_Galileo's theory of Motion—Extracts from the Dialogues._
DURING Galileo's residence at Sienna, when his recent persecution had
rendered astronomy an ungrateful, and indeed an unsafe occupation for
his ever active mind, he returned with increased pleasure to the
favourite employment of his earlier years, an inquiry into the laws and
phenomena of motion. His manuscript treatises on motion, written about
1590, which are mentioned by Venturi to be in the Ducal library at
Florence, seem, from the published titles of the chapters, to consist
principally of objections to the theory of Aristotle; a few only appear
to enter on a new field of speculation. The 11th, 13th, and 17th
chapters relate to the motion of bodies on variously inclined planes,
and of projectiles. The title of the 14th implies a new theory of
accelerated motion, and the assertion in that of the 16th, that a body
falling naturally for however great a time would never acquire more than
an assignable degree of velocity, shows that at this early period
Galileo had formed just and accurate notions of the action of a
resisting medium. It is hazardous to conjecture how much he might have
then acquired of what we should now call more elementary knowledge; a
safer course will be to trace his progress through existing documents in
their chronological order. In 1602 we find Galileo apologizing in a
letter addressed to his early patron the Marchese Guido Ubaldi, for
pressing again upon his attention the isochronism of the pendulum, which
Ubaldi had rejected as false and impossible. It may not be superfluous
to observe that Galileo's results are not quite accurate, for there is a
perceptible increase in the time occupied by the oscillations in larger
arcs; it is therefore probable that he was induced to speak so
confidently of their perfect equality, from attributing the increase of
time which he could not avoid remarking to the increased resistance of
the air during the larger vibrations. The analytical methods then known
would not permit him to discover the curious fact, that the time of a
total vibration is not sensibly altered by this cause, except so far as
it diminishes the extent of the swing, and thus in fact, (paradoxical as
it may sound) renders each oscillation successively more rapid, though
in a very small degree. He does indeed make the same remark, that the
resistance of the air will not affect the time of the oscillation, but
that assertion was a consequence of his erroneous belief that the time
of vibration in all arcs is the same. Had he been aware of the
variation, there is no reason to think that he could have perceived that
this result is not affected by it. In this letter is the first mention
of the theorem, that the times of fall down all the chords drawn from
the lowest point of a circle are equal; and another, from which Galileo
afterwards deduced the curious result, that it takes less time to fall
down the curve than down the chord, notwithstanding the latter is the
direct and shortest course. In conclusion he says, "Up to this point I
can go without exceeding the limits of mechanics, but I have not yet
been able to demonstrate that all arcs are passed in the same time,
which is what I am seeking." In 1604 he addressed the following letter
to Sarpi, suggesting the false theory sometimes called Baliani's, who
took it from Galileo.
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"Returning to the subject of motion, in which I was entirely without a
fixed principle, from which to deduce the phenomena I have observed, I
have hit upon a proposition, which seems natural and likely enough; and
if I take it for granted, I can show that the spaces passed in natural
motion are in the double proportion of the times, and consequently that
the spaces passed in equal times are as the odd numbers beginning from
unity, and the rest. The principle is this, that the swiftness of the
moveable increases in the proportion of its distance from the point
whence it began to move; as for instance,—if a heavy body drop from A
towards D, by the line ABCD, I suppose the degree of velocity which it
has at B to bear to the velocity at C the ratio of AB to AC. I shall be
very glad if your Reverence will consider this, and tell me your opinion
of it. If we admit this principle, not only, as I have said, shall we
demonstrate the other conclusions, but we have it in our power to show
that a body falling naturally, and another projected upwards, pass
through the same degrees of velocity. For if the projectile be cast up
from D to A, it is clear that at D it has force enough to reach A, and
no farther; and when it has reached C and B, it is equally clear that it
is still joined to a degree of force capable of carrying it to A: thus
it is manifest that the forces at D, C and B decrease in the proportion
of AB, AC, and AD; so that if, in falling, the degrees of velocity
observe the same proportion, that is true which I have hitherto
maintained and believed."
