. But because that difference was small (for that place of the tower from which it was let fall did not exceed 80 feet), therefore he [Cabeo] could never be brought to admit any inequality, or difference in the fall.

[Margin: Of Arriaga.]

At length, also, Arriaga (disput. 4 On Generation, sect. 5, subsect. 3), and those subscribing to him—Bartolomeo Mastrio and Bonaventura Belluto (disput. 3 On the heaven, q. 3, from no. 52 to 57)—assert that any two heavy bodies whatever, whether of the same or of diverse kind, or bulk, or figure, however much differing in gravity, descend in the same time from the same altitude, and slip down with an equal velocity in themselves. But the foundation of so great an assertion is Arriaga’s experiment, by which he says that he often let fall together, from the same table, a dry crust of bread two fingers [thick], and the pen with which he was writing, and a huge rock which he could scarcely sustain with his hands, and noticed that these struck the pavement together at the same moment, and were moved equally swiftly; whence he exclaims and complains that, in a matter so easy, the experience of no philosophers ever proved this descent [to be unequal], but almost all remained in the faith of [their] forefathers.

[Margin: 4th Error. — Of Aristotle.]

Fourthly, others asserted that the heavier [bodies] descend sooner and are moved more swiftly, but affirmed too great a difference in the descent and velocity of two [bodies] not equally heavy; for they said that velocity is to velocity as gravity is to gravity—so much so that if of two globes one be of one pound, the other of ten pounds, and each be let fall together from the same height, the one which is of ten pounds reaches the earth ten times sooner than the other, or in the same time measures out ten times the space. So Aristotle taught; for not only did he teach (bk. 2 On the heaven, text 46), “That which is greater is borne more swiftly,” and (text 77, 79), “A small particle of earth, if raised and let fall, is borne downward, and the greater it is, the more swiftly it is moved”;

[…continues on p. 383 (PDF 418) with the catchword “[mo]ueri”: “…the more swiftly it is moved”—the rest of the fourth error, then the fifth, and the first class of experiments.]

(printed p. 383 — within Chapter XVI. The survey of errors closes with the 5th — the disputed law of acceleration, opposing Galileo’s odd-number/squares-of-times law to Cabeo’s and Baliani’s arithmetic progression and Chiaramonti’s decreasing increment — and Riccioli announces he will settle the matter by experiment. The First Class of Experiments then opens, on levity as distinct from gravity: glass-tube experiments with a rising air-bubble and oil drops, proving levity a real positive upward force, not the mere privation of gravity.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 383]

[the greater it is, the more swiftly it is] moved; but besides (bk. 3 On the heaven, text 27): as a body of the same kind is [related to itself], or as gravity is to gravity in two bodies of the same kind, so is velocity to velocity, and space to space traversed in equal time. To which afterward many of the Peripatetics assented.

[Margin: 5th Error. — Of Cabeo. — Of Baliani. — Of Chiaramonti.]

Fifthly, finally, some erred from the truth in establishing the increment of the velocity of heavy bodies and its proportion—but in diverse ways. For when Galileo (dialogue 2 On the two systems of the world, Latin p. 163, Italian p. 217) had asserted, from his experiments, that this increment happens according to the odd numbers counted from unity—or, what comes to the same, that the spaces traversed by a heavy [body] in natural descent are to one another as the squares of the times, or have to each other the doubled proportion of that which the times have to each other in which that transit is accomplished; so that if a heavy [body] in the first scruple [instant] of time completes one palm, in two scruples it completes 4, and in three 9, and in four 16, etc.; wherefore, the scruples taken separately, to the first [there corresponds] 1 palm, to the second 3, to the third 5, to the fourth 7 palms—this, I say, when Galileo had asserted from his [experiments], and Gassendi had subscribed to it (epistle 2 On motion impressed [and] transferred by a mover)—Cabeo, nevertheless (bk. 1 Meteor., text 17, q. 3), not only calls it into doubt, but says that more probably this increment (if indeed it be admitted that heavy bodies are moved with unequal velocity and uniformly-difformly) happens by an Arithmetic progression: so that if in the first scruple of time the velocity was as 1, in the second it is 2, in the third 3, etc., according to the Arithmetic progression—which also Baliani thought much more true (bk. 4 On the motion of Heavy bodies, p. 110). But Scipio Chiaramonti (bk. 12 On the Universe, ch. 28), in order to show that the increment of this velocity does not grow to infinity, but at length so decreases that the velocity turns out equal [uniform], says that he divided the line AB (of about 1½ perches) into three equal parts—namely into AC, CD, DB—and used, for measuring the time, both a pulse-clock [pendulum] and water gently flowing from the little siphon of a vessel (and always of the same weight); and [says] that the weight reached, from A to B, in the time of 514 particles; but from C to B, in the time of 451; and from D to B, in the time of 320.

[Translator’s note — a small engraved diagram accompanies this passage: a vertical line from A (top) down to B (bottom), with marks at C and D dividing it into three equal segments AC, CD, DB.]

When, therefore (says Chiaramonti), the time from C to D is equal to the time from D to B (understand: if it be let fall now from C, now from D)—if from the time CB (451) you subtract DB (320), there remain 131 parts [for CD]. Again, because from A to D the whole time is as much as from C to B—that is, 451 parts—and AB is 514, if you subtract AD from AB, there is left DB, [whose] time [is] 63. Finally, because the time AC is equal to the time DB—if the weight be let fall from A to C, or from D to B—and AC (or DB) is 320: therefore, for traversing the equal parts, these times are required, in order: for the 1st [part], AC, a time of 320; for the 2nd, CD, of 131; for the 3rd, DB, a time of 63. But the proportion of 320 to 131 is greater than [that] of 131 to 63; therefore the proportion of the increment of velocity decreases in the progress of the descent.

Thus far concerning the—I do not say opinions, but errors—of others; because the opposite of them, to us and to others who were present as witnesses to our observations (soon to be reported), is now not only probable, but evident and certain. And therefore we shall proceed—not from likely conjectures, but from infallible Physico-Mathematical knowledge—from the following experiments to certain conclusions.

The First Class of Experiments, for Lightness as distinct from Gravity

[Margin: 1st Experiment, of air perforating water so that you make it tend upward.]

[III.] A glass fistula, or tube, AB (as we sketch it in the following first figure), at least three feet long and as straight as possible, and closed at one end A by continuous glass—fill almost entirely with water, leaving only a small space AM, as wide as the breadth of the smallest, or “ear,” finger; then close the mouth of the fistula with a wooden stopper, but smear the air-holes round about with bitumen, or with a wax-salve composed of virgin wax, pitch, and powder of Armenian bole, or of red rock-flower, so that no more air can enter—nor can the water go out, even drop by drop. Which being done, turn downward that part of the fistula whose top AM the enclosed air has now occupied, erecting the fistula perpendicular to the horizon; for you will see the air—which, in B, the bottom of the fistula, had had the form of a cylinder occupying the whole cavity of the fistula breadthwise—soon ascend, and so ascend that, contracting itself, it extends lengthwise; and [you will see] the upper surface of that little cylinder—which was flat, in the manner of a little disk—now made conoid, as you discern in C; namely, that it may more easily pervade the water, and as it were perforate it by that figure, and yield place all around (by its contraction) to that [water] tending downward. You will notice, besides, that this obtuse cone—or rather aerial pyramid—in [its] ascent is more pointed and narrowed, and accordingly that more of the descending water passes between its convexity and the cavity of the glass than before; and at length, when it has flown up to the top of the water, that it restores itself to its pristine cylindrical figure, if the cavity of the fistula be such.

These being laid down (which are most true, and I have often shown to many), it follows—philosophizing, however, sincerely and prudently—that although the ascent of the air and the yielding of the water begin at the same moment, yet by nature and causality the air is moved upward prior to the water’s flowing to fill the place which is deserted by the air; since it is the air which, by swelling, enters upward into the water and compels [it] to yield to itself, as if a wedge were driven into its middle. Otherwise—if the air tended upward [only] because it is thrust into the upper [parts] by the water tending downward—the water would rather pervade the air wedge-wise (just as water falling outside the fistula bursts through the air beneath, but does not, surrounding it, receive it within itself), and this the more easily, the less the air, by its rarity, is fit to resist the impeller; or at least [the water] would blunt the swelling above (by which [the air] swells upward), and, compressing it below as if with pincers, would constrict its lowest part to a conoid figure. Now if the Air is prior in this motion, and behaves actively, it surely does this by some motive force upward; and this [force] Philosophers and non-Philosophers commonly call levity. There is given, therefore, a levity distinct from gravity; nor does levity consist in the privation of greater gravity.

[Margin: 2nd Experiment.]

[IV.] But take another experiment, perhaps more evident to you—or experience it yourself. Fill the aforesaid fistula, or another DE (in the 2nd figure), almost entirely with water, but pour upon it three or four drops of oil, leaving still a very small space for a little portion of air DL; then close it well, as above, and—turning the part D downward—erect it perpendicular to the horizon; for you will see the air [borne] most swiftly, the oil most slowly, upward—so much so that the air which was in E, penetrating the oil F, is now in H, when the oil is still a little above F; and when the air has obtained the top D of the fistula, the oil will not yet have drained off half the way. Therefore, the oil struggling upward around the middle of the fistula, turn again the lower part of the fistula upward; for at once the air H will fly upward, pervading the oil G wedge-wise, and will fly across into K. But sometimes it so pervades upward through the perforated oil that, nevertheless, a little portion of air, formed into a globule I, adheres to the oil above; and then the oil G will tend upward far more swiftly than it otherwise would have done in the same position—because it enjoys a double motion: one from [its] intrinsic levity, the other of traction, by which it is drawn by the air I. It is a sign, therefore, that the air is borne upward by its own levity; otherwise the water, by pressing the oil G, would more easily compress, or scatter to the sides, the air I. Nor can a physical and solid reason be rendered why that thing conjoined from oil and air should be moved more swiftly than the oil alone. For it cannot be said that the air is thrust upward by the oil G descending (as the less heavy by the heavier), since the eye evidently sees the oil G continually borne upward, and not slip down interruptedly so as to drive the air [up]; nor [can it be said] that the water, by descending faster, also expels the oil upward faster—since the bulk of water above the oil is not greater than before, and would most easily compress the airy bubble I; nor, finally, can it be said that the [thing] conjoined from G [and] I is something lighter than an equal [amount] of water in the same bulk, and that it therefore exceeds water in gravity more than the oil taken separately [does], and so wishes to occupy its place sooner; for if levity is not granted, and the little portion of air I has something of gravity, then rather from it and the oil a heavier body has been made than the oil alone would be.

[…continues on p. 384 (PDF 419) with the catchword “De-”: ¶V begins—with the figures of the fistulae and further experiments on lightness and gravity.]

(printed p. 384 — within Chapter XVI. ¶V gives the third levity experiment (a submerged wooden cylinder rises with positive upward force). The Second Class of Experiments then opens, proving the accelerated descent of heavy bodies in air by sound (louder impacts from greater heights), by touch (the harder blow of a ball dropped from higher), and by sight (¶VII); the page carries the engraved figures of the levity-apparatus.)

[Header: BOOK IX. SECTION IV. — 384]

[Translator’s note — three engraved figures accompany this and the preceding paragraphs: Fig. 1, the glass fistula with water (closed top A, air-space M, air-bubble C); Fig. 2, the fistula with oil and water (top D, oil-band L, air-bubbles K, I, O, G); Fig. 3, the cylindrical vessel of ¶V (the floating wooden cylinder, a hand S holding the rod, the vessel ZNVO).]

[V.] Finally, let a cylindrical vessel ONPQ be made, whose bottom OP is parallel to the horizon, and on the bottom set a wooden cylinder TOPX—but such that it floats in water; [press that wood down with a rod SR, so that the whole of it is immersed in the water; and when, the hand releasing it, it is let go, you will see the wood rise swiftly toward the top, occupying the place of the water—which yields to it]. But what will the adversaries say if in the bottom of the vessel there were a wide hole Y (narrower, however, than the cylinder, and closed) which were opened at the same moment in which the hand raises the rod RS? Certainly the water would flow out downward, and yet that cylindrical wood would tend upward. Let them acknowledge, therefore, in that wood a levity—some [force] which, by producing impetus upward, by nature moves and drives the water prior [to its descent], and is the cause why the water (a fluid body) so yields to it that it [the water] enters into its place, lest a vacuum be given; and that [the water] does not exercise gravity in act, but the upper parts of the water are impelled by the wooden cylinder, and yield place to it by departing to the sides, that they may fill the place of those parts which from below enter into the place of the cylinder.

The Second Class of Experiments, for the Unequal motion of Heavy bodies descending in Air more and more swiftly, the nearer they approach to the terminus toward which they tend

[Margin: 1st Experiment, from sound and hearing.]

