of the resistance of the air, a statement of their rates of fall is of
interest. In one second a freely falling body near the earth is found to
drop 16 feet. In two seconds it drops 64 feet altogether, viz. 16 feet
in the first, and 48 feet in the next second; because at the beginning
of every second after the first it has the accumulated velocity of
preceding seconds. The height fallen by a dropped body is not
proportional to the time simply, but to what is rather absurdly called
the square of the time, _i.e._ the time multiplied by itself.
For instance, in 3 seconds it drops 9 x 16 = 144 feet; in 4 seconds 16 x
16, or 256 feet, and so on. The distances travelled in 1, 2, 3, 4, &c.,
seconds by a body dropped from rest and not appreciably resisted by the
air, are 1, 4, 9, 16, 25, &c., respectively, each multiplied by the
constant 16 feet.
Another way of stating the law is to say that the heights travelled in
successive seconds proceed in the proportion 1, 3, 5, 7, 9, &c.; again
multiplied by 16 feet in each case.
[Illustration: FIG. 35.--Curve described by a projectile, showing how it
drops from the line of fire, _O D_, in successive seconds, the same
distances _AP_, _BQ_, _CR_, &c., as are stated above for a dropped
body.]
All this was experimentally established by Galileo.
A body takes half a second to drop 4 feet; and a quarter of a second to
drop 1 foot. The easiest way of estimating a quarter of a second with
some accuracy is to drop a bullet one foot.
A bullet thrown or shot in any direction falls just as much as if merely
dropped; but instead of falling from the starting-point it drops
vertically from the line of fire. (See fig. 35).
The rate of fall depends on the intensity of gravity; if it could be
doubled, a body would fall twice as far in the same time; but to make it
fall a given distance in half the time the intensity of gravity would
have to be quadrupled. At a place where the intensity of gravity is
1/3600 of what it is here, a body would fall as far in a minute as it
now falls in a second. Such a place occurs at about the distance of the
moon (_cf._ page 177).
The fact that the height fallen through is proportional to the square
of the time proves that the attraction of the earth or the intensity of
gravity is sensibly constant throughout ordinary small ranges. Over
great distances of fall, gravity cannot be considered constant; so for
things falling through great spaces the Galilean law of the square of
the time does not hold.
The fact that things near the earth fall 16 feet in the first second
proves that the intensity of ordinary terrestrial gravity is 32 British
units of force per pound of matter.
The fact that all bodies fall at the same rate (when the resistance of
the air is eliminated), proves that weight is proportional to mass; or
more explicitly, that the gravitative attraction of the earth on matter
near its surface depends on the amount of that matter, as estimated by
its inertia, and on nothing else.
LECTURE IV
GALILEO AND THE INVENTION OF THE TELESCOPE
Contemporary with the life of Kepler, but overlapping it at both ends,
comes the great and eventful life of Galileo Galilei,[5] a man whose
influence on the development of human thought has been greater than that
of any man whom we have yet considered, and upon whom, therefore, it is
necessary for us, in order to carry out the plan of these lectures, to
bestow much time. A man of great and wide culture, a so-called universal
genius, it is as an experimental philosopher that he takes the first
rank. In this capacity he must be placed alongside of Archimedes, and it
is pretty certain that between the two there was no man of magnitude
equal to either in experimental philosophy. It is perhaps too bold a
speculation, but I venture to doubt whether in succeeding generations we
find his equal in the domain of purely experimental science until we
come to Faraday. Faraday was no doubt his superior, but I know of no
other of whom the like can unhesitatingly be said. In mathematical and
deductive science, of course, it is quite otherwise. Kepler, for
instance, and many men before and since, have far excelled Galileo in
mathematical skill and power, though at the same time his achievements
in this department are by no means to be despised.
Born at Pisa three centuries ago, on the very day that Michael Angelo
lay dying in Rome, he inherited from his father a noble name, cultivated
tastes, a keen love of truth, and an impoverished patrimony. Vincenzo de
Galilei, a descendant of the important Bonajuti family, was himself a
mathematician and a musician, and in a book of his still extant he
declares himself in favour of free and open inquiry into scientific
matters, unrestrained by the weight of authority and tradition.