We have no means of knowing how early Galileo discovered the fallacy of
this reasoning. In his Dialogues on Motion, which contain the correct
theory, he has put this erroneous supposition in the mouth of Sagredo,
on which Salviati remarks, "Your discourse has so much likelihood in it,
that our author himself did not deny to me when I proposed it to him,
that he also had been for some time in the same mistake. But that which
I afterwards extremely wondered at, was to see discovered in four plain
words, not only the falsity, but the impossibility of a supposition
carrying with it so much of seeming truth, that although I proposed it
to many, I never met with any one but did freely admit it to be so; and
yet it is as false and impossible as that motion is made in an instant:
for if the velocities are as the spaces passed, those spaces will be
passed in equal times, and consequently all motion must be
instantaneous." The following manner of putting this reasoning will
perhaps make the conclusion clearer. The velocity at any point is the
space that would be passed in the next moment of time, if the motion be
supposed to continue the same as at that point. At the beginning of the
time, when the body is at rest, the motion is none; and therefore, on
this theory, the space passed in the next moment is none, and thus it
will be seen that the body cannot begin to move according to the
supposed law.
A curious fact, noticed by Guido Grandi in his commentary on Galileo's
Dialogues on Motion, is that this false law of acceleration is precisely
that which would make a circular arc the shortest line of descent
between two given points; and although in general Galileo only declared
that the fall down the arc is made in less time than down the chord (in
which he is quite correct), yet in some places he seems to assert that
the circular arc is absolutely the shortest line of descent, which is
not true. It has been thought possible that the law, which on reflection
he perceived to be impossible, might have originally recommended itself
to him from his perception that it satisfied his prejudice in this
respect.
John Bernouilli, one of the first mathematicians in Europe at the
beginning of the last century, has given us a proof that such a reason
might impose even on a strong understanding, in the following argument
urged by him in favour of Galileo's second and correct theory, that the
spaces vary as the squares of the times. He had been investigating the
curve of swiftest descent, and found it to be a cycloid, the same curve
in which Huyghens had already proved that all oscillations are made in
accurately equal times. "I think it," says he, "worthy of remark that
this identity only occurs on Galileo's supposition, so that this alone
might lead us to presume it to be the real law of nature. For nature,
which always does everything in the very simplest manner, thus makes one
line do double work, whereas on any other supposition, we must have had
two lines, one for equal oscillations, the other for the shortest
descent."[140]
Venturi mentions a letter addressed to Galileo in May 1609 by Luca
Valerio, thanking him for his experiments on the descent of bodies on
inclined planes. His method of making these experiments is detailed in
the Dialogues on Motion:—"In a rule, or rather plank of wood, about
twelve yards long, half a yard broad one way, and three inches the
other, we made upon the narrow side or edge a groove of little more than
an inch wide: we cut it very straight, and, to make it very smooth and
sleek, we glued upon it a piece of vellum, polished and smoothed as
exactly as possible, and in that we let fall a very hard, round, and
smooth brass ball, raising one of the ends of the plank a yard or two at
pleasure above the horizontal plane. We observed, in the manner that I
shall tell you presently, the time which it spent in running down, and
repeated the same observation again and again to assure ourselves of the
time, in which we never found any difference, no, not so much as the
tenth part of one beat of the pulse. Having made and settled this
experiment, we let the same ball descend through a fourth part only of
the length of the groove, and found the measured time to be exactly half
the former. Continuing our experiments with other portions of the
length, comparing the fall through the whole with the fall through half,
two-thirds, three-fourths, in short, with the fall through any part, we
found by many hundred experiments that the spaces passed over were as
the squares of the times, and that this was the case in all inclinations
of the plank; during which, we also remarked that the times of descent,
on different inclinations, observe accurately the proportion assigned to
them farther on, and demonstrated by our author. As to the estimation of
the time, we hung up a great bucket full of water, which by a very small
hole pierced in the bottom squirted out a fine thread of water, which we
caught in a small glass during the whole time of the different descents:
then weighing from time to time, in an exact pair of scales, the
quantity of water caught in this way, the differences and proportions of
their weights gave the differences and proportions of the times; and
this with such exactness that, as I said before, although the
experiments were repeated again and again, they never differed in any
degree worth noticing." In order to get rid of the friction, Galileo
afterwards substituted experiments with the pendulum; but with all his
care he erred very widely in his determination of the space through
which a body would fall in 1´´, if the resistance of the air and all
other impediments were removed. He fixed it at 4 _braccia_: Mersenne has
engraved the length of the '_braccia_' used by Galileo, in his "Harmonie
Universelle," from which it appears to be about 23½ English inches, so
that Galileo's result is rather less than eight feet. Mersenne's own
result from direct observation was thirteen feet: he also made
experiments in St. Peter's at Rome, with a pendulum 325 feet long, the
vibrations of which were made in 10´´; from this the fall in 1´´ might
have been deduced rather more than sixteen feet, which is very close to
the truth.