[VI.] The First Experiment is taken from sound. For let fall, from a height of 10 feet, a globe—of wood, or of bone, or of metal—into a basin placed beneath, and attend to the ring [tinnitus] arising from the percussion. Then let that same globe fall from a height of 20 feet, or two perches; for you will feel [hear] a far greater and more widely diffused sound. Afterward raise the basin to a height of 10 feet, and into it let the same globe fall from a height of 10 feet; for you will feel a ring similar to the first. Therefore that globe, in the second case, acquired a greater impetus from the fall from the greater altitude than from the lesser in the first and third cases; and in the second half of the downward journey it acquired more impetus than in the first half; and accordingly it was moved downward with unequal velocity—and so unequal that it descended faster, in the second case, through the 10 latter than through the 10 prior feet. Since by perpetual experiments it is manifest that a greater impetus is followed by a swifter motion of the movable. But if you wish a more domestic experiment: pour water into a cup from a vessel distant two or three fingers from the cup—you will feel no noise; raise the vessel to two or three feet, and you will feel a noise from the falling water. Hence those who dwelt at the cataracts, or waterfalls, of the Nile are said by Cicero (in the Dream of Scipio) to have become deaf, on account of the crash of the waters rushing down precipitously from on high.

[Margin: 2nd Experiment, estimable by touch itself.]

The Second [experiment] is taken from percussion, perceptible by touch itself. Place your hand under a tennis-ball while it is let fall by another from a height of 10 feet—you will experience a very light percussion; but if it be let fall from a height of 50 feet or more, you will feel your hand struck not without some pain. It conceived, therefore, a greater impetus from the higher fall. Which greater impetus, as I was saying, the wretched Aeschylus felt from the tortoise cast down from on high by the Eagle onto his head—and the brute bird itself foresaw [it], doubting nothing of the future success. Elpenor felt [it], fallen from a tower; whence that Ovidian comparison (bk. 1 of the Tristia):

“He who falls on the level (yet this scarcely happens itself) / so falls that, the ground being touched, he can rise again. / But the wretched Elpenor, slipped from a high roof, / met his King, a lamentable shade.”

And hence that adage, and most well-known warning of the Poet:

”…They are raised on high / that they may rush down with a heavier fall…”

Finally, do not those who run down from some slope conceive so much impetus that, however much they wish, they cannot afterward check the course which at the beginning they could most easily have checked?

[Margin: 3rd Experiment, from percussion, estimable by the eyes.]

[VII.] The Third Experiment is taken from the greater percussion of [bodies] falling from on high, but estimable by the eyes. For a clay globe, which—let fall from a small height—is not itself broken, or cannot break an egg-shell or a nut-shell placed perpendicularly beneath it, or raise a weight placed on a wooden balance, or penetrate a palm’s depth of water—if it be let fall from a higher place, at length accomplishes all those: for it breaks, and is broken, and raises that weight. So a wooden globe, or a tennis-ball, falling from a low height into a cistern, or a great vessel full of water, is submerged a few fingers below the water; but if it fall from a very lofty place, it penetrates many feet below the water, and sometimes even to the bottom. And by other innumerable experiments of this kind it becomes evident that a heavy body, naturally falling from a higher place, has acquired more and more impetus at the end of [its] motion.

[…continues on p. 385 (PDF 420) with the catchword “Quar-”: ¶—the fourth experiment (the rebound of a tennis-ball), then the great measured free-fall experiment from the tower of the Asinelli.]

(printed p. 385 — within Chapter XVI, the climax of the free-fall experiments. After a rebound experiment, ¶VIII describes the great measured trials at Bologna (1640): clay balls dropped from the city’s towers, above all the Asinelli, timed by a tiny pendulum calibrated against stellar transits, with Grimaldi and Cassiani assisting. ¶X gives the measured fall-distances — 10, 40, 90, 160, 250 ft in 5, 10, 15, 20, 25 vibrations — confirming that distances grow as the squares of the times. Includes the engraved Asinelli-tower figure.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 385]

[Margin: 4th Experiment, from the rebound of a tennis-ball.]

[The Fourth experiment] is taken from the rebound, or recoil, of a tennis-ball into the air. For a leathery ball of this kind, very hard and no bigger than an egg-yolk, we ordered to be let fall (so that it might strike the ground at a more acute angle) from a height of 37 feet onto a stone-paved floor, and it rebounded up to 7½ feet; but when it was let fall from a height of 73 feet, it rebounded to 11¼ feet. Another tennis-ball, however—larger, and striking the floor at a more obtuse angle—let fall from a height of 37 feet, rebounded to nearly 6 feet; [and] let fall from a height of 73 feet, rebounded to 7½ feet. But I seem to myself to be playing, as long as I do not proceed to nobler experiments, and [ones] most evidently demonstrating not only the inequality in the motion of heavy bodies, but also the increment of their velocity, uniformly-difformly augmented toward the end of the motion.

[Margin: 5th Experiment. — A Pendulum applied to the most subtle measure of times.]

[VIII.] The Fifth Experiment, therefore—taken by us very often—was the measurement of the space which some heavy [body] completes by natural descent in equal times. I had indeed attempted this at Ferrara, with Fr. Cabeo, in the Year 1634; but [it was] from a tower of our church not exceeding a hundred feet, and with a Pendulum whose exact times [in relation to] the Prime Mobile I did not yet know. But when I was at Bologna in the Year 1640, and had now examined Pendulums of diverse length by the transit of the Fixed [stars] through the middle of the heaven, I selected, for this experiment, that smallest [pendulum]—which, from the apex of [its] motion to the center of the little ball, has a length of one ounce [inch] and yet fifteen hundredths of an ounce-particle of the ancient Roman foot, and which (as I showed and set forth in bk. 2, ch. 21) is, by a single simple vibration, equated to 10‴ [thirds]; by the thirds [third-scruples] of the Prime Mobile, and by six vibrations, it most exactly equals one Second [scruple]; and so one of its simple vibrations is most nearly equal to that time which Musicians are wont to designate by the note of a semichroma [semiquaver], if the raising and depressing of the hand of the choir-master, or conductor moderating the voices, be done in the ordinary manner.

But because the oscillations, or vibrations, of so short a Pendulum are most swift and most frequent, and I judged that not even the error of one was to be admitted (lest any confusion and fallacy in numbering thrust itself upon the eye), after each decade of vibrations—marked by the raised finger of the hand, now closed—we were wont to begin the numeration again from unity, and (the nomenclature of the numbers which are below a decade being contracted) most expeditiously to pronounce one after another; of which kind are these Italian words, but as here at Bologna they are commonly pronounced more briefly: “Vn, dù, tri, quatr, cinq, sei, sett, ott, nou, dies” [one, two, three, four, five, six, seven, eight, nine, ten]. To which if you subjoin the musical semichromata (as I said), and follow the ordinary measure of musical beating, you will have a specimen most near to the times which one simple vibration of this our Pendulum measures.

To this numeration, moreover, we accustomed others also—especially the Fathers Francesco Maria Grimaldi and Giorgio Cassiani—whom I used very much in the experiment soon to be set forth. And truly it is marvelous to tell that, although those [Fathers] and I used two such Pendulums (they indeed stationed on the cornice of the Tower of the Asinelli, I on the pavement of the platform beneath), and although each of us had separately marked his own vibrations on a sheet—[those vibrations] by which the heavy [body] descended thence to the pavement—yet, with repeated experiments, there was never a difference between us of one whole little-vibration. Which I know will scarcely be believed by some; and yet I testify that it was most truly so, and the aforesaid Fathers of our Society will always attest [it]. Thus far concerning the Pendulum and the measure of times.

[Margin: The height of the Bolognese towers.]

After these [preparations] we prepared a very large basket full of clay globes, each of which weighed eight ounces; and for the shorter intervals we used the windows of our College, but for the higher [intervals] diverse towers—not indeed whole [towers], but the windows or little windows of them: namely, the tower of St. Francis, which is 150 Roman feet high; and of St. James [S. Giacomo], which [is] 189 feet; and of St. Petronius, which [is] 200 feet; and of St. Peter, which [is] 208 feet; but especially and more frequently the tower of the Asinelli, which is 312 feet high in all, but from the cornice to the base, or platform, only 280 feet. Which, because it is most convenient for experiments of this kind—just as if it had been constructed for this end—it pleases [me] to set before the eyes [in a figure].

[Margin: Description of the Tower of the Asinelli.]

[IX.] Let, in the following diagram, the thicker trunk of the Tower be IBCD, upon a nearly cubic base VYZX (much thicker than the trunk of the tower [is]), from which base projects the platform YZH—fenced about with broader stone railings, so that at least six men at once can safely walk around the tower along it, in the same row, between the railings and the tower. But above projects the cornice BC, fenced about with beaked stone railings, so that from the railings G, Φ, O—as from windows—any man of ordinary stature can safely look down, and, by a plumb-line let down thence to the pavement of the platform ID, measure (as we did more than once) the altitude GI, which (as I said) we found [to be] 280 ancient Roman feet. The rest [of the height] we measured, both with measuring-rods and also the whole [height] by Altimetry, with Fr. Grimaldi.

[Margin: The convenience of the Asinelli Tower for these experiments.]

This Tower, therefore, has the greatest conveniences for that business; for the globes let fall from the windows G, Φ, O fall perpendicularly onto the pavement ID, neither striking the foot of the tower, nor falling outside the railings YZ. Then there is no need to forbid anyone from passing through the square spread around its base while the globes are let fall from the cornice, but they can be let fall again and again without any danger to anyone. It has, besides, certain iron bands around F and T, with iron clasps binding the opposite walls together, which we used as the termini for measuring the residual interval to be completed by the globe when it had arrived at T, or at F. Moreover, the line NH supplies the place of the other towers and altitudes which we used.

[Translator’s note — engraved figure of the Tower of the Asinelli: the tall brick shaft on its cubic base VYZX, the platform YZH (pavement points I, D) with railings, the cornice BC with the windows G, Φ, O and the weathervane-topped dome; iron-band points R, S (and F, T); a door in the base. Down the right side runs the vertical fall-line with the lettered marks H (the pavement, with β just above it), K, L, M, and N, marking the fall-heights tabulated below.]

[X.] In the Year 1640, therefore, in the month of May—and afterward at other times—we inquired [for] the altitude Hβ, such [a height] that a clay globe of eight ounces, let fall, would strike the pavement after exactly, precisely, five vibrations of the aforesaid pendulum, or in the time of 50 thirds; and we found it to be 10 Roman feet, the experiment being often repeated. Then we inquired the altitude necessary for the descent of another globe of the same kind and weight, in double the time, or ten vibrations; and we found it to be 40 feet, which interval KH designates. Hence, ascending higher, we inquired the altitude due to a triple-longer time, or exactly 15 vibrations, for the descent of such a globe, and we detected LH [to be] 90 feet. So, for a time of 20 vibrations, we found the altitude MH [to be] 160 feet; and for 25 vibrations, NH [to be] 250 feet. Finally, because we could not climb to so great an altitude as 30 vibrations require, the [globe] was nevertheless let fall…

The measured fall-heights confirm the law that the spaces are as the squares of the times (10 · n²):

[…continues on p. 386 (PDF 421) with the catchword “tamen”: “…was nevertheless [let fall to test the 30-vibration interval]“—the rest of the free-fall data and the conclusions drawn from it.]

(printed p. 386 — within Chapter XVI. ¶X closes the Asinelli-tower measurement: the successive equal-time intervals cover 10, 30, 50, 70, 90 ft — the odd-number ratio — so the fall is uniformly accelerated. The Third Class of Experiments (¶XI) recounts Riccioli’s earlier pendulum work, his reading of Galileo’s banned Dialogo, and his 1640 trials that confirmed Galileo’s law; he brings the result to the delighted Cavalieri. ¶XII states the law mathematically: the spaces are as the squares of the times.)

[Header: BOOK IX. SECTION IV. — 386]

[the globe was] nevertheless often [let fall] from the cornice of the Asinelli tower—namely from G to I, an interval of 280 feet—and it struck the pavement when both I (at I) and the Fathers Grimaldi and Cassiani (at G), standing, counted 26 vibrations, as appeared when the sheets were mutually communicated. Let us now imagine the intervals of the line NH transferred to the intervals of the line OT; and [let] the interval of the first five vibrations, or of the first time, traversed by the globe, be OC—for it will be 10 feet, as much namely as βH was; but the interval of the second time, OQ, will be equal to KH, 40 feet; and of the third time, OR, equal to LH, 90 feet; and of the fourth time, OS, equal to the space MH, 160 feet. Finally, the whole interval OT of the fifth time, comprehended in 25 vibrations, will be equal to the whole NH, 250 feet. Note, however, that here, for T, we take some sign on the face of the tower between S and the pavement.

[Margin: The increment of the velocity of heavy bodies.]