In all probability the son imbibed these precepts: certainly he acted on
them.
Vincenzo, having himself experienced the unremunerative character of
scientific work, had a horror of his son's taking to it, especially as
in his boyhood he was always constructing ingenious mechanical toys, and
exhibiting other marks of precocity. So the son was destined for
business--to be, in fact, a cloth-dealer. But he was to receive a good
education first, and was sent to an excellent convent school.
Here he made rapid progress, and soon excelled in all branches of
classics and literature. He delighted in poetry, and in later years
wrote several essays on Dante, Tasso, and Ariosto, besides composing
some tolerable poems himself. He played skilfully on several musical
instruments, especially on the lute, of which indeed he became a master,
and on which he solaced himself when quite an old man. Besides this he
seems to have had some skill as an artist, which was useful afterwards
in illustrating his discoveries, and to have had a fine sensibility as
an art critic, for we find several eminent painters of that day
acknowledging the value of the opinion of the young Galileo.
Perceiving all this display of ability, the father wisely came to the
conclusion that the selling of woollen stuffs would hardly satisfy his
aspirations for long, and that it was worth a sacrifice to send him to
the University. So to the University of his native town he went, with
the avowed object of studying medicine, that career seeming the most
likely to be profitable. Old Vincenzo's horror of mathematics or science
as a means of obtaining a livelihood is justified by the fact that while
the University Professor of Medicine received 2,000 scudi a year, the
Professor of Mathematics had only 60, that is L13 a year, or 7-1/2_d._ a
day.
So the son had been kept properly ignorant of such poverty-stricken
subjects, and to study medicine he went.
But his natural bent showed itself even here. For praying one day in the
Cathedral, like a good Catholic as he was all his life, his attention
was arrested by the great lamp which, after lighting it, the verger had
left swinging to and fro. Galileo proceeded to time its swings by the
only watch he possessed--viz., his own pulse. He noticed that the time
of swing remained as near as he could tell the same, notwithstanding the
fact that the swings were getting smaller and smaller.
By subsequent experiment he verified the law, and the isochronism of the
pendulum was discovered. An immensely important practical discovery
this, for upon it all modern clocks are based; and Huyghens soon applied
it to the astronomical clock, which up to that time had been a crude and
quite untrustworthy instrument.
The best clock which Tycho Brahe could get for his observatory was
inferior to one that may now be purchased for a few shillings; and this
change is owing to the discovery of the pendulum by Galileo. Not that he
applied it to clocks; he was not thinking of astronomy, he was thinking
of medicine, and wanted to count people's pulses. The pendulum served;
and "pulsilogies," as they were called, were thus introduced to and used
by medical practitioners.
The Tuscan Court came to Pisa for the summer months, for it was then a
seaside place, and among the suite was Ostillio Ricci, a distinguished
mathematician and old friend of the Galileo family. The youth visited
him, and one day, it is said, heard a lesson in Euclid being given by
Ricci to the pages while he stood outside the door entranced. Anyhow he
implored Ricci to help him into some knowledge of mathematics, and the
old man willingly consented. So he mastered Euclid and passed on to
Archimedes, for whom he acquired a great veneration.
His father soon heard of this obnoxious proclivity, and did what he
could to divert him back to medicine again. But it was no use.
Underneath his Galen and Hippocrates were secreted copies of Euclid and
Archimedes, to be studied at every available opportunity. Old Vincenzo
perceived the bent of genius to be too strong for him, and at last gave
way.
[Illustration: FIG. 36.--Two forms of pulsilogy. The string is wound up
till the swinging weight keeps time with the pulse, and the position of
a bead or of an index connected with the string is then read on a scale
or dial.]
With prodigious rapidity the released philosopher now assimilated the
elements of mathematics and physics, and at twenty-six we find him
appointed for three years to the University Chair of Mathematics, and
enjoying the paternally dreaded stipend of 7-1/2_d._ a day.
Now it was that he pondered over the laws of falling bodies. He
verified, by experiment, the fact that the velocity acquired by falling
down any slope of given height was independent of the angle of slope.