From another letter also written in the early part of 1609, we learn
that Galileo was then busied with examining the strength and resistance
"of beams of different sizes and forms, and how much weaker they are in
the middle than at the ends, and how much greater weight they can
support laid along their whole length, than if sustained on a single
point, and of what form they should be so as to be equally strong
throughout." He was also speculating on the motion of projectiles, and
had satisfied himself that their motion in a vertical direction is
unaffected by their horizontal velocity; a conclusion which, combined
with his other experiments, led him afterwards to determine the path of
a projectile in a non-resisting medium to be parabolical.
Tartalea is supposed to have been the first to remark that no bullet
moves in a horizontal line; but his theory beyond this point was very
erroneous, for he supposed the bullet's path through the air to be made
up of an ascending and descending straight line, connected in the middle
by a circular arc.
Thomas Digges, in his treatise on the Newe Science of Great Artillerie,
came much nearer the truth; for he remarked[141], that "The bullet
violentlye throwne out of the peece by the furie of the poulder hath two
motions: the one violent, which endeuoreth to carry the bullet right out
in his line diagonall, according to the direction of the peece's axis,
from whence the violent motion proceedeth; the other naturall in the
bullet itselfe, which endeuoreth still to carrye the same directlye
downeward by a right line perpendiculare to the horizon, and which dooth
though insensiblye euen from the beginning by little and little drawe it
from that direct and diagonall course." And a little farther he observes
that "These middle curve arkes of the bullet's circuite, compounded of
the violent and naturall motions of the bullet, albeit they be indeed
mere helicall, yet have they a very great resemblance of the Arkes
Conical. And in randons above 45° they doe much resemble the Hyperbole,
and in all vnder the Ellepsis. But exactlye they neuer accorde, being
indeed Spirall mixte and Helicall."
Perhaps Digges deserves no greater credit from this latter passage than
the praise of a sharp and accurate eye, for he does not appear to have
founded this determination of the form of the curve on any theory of the
direct fall of bodies; but Galileo's arrival at the same result was
preceded, as we have seen, by a careful examination of the simplest
phenomena into which this compound motion may be resolved. But it is
time to proceed to the analysis of his "Dialogues on Motion," these
preliminary remarks on their subject matter having been merely intended
to show how long before their publication Galileo was in possession of
the principal theories contained in them.
Descartes, in one of his letters to Mersenne, insinuates that Galileo
had taken many things in these Dialogues from him: the two which he
especially instances are the isochronism of the pendulum, and the law of
the spaces varying as the squares of the times.[142] Descartes was born
in 1596: we have shown that Galileo observed the isochronism of the
pendulum in 1583, and knew the law of the spaces in 1604, although he
was then attempting to deduce it from an erroneous principle. As
Descartes on more than one occasion has been made to usurp the credit
due to Galileo, (in no instance more glaringly so than when he has been
absurdly styled the forerunner of Newton,) it will not be misplaced to
mention a few of his opinions on these subjects, recorded in his letters
to Mersenne in the collection of his letters just cited:—"I am
astonished at what you tell me of having found by experiment that bodies
thrown up in the air take neither more nor less time to rise than to
fall again; and you will excuse me if I say that I look upon the
experiment as a very difficult one to make accurately. This proportion
of increase according to the odd numbers 1, 3, 5, 7, &c., which is in
Galileo, and which I think I wrote to you some time back, cannot be
true, as I believe I intimated at the same time, unless we make two or
three suppositions which are entirely false. One is Galileo's opinion,
that motion increases gradually from the slowest degree; and the other
is, that the air makes no resistance." In a later letter to the same
person he says, apparently with some uneasiness, "I have been revising
my notes on Galileo, in which I have not said expressly, that falling
bodies do not pass through every degree of slowness, but I said that
this cannot be determined without knowing what weight is; _which comes
to the same thing_. As to your example, I grant that it proves that
every degree of velocity is infinitely divisible, but not that a falling
body actually passes through all these divisions.—It is certain that a
stone is not equally disposed to receive a new motion or increase of
velocity, when it is already moving very quickly, and when it is moving
slowly. But I believe that I am now able to determine in what proportion
the velocity of a stone increases, not when falling in a vacuum, but in
this substantial atmosphere.—However I have now got my mind full of
other things, and I cannot amuse myself with hunting this out, _nor is
it a matter of much utility_." He afterwards returns once more to the
same subject:—"As to what Galileo says, that falling bodies pass
through every degree of velocity, I do not believe that it generally
happens, but I allow it is not impossible that it may happen
occasionally." After this the reader will know what value to attach to
the following assertion by the same Descartes:—"I see nothing in
Galileo's books to envy him, and hardly any thing which I would own as
mine;" and then may judge how far Salusbury's blunt declaration is borne
out, "Where or when did any one appear that durst enter the lists with
our Galileus? save only one bold and unfortunate Frenchman, who yet no
sooner came within the ring but he was hissed out again."[143]
The principal merit of Descartes must undoubtedly be derived from the
great advances he made in what are generally termed Abstract or Pure
Mathematics; nor was he slow to point out to Mersenne and his other
friends the acknowledged inferiority of Galileo to himself in this
respect. We have not sufficient proof that this difference would have
existed if Galileo's attention had been equally directed to that object;
the singular elegance of some of his geometrical constructions indicates
great talent for this as well as for his own more favourite
speculations. But he was far more profitably employed: geometry and pure
mathematics already far outstripped any useful application of their
results to physical science, and it was the business of Galileo's life
to bring up the latter to the same level. He found abstract theorems
already demonstrated in sufficient number for his purpose, nor was there
occasion to task his genius in search of new methods of inquiry, till
all was exhausted which could be learned from those already in use. The
result of his labours was that in the age immediately succeeding
Galileo, the study of nature was no longer in arrear of the abstract
theories of number and measure; and when the genius of Newton pressed it
forward to a still higher degree of perfection, it became necessary to
discover at the same time more powerful instruments of investigation.