Therefore the aforesaid globe descended more and more swiftly, the farther it receded from O and the nearer it approached to D; and, the single intervals being separated (each corresponding to the measures of the equal times), in the first time it completed OC, 10 feet; in the second, CQ, 30 feet; in the third, QR, 50 feet; in the fourth, RS, 70 feet; and in the fifth, ST, 90 feet—which numbers, taken together, make 250. Which increment is notable and worthy of note, and to be confirmed by the following experiment. For here we only inquire whether Heavy bodies, naturally gliding down through the air along a straight line perpendicular to the horizon, descend uniformly, or rather difformly-uniformly [with uniform acceleration]; and whether with a decrement, or rather with an increment, of velocity.

The Third Class of Experiments, for the Proportion of the Increment of the velocity of Heavy bodies descending through the Air

[Margin: The Proportion of the Increment of velocity asserted by Galileo.]

[XI.] Although in the Year 1629—when I first began (with Fr. Daniele Bartoli, and Lord Alfonso Iseo, a distinguished Geometer), on another occasion, to examine two pendulums of the same length and weight let fall together from the same terminus, [to see] whether each always proceeded with equal pace through its arc—[I came] to notice that the oscillations of the same pendulum are all, among themselves, sensibly equal in time, or synchronous; and although afterward, at Ferrara in the Year 1634, with Fr. Cabeo, I most certainly detected this very thing—I had not yet understood or known the proportion of the increment of the velocity of Heavy bodies handed down by Galileo (dialogue 2 On the System of the World), and asserted to be according to the odd numbers begun from unity. Nay, then, from my cruder experiments, I had suspected it to be continually triple, namely according to these numbers 1, 3, 9, 27. But afterward, the faculty being granted to me of reading those dialogues which the Sacred Congregation of the Index had forbidden, marked with censures, I found in them (Italian p. 217, Latin p. 163) that the aforesaid increment was detected by him, from experiments, to be according to the odd numbers countable from unity—namely as 1, 3, 5, 7, 9, 11, etc. Yet I suspected that something of fallacy lurked in his experiments, because in the same dialogue 2 (Italian p. 219, Latin p. 164) he asserts that an iron globe of a hundred pounds, let fall from a height of a hundred cubits, reached the earth in five Seconds of time; since [for] me a clay globe of 8 ounces descended from a much greater altitude—namely from GI, 280 feet, which make 187 cubits—in precisely 26 vibrations of my pendulum, which make a time of 4 Seconds and 20 Thirds of the Prime Mobile [sidereal time]. And I was certain that in my numeration of time there had been no sensible error; and the error of Galileo—which lurked in a time not exact to the time of the Prime Mobile and the transit of the Fixed [stars] through the middle of the heaven—I transferred to the intervals completed in the descent of that globe. But I also scarcely believed that he could have used an iron globe of so great a weight, especially since he did not even name the tower from which he had ordered it to be let fall. And so, full of this suspicion, I began (as I said), already from the Year 1640, the measurement of this increment with all subtlety, hoping that I might find some other [proportion], perhaps nearer to mine. But in truth I detected that it was rather true, as Galileo had asserted.

[Margin: 1st Experiment. — 2nd Experiment. — The same proportion confirmed by us.]

For, from what was said in the preceding experiment set forth at number 10, I recognized that increment to have been according to the proportion of feet 10, 30, 50, 70, 90—which is just as if it were expressed in the smallest numbers 1, 3, 5, 7, 9. But not yet fully acquiescing in that [result], the time being changed and 6 vibrations of the pendulum being assumed (which make one whole Second of the prime mobile), I inquired, with Fr. Grimaldi, the altitude due to them such that a clay globe of 8 ounces, let fall, would reach the pavement; and I found βH, 15 feet; then at the end of the second Second, or precisely 12 vibrations, I obtained the altitude KH, 60 feet; and at the end of 18 vibrations, that is the third Second, LH was 135 feet; but at the end of 24 vibrations, that is the fourth Second, MH was 240 feet. But 280 feet, in the Tower of the Asinelli, a similar globe traversed again and more often in 26 vibrations, that is 4 Seconds and 20 Thirds. For although, in greater distances, one or another foot, taken beyond or short, does not bring a difference of one whole vibration, yet I detected that the aforesaid number of vibrations most exactly corresponds to the aforesaid intervals. Wherefore, by this experiment likewise—separating the intervals severally due to equal times—I was certain that this proportion was preserved in the feet 15, 45, 75, 105, which is wholly such as [is] among the numbers 1, 3, 5, 7. For as 1 to 3, so 15 to 45, etc. And I found the same by some other experiments, which for brevity’s sake (and because they are involved with fractional numbers) I here omit; yet I shall set forth some third [experiment] in the following table.

[Margin: 3rd Experiment.]

Therefore I betook myself, with Fr. Grimaldi, to Fr. Bonaventura Cavalieri—the foremost Professor of Mathematics in the University of Bologna, and once a pupil of Galileo—and narrated to him the agreement of my experiments with the experiments of Galileo, as to this proportion at least; for he, fixed to his couch or little chair by gout in the hands and feet at once, could not be present at them. But it is incredible to tell how greatly he was exhilarated by this our attestation.

[Margin: The proportion of the spaces completed by Heavy bodies in descent.]

[XII.] Those not ignorant of Geometry now recognize that the spaces completed in equal compounded times by such heavy bodies, naturally descending, are to one another in the doubled ratio of their times—or have to one another as the squares of the compounded times. Which very thing Galileo gathers (in that dialogue 2, the same page 217, or 163). For in the first experiment, the vibrations were, in order, these: 5, 10, 15, 20, 25, whose square numbers (that is, born from the multiplication of the same number into itself) are 25, 100, 225, 400, 625; but the spaces completed were, in order, feet 10, 40, 90, 160, 250. Now as 25 to 100, so 10 to 40; and as 100 to 225, so 40 to 90; and as 225 to 400, so 90 to 160; and finally, as 400 to 625, so 160 to 250. So in the 2nd experiment, the order of vibrations was 6, 12, 18, 24, 26, whose squares are 36, 144, 324, 576, 676; but the intervals traversed and compounded were, in order, feet 15, 60, 135, 240, 280. And as 36 to 144, so 15 to 60; and as 144 to 324, so 60 to 135; and again, as 324 to 576, so 135 to 240; and finally, as 576 to 676, so 240 to 280. Therefore the aforesaid spaces are to one another as the squares of the vibrations, or of the times. For ease’s sake, however, those equal times can be reduced to the smallest numbers, so that the first time is worth 1, the second 2, the third 3, the fourth 4, and the fifth 5; and accordingly these squares 1, 4, 9, 16, 25 can be applied to examine the said proportion continually.

In the third experiment, however (which you have expressed in the table soon to be exhibited), they ought to have been 288 feet, that the aforesaid proportion might be exactly preserved; but it was not permitted us conveniently to test it, except from a height of 280 feet. It pleases [me], moreover, to gather what we have hitherto said for so beautiful a proportion, and its foundations, into one table, and to propose [them] to the Reader, to be tasted in a brief synopsis. Yet I add that this proportion (as I once hoped) was not found by us in the weight liftable by a falling globe, although it inclines toward this proportion; for if a globe falling from a height of one foot lifts a one-ounce weight, it will not lift a double weight from a height of 4 feet, or a quadruple weight from a height of 9 feet, etc. See below, the 12th class of Experiments.

[…continues on p. 387 (PDF 422) with the catchword “Ordo”: the synopsis table of the free-fall experiments, then the Fourth Class (the unequal descent of two bodies of diverse weight).]

(printed p. 387 — within Chapter XVI. The page opens with the synopsis table of Riccioli’s three free-fall experiments, confirming the squares-of-times and odd-number laws. The Fourth Class of Experiments (¶XIII) then begins, on the unequal descent of two bodies of diverse weight from the same altitude: paired drops from 280 ft found the heavier ball always faster, repeated twelve times, with a table of six such paired drops.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 387]

Synopsis table of the descent-experiments (pendulum: 1²⁰⁄₁₀₀-inch length; the “simple vibrations,” the prime-mobile time in Seconds ″ and Thirds ‴, the square of the vibration-count, the space the 8-oz clay globe completes by the end of the time, the space completed in each separate time, and the minimal-number proportion of the velocity-increment):

(The final row of experiments II and III is the full 280-ft Asinelli-tower drop at 26 vibrations, which does not complete an even last interval; hence those last per-interval and ratio figures break the regular 1, 3, 5, 7, 9 sequence. The square-number column equals the cumulative-feet column up to a constant factor — confirming that distance ∝ time².)

The Fourth Class of Experiments, for the Unequal Descent of Two Heavy bodies of diverse weight from the same altitude through the Air

[Margin: 1st Experiment.]

[XIII.] This experiment is very fallacious unless it be carried out with great diligence and circumspection, and unless the conclusions be prudently drawn from it, all the things being considered which can concur to incite or retard the motion. For the best comparison of all, and one to remove every doubt, would be if, from the same altitude and at the same moment of time, two globes of the same bulk and the same kind, yet of diverse weight, were let fall: namely, we could [take] two clay or metallic globes, of which one were a palm in diameter and weighed one pound, the other of two palms’ diameter and weighed eight pounds. The palpable drawback in that experiment is that a clay globe [of such size]… [the lighter] descends to the earth in whatever altitude in the same physical moment of time. But they at once laid this opinion aside. For a clay globe, lighter—namely of 10 ounces—let fall from G at the same moment in which another clay globe of the same bulk, but of 20 ounces, was let fall from O, appeared still at F, distant from the pavement I at least 15 feet, at the moment in which the heavier had struck the same pavement at D, and had already burst into six hundred fragments. And although, even before the arrival of each at the pavement, the [globe] now manifestly lighter—which, for the sake of distinction, by agreement Fr. Grimaldi always let fall with his left hand—appeared, greatly separated from the heavier, a little below the middle of the tower. But also, the time being numbered by aid of the pendulum, the heavier descended in exactly 23 vibrations, that is 3 Seconds and 30 Thirds; but the lighter in 26 vibrations, namely 4 Seconds and 20 Thirds, of the prime mobile. And this experiment, repeated twelve times, always had a similar result.

Since, therefore, the difference was easier to observe, and greater, in the interval-difference FI than in the time-difference, it seemed advisable, in the descent of the other globes, to attend rather to the residual interval of the slower-descending [globe] than to the time-difference. But what succeeded for us, the following table will teach more briefly and clearly—in which the earlier experiments regard two globes of the same bulk, the later ones globes of diverse bulk:

Globes let fall at the same moment from a height of 280 feet:

(The 1st experiment was repeated twelve times, the 2nd twice. In every paired drop the heavier globe reached the pavement first; the column at right gives how far above the pavement the slower globe still was at that instant.)

[…continues on p. 388 (PDF 423) with the catchword “Al-”: the further unequal-descent trials and the conclusions Riccioli draws about the role of weight and air-resistance in fall.]

(printed p. 388 — within Chapter XVI, the Fourth Class continues. A further table of paired drops of unequal-sized globes shows that air-resistance, not weight alone, governs the outcome, since in some pairs the denser-though-lighter ball wins. ¶XIV draws five corollaries distinguishing density from absolute weight, Riccioli insisting it is “better to report the experiments faithfully than to trim them.”)

[Header: BOOK IX. SECTION IV. — 388]

The other part of the preceding Table

Globes let fall at the same moment from a height of 280 feet (of unequal size):

(In experiment 12 the two weights are equal but the leaden ball — denser and smaller — wins; in 17 and 18 the lighter walnut and the tiny lead ball beat the heavier clay, because their density gives them less air-resistance.)

Corollaries from the preceding Table

[XIV.] I suppose here that “heavier in species” is that which, in an equal bulk naturally suited to it, is of greater weight—as lead is said to be heavier than wax, because if two palm-sized globes were weighed, one of lead, the other of wax, the leaden one will weigh more pounds or ounces or drachms, etc., than the waxen. But “heavier in the individual” is that which is of greater weight absolutely—whether it be of the same species or of a diverse [species] with respect to the other—as a waxen globe of a hundred pounds is said to be heavier than a leaden globe of one ounce. Hence arise five useful combinations.

For some are equally heavy in species and in individual, as are two leaden spheres, each of one pound, and these are naturally of the same bulk; for nothing forbids that, supernaturally or preternaturally, one of them, rarefied, should turn out of greater bulk. But some are equally heavy in species, but not in individual; as are two leaden globes, of which one weighs one pound, the other two pounds; and among these, if they be solid, that which is heavier in the individual is also greater [in bulk]—granted that by supernatural or preternatural condensation it could turn out equal to or less than the other; but if one of them be hollow inside, or full of air or of a lighter body, it can—though lighter in the individual—equal the bulk of the other. Again, some are equally heavy in individual, but not in species, of which kind are a leaden globe and a waxen globe, each of one pound, and naturally that which is lighter in species (the wax) is the larger—granted that it could be supernaturally or preternaturally condensed so as not to be larger than the other. Again, some are heavier than others, either in species and individual (as if a leaden globe be of 10 pounds and a waxen of one, and they can be equal in bulk to the others, or less, or greater). But some are heavier in species, yet lighter in individual—as a little leaden globe of one ounce is denser than a waxen globe of a hundred pounds, and these are naturally smaller than the others. In this combination, by the correlative term, is contained the combination of heavier-in-individual but lighter-in-species. For if lead of one ounce, heavier as to species, is lighter in the individual than wax of a hundred pounds, conversely wax of a hundred pounds is heavier in the individual, but lighter in species, than lead of one ounce. These being laid down, the corollaries written below follow from the preceding table.