Also, that the height fallen through was proportional to the square of
the time.
Another thing he found experimentally was that all bodies, heavy and
light, fell at the same rate, striking the ground at the same time.[6]
Now this was clean contrary to what he had been taught. The physics of
those days were a simple reproduction of statements in old books.
Aristotle had asserted certain things to be true, and these were
universally believed. No one thought of trying the thing to see if it
really were so. The idea of making an experiment would have savoured of
impiety, because it seemed to tend towards scepticism, and cast a doubt
on a reverend authority.
Young Galileo, with all the energy and imprudence of youth (what a
blessing that youth has a little imprudence and disregard of
consequences in pursuing a high ideal!), as soon as he perceived that
his instructors were wrong on the subject of falling bodies, instantly
informed them of the fact. Whether he expected them to be pleased or not
is a question. Anyhow, they were not pleased, but were much annoyed by
his impertinent arrogance.
It is, perhaps, difficult for us now to appreciate precisely their
position. These doctrines of antiquity, which had come down hoary with
age, and the discovery of which had reawakened learning and quickened
intellectual life, were accepted less as a science or a philosophy, than
as a religion. Had they regarded Aristotle as a verbally inspired
writer, they could not have received his statements with more
unhesitating conviction. In any dispute as to a question of fact, such
as the one before us concerning the laws of falling bodies, their method
was not to make an experiment, but to turn over the pages of Aristotle;
and he who could quote chapter and verse of this great writer was held
to settle the question and raise it above the reach of controversy.
It is very necessary for us to realize this state of things clearly,
because otherwise the attitude of the learned of those days towards
every new discovery seems stupid and almost insane. They had a
crystallized system of truth, perfect, symmetrical--it wanted no
novelty, no additions; every addition or growth was an imperfection, an
excrescence, a deformity. Progress was unnecessary and undesired. The
Church had a rigid system of dogma, which must be accepted in its
entirety on pain of being treated as a heretic. Philosophers had a
cast-iron system of truth to match--a system founded upon Aristotle--and
so interwoven with the great theological dogmas that to question one was
almost equivalent to casting doubt upon the other.
In such an atmosphere true science was impossible. The life-blood of
science is growth, expansion, freedom, development. Before it could
appear it must throw off these old shackles of centuries. It must burst
its old skin, and emerge, worn with the struggle, weakly and
unprotected, but free and able to grow and to expand. The conflict was
inevitable, and it was severe. Is it over yet? I fear not quite, though
so nearly as to disturb science hardly at all. Then it was different; it
was terrible. Honour to the men who bore the first shock of the battle!
Now Aristotle had said that bodies fell at rates depending on their
weight.
A 5 lb. weight would fall five times as quick as a 1 lb. weight; a 50
lb. weight fifty times as quick, and so on.
Why he said so nobody knows. He cannot have tried. He was not above
trying experiments, like his smaller disciples; but probably it never
occurred to him to doubt the fact. It seems so natural that a heavy body
should fall quicker than a light one; and perhaps he thought of a stone
and a feather, and was satisfied.
Galileo, however, asserted that the weight did not matter a bit, that
everything fell at the same rate (even a stone and a feather, but for
the resistance of the air), and would reach the ground in the same time.
And he was not content to be pooh-poohed and snubbed. He knew he was
right, and he was determined to make every one see the facts as he saw
them. So one morning, before the assembled University, he ascended the
famous leaning tower, taking with him a 100 lb. shot and a 1 lb. shot.
He balanced them on the edge of the tower, and let them drop together.
Together they fell, and together they struck the ground.
The simultaneous clang of those two weights sounded the death-knell of
the old system of philosophy, and heralded the birth of the new.
But was the change sudden? Were his opponents convinced? Not a jot.
Though they had seen with their eyes, and heard with their ears, the
full light of heaven shining upon them, they went back muttering and
discontented to their musty old volumes and their garrets, there to
invent occult reasons for denying the validity of the observation, and
for referring it to some unknown disturbing cause.
They saw that if they gave way on this one point they would be letting
go their anchorage, and henceforward would be liable to drift along with
the tide, not knowing whither. They dared not do this. No; they _must_
cling to the old traditions; they could not cast away their rotting
ropes and sail out on to the free ocean of God's truth in a spirit of
fearless faith.