This alternating process has been successfully continued to the present
time; the analyst acts as the pioneer of the naturalist, so that the
abstract researches, which at first have no value but in the eyes of
those to whom an elegant formula, in its own beauty, is a source of
pleasure as real and as refined as a painting or a statue, are often
found to furnish the only means for penetrating into the most intricate
and concealed phenomena of natural philosophy.
Descartes and Delambre agree in suspecting that Galileo preferred the
dialogistic form for his treatises, because it afforded a ready
opportunity for him to praise his own inventions: the reason which he
himself gave is, the greater facility for introducing new matter and
collateral inquiries, such as he seldom failed to add each time that he
reperused his work. We shall select in the first place enough to show
the extent of his knowledge on the principal subject, motion, and shall
then allude as well as our limits will allow to the various other points
incidentally brought forward.
The dialogues are between the same speakers as in the "System of the
World;" and in the first Simplicio gives Aristotle's proof,[144] that
motion in a vacuum is impossible, because according to him bodies move
with velocities in the compound proportion of their weights and the
rarities of the mediums through which they move. And since the density
of a vacuum bears no assignable ratio to that of any medium in which
motion has been observed, any body which should employ time in moving
through the latter, would pass through the same distance in a vacuum
instantaneously, which is impossible. Salviati replies by denying the
axioms, and asserts that if a cannon ball weighing 200 lbs., and a
musket ball weighing half a pound, be dropped together from a tower 200
yards high, the former will not anticipate the latter by so much as a
foot; "and I would not have you do as some are wont, who fasten upon
some saying of mine that may want a hair's breadth of the truth, and
under this hair they seek to hide another man's blunder as big as a
cable. Aristotle says that an iron ball weighing 100 lbs. will fall from
the height of 100 yards while a weight of one pound falls but one yard:
I say they will reach the ground together. They find the bigger to
anticipate the less by two inches, and under these two inches they seek
to hide Aristotle's 99 yards." In the course of his reply to this
argument Salviati formally announces the principle which is the
foundation of the whole of Galileo's theory of motion, and which must
therefore be quoted in his own words:—"A heavy body has by nature an
intrinsic principle of moving towards the common centre of heavy things;
that is to say, to the centre of our terrestrial globe, with a motion
continually accelerated in such manner that in equal times there are
always equal additions of velocity. This is to be understood as holding
true only when all accidental and external impediments are removed,
amongst which is one that we cannot obviate, namely, the resistance of
the medium. This opposes itself, less or more, accordingly as it is to
open more slowly or hastily to make way for the moveable, which being by
its own nature, as I have said, continually accelerated, consequently
encounters a continually increasing resistance in the medium, until at
last the velocity reaches that degree, and the resistance that power,
that they balance each other; all further acceleration is prevented, and
the moveable continues ever after with an uniform and equable motion."