[Margin: 1st Corollary.]

First, two spheres equally heavy in species and in individual naturally descend equally swiftly through the same medium (understand always, if they be let fall together from the same altitude); for they are of equal bulk, and so all things are equal. If, however, supernaturally or preternaturally one of them turned out larger by rarefaction or smaller by condensation, the smaller would descend faster, because the angle of contact (the interior one) by which it would strike the plane of the air beneath would be more acute.

[Margin: 2nd Corollary.]

Secondly, of two spheres equally heavy in species but not in individual, that one naturally descends faster through the air which is heavier in the individual—whether they be of equal bulk (because one of them is hollow inside, as appears from experiment 1, repeated twelve times, and experiment 2), or of diverse bulk (as appears from experiments 7, 8, 9, 13, 14, 15, 16, 21). Of which there are some in which I would believe the proportion in velocity was not preserved with the excess in individual weight, similar to that which appeared in others—either on account of an excessive excess of magnitude, which would make an obtuse angle, or because the globes were not everywhere perfectly rounded. But it is better to report the experiments faithfully than to trim them by limiting.

[Margin: 3rd Corollary.]

Thirdly, of two spheres equally heavy in the individual but not in species, that one naturally descends faster through the air which is heavier in species. Which is established by experiment 12. And the reason is that, naturally, that which is heavier in species is of less bulk than the other equally heavy in the individual; wherefore it surpasses it not only by specific gravity, but also by the acuteness of the figure, or angle of contact, arising from the smallness of the sphere—besides that, in our case, wood is more porous than lead, and, on account of the levity of the air or vapors lurking within the pores, less apt for gravitating.

[Margin: 4th Corollary.]

Fourthly, of two spheres, of which one is heavier both in species and in the individual, that one naturally descends faster through the air which is heavier—whether it be larger than the other (as appeared from experiment 10), or equal to it (as was established from experiments 3, 4, 5, 6), or even smaller (as we saw from experiment 11).

[Margin: 5th Corollary.]

Fifthly, of two spheres, of which one is heavier in species but not in the individual, that which is heavier in species can (naturally speaking) descend faster through the air; it can also [descend] equally fast; it can also [descend] slower. An example of the first case we had in experiments 12, 17, and 18; but an example of the third case in experiments 19 and 20. But of the second and third we have a sufficient argument by comparing experiment 8 with 18. For in each, the clay globe was of 11 ounces (the drachms being meanwhile neglected); but the clay globe of 11 ounces, let fall with the clay globe of 21 ounces (experiment 8), was so much slower that it was 12 feet from the pavement when the clay globe of 21 ounces struck it; whereas the clay globe likewise of 11 ounces, let fall (experiment 18) with a leaden globe of nearly 2 ounces, was not slower except by as much as the interval of one foot designates. Therefore, if the clay globe of 21 ounces had been let fall with the leaden globe of nearly 2 ounces, it would have left the leaden behind it by an interval of 11 feet—or at least of more than one—and would have turned out faster in

[…continues on p. 389 (PDF 424) with the catchword “descen[su]”: “…in [its] descent”—the close of the corollaries, then the experiments on descent through water.]

(printed p. 389 — within Chapter XVI. The sixth corollary gives Riccioli’s conclusion: simply speaking, heavier bodies descend faster, at least in our air. The inquiry then moves to water: the Fifth Class of Experiments (¶XV) times globes falling through water-filled glass tubes, finding acceleration there too, and the Sixth Class (¶XVI) finds the data approaching the same squares-of-times proportion as in air.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 389]

[and would have turned out faster in its] descent; since therefore, the individual gravity being increased, it [the clay] would have turned from slower to faster, surely the individual gravity of the clay could have been increased by [added] gravity so little that it would descend equally fast with the lead. I said “at least of more than one [foot],” for experiments 19 and 20 move me—in which the leaden globule of one ounce, on account of [its] smallness, acquired more velocity than it lost (by reason of weight) from [comparison with] the heavier clay, although much greater; so that it was indeed slower than the clay, but a little slower; to which also contributed the more perfect roundness of the leaden [globes] than of the clay ones.

[Margin: 6th Corollary.]

Sixthly, of two spheres let fall together from the same altitude through the same air, that one which is in any way lighter never naturally descends faster, or equally fast; for [the faster] is either heavier in species and individual, or heavier in individual, or heavier in species. Wherefore, simply speaking, those which are heavier descend faster, naturally—in our air at least. But it rarely happens that they be of diverse weight, and the defect of individual gravity be compensated by the acuteness of smallness in bulk, so that they descend equally fast.

The Fifth Class of Experiments, for the Unequal [accelerated] descent of Heavy bodies through Water

[Margin: 1st Experiment.]

[XV.] Although by no means necessary, it is yet very useful—both for this argument which is in hand, and for the integrity of physical speculations—to explore whether, and by what proportion, the velocity of Heavy bodies naturally descending through water continually grows. First, therefore, we used two glass tubes, of which one was 3⅓ Roman feet high, the other 4 feet and 2 inches; and these being divided in two by a mark, and filled with water, and erected to the perpendicular, we let fall through them various globules. A leaden [globe] of 4 drachms and 10 grains absolved the first half of the [first] tube in nearly 5 little-vibrations of our smallest pendulum—that is, 50 Thirds—but the remaining half in two vibrations, that is 20 Thirds; wherefore it absolved the whole height of the tube in 7 vibrations, or one Second of the prime mobile and besides 10 Thirds. It was therefore far swifter at the end of the motion than at the beginning. A glass globule, too, whose weight was 15 wheat-grains, measured the whole tube in its descent in precisely 2 Seconds, that is 12 vibrations of the smallest pendulum—but the first half in 8 or 8½ vibrations, and the second in nearly 4. A bone globe, finally, weighing 2 drachms and 6 grains, descended through the whole tube in 10 vibrations, namely 1 Second and 40 Thirds; but the first half of the tube in 7 vibrations, namely 1 Second and 10 Thirds, and the second in 3 vibrations, or 30 Thirds. Altogether, therefore, the heavy [bodies] of these three kinds descended much more swiftly at the end than at the beginning of the motion. See the rest in the following table, which will be more confirmed by the table to be exhibited at number 17.

Time of descent through each half of the tube (the second half always swifter than the first — the body accelerates in water):

(The last two rows, of ebony, complete the table in the source. In every case the denser/heavier globe traverses the tube faster than the lighter, and every globe traverses the second half-tube in less time than the first — confirming that descent through water, too, is accelerated.)

The Sixth Class of Experiments, for the Increment of the Velocity of Heavy bodies descending through Water

[Margin: Examination of the experiments.]

[XVI.] First, from the experiments of the immediately preceding table one may sniff out the proportion which we are investigating; for it approaches closely to that which we found in Air and explained at numbers 11 and 12—namely, the proportion of the spaces with the squares of the times. For in Experiment 1, the leaden globe absolved the first half of the tube in 50 Thirds, and the second in 20 Thirds; the whole tube, therefore, in 70 Thirds. But as Space 1 is to a double-longer [space], that is to 2—or as the half is to the whole—so [is] the square of the first time, 50‴, [namely] 2500, to the square of the whole time, 70‴, [namely] 4900: most nearly; for 2500 is a little greater than the half of 4900. Again, in the 2nd Experiment, the leaden globe completed the first half in 70 Thirds, and the second in 30 Thirds; wherefore the whole time of the tube traversed was 100 Thirds. But as 1 to 2, so nearly is the square of the number 70, [namely] 4900, to the square of the number 100, [namely] 10,000; for 4900 is a little below 5000, namely the half of 10,000. In the 3rd Experiment, the whole time of descent of the larger glass [globe] was 125 Thirds, and the time of the first half of the space was 85 Thirds (for we resolved [the Seconds into Thirds]); whose square, 7225, compared to the square of 125, namely 15,625, is still much less than the half. And so, the rest being examined, the squares of the two times—of the first time for the half-space, and of the whole time for the whole space—are found to have been to each other nearly as 1 to 2; which the following table sets before the eyes.

[…continues on p. 390 (PDF 425) with the catchword “Expe-”: the table of the water-descent experiments (the table promised “at number 17”), and the further analysis of descent through water.]

(printed p. 390 — within Chapter XVI, the descent-through-water experiments. A table confirms the squares-of-times law in water; ¶XVII tests it further with three tubes whose heights are as 1:4:9 and with a well and cistern, and ¶XVIII finds the law confirmed — heavy bodies in water have the same velocity-increment as in air, at least for bodies denser than ivory. The Seventh Class (¶XIX) then begins: in water, even more evidently than in air, the heavier of two like globes descends faster.)

[Header: BOOK IX. SECTION IV. — 390]

Table (number 17) — the squares of the descent-times (for each tube-trial: the square of the first-half time, with its root, and the square of the whole time, with its root):

The spaces completed in equal compounded times are to one another as the squares of the compounded times. [In each row the whole-time square is approximately double the first-half-time square — since the first half is half the space.]

[XVII.] But that we might investigate, by surer experiments, whether the aforesaid proportion was true and continuous, we used three tubes whose height preserved the order of the squares—so that the first was as 1, the second as 4, the third as 9; that is, the first of 18 inches, the second of 72 inches, the third of 162 inches—which is just as if we should say the first was 1½ feet, the second 6 feet, the third 13½ feet. [Such] that, if the time of completing the second space were double the time of the first, and the time of the third space (or tube) triple the time of the first tube traversed by the same globe (let fall through the aforesaid tubes full of water and perpendicularly erected), we should be certain of the aforesaid proportion. Moreover, because the aforesaid tubes were unsuited for globes of greater bulk on account of their narrowness—and in the descent, the water fluctuating and passing through narrowly, [the globes] struck alternately against the sides of the tubes—we used a well, whose water was 12 feet deep, and a cistern, whose water was exactly 3 feet deep; wherefore the water-height of the cistern was to the height of the well as 1 to 4, that is, as the square of unity to the square of the binary. [So] that in this way too it might be established for us whether the time of descent through the 12-foot well-water was double the time of descent of the same globe through the 3-foot cistern-water, as it ought to be if the increment of velocity were according to the squares of the times. But what succeeded for us will appear from select experiments faithfully exhibited in the following table.

Time of descent through water (in the three tubes) — depths 1½, 6, 13½ ft (= 1 : 4 : 9), weight in drachms · scruples · grains, time in Seconds ″ and Thirds ‴:

(For depths in the ratio 1 : 4 : 9 the descent-times come out close to the ratio 1 : 2 : 3 — the ebony of row 1 (3″, 6″, 9″) being exact — confirming that depth ∝ time². The ivory’s deepest trial is wanting in the source.)

Time of descent through water (well and cistern) — depths 3 ft and 12 ft (= 1 : 4), weight in pounds · ounces:

(For the depth-ratio 1 : 4 the times stand close to 1 : 2 — again confirming depth ∝ time².)

[XVIII.] Of the [experiments] which favor the aforesaid [proportion] exactly there are 8, 11, 14, 15; for the spaces completed in equal compounded times have to one another as the squares of the compounded times. Yet experiments 1, 2, 4, 6, 7, 9, 10, 12, 16 also most nearly support the same—when, by one or at most two vibrations of the pendulum (that is, 10 or 20 Thirds), they exceed or fall short of the desired measure of time. The rest exceed [it] by somewhat more. We, nevertheless, value more highly experiments 14 and 15, because the width of the well, and the most free descent of the globes, and the great force of being borne [downward] to overcome the resistance of the water, exhibited to us a more evident and more often repeated observation. Wherefore it seems more certain to us that Heavy bodies descending naturally even through water have the same increment of velocity which they have in air—if the corrected and select observations be regarded; speaking [that is] of bodies whose specific gravity to the gravity of water is such as is in iron and similar bodies, and is greater than the gravity of ivory.

The Seventh Class of Experiments, for the Unequal Descent through Water of Two Heavy bodies of diverse weight

[XIX.] Now from the Tables exhibited at numbers 15, 16, and 17, it appears that of two globes of the same species, that one which was heavier in the individual, and greater in bulk than the other, always descended faster from the same altitude to the same bottom of the water. For a leaden globe of 4 drachms and 10 grains traversed the first tube in 70 Thirds, which [tube] a leaden [globe] nearly half lighter—or of 2 drachms 6 grains—did not absolve except in 100 Thirds; and so of the other comparisons in the table of number 15. So in the table of number 17 you will see pairs of globes of the same species (or nearly the same) always to have so descended that the heavier descended faster—which was so evident in water that several of our [men] who were present, even the more exact numeration of time being neglected, but weighing the time of the fall by common estimation,

[…continues on p. 391 (PDF 426) with the catchword “pen[sitantes]”: “…weighing [the time], would certainly discern that the lighter globe descended somewhat slower”—then the comparison of same-bulk globes, and the experiments on the ascent of light bodies through water.]