[Illustration: FIG. 37.--Tower of Pisa.]
Yet they had received a shock: as by a breath of fresh salt breeze and
a dash of spray in their faces, they had been awakened out of their
comfortable lethargy. They felt the approach of a new era.
Yes, it was a shock; and they hated the young Galileo for giving it
them--hated him with the sullen hatred of men who fight for a lost and
dying cause.
We need scarcely blame these men; at least we need not blame them
overmuch. To say that they acted as they did is to say that they were
human, were narrow-minded, and were the apostles of a lost cause. But
_they_ could not know this; _they_ had no experience of the past to
guide them; the conditions under which they found themselves were novel,
and had to be met for the first time. Conduct which was excusable then
would be unpardonable now, in the light of all this experience to guide
us. Are there any now who practically repeat their error, and resist new
truth? who cling to any old anchorage of dogma, and refuse to rise with
the tide of advancing knowledge? There may be some even now.
Well, the unpopularity of Galileo smouldered for a time, until, by
another noble imprudence, he managed to offend a semi-royal personage,
Giovanni de Medici, by giving his real opinion, when consulted, about a
machine which de Medici had invented for cleaning out the harbour of
Leghorn. He said it was as useless as it in fact turned out to be.
Through the influence of the mortified inventor he lost favour at Court;
and his enemies took advantage of the fact to render his chair
untenable. He resigned before his three years were up, and retired to
Florence.
His father at this time died, and the family were left in narrow
circumstances. He had a brother and three sisters to provide for.
He was offered a professorship at Padua for six years by the Senate of
Venice, and willingly accepted it.
Now began a very successful career. His introductory address was marked
by brilliant eloquence, and his lectures soon acquired fame. He wrote
for his pupils on the laws of motion, on fortifications, on sundials, on
mechanics, and on the celestial globe: some of these papers are now
lost, others have been printed during the present century.
Kepler sent him a copy of his new book, _Mysterium Cosmographicum_, and
Galileo in thanking him for it writes him the following letter:--[7]
"I count myself happy, in the search after truth, to have so great
an ally as yourself, and one who is so great a friend of the truth
itself. It is really pitiful that there are so few who seek truth,
and who do not pursue a perverse method of philosophising. But this
is not the place to mourn over the miseries of our times, but to
congratulate you on your splendid discoveries in confirmation of
truth. I shall read your book to the end, sure of finding much that
is excellent in it. I shall do so with the more pleasure, because
_I have been for many years an adherent of the Copernican system_,
and it explains to me the causes of many of the appearances of
nature which are quite unintelligible on the commonly accepted
hypothesis. _I have collected many arguments for the purpose of
refuting the latter_; but I do not venture to bring them to the
light of publicity, for fear of sharing the fate of our master,
Copernicus, who, although he has earned immortal fame with some,
yet with very many (so great is the number of fools) has become an
object of ridicule and scorn. I should certainly venture to publish
my speculations if there were more people like you. But this not
being the case, I refrain from such an undertaking."
Kepler urged him to publish his arguments in favour of the Copernican
theory, but he hesitated for the present, knowing that his declaration
would be received with ridicule and opposition, and thinking it wiser to
get rather more firmly seated in his chair before encountering the
storm of controversy.
The six years passed away, and the Venetian Senate, anxious not to lose
so bright an ornament, renewed his appointment for another six years at
a largely increased salary.
Soon after this appeared a new star, the stella nova of 1604, not the
one Tycho had seen--that was in 1572--but the same that Kepler was so
much interested in.
Galileo gave a course of three lectures upon it to a great audience. At
the first the theatre was over-crowded, so he had to adjourn to a hall
holding 1000 persons. At the next he had to lecture in the open air.
He took occasion to rebuke his hearers for thronging to hear about an
ephemeral novelty, while for the much more wonderful and important
truths about the permanent stars and facts of nature they had but deaf
ears.