That such a limiting velocity is not greater than some which may be
exhibited may be proved as Galileo suggested by firing a bullet upwards,
which will in its descent strike the ground with less force than it
would have done if immediately from the mouth of the gun; for he argued
that the degree of velocity which the air's resistance is capable of
diminishing must be greater than that which could ever be reached by a
body falling naturally from rest. "I do not think the present occasion a
fit one for examining the cause of this acceleration of natural motion,
on which the opinions of philosophers are much divided; some referring
it to the approach towards the centre, some to the continual diminution
of that part of the medium remaining to be divided, some to a certain
extrusion of the ambient medium, which uniting again behind the moveable
presses and hurries it forwards. All these fancies, with others of the
like sort, we might spend our time in examining, and with little to gain
by resolving them. It is enough for our author at present that we
understand his object to be the investigation and examination of some
phenomena of a motion so accelerated, (no matter what may be the cause,)
that the momenta of velocity, from the beginning to move from rest,
increase in the simple proportion in which the time increases, which is
as much as to say, that in equal times are equal additions of velocity.
And if it shall turn out that the phenomena demonstrated on this
supposition are verified in the motion of falling and naturally
accelerated weights, we may thence conclude that the assumed definition
does describe the motion of heavy bodies, and that it is true that their
acceleration varies in the ratio of the time of motion."
When Galileo first published these Dialogues on Motion, he was obliged
to rest his demonstrations upon another principle besides, namely, that
the velocity acquired in falling down all inclined planes of the same
perpendicular height is the same. As this result was derived directly
from experiment, and from that only, his theory was so far imperfect
till he could show its consistency with the above supposed law of
acceleration. When Viviani was studying with Galileo, he expressed his
dissatisfaction at this chasm in the reasoning; the consequence of which
was, that Galileo, as he lay the same night, sleepless through
indisposition, discovered the proof which he had long sought in vain,
and introduced it into the subsequent editions. The third dialogue is
principally taken up with theorems on the direct fall of bodies, their
times of descent down differently inclined planes, which in planes of
the same height he determined to be as the lengths, and with other
inquiries connected with the same subject, such as the straight lines of
shortest descent under different data, &c.
[Illustration]
The fourth dialogue is appropriated to projectile motion, determined
upon the principle that the horizontal motion will continue the same as
if there were no vertical motion, and the vertical motion as if there
were no horizontal motion. "Let AB represent a horizontal line or plane
placed on high, on which let a body be carried with an equable motion
from A towards B, and the support of the plane being taken away at B,
let the natural motion downwards due to the body's weight come upon it
in the direction of the perpendicular BN. Moreover let the straight line
BE drawn in the direction AB be taken to represent the flow, or measure,
of the time, on which let any number of equal parts BC, CD, DE, &c. be
marked at pleasure, and from the points C, D, E, let lines be drawn
parallel to BN; in the first of these let any part CI be taken, and let
DF be taken four times as great as CI, EH nine times as great, and so
on, proportionally to the squares of the lines BC, BD, BE, &c., or, as
we say, in the double proportion of these lines. Now if we suppose that
whilst by its equable horizontal motion the body moves from B to C, it
also descends by its weight through CI, at the end of the time denoted
by BC it will be at I. Moreover in the time BD, double of BC, it will
have fallen four times as far, for in the first part of the Treatise it
has been shewn that the spaces fallen through by a heavy body vary as
the squares of the times. Similarly at the end of the time BE, or three
times BC, it will have fallen through EH, and will be at H. And it is
plain that the points I, F, H, are in the same parabolical line BIFH.
The same demonstration will apply if we take any number of equal
particles of time of whatever duration."
The curve called here a Parabola by Galileo, is one of those which
results from cutting straight through a Cone, and therefore is called
also one of the Conic Sections, the curious properties of which curves
had drawn the attention of geometricians long before Galileo thus began
to point out their intimate connexion with the phenomena of motion.
After the proposition we have just extracted, he proceeds to anticipate
some objections to the theory, and explains that the course of a
projectile will not be accurately a parabola for two reasons; partly on
account of the resistance of the air, and partly because a horizontal
line, or one equidistant from the earth's centre, is not straight, but
circular. The latter cause of difference will, however, as he says, be
insensible in all such experiments as we are able to make. The rest of
the Dialogue is taken up with different constructions for determining
the circumstances of the motion of projectiles, as their range, greatest
height, &c.; and it is proved that, with a given force of projection,
the range will be greatest when a ball is projected at an elevation of
45°, ranges of all angles equally inclined above and below 45°
corresponding exactly to each other.
[Illustration]
One of the most interesting subjects discussed in these dialogues is the
famous notion of Nature's horror of a vacuum or empty space, which the
old school of philosophy considered as impossible to be obtained.