(printed p. 391 — within Chapter XVI. ¶XIX continues, comparing globes of the same bulk but different material in water, with a table showing the denser always descends faster. The Eighth Class (¶XX) then begins: light bodies rising through water also accelerate, demonstrated by tube experiments with an air-filled glass sphere and a 14-ft-well apparatus.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 391]

[several of our men, weighing the time of the fall by common estimation,] would certainly discern that the lighter globe descended somewhat slower. I said “nearly of the same species,” for the ivory and bone were not of the same lowest species; but I did not have two boxwood globes which would descend below the water—for boxwood is rarely found which, if it be not knotty, descends to the bottom of the water.

But since hitherto we have compared globes of diverse bulk among themselves, let us also compare globes of the same bulk. When, on account of the space-difference (or the opacity of the tube), it was not permitted to attend [to the space], or on account of their narrowness it was not easy to let fall two globes at once, we used the difference of time—which, by the slowness of the motion through a denser medium (namely through water), was more evidently observable to us. Among globes of the same bulk, we made two pairs of nearly the same species—namely, one solid clay and the other hollow inside; the rest were of diverse species. The difference of time, therefore, was such as you see in the table soon following.

Time of descent of globes of the same bulk (weight in ounces · drachms · scruples · grains; time in Seconds ″ and Thirds ‴; through water of the depth shown):

The stone [globes], moreover, were of powder of baked brick, agglutinated, and therefore of far greater bulk than all the rest. (In every pair the denser/heavier globe reaches the bottom in the shorter time.)

The Eighth Class of Experiments, for the Unequal Ascent of Light bodies through Water, taken separately

[XX.] Now at numbers 3 and 4 we established positive Levity by experiments taken from air, and [now we establish it from water]. (12) But the cause why the air—which ought to have been faster—was nevertheless slower than the oil [in the earlier fistula experiment] was this: that the air alone, retaining its own rarity, ascended through the narrow channel of the fistula in such a way as to leave, for the water succeeding into its place all around it, a most thin path; wherefore, since the water had to be attenuated, it required little delays of time. But the oil F, impelled upward by the air E penetrating, decreased from its pristine bulk during the ascent—some part of it adhering to the air below, and flying up with the air; hence it came about that around the oil the path was wider for the water succeeding into its place, and not so much delay was necessary, since the water needed almost no attenuation. But, these rudiments of experiments (begun in too-narrow tubes) being dismissed, let us come to other, more certain [ones].

[Margin: 2nd Experiment.]

Secondly, therefore, we used tubes full of water, having a height of 4 or 6 feet, and—releasing from the bottom various globes lighter than water, and then releasing the same upward from the middle of the tube—we always detected that light [bodies] are moved faster toward the end of the motion. For example, a little glass sphere full of air inside, released from the middle of the tube, ascended upward through a height of 3 feet in the time of one Second and 20 Thirds; the same, released from the bottom of the tube, ascended through a height of 6 feet in the time of 2 Seconds and 10 Thirds. But if it were moved uniformly, it ought to have absolved the double space in a double-longer time, namely in 2 Seconds and 40 Thirds. Which I say of this [globe]; we experienced [the same] by very many other experiments of various globules lighter than water.

But because the narrowness of the Tubes did not permit so convenient an operation, it pleased [us] to recur to a well, whose water was 14 feet deep. We let down, therefore, to its bottom a pole composed of partial shafts clasped together, at whose lowest [end] was a cylindrical vessel with a lid that could be tipped back at the [right] moment by a pulling-cord; and in the vessel itself was enclosed a sphere lighter than water, which sphere the weight of the lid held within the vessel. [The lid being tipped, the light sphere ascends through the water—and again it traverses the first] half of the space in a longer time than the latter [half]. For example, a Beech sphere, ascending [through] the first half of the water—that is, the seven-foot [half]—traverses

[…continues on p. 392 (PDF 427) with the catchword “pertran[sit]”: “…traverses [it in a longer time than the upper half]“—the rest of the ascent-through-water experiments and their analysis.]

(printed p. 392 — within Chapter XVI. ¶XX closes the Eighth Class (a beech sphere’s accelerating ascent through water). The Ninth Class (¶XXI) finds the increment of ascent-velocity inconstant and smaller than in falling, because the buoyancy-excess of water over a light body is far less than the weight-excess of metal over water. The Tenth Class (¶XXII) shows that of two light bodies the lighter ascends faster, and the Eleventh Class (¶XXIII) opens the inclined-plane experiments with a 35-ft channel.)

[Header: BOOK IX. SECTION IV. — 392]

[For example, a Beech sphere, ascending through the first half of the water — that is, the seven-foot — traverses it] in 16 vibrations; but the whole water, double-higher (or of 14 feet), in 28 vibrations; therefore it absolved the latter half in 12 vibrations, although it ought [to have taken] 16 if it had been moved uniformly; and so of the rest. The cause why the waxen [sphere] ascended so slowly is, both the defect of the most perfect roundness, and the small difference of specific gravity between water and wax. For since the waxen sphere and the glass-airy one each had an equal diameter (namely of 2⅓ inches), and the waxen one was, as to weight, of 3 ounces, but the glass-airy one of 7 drachms—we filled that same glass [sphere] with water, and its weight was 3 ounces, 5 drachms, and one scruple; the weight of the glass alone (which was nearly 7 drachms) being therefore subtracted, there remains the weight of the water [as] 2 ounces, 6 drachms, and one scruple—but of water having so much smaller a diameter as was twice the thickness of the glass sphere (which indeed was small). Hence we gather that, if a watery sphere be equal to the waxen sphere, the watery one is a little heavier—which also appears from the immersion itself, for the waxen sphere floats on water in such a way that, although it does not descend to the bottom, yet nearly the whole is submerged below the surface.

The Ninth Class of Experiments, for the Increment of the Velocity of Light bodies ascending through Water

[XXI.] [The square of the whole-ascent time of the beech sphere being 784 (28 vibrations), the time in which the same traverses 7 feet was 16 vibrations, whose square is] 256; but 784 to 256 is more than as 2 to 1, since it is as 3 to 1 and beyond; and so you will find concerning the other times answering to the whole space and the half, the squares of which times see in the following table, in the same order which we preserved in the preceding [one]:

And so the proportion which such bodies preserve is inconstant; for in some, the time of the first equal space seems to be to the time of the second equal space as 4 to 3; in some, as 5 to 4; in some, nearer to equality, or as 10 to 9. For in the whole space of 14 feet, in the first experiment, 28 vibrations agree; but to its first half, 16 vibrations—which being subtracted from 28, there remain 12; but as 16 to 12, so 4 to 3. So, 22 being subtracted from 37, there remain 15; but as 22 to 15, so nearly 4 to 3. But not so in the other experiments, as you will learn if you subtract the time ascribed to the first half from the time of the whole space.

[Margin: Why is the increment of velocity less in Light bodies than in Heavy bodies?]

But the cause why such bodies do not have so great an increment of velocity in water is not only heterogeneity (as in glass spheres made of glass—which is specifically heavier than water—and of air; or in wooden ones, drinking water into their pores the more, the more often the experiment is repeated within a short time), but the one or chief [cause] is that the excess of the gravity of water over the gravity of the aforesaid [light] bodies is far less than the excess of the gravity of a metallic or stone sphere over the gravity of water, or [over] the slight gravity of our air. But if bodies could be found as much lighter than water as metallic or stone bodies are heavier than water—not to say than air—then it is very probable that the increment of velocity would be according to the squares of the times: so that if a lighter sphere ascended through a space of 7 feet in the time of 5 vibrations (whose square is 25), it would complete a double-greater space in 7 vibrations (whose square 49 is to 25 most nearly double).

The Tenth Class of Experiments, for the Unequal Ascent of two Light bodies through Water, but of differing Levity

[XXII.] From the same table of number 20 it sufficiently appears that those which are lighter—either in species and in individual at once, or in species at least—ascend faster through water. For, comparing three spheres of diverse species among themselves—namely a smaller beech, a waxen, and a glass-airy [air-filled glass], but of the same bulk (lest inequality be referred to figure or bulk)—it appears that the lightest, namely the glass-airy of 7 drachms, ascended in 33 vibrations; but the beech of 22 drachms in 37 vibrations; and the waxen of 24 drachms in 88 vibrations. Likewise, comparing the larger beech with the larger glass-airy of the same bulk (or of 3 inches in diameter), the glass-airy of 12 drachms ascended in 26 vibrations, but the beech of 5 inches (or 40 drachms) in 28 vibrations; and these for the lighter in species and individual at once. But for the lighter in species only, compare the larger beech with the waxen. The little bladder, however, was neither most exactly spherical (as we already noted), and besides was so immediately wetted in the water and imbued with moisture that, now made heavier, it exceeded the weight assigned to it in the table.

The Eleventh Class of Experiments, for the fall of various globes down a plane inclined to the Horizon at diverse degrees of inclination

[XXIII.] We prepared a wooden tube, or channel, of 35 feet, which—raising it from the plane of the Horizon—we [used to] number the time in which both water and various globes descended through the whole declivity of the channel; which we repeated at diverse times, before some Fathers of our Society. We shall represent the experiments by the following figure, and a table pertaining to it; in which AB is a perpendicular line, and AC equidistant from the Horizon [horizontal], above which—in AB, but moved toward C—the diverse heights of the right extremity of the Channel were measured; further, the angles ACI, ACK, etc., are the angles [of inclination]

[…continues on p. 393 (PDF 428) with the inclined-plane figure and its table—the times of descent down the channel at the various angles of inclination.]

(printed p. 393 — within Chapter XVI. The page opens with the inclined-plane table for the 35-ft channel: descent-times of water and wooden and clay globes at increasing slopes. The Twelfth Class (¶XXIV) then describes Riccioli’s special bell-fitted balance for measuring the impetus of a falling one-ounce ball, with Grimaldi assisting: the drop-heights needed to lift 2, 3, or 4 ounces yield two corollaries — the higher the drop the greater the impetus, and the impetus very probably grows by the odd-number proportion.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 393]

Experiments through the Channel of 35 feet — the far end raised to the heights and angles shown; descent-times in Seconds ″ and Thirds ‴:

[Translator’s note — the figure (referenced at ¶XXIII): AB is a vertical line, AC horizontal (level with the horizon); the 35-ft channel is hinged at C, and its far end is raised to the successive positions D, E, F, G, H, I, K, L, B — whose heights are measured up the vertical (AD, AE, … AB) and whose inclinations are the angles ACD, ACE, … ACB. The “course” CD, CE, … CB is the channel itself at each elevation.]

Do not wonder if the clay globe, heavier than the wooden, was slower; for, by a more continuous friction, it descended through the channel, whereas the wooden one, leaping, was carried further forward by its lightness. Now, if water—raised one foot from the Horizon [at the far end]—runs down 35 feet (or 7 paces) in the time of 15 Seconds, it would seem [that it would] run 28 paces in one Minute, and 1680 paces in one hour, if it ran with uniform velocity. But from a raising of 5 feet (or one pace) it completed 7 paces in about 6 Seconds, therefore 4200 paces in one hour. Yet, on account of the great force of waters—notwithstanding the obliquity and winding of the waves—a river, whose fall is one pace per mile, is wont to complete about 2000 paces in one hour, as we said (bk. 2, ch. 13, no. 8).

The Twelfth Class of Experiments, for the Elevation of greater and greater weight by a Heavy body naturally descending from a greater and greater Altitude

[Margin: A balance of structure apt for this.]

[XXIV.] For testing this, I had a balance made, whose pans were not hung but fixed upon the beam (or upon its arms), lest cords or little chains should impede the perpendicular descent of the ball onto the very center of the pans. The beam itself I supported in its middle by a well-firm fulcrum having a large pedestal; and, the exact equilibrium of the pans being found, beneath the right pan I placed a second fulcrum, which would keep that pan—loaded with any superimposed weight—in equilibrium [held up], until the ball, let fall from a sufficient height onto the left pan, should by its impact raise the right pan (with its weight) to a height of four fingers, and make the [now] raised pan strike a little bell. Upon the other pan, that is the left, I gently laid in the middle a scruple—that is, the twenty-fourth part of an ounce—and by the force of this the right pan was raised to a height of 4 fingers in breadth, and no more, which I have often found by trial. This height, therefore, being chosen, I fitted a little bell above, distant 4 fingers from the right pan, so that the pan, when raised, might by its ringing make the elevation manifest, and by the failure of the ringing, the failure of the elevation—more surely to the ears than to the eye. Moreover, I chose a leather playing-ball, well hard, and—with lead inserted among its stuffing—reduced to the weight of one single ounce, lest a globe of another kind, by falling and striking the left pan, should either be bruised by it, or bruise or break it.