But the main point he brought out concerning the new star was that it
upset the received Aristotelian doctrine of the immutability of the
heavens. According to that doctrine the heavens were unchangeable,
perfect, subject neither to growth nor to decay. Here was a body, not a
meteor but a real distant star, which had not been visible and which
would shortly fade away again, but which meanwhile was brighter than
Jupiter.
The staff of petrified professorial wisdom were annoyed at the
appearance of the star, still more at Galileo's calling public attention
to it; and controversy began at Padua. However, he accepted it; and now
boldly threw down the gauntlet in favour of the Copernican theory,
utterly repudiating the old Ptolemaic system which up to that time he
had taught in the schools according to established custom.
The earth no longer the only world to which all else in the firmament
were obsequious attendants, but a mere insignificant speck among the
host of heaven! Man no longer the centre and cynosure of creation, but,
as it were, an insect crawling on the surface of this little speck! All
this not set down in crabbed Latin in dry folios for a few learned
monks, as in Copernicus's time, but promulgated and argued in rich
Italian, illustrated by analogy, by experiment, and with cultured wit;
taught not to a few scholars here and there in musty libraries, but
proclaimed in the vernacular to the whole populace with all the energy
and enthusiasm of a recent convert and a master of language! Had a
bombshell been exploded among the fossilized professors it had been less
disturbing.
But there was worse in store for them.
A Dutch optician, Hans Lippershey by name, of Middleburg, had in his
shop a curious toy, rigged up, it is said, by an apprentice, and made
out of a couple of spectacle lenses, whereby, if one looked through it,
the weather-cock of a neighbouring church spire was seen nearer and
upside down.
The tale goes that the Marquis Spinola, happening to call at the shop,
was struck with the toy and bought it. He showed it to Prince Maurice of
Nassau, who thought of using it for military reconnoitring. All this is
trivial. What is important is that some faint and inaccurate echo of
this news found its way to Padua, and into the ears of Galileo.
The seed fell on good soil. All that night he sat up and pondered. He
knew about lenses and magnifying glasses. He had read Kepler's theory of
the eye, and had himself lectured on optics. Could he not hit on the
device and make an instrument capable of bringing the heavenly bodies
nearer? Who knew what marvels he might not so perceive! By morning he
had some schemes ready to try, and one of them was successful.
Singularly enough it was not the same plan as the Dutch optician's, it
was another mode of achieving the same end.
He took an old small organ pipe, jammed a suitably chosen spectacle
glass into either end, one convex the other concave, and behold, he had
the half of a wretchedly bad opera glass capable of magnifying three
times. It was better than the Dutchman's, however; it did not invert.
It is easy to understand the general principle of a telescope. A
general knowledge of the common magnifying glass may be assumed.
Roger Bacon knew about lenses; and the ancients often refer to
them, though usually as burning glasses. The magnifying power of
globes of water must have been noticed soon after the discovery of
glass and the art of working it.
A magnifying glass is most simply thought of as an additional lens
to the eye. The eye has a lens by which ordinary vision is
accomplished, an extra glass lens strengthens it and enables
objects to be seen nearer and therefore apparently bigger. But to
apply a magnifying glass to distant objects is impossible. In order
to magnify distant objects, another function of lenses has also to
be employed, viz., their power of forming real images, the power on
which their use as burning-glasses depends: for the best focus is
an image of the sun. Although the object itself is inaccessible,
the image of it is by no means so, and to the image a magnifier can
be applied. This is exactly what is done in the telescope; the
object glass or large lens forms an image, which is then looked at
through a magnifying glass or eye-piece.
Of course the image is nothing like so big as the object. For
astronomical objects it is almost infinitely less; still it is an
exact representation at an accessible place, and no one expects a
telescope to show distant bodies as big as they really are. All it
does is to show them bigger than they could be seen without it.
But if the objects are not distant, the same principle may still be
applied, and two lenses may be used, one to form an image, the
other to magnify it; only if the object can be put where we please,
we can easily place it so that its image is already much bigger
than the object even before magnification by the eye lens. This is
the compound microscope, the invention of which soon followed the
telescope. In fact the two instruments shade off into one another,
so that the reading telescope or reading microscope of a laboratory
(for reading thermometers, and small divisions generally) goes by
either name at random.