Galileo's notions of it were very different; for although he still
unadvisedly adhered to the old phrase to denote the resistance
experienced in endeavouring to separate two smooth surfaces, he was so
far from looking upon a vacuum as an impossibility, that he has
described an apparatus by which he endeavoured to measure the force
necessary to produce one. This consisted of a cylinder, into which is
tightly fitted a piston; through the centre of the piston passes a rod
with a conical valve, which, when drawn down, shuts the aperture
closely, supporting a basket. The space between the piston and cylinder
being filled full of water poured in through the aperture, the valve is
closed, the vessel reversed, and weights are added till the piston is
drawn forcibly downwards. Galileo concluded that the weight of the
piston, rod, and added weights, would be the measure of the force of
resistance to the vacuum which he supposed would take place between the
piston and lower surface of the water. The defects in this apparatus for
the purpose intended are of no consequence, so far as regards the
present argument, and it is perhaps needless to observe that he was
mistaken in supposing the water would not descend with the piston. This
experiment occasions a remark from Sagredo, that he had observed that a
lifting-pump would not work when the water in the cistern had sunk to
the depth of thirty-five feet below the valve; that he thought the pump
was injured, and sent for the maker of it, who assured him that no pump
upon that construction would lift water from so great a depth. This
story is sometimes told of Galileo, as if he had said sneeringly on this
occasion that Nature's horror of a vacuum does not extend beyond
thirty-five feet; but it is very plain that if he had made such an
observation, it would have been seriously; and in fact by such a
limitation he deprived the notion of the principal part of its
absurdity. He evidently had adopted the common notion of suction, for he
compares the column of water to a rod of metal suspended from its upper
end, which may be lengthened till it breaks with its own weight. It is
certainly very extraordinary that he failed to observe how simply these
phenomena may be explained by a reference to the weight of the elastic
atmosphere, which he was perfectly well acquainted with, and endeavoured
by the following ingenious experiment to determine:—"Take a large glass
flask with a bent neck, and round its mouth tie a leathern pipe with a
valve in it, through which water may be forced into the flask with a
syringe without suffering any air to escape, so that it will be
compressed within the bottle. It will be found difficult to force in
more than about three-fourths of what the flask will hold, which must be
carefully weighed. The valve must then be opened, and just so much air
will rush out as would in its natural density occupy the space now
filled by the water. Weigh the vessel again; the difference will show
the weight of that quantity of air."[145] By these means, which the
modern experimentalist will see were scarcely capable of much accuracy,
Galileo found that air was four hundred times lighter than water,
instead of ten times, which was the proportion fixed on by Aristotle.
The real proportion is about 830 times.
The true theory of the rise of water in a lifting-pump is commonly dated
from Torricelli's famous experiment with a column of mercury, in 1644,
when he found that the greatest height at which it would stand is
fourteen times less than the height at which water will stand, which is
exactly the proportion of weight between water and mercury. The
following curious letter from Baliani, in 1630, shows that the original
merit of suggesting the real cause belongs to him, and renders it still
more unaccountable that Galileo, to whom it was addressed, should not at
once have adopted the same view of the subject:—"I have believed that a
vacuum may exist naturally ever since I knew that the air has sensible
weight, and that you taught me in one of your letters how to find its
weight exactly, though I have not yet succeeded with that experiment.
From that moment I took up the notion that it is not repugnant to the
nature of things that there should be a vacuum, but merely that it is
difficult to produce. To explain myself more clearly: if we allow that
the air has weight, there is no difference between air and water except
in degree. At the bottom of the sea the weight of the water above me
compresses everything round my body, and it strikes me that the same
thing must happen in the air, we being placed at the bottom of its
immensity; we do not feel its weight, nor the compression round us,
because our bodies are made capable of supporting it. But if we were in
a vacuum, then the weight of the air above our heads would be felt. It
would be felt very great, but not infinite, and therefore determinable,
and it might be overcome by a force proportioned to it. In fact I
estimate it to be such that, to make a vacuum, I believe we require a
force greater than that of a column of water thirty feet high."[146]
[Illustration]
This subject is introduced by some observations on the force of
cohesion, Galileo seeming to be of opinion that, although it cannot be
adequately accounted for by "the great and principal resistance to a
vacuum, yet that perhaps a sufficient cause may be found by considering
every body as composed of very minute particles, between every two of
which is exerted a similar resistance." This remark serves to lead to a
discussion on indivisibles and infinite quantities, of which we shall
merely extract what Galileo gives as a curious paradox suggested in the
course of it. He supposes a basin to be formed by scooping a hemisphere
out of a cylinder, and a cone to be taken of the same depth and base as
the hemisphere. It is easy to show, if the cone and scooped cylinder be
both supposed to be cut by the same plane, parallel to the one on which
both stand, that the area of the ring CDEF thus discovered in the
cylinder is equal to the area of the corresponding circular section AB
of the cone, wherever the cutting plane is supposed to be.[147] He then
proceeds with these remarkable words:—"If we raise the plane higher and
higher, one of these areas terminates in the circumference of a circle,
and the other in a point, for such are the upper rim of the basin and
the top of the cone. Now since in the diminution of the two areas they
to the very last maintain their equality to one another, it is in my
thoughts proper to say that the highest and ultimate terms[148] of such
diminutions are equal, and not one infinitely bigger than the other. It
seems therefore that the circumference of a large circle may be said to
be equal to one single point. And why may not these be called equal if
they be the last remainders and vestiges left by equal magnitudes[149]?"