These things being prepared, I approached a certain square column, having the figure of a parallelogram (of which kind we have two in our Solarium [sundial-room]), and placed upon the right pan a weight of two ounces, letting the ball drop perpendicularly onto the center of the left pan along the side of the column; by the help of a certain gnomon exhibiting this position, with Fr. Francesco Maria Grimaldi observing whether the ball fell upon the very center of the pan, I sought out, by experiments often repeated, how great a height was required that the aforesaid ball should raise a weight of two ounces; then how great, that it should raise a weight of three ounces; and lastly how great, that it should raise a weight of four ounces—raise [it], I say, in such a way that here, from a ringing (but a very moderate and almost least one), and there, from a failure of the ringing if the height were a little less, the height required for raising the right pan up to a height of 4 fingers might be made certain. Now for the elevation of two ounces, there was required a height of 2 Roman feet and nearly a dodrans [three-quarters of a foot]—that is, of about 33 or 32 inches; for it stood about this terminus, so much so that from a height of 31 inches the right pan was not raised up to the striking of the bell (although most nearly so), but from a height of 34 inches the striking was already great, etc. To raise a weight of 3 ounces, a height of 84 inches was required; and to raise a weight of 4 ounces, a height of 116 inches, or a little more—as you see in the following table:

A leather ball of one single ounce, that it may raise the weights written below to a height of 4 fingers, must be dropped perpendicularly from the height written below (in inches of the Roman foot):

[Margin: 1st Corollary.]

[1st Corollary.] These things being laid down, it now appears that a Heavy body, the higher the place from which it is let fall, acquires the greater impetus—since a one-ounce ball, of itself unable to raise a pan loaded with a two-ounce weight (a sextans), can nevertheless do so if it be let fall from a height of 32 inches, etc.; and raises a weight of 3 ounces from a height of 84 inches; and a weight of 4 ounces from a height of 116 inches.

[Translator’s note: the print reads “dextante” (a dextans = 10 ounces), but with “biunciali” (two-ounce) and the whole experiment the intended unit is the sextans (= 2 ounces); the printed “d” is almost certainly an error for “s”.]

[Margin: 2nd Corollary.]

[2nd Corollary.] It appears, besides, to be not improbable that the acquired gravity, or impetus, of a Heavy body naturally descending—[the impetus] acquired at the end of [a stretch of] equal times—is to the impetus acquired at the end of the following equal time as the space is to the space traversed in equal times taken separately; or that it keeps the proportion which the odd numbers, counted in order from unity, have—which is surely a most beautiful thing and most worthy to be known, if it can be confirmed by continued experiments. For the leather ball has of itself a gravitation, or impetus, as 1, or of one ounce, since by its force it counterbalances one ounce; but with one scruple added, it raises an ounce up to 4 fingers. Now from a height of 32 inches it alone raises, to the same height of 4 fingers, a weight of 2 ounces; therefore it acquired an impetus of one ounce and one scruple. But if it had had to raise [only] to the least of sensible heights, a gain of one ounce, or a one-ounce impetus, would have sufficed; and for this gain a height of about 30 inches would have sufficed, most nearly. But when it descended from a height of 116 inches—which is nearly quadruple the height of 30 inches—the time completed was double-longer than that in which it descended from a height of 30 inches. For from the earlier experiments of the 3rd Class it appears that, if a Heavy body at the end of the first time traverses a space as 1, at the end of the second equal time it traverses a space as 4; and therefore, arguing conversely, if the first space is to the second as 1 to 4, the first time is equal to the second time taken separately. Now at the end of this second time it raised a weight of 4 ounces; therefore it acquired a quadrantal impetus, that is, of three ounces

[…continues on p. 394 (PDF 429) with the catchword “vncia[rum]”: “…of three ounces” (vnciarum 3; a quadrans = 3 oz, the ball’s own 1 oz plus the 2 oz gained)—the conclusion of the weight-lifting analysis and its bearing on the odd-number law of acquired impetus.]

(printed p. 394 — within Chapter XVI. The page closes the weight-lifting experiment’s second corollary, finishes the Thirteenth Class of experiments (a globe’s perforation of water from ever-greater heights), and opens the “Selected Theorems from the foregoing Experiments”: Theorem 1, levity is a positive quality, not mere privation of gravity; Theorem 2, heavy and light bodies move with accelerating velocity; Theorem 3, the velocity-increment follows the odd numbers (spaces as squares of times), against the arithmetic-progression rule of Chiaramonti, Cabeo, and Baliani.)

[Header: BOOK IX. SECTION IV. — 394]

…of three ounces; but in the second time taken separately it acquired an impetus of 2 ounces. But if the space traversed in the first time be taken as 1, the space traversed in the second time taken separately was as 3. Therefore the impetuses acquired at the end of equal times stand to one another as the spaces traversed in equal times—yet taken separately. Would that it had been permitted us to establish or examine this proportion by experiments continued to greater heights! But it was not permitted, because from a height of

[Margin: How great is the impetus acquired in the motion of Heavy bodies?]

270 inches—such as was necessary that the [ball] should descend in a time triple [the first]—it could not be let fall in such a way that it would fall again and again exactly onto the center of the empty pan, as is needful for certifying the experiment; for that is too difficult, nor is it done without uncertainty. But what I have suspected from the first trace of this proportion, I preferred to communicate to the Reader rather than to keep wholly silent.

The Thirteenth Class of Experiments, for the Perforation of Water made by a globe falling from a greater and greater Altitude

We employed a wooden globe of 2 ounces, which—let fall perpendicularly above a bucket of water 7 inches of the Roman foot deep—from a distance less than 7 inches [above the surface] did not reach the bottom of the bucket; but from a distance of 7 inches it now reached it with a very slight stroke, and from a greater distance with a greater stroke. We then ordered a wooden two-congius vessel [bicongium] to be filled with so much water that its depth was 14 inches; and yet the same globe did not penetrate that whole depth, nor strike the bottom, except from a height of 10 feet taken from the bottom of the water—but [reckoned] from the surface, of 8 feet and 10 inches, that is, of 106 inches—which may surely seem wondrous. But perhaps it happens by reason of the resistance of the deeper water, and of its agitation already produced; or rather because that impetus acquired in air is more and more diminished until it is overcome, and the wooden globe flies back upward, inasmuch as it is lighter than water. And since this return happens at once, it is a sign of the very swift extinction of that impetus—namely by [the body’s] levity, which more and more resists by producing an impetus upward, until this latter overcomes the prior impetus.

Selected Theorems from the foregoing Experiments

[Margin: 1st Theorem.]

[XXIV.] Levity does not consist in the privation of greater gravity, but is a quality positively distinct from gravity.

[Translator’s note: the source numbers this paragraph XXIV—the same numeral it used for the balance experiment (¶XXIV, p. 393); here it opens the First Theorem. The two following theorems are introduced by their italic enunciations with marginal labels only, without a running paragraph number.]

This is against Epicurus and Strato, indeed also against Plato (in Simplicius, On the Heavens I, text 89, comment 87) and Galileo (in the treatise On the things that float upon water, p. 22); the first of whom Aristotle there refutes—[arguing] that if air or fire were thrust upward by an external force from water and earth (which are heavier), they would not be moved faster at the end but slower, as happens to every movable thing moved by an external force; and moreover that a greater fire would be moved upward more slowly than a smaller fire. But since these things which Aristotle supposes—about the greater velocity of the natural motion of the elements at the end, and of a greater rather than a smaller fire—could be denied by someone, it pleased us to confirm these very things by experiments in air and oil (which are bodies lighter than water), as appears from what was said at numbers 20, 21, 22; and besides, to establish this Theorem with physical evidence by new arguments and experiments, as we think was done at numbers 3, 4, and 5. As for what Galileo says—that fiery exhalations ascend faster through water than through air, although, if they were positively light, they ought to ascend faster through air, as the thinner medium—this arises from the comparison and greater excess which the levity of fire has over the partial levity of water, [as compared] than to the levity of air.

[Margin: 2nd Theorem.]

Heavy bodies descending, and Light bodies ascending, by natural motion, are moved with a uniformly difform velocity, and that greater and greater toward the end of the motion.

This is, first, against Arriaga (Disputations 4, On Generation, sect. 5, subs. 3), asserting this motion to be even apparently uniform; and against Simplicius (On the Heavens I, text 88) and Cabeo (Meteors I, text 17, qu. 3), who—concerning its inequality, though so commonly asserted—nevertheless doubted. Next it is against Galileo (Dialogue, Day 2, On the two Chief Systems of the World) and Bullialdus (in his Philolaus, bk. 1, ch. 4), who indeed concede that the motion of Heavy bodies appears to us unequal, as it seems to occur along a straight perpendicular line, but [hold that] it really occurs along one or more circular lines, so that it is uniform, although on account of the motion of the Earth it appears rectilinear and unequal. But against both [parties] the experiments delivered from number 6 to 13 for Heavy bodies, and from number 20 for Light bodies, manifestly militate—all of them indeed against the former opinion or suspicion; but against the latter, those [experiments] which rest not on the mere measurement of the unequal space traversed in equal times, but [on the fact] that [the strokes are] made with greater sound, percussion, and elevation, [all] produced by a greater impetus; for if they were moved uniformly, from however great a distance they had begun their motion, they would acquire no greater force whereby they could effect a greater sound or percussion.

[Margin: 3rd Theorem.]

The increment of the velocity of Heavy bodies naturally descending through air is according to the odd numbers counted in order from unity. Or: the Spaces completed in equal times by the aforesaid motion have to one another the doubled [squared] proportion of the compounded times at the end of which they are traversed. Or: the aforesaid Spaces are to one another as the squares of the compounded times.

I said “naturally,” not only to exclude violent or supernatural motions, but also any preternatural ones whatsoever—of which kind is that which arises from the agitation of our [local] air; for such movables must be such as to overcome this ordinary agitation.

This is against Chiaramonti, who (On the Universe, bk. 12, ch. 28) judged that this increment is not increased but diminished, fearing for himself [the consequence] of an infinite velocity, as I said at number 2; and against Cabeo, who (Meteors I, text 17, qu. 3) judged it more probable that this increment occurs according to an Arithmetic progression—so that if in the first time a Heavy body completed 1 foot, in the second equal time it would complete, separately, 2; and in the third time, 3; and in the fourth,

[Margin: Baliani’s later opinion is worse than his earlier.]

4; and so that, if at the end of the fourth time the motion should cease, the sum of the compounded feet would be 10. Baliani too judged this same progression more probable (On the natural motion of Heavy bodies, bk. 4, pp. 110–113)—though he had agreed with us in bk. 1, proposition 6. But why did he afterward depart from himself? He answers, himself, that the Sixth Proposition rests on experiments liable to the deception of the senses, by which an insensible error cannot be detected; which happens here from this: that the portions of time equal to that first [portion]—in which is completed the first portion of motion, [the portion] independent of impetus—cannot be perceived, being insensible, just as the said first portion of motion is insensible; which, if they were perceived, we should see the motion increased according to the natural progression. Then (p. 113) he gives this rule for finding the spaces completed after equal times: Multiply the number of times—if it be even—by its half, and add the half; if odd, [multiply] by the portion greater than the half, and there will result the sum of the spaces accomplished in the given time. Let 4 times be given: multiply by 2; to the product 8 add the half, 2; it becomes 10, the sum of the spaces. Let 9 times be given: multiply by 5; the product 45 is the sum of the spaces. Therefore the motion is increased—unless I am mistaken [as to their view]—according to an arithmetic progression: not [according to] the odd numbers from unity believed hitherto, but the natural [progression]. But nonetheless, since nearly the same effects follow—on account of the insensible discrepancy—it is no wonder that it has been believed that the spaces are in the doubled ratio of the times; since, even if perhaps it is not precisely true, it is nevertheless so near to the truth that the sense could not perceive the [departure from] truth in the experiments employed. But Baliani did prudently in not affirming this his opinion for certain, but limiting it with formulas of doubting. For not only Galileo (as he himself [does] in the 2nd Dialogue on the System of the world—to which Gassendi subscribed in his Epistles on motion, printed), but we too, by most certain experiments, through sensible intervals of time, detect the aforesaid proportion of the odd numbers from unity. But how false is the rule here delivered by Baliani and Cabeo, and how differing the effects that follow, appears from the experiment of which we spoke at number 12: for in the first equal time of one Second the clay globe completed 15 feet, and at the end of the fourth Second it precisely completed 240 feet; but according to the rule of Baliani and Cabeo it would have completed only 150 feet—that is, only tenfold the prior space. Let Baliani therefore recall his earlier

[…continues on p. 395 (PDF 430) with the catchword “rem” (priorem): ”…[his] earlier [opinion]“—Riccioli’s continued case for the odd-number / squares law against the arithmetic-progression rule of Baliani and Cabeo.]