We think no one can refuse to admit the probability, that Newton may
have found in such passages as these the first germ of the idea of his
prime and ultimate ratios, which afterwards became in his hands an
instrument of such power. As to the paradoxical result, Descartes
undoubtedly has given the true answer to it in saying that it only
proves that the line is not a greater area than the point is. Whilst on
this subject, it may not be uninteresting to remark that something
similar to the doctrine of fluxions seems to have been lying dormant in
the minds of the mathematicians of Galileo's era, for Inchoffer
illustrates his argument in the treatise we have already mentioned, that
the Copernicans may deduce some true results from what he terms their
absurd hypothesis, by observing, that mathematicians may deduce the
truth that a line is length without breadth, from the false and
physically impossible supposition that a point flows, and that a line is
the fluxion of a point.[150]
A suggestion that perhaps fire dissolves bodies by insinuating itself
between their minute particles, brings on the subject of the violent
effects of heat and light; on which Sagredo inquires, whether we are to
take for granted that the effect of light does or does not require time.
Simplicio is ready with an answer, that the discharge of artillery
proves the transmission of light to be instantaneous, to which Sagredo
cautiously replies, that nothing can be gathered from that experiment
except that light travels more swiftly than sound; nor can we draw any
decisive conclusion from the rising of the sun. "Who can assure us that
he is not in the horizon before his rays reach our sight?" Salviati then
mentions an experiment by which he endeavoured to examine this question.
Two observers are each to be furnished with a lantern: as soon as the
first shades his light, the second is to discover his, and this is to be
repeated at a short distance till the observers are perfect in the
practice. The same thing is to be tried at the distance of several
miles, and if the first observer perceive any delay between shading his
own light and the appearance of his companion's, it is to be attributed
to the time taken by the light in traversing twice the distance between
them. He allows that he could discover no perceptible interval at the
distance of a mile, at which he had tried the experiment, but recommends
that with the help of a telescope it should be tried at much greater
distances. Sir Kenelm Digby remarks on this passage: "It may be objected
(if there be some observable tardity in the motion of light) that the
sunne would never be truly in that place in which unto our eyes he
appeareth to be; because that it being seene by means of the light which
issueth from it, if that light required time to move in, the sunne
(whose motion is so swifte) would be removed from the place where the
light left it, before it could be with us to give tidings of him. To
this I answer, allowing peradventure that it may be so, who knoweth the
contrary? Or what inconvenience would follow if it be admitted[151]?"
The principal thing remaining to be noticed is the application of the
theory of the pendulum to musical concords and dissonances, which are
explained, in the same manner as by Kepler in his "Harmonices Mundi," to
result from the concurrence or opposition of vibrations in the air
striking upon the drum of the ear. It is suggested that these vibrations
may be made manifest by rubbing the finger round a glass set in a large
vessel of water; "and if by pressure the note is suddenly made to rise
to the octave above, every one of the undulations which will be seen
regularly spreading round the glass, will suddenly split into two,
proving that the vibrations that occasion the octave are double those
belonging to the simple note." Galileo then describes a method he
discovered by accident of measuring the length of these waves more
accurately than can be done in the agitated water. He was scraping a
brass plate with an iron chisel, to take out some spots, and moving the
tool rapidly upon the plate, he occasionally heard a hissing and
whistling sound, very shrill and audible, and whenever this occurred,
and then only, he observed the light dust on the plate to arrange itself
in a long row of small parallel streaks equidistant from each other. In
repeated experiments he produced different tones by scraping with
greater or less velocity, and remarked that the streaks produced by the
acute sounds stood closer together than those from the low notes. Among
the sounds produced were two, which by comparison with a viol he
ascertained to differ by an exact fifth; and measuring the spaces
occupied by the streaks in both experiments, he found thirty of the one
equal to forty-five of the other, which is exactly the known proportion
of the lengths of strings of the same material which sound a fifth to
each other.[152]
Salviati also remarks, that if the material be not the same, as for
instance if it be required to sound an octave to a note on catgut, on a
wire of the same length, the weight of the wire must be made four times