(printed p. 395 — within Chapter XVI, continuing the “Selected Theorems.” The page lays out Theorems 4–13 on free fall: the odd-number law holds through water as through air; and in various pairings of species, bulk, figure, and weight, the individually heavier body falls faster — against Galileo’s and Baliani’s equal-fall claims — culminating in the compendium that the heavier descends faster (Theorem 12), yet the velocity-difference is far smaller than the weight-difference (Theorem 13, against Aristotle).)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 395]

…[his] earlier opinion, asserted in his book 1, proposition 6, as being better than this later one, and proved by experiments by no means fallacious.

[Margin: 4th Theorem.]

Of certain Heavy bodies naturally descending through water, the increment of velocity is, in itself, as great as [that] of the same bodies descending through air, although the descent itself is slower through water than through air. That is, their velocity grows according to the odd numbers; or, in completing the space, it keeps the doubled proportion of the times; or, the watery spaces traversed in equal times are to one another as the squares of the aggregated times.

[Translator’s note: the print reads “triplicatam” (tripled) proportion, but the clause it explains—“as the squares of the times”—and the parallel Theorem 3 (p. 394) require “duplicatam” (doubled = squared); an evident misprint.]

This appears from the experiments delivered at numbers 16, 17, and 18. I said “of certain”—namely, of metallic, stone, and similar bodies; for boxwood bodies, or [those] of ebony or holm-oak, although heavier than water, nevertheless descend more slowly, and do not acquire so great an increment of velocity in water.

[Margin: 5th Theorem.]

Of most Light bodies naturally ascending through water, the increment of velocity—[which is] per accidens—is not as great as [that] of heavy metallic or stone bodies descending through air or through water. This appears from the experiments delivered at number 20 and examined at number 21, where also the reason for the disparity was indicated; and what would happen of itself, with the impediments removed which force themselves upon the experiments, I have said in the same place.

[Margin: 6th Theorem.]

Of two Heavy bodies of the same species, bulk, and figure, but of different weight, let fall together at the same instant from the same altitude through air, that one descends faster to the same pavement or ground which is heavier in the individual. This theorem, and the following, is against Nicolaus Cabeo, Rodrigo de Arriaga, Bartholomaeus Mastrius, and Bonaventura Bellutus (in the passages adduced at number 2, under the fifth error)—who, from rude and almost laughable little experiments, did not hesitate to assert universally that any two heavy bodies let fall together descend equally fast, and strike the ground at the same instant, without any sensible difference. Now this sixth Theorem is established by 11 experiments, which hold a place at number 13 in tables 1 and 2, and concerning which [see] number 14, corollary 2.

[Margin: 7th Theorem.]

Of two Heavy bodies of the same bulk and figure, but of different species, let fall through air from the same altitude at the same instant, that one descends faster which is heavier in species, and so [also] in the individual. This is against Cabeo, Arriaga, Mastrius, and Bellutus, [as] above. But not against Baliani, since he admits that of two globes of the same bulk—one of wax, the other of lead—let fall from the citadel of Savona, of a height of fifty feet, the waxen one was somewhat slower, so that it was about one foot from the ground when the leaden one touched the level below. Now our theorem appears from the experiments delivered in the table of number 13—namely experiments 3, 4, 5, 6, repeated, however, quite often. And reasoning a priori plainly teaches it: for since all else is equal, and one cannot have recourse to a resistance arising from a more obtuse angle (with which a greater mass touches the flat of the air), nor to a figure less apt for penetrating the medium—seeing that both are of the same bulk and figure—while on the other hand that which is heavier in species is so also in the individual (being, when compared with a body of equal bulk but of different and therefore lighter species); it is necessary that the greater gravity, in species and in the individual, should produce its proportioned effect: that is, weigh more, and move with a greater impetus the body to which it is naturally implanted. For it cannot be said that the defect of specific gravity in the one is compensated by an excess of individual gravity; for then (naturally speaking) that which is of lighter species would have to be of greater bulk—which is against the hypothesis.

[Margin: 8th Theorem.]

Of two Heavy bodies of the same species and figure, but of different bulk, and so of different individual gravity, let fall from the same altitude at the same instant, that one descends faster which is heavier in the individual. This is against not only Cabeo, Arriaga, Mastrius, and Bellutus, but also against Galileo (Dialogue, Day 2, On the System of the World) and Giovanni Battista Baliani (On the motion of heavy solid and liquid bodies, bk. 1)—of whom Galileo affirms that two iron globes let fall together from the same altitude (even were it the concave [vault] of the Lunar heaven) descend to the earth in the same time; while Baliani [affirms] that heavy bodies, whether of the same or of different species, very unequal in bulk and gravity, descend naturally through the same space in equal time through air, and strike the ground below at the same indivisible instant of time.

[Margin: The occasion of the errors of Galileo and Baliani.]

But the occasion of this erroneous opinion was the too-short space which they used; for they confess (in the passages reviewed above) that they used—Galileo, a height of only one hundred cubits, that is, 150 feet; and Baliani, a much smaller height, namely of 50 feet. Now when two heavy bodies of the same species, but of different bulk and so of different individual gravity, are employed, less difference appears in their fall; and therefore in these a greater altitude than in the rest is required, that the difference may be evident to sense; and far greater attention is required to discern the sound and stroke of the one from the other. Perhaps, too, Galileo denied this inequality lest it harm the motion of the earth, as we shall say in chapter 17, number 12. But the proposed theorem appears sufficiently from the experiments exhibited in the table at number 13, and gathered there at corollary 2—namely, experiments 7, 8, 9, 13, 14, 15, 16, 21 of that table.

[Margin: 9th Theorem.]

Of two Heavy bodies, of the same figure and the same individual gravity [equal weight], let fall together through air from the same altitude, that one descends faster which is heavier in species. This is against Cabeo, Arriaga, Mastrius, Bellutus, and Baliani, [as] above. But what Galileo thought of these is not yet certainly established, for he seems only to have compared together globes of the same species. Yet for this Theorem stands experiment 12 in the table of number 13, and corollary 3 placed at number 14.

[Margin: 10th Theorem.]

Of two Heavy bodies of the same figure, but of different species, and of different gravity both specific and individual, let fall together through air from the same altitude, that one descends faster which is heavier—whether it be of greater, of smaller, or of equal magnitude. This is against Cabeo, Arriaga, Mastrius, and Bellutus; and, as to the last part of the theorem (concerning equal magnitude), against Baliani. But the whole theorem is sufficiently proved by the experiments adduced at number 13, and at number 14 for corollary 4.

[Margin: 11th Theorem.]

Of two Heavy bodies of the same figure, of which one is heavier in species, the other heavier in the individual, let fall together through air from the same altitude: sometimes both [descend] equally fast; sometimes that which is heavier in species descends faster; sometimes that which is heavier in the individual. See what we said at number 14, corollary 5, where we also indicated the causes of this variety—especially the air enclosed within the pores of the one [body] that is less dense or compacted; then the magnitude of the sphere (striking the air with a more obtuse angle); and then the roughness of the surface. As to what Baliani says (On the motion of heavy bodies, bk. 1)—that if all impediments be removed, heavy bodies descend equally, because although one be lighter and the other heavier, yet in that which is heavier the greater gravity must move a greater [quantity of] matter too—[this] is neither true nor consistent: not true, because it contradicts our experiments; not consistent, because if it is heavier, it will also be greater in bulk (for he speaks of absolute and individual gravity), and so it will have an impediment from the multitude of air to be penetrated, and from the more obtuse angle with which it tries to strike it.

[Margin: 12th Theorem.]

Of two Heavy bodies of different weight let fall together from the same altitude, for the most part the one descends faster than the other; and that which descends faster is always in some respect heavier—namely either in species and individual, or in the individual, or in species. Wherefore, if it must be pronounced absolutely, it is to be asserted that that which is heavier descends faster. This is, as it were, the compendium of all the preceding Theorems, gathered into one; and it is sufficiently established from what was said at number 14, corollary 6. And the same is to be said of Light bodies in ascent, according to what was said at number 22.

[Margin: 13th Theorem.]

Of two Heavy bodies descending unequally from the same altitude, the difference in velocity is not proportional to the difference of gravity alone, but is much smaller. This is against Aristotle and some Peripatetics in On the Heavens III, text 27, where, among other things, he says of two bodies of unequal bulk: “For the velocity of the smaller will be to that of the greater as the greater body is to the smaller body”; wherefore, comparing two bodies of the same species, he thought that the one heavier than the other in the individual is moved naturally as much faster as it is heavier—so that if one be double the weight of the other, it would also complete a double-greater space in equal time. But this is deservedly refut-

[…continues on p. 396 (PDF 431) with the catchword “redar[guitur]”: ”…[it] is refuted”—Riccioli’s rebuttal of the Aristotelian velocity-∝-weight rule.]

(printed p. 396 — within Chapter XVI. The page finishes Theorem 13, refuting Aristotle’s velocity-proportional-to-weight rule with the Asinelli-tower data, and explains why velocity does not track weight. It then opens the “Selected Problems from the foregoing Theorems,” applying the squares-of-times law: Problem 1 finds fall-times over other heights (with worked examples up to a fall from the Moon’s heaven), Problem 2 the inverse, and Problem 3 begins gauging the density-ratio of water to air.)

[Header: BOOK IX. SECTION IV. — 396]

[This is] refuted by Galileo (Dialogue 2, On the System of the World, Latin p. 164) and by Baliani (On the natural motion of heavy bodies, bk. 1, p. 5), and—what is the chief point—by the most evident experiments performed by us and cast into the table at number 13. For whether you regard the difference in time, or in space, it was never as great as the difference in weight. For a clay globe, hollow inside and of 10 ounces, measured the height from the cornice to the lowest base of the Asinelli tower in 4 Seconds and 20 Thirds [of the primum mobile]; but a clay globe of the same magnitude, yet of 20 ounces, accomplished the same height in 3 Seconds and 30 Thirds; and when this [heavier one] was now striking the pavement, the lighter was not distant from the pavement except by about 15 feet—out of those of which the aforesaid height was 280. But if the velocity of the heavier had been as much greater as its gravity is greater, the globe of 20 ounces would have descended in a time double-smaller than the globe of 10 ounces, that is, in 2 Seconds and 10 Thirds, and the lighter would have been 140 feet from the pavement when the heavier struck the pavement. So in all the other experiments the excess of the space completed by the heavier over the space completed by the lighter body was found far less than the excess of gravity over gravity.

Wherefore that proportion which at first sight seemed probable to Aristotle and his followers, recalled to experiment, has utterly vanished.

[Margin: Why velocity is not to velocity as gravity is to gravity.]

But since the reason a priori can hardly be rendered why the effect [velocity to velocity] does not keep the proportion which the cause [gravity to gravity] has, hence it came about that not a few of those already named thought that any two heavy bodies whatsoever, however different in weight, descend equally of themselves, if the things which per accidens retard one of them be removed. But we say that no case can be given naturally in which two bodies are of different weight, and yet all impediments retarding one of them be removed from them: for that which is heavier will either be greater in bulk—on account of which it will bore through the air with a more obtuse angle; or it will be lighter in species, and enclose within its pores little portions of air or fire; or it will have a rougher surface, or be allotted a figure less apt for penetrating the air—on account of which the force of the greater gravity in moving will be blunted. But if there be given two bodies of the same species, bulk, figure, and smoothness, and yet one be double lighter than the other in the individual—but, on account of a greater rarity preternaturally or supernaturally acquired, has obtained an extension and magnitude equal with the other—nevertheless that which is double heavier will not therefore descend double faster; perhaps because the principal mover is the very substance of the movable body, which uses gravity as a merely instrumental cause. But the substance of heavy bodies of the same species requires that they arrive as quickly as possible at their natural place—with a determinate increment of velocity indeed, but a very similar one [across them]; wherefore that which is lighter in the individual is moved faster by the force of the substantial principle than it otherwise could be, were mere gravity regarded.

I come now to certain selected Problems, whose practice is not only desirable in itself, but will also be of use to us for the arguments to be adduced against the motion of the earth, or for the motion of the earth, in chapter 17.

Selected Problems from the foregoing Theorems

[Margin: 1st Problem.]

[XXV.] Given the time in which some Heavy body, descending naturally, traversed a given altitude: to find the time in which it traversed a smaller, or would traverse a greater, given altitude.

By examples the Problem will be explained, and at the same time its practice taught. We said at number 12 that a clay globe of 8 ounces descended from the cornice of the Asinelli tower through a space of 280 old Roman feet in the time of 4 Seconds and 20 Thirds; that is, if you reduce the time to Thirds alone, it traversed it in the time of

[Margin: 1st Example.]

260 Thirds. Now let someone ask: in how much time did it traverse the half of that space, namely 140 feet? That this problem may be solved, [proceed] from the foundations of Theorem 3. Let it be made: as the greater space to the lesser space sought, so the square of the known time to the square of the sought time; for the root of this square will be the sought time. Now, of the given time of 260 Thirds, the squared number is 67,600. Let it therefore be made: as space 280 to space 140, so 67,600 to 33,800; the nearest square root of which last number is 184. Therefore in 184 Thirds such a globe traversed [the first] 140 feet, and the remaining 140 in 76 Thirds.