as great, and so for other intervals. "The immediate cause of the forms
of musical intervals is neither the length, the tension, nor the
thickness, but the proportion of the numbers of the undulations of the
air which strike upon the drum of the ear, and make it vibrate in the
same intervals. Hence we may gather a plausible reason of the different
sensations occasioned to us by different couples of sounds, of which we
hear some with great pleasure, some with less, and call them accordingly
concords, more or less perfect, whilst some excite in us great
dissatisfaction, and are called discords. The disagreeable sensation
belonging to the latter probably arises from the disorderly manner in
which the vibrations strike the drum of the ear; so that for instance a
most cruel discord would be produced by sounding together two strings,
of which the lengths are to each other as the side and diagonal of a
square, which is the discord of the false fifth. On the contrary,
agreeable consonances will result from those strings of which the
numbers of vibrations made in the same time are commensurable, "to the
end that the cartilage of the drum may not undergo the incessant torture
of a double inflexion from the disagreeing percussions." Something
similar may be exhibited to the eye by hanging up pendulums of different
lengths: "if these be proportioned so that the times of their vibrations
correspond with those of the musical concords, the eye will observe with
pleasure their crossings and interweavings still recurring at
appreciable intervals; but if the times of vibration be incommensurate,
the eye will be wearied and worn out with following them."
The second dialogue is occupied entirely with an investigation of the
strength of beams, a subject which does not appear to have been examined
by any one before Galileo beyond Aristotle's remark, that long beams are
weaker, because they are at once the weight, the lever, and the fulcrum;
and it is in the development of this observation that the whole theory
consists. The principle assumed by Galileo as the basis of his inquiries
is, that the force of cohesion with which a beam resists a cross
fracture in any section may all be considered as acting at the centre of
gravity of the section, and that it breaks always at the lowest point:
from this he deduced that the effect of the weight of a prismatic beam
in overcoming the resistance of one end by which it is fastened to a
wall, varies directly as the square of the length, and inversely as the
side of the base. From this it immediately follows, that if for instance
the bone of a large animal be three times as long as the corresponding
one in a smaller beast, it must be nine times as thick to have the same
strength, provided we suppose in both cases that the materials are of
the same consistence. An elegant result which Galileo also deduced from
this theory, is that the form of such a beam, to be equally strong in
every part, should be that of a parabolical prism, the vertex of the
parabola being the farthest removed from the wall. As an easy mode of
describing the parabolic curve for this purpose, he recommends tracing
the line in which a heavy flexible string hangs. This curve is not an
accurate parabola: it is now called a catenary; but it is plain from the
description of it in the fourth dialogue, that Galileo was perfectly
aware that this construction is only approximately true. In the same
place he makes the remark, which to many is so paradoxical, that no
force, however great, exerted in a horizontal direction, can stretch a
heavy thread, however slender, into an accurately straight line.
The fifth and sixth dialogues were left unfinished, and annexed to the
former ones by Viviani after Galileo's death: the fragment of the fifth,
which is on the subject of Euclid's Definition of Ratio, was at first
intended to have formed a part of the third, and followed the first
proposition on equable motion: the sixth was intended to have embodied
Galileo's researches on the nature and laws of Percussion, on which he
was employed at the time of his death. Considering these solely as
fragments, we shall not here make any extracts from them.
FOOTNOTES:
[140] Joh. Bernouilli, Opera Omnia, Lausannæ, 1744. tom. i. p. 192.
[141] Pantometria, 1591.
[142] Lettres de Descartes. Paris, 1657.
[143] Math. Coll. vol. ii.
[144] Phys. Lib. iv. c. 8.
[145] It has been recently proposed to determine the density of
high-pressure steam by a process analogous to this.
[146] Venturi, vol. ii.
[147] Galileo also reasons in the same way on the equality of the solids
standing on the cutting plane, but one is sufficient for our present
purpose.
[148] Gli altissimi e ultimi termini.
[149] Le ultime reliquie e vestigie lasciate da grandezze eguali.
[150] Punctum fluere, et lineam esse fluxum puncti. Tract. Syllept.
Romæ, 1633.
[151] "Treatise of the Nature of Bodies. London, 1665."
[152] This beautiful experiment is more easily tried by drawing the bow
of a violin across the edge of glass strewed with fine dry sand. Those
who wish to see more on the subject may consult Chladni's 'Acoustique.'