[Margin: 2nd Example, for one mile.]

Again, let another ask in how much time such a globe would traverse a whole mile, which contains 5000 feet. Let it therefore be made: as the lesser space, which was of 280 feet, to the greater space, that is, of 5000 feet, so the square of the time of 260 Thirds, 67,600, to the square 1,207,107 Thirds; the nearest root of which is 1098 Thirds—that is, 18 Seconds and 18 Thirds. And by the same device you will find that such a Heavy body, if it descended through a medium of the kind nearest to us (such as air), would traverse one Semidiameter of the Earth (which for us contains 4139 Italian miles) in the time of 70,685 Thirds; for as 1 mile to 4139, so the square 1,207,107 Thirds to the square 4,996,015,873, whose nearest root is 70,685 Thirds, which make 19 Minutes, 38 Seconds, and 5 Thirds.

[Margin: 3rd Example, for the semidiameter of the Earth.]

In so great a time, then, the aforesaid Heavy body would reach the center of the Earth, if a well were bored all the way to there and were full of air. But that we may proceed further—at least as far as the Moon, since the upper region of air and the [region] of fire, if it exists, is a less-resisting medium—you will easily grant that the aforesaid globe (or at least a larger one, namely of one pound) would traverse the first semidiameter of the Earth, the one nearest the Lunar heaven, in 19 Minutes, whose square is 361.

[Margin: 4th Example, for the Lunar distance.]

To one asking, therefore, in how much time a clay globe of one pound would descend from the summit of the Lunar heaven (assumed to be distant 70 semidiameters—though let no one assign it so great a distance), I answer with confidence that it would descend in 159 minutes of an hour, that is, in 2 hours and 39 minutes; for as 1 semidiameter of the Earth to 70 semidiameters, so the square 361 (answering to the root of 19 minutes) to the square 25,279, whose nearest root gives 159 minutes, that is, 2 hours 39 minutes. Yet elsewhere (bk. 2, ch. 21, and bk. 7, sect. 6, ch. 5), proceeding a little differently, we gathered that such a globe would measure the Lunar distance in about 3 hours. But for the distance of the Sun, of Saturn, and of the Fixed [stars], see the table set out at bk. 7, sect. 6, ch. 5—in which, however, we neglected the small fractions, for the reasons adduced in the same place.

[Margin: 2nd Problem.]

[XXVI.] Given the time in which some Heavy body, descending naturally, traversed a given altitude: to investigate another altitude, smaller or greater, which it would traverse in another given time. Let it be made: as the square of the greater time to the square of the lesser time, so the given greater space to the lesser space sought. Or, if the space sought must be greater: let it be made as the square of the shorter given time to the square of the longer time, so the given lesser space to the greater space sought.

[Margin: Example.]

For example, since a clay globe of 8 ounces ran through 280 feet in 260 Thirds: if you ask how much space it will run through in one hour—that is, in 216,000 Thirds—let it be made: as the square of 260, [namely] 67,600, to the squared number of 216,000, [namely] 46,656,000,000, so 280 feet to the feet sought; they will be 153,308,875 feet, that is, 30,661,775 paces, that is, 30,662 Italian miles. Therefore in 24 hours (whose square is 576) it would traverse 30,662 multiplied 576 times, that is, 17,661,212 miles.

[Margin: 3rd Problem.]

[XXVII.] Given the space which the same Heavy body completed in air and in water, and the time corresponding to each space: to investigate the probable difference between the gravity—or at least the density—of water and of air. First, by Problem 1, inquire in how much time that heavy body would complete an aerial space equal to the watery [space] traversed by it; then compare the time found with the time in which the same heavy body completed just as much watery space: for by how much greater time it spent in the watery [space] than in the aerial, by so much is the resistance of water probably greater than that of air—whether it be said to arise from gravity, or rather from greater density; unless also humidity, making viscosity together with density, helps the resistance that must be overcome by the movable.

[Margin: Example.]

Example. For me, a little lead globule of 4 drachms, or half an ounce, measured in air 280 feet in 24 vibrations of the pendulum, but in water measured 4 feet in 7 vibrations of the same pendulum—that is, there [in air] in 4 Seconds, here [in water] in 1 Second and 10 Thirds. I ask in how much time it would have traversed the watery

[…continues on p. 397 (PDF 432) with the catchword “aqueos”: “…the watery feet [in air]“—completing the water-vs-air resistance example.]

(printed p. 397 — within Chapter XVI, closing the “Selected Problems” and the chapter itself. Problem 3 concludes the water-to-air density ratio; Problem 4 gives a float-and-tube method to sound the depth of the sea, restoring Alberti’s problem; Problem 5 computes the enormous lag between unequal bodies over hours of continuous fall; Problem 6 finds how slow a body must begin in order to reach the Earth’s center in six hours. Chapter XVI ends here.)

[Header: ON THE SYSTEM OF THE MOVED EARTH — 397]

280 feet of water. Let it be made: as 4 feet to 280 feet, so the square of 7 vibrations, 49, to 3430—the square, namely, nearest [to that] of 59 vibrations. Therefore, as 59 to 24, so probably is the resistance and density of well-water to the resistance and density of our [local] air and [the air] nearest the earth. Unless someone should wish density to be to density as the square of time to the square of time; and so, in our case, the density of water to that of our air as 576 (which is the square of 24 vibrations) to 3430 (which is the square of 59 vibrations).

[Margin: 4th Problem, on investigating the depth of the Sea.]

[XXVIII.] To search out the depth of the Sea, or of any water, the per-accidens impediments removed. First, in a glass tube of (say) 10 feet, full of water of the kind whose depth you wish to measure, and erected perpendicular to the horizon, let down a cork float—or a gall-nut—D, into whose bottom, through a little handle or loose ring, is inserted a lead hook A·B·F (whose greater part is BF); and with a pendulum measure precisely the time in which the whole aggregate of gall-nut and hook reaches the bottom. Then, as soon as the hook, tipping over, has freed the float D so that it tends upward, measure likewise the time in which the float D ascends to the top of the water, and keep each time separately, and also their sum. Secondly, let down the same aggregate of float and hook into the Sea, and measure the time which intervenes between the release and the return of the float to the top of the water; distribute this sum, as a common gain, to the aggregate of float and hook—or to the descent alone—by the Rule of Society [partnership] or the Rule of Three, and you will have the distinct time in which the float with the hook descends to the bottom of the sea. Thirdly, therefore, by the second problem delivered a little before, inquire the desired depth of the sea. I said “the per-accidens impediments removed”—such as would be if that weight should strike upon some point, or a hidden rock, or upon the back of a fish then swimming beneath, or if the sea were stirred from the bottom by the force of winds.

[Translator’s note — the engraved figure (left column): at left, the sounding device — a float D (a cork or gall-nut, drawn as a dark disk) with the lead hook hanging from its base, its short cross-piece A·B and the longer descending arm B·F (hook at F); at right, the graduated glass test-tube G…K (a tall narrow vessel, G at the rim, K at the bottom point) carrying a beam or scale M·E·R above it (E the central suspension), marked with the depth-graduations “3” and “2½”.]

[Margin: Example.]

Example. Let the aggregate of float and hook have traversed 10 feet of sea-water in the time of six vibrations of our smallest pendulum, or 60 Thirds of the primum mobile; and let the float have ascended to the same height in 10 vibrations, or one Second and 40 Thirds. The sum, therefore, of ascent and descent is 2 Seconds and 40 Thirds, or, resolved, 160 Thirds. Then suppose the float with the hook, let down into the sea, to have reached the bottom of the sea and returned to the top in the time of one Minute, or 60 Seconds, which are 3600 Thirds. For if to the whole time of 160 Thirds [in the tube] there correspond [in the sea] 3600 Thirds, then to the time alone in which the hook with the float descended through the 10 feet of water—which was 60 Thirds—there correspond 1350 Thirds, in which, namely, that hook descended to the bottom of the sea. Finally, as the square of the time of 60 [Thirds], 3600, to the square of the time of 1350 Thirds, 1,822,500, so 10 feet to 5062 feet—that is, to one Italian mile and 62 feet besides, since a pace consists of five feet. So great, then, will be the depth of that sea. Thus you see corrected, and restored in a much nobler manner, the problem of Leon Battista Alberti, of which we spoke in bk. 2, ch. 5, no. 11.

[Margin: 5th Problem.]

[XXIX.] Given the time in which one heavy body, by natural descent, measures a given altitude, and the difference of interval by which another heavy body has measured [its fall] more slowly than the first: to investigate the difference of interval that would intervene in a given greater time. Wherefore, by Problem 2, [find] the interval due to each of the two heavy bodies for the second given or assumed time; for the difference of these intervals will be that very [difference] which you seek.

[Margin: 2nd Example.]

Example. We said at number 13, in the table at experiment 21, that a lead globule of one ounce, let fall from the cornice of the Asinelli tower, touched the pavement of the base (which is distant 280 feet thence) when another globule of 4 drachms, let fall from the same place at the same time, was 5 feet from the pavement. Now the one-ounce sphere, while I measured the time, descended in 24 vibrations, or 4 Seconds. It is asked how far the half-ounce sphere would have been from the pavement if such motion had been continuous for three hours, that is, for 10,800 Seconds. First, then, let it be made: as the square of the time of 4 Seconds (which is 16) to the square of 10,800 Seconds (which is 116,640,000), so 280 feet to 2,041,200,000 feet—which are 408,240,000 paces, or 408,240 Italian miles. Again, let it be made: as 16 to 116,640,000, so the 275 feet which the half-ounce globule completed [in 4 Seconds] to 2,006,625,000 feet; which, subtracted from the 2,041,200,000 feet, leave a difference of 34,585,000 feet—namely 6,917,000 paces, which make 6917 Italian miles. Now in the 408,240 Italian miles the Earth’s semidiameter (which for us is 4139 miles) is contained 91 times. Therefore, if from the height of 91 terrestrial semidiameters—which is about one-and-a-half times the distance of the Moon—the two aforesaid (or similar) globes of the same species were let fall together, one being double heavier than the other, and the medium were such as our air, then, when at the end of three hours the heavier struck the Earth, the lighter would still be 6917 Italian miles from the earth. But if you enter the reckoning for six hours—that is, 21,600 Seconds, whose square is 266,560,000—you will find that the one-ounce sphere would traverse 929,210 miles, and the half-ounce 916,300; whose difference at the end of six hours would be 12,910 miles, that is, greater than three Earth-semidiameters.

[Translator’s note: the print gives 21,600² as “266,560,000” here, but the square is 466,560,000 (correctly so given by Riccioli himself at Problem 6 below); an evident misprint.]

But it is worth weighing another notable case, of which we spoke at number 13, experiment 12. For a lead globe of 2½ ounces measured 56 paces, or 280 feet, in 4 Seconds very nearly; and at the moment in which it struck the pavement, a wooden globe of greater bulk but equal weight—namely of 2½ ounces—was manifestly 40 feet, that is 8 paces, from the pavement; wherefore the wooden one had then completed 48 paces. I ask how great the difference of interval would be, if both had moved for six hours. The square of 4 Seconds is 16, and the square of 21,600 Seconds (of which six hours consist) is 266,560,000. Let it therefore be made: as 16 to 266,560,000, so 56 paces to 929,210,000 paces—that is, 929,210 Italian miles. Again, let it be made: as 16 to 266,560,000, so 48 paces to 799,680,000—that is, 799,680 miles; the difference of which miles is 129,530. Therefore, if the aforesaid globes were let fall together from a height of 929,210 miles—that is, of 224½ terrestrial semidiameters—and the air, or medium, were such as our air, then, when the lead globe of 2½ ounces struck the earth, the wooden one of 2½ ounces would be distant from the earth by 129,530 miles, that is, by about 31¼ terrestrial semidiameters.

[Margin: 6th Problem.]

[XXX.] Given a Heavy body which in six hours would reach from the surface of the earth to the center of the earth through a well full of air or water: to investigate its slowness—that is, in how much time it ought to traverse the first pace. I suppose, moreover, that in the Earth’s semidiameter are contained 4,139,000 paces. Therefore, by Problem 1, let it be made: as 4,139,000 paces to 1 pace, so the square of 21,600 Seconds (which make six hours), 466,560,000, to the square 112¾; whose nearest square root is 10½. Such a Heavy body, then, would have to consume, in naturally traversing the first single pace (or five feet), about 10½ Seconds—that is, 60 vibrations of our smallest pendulum. But such a body will hardly be found; and among the many experiments made in water, the body scarcely nearest fit for this was an ebony globe, of which [we spoke] at number 15, experiment 9 or 10 in the table; for it, in the time of 7 Seconds, completed 4⅙ feet. But in air this is much more difficult.

(Here ends Chapter XVI. The page closes with a printer’s ornament; the catchword “CA-” opens Chapter XVII on p. 398.)

[…continues on p. 398 (PDF 433) with the catchword “CA-” (CAPVT XVII): the next chapter begins.]