_Correspondence on Longitudes.—Pendulum Clock._
IN the spring of 1636, having finished his Dialogues on Motion, Galileo
resumed the plan of determining the longitude by means of Jupiter's
satellites. Perhaps he suspected something of the private intrigue which
thwarted his former expectations from the Spanish government, and this
may have induced him on the present occasion to negotiate the matter
without applying for Ferdinand's assistance and recommendation.
Accordingly he addressed himself to Lorenz Real, who had been Governor
General of the Dutch possessions in India, freely and unconditionally
offering the use of his theory to the States General of Holland. Not
long before, his opinion had been requested by the commissioners
appointed at Paris to examine and report on the practicability of
another method proposed by Morin,[153] which consisted in observing the
distance of the moon from a known star. Morin was a French philosopher,
principally known as an astrologer and zealous Anti-Copernican; but his
name deserves to be recorded as undoubtedly one of the first to
recommend a method, which, under the name of a Lunar distance, is now in
universal practice.
The monthly motion of the moon is so rapid, that her distance from a
given star sensibly varies in a few minutes even to the unassisted eye;
and with the aid of the telescope, we can of course appreciate the
change more accurately. Morin proposed that the distances of the moon
from a number of fixed stars lying near her path in the heavens should
be beforehand calculated and registered for every day in the year, at a
certain hour, in the place from which the longitudes were to be
reckoned, as for instance at Paris. Just as in the case of the eclipses
of Jupiter's satellites, the observer, when he saw that the moon had
arrived at the registered distance, would know the hour at Paris: he
might also make allowance for intermediate distances. Observing at the
same instant the hour on board his ship, the difference between the two
would show his position in regard of longitude. In using this method as
it is now practised, several modifications are to be attended to,
without which it would be wholly useless, in consequence of the
refraction of the atmosphere, and the proximity of the moon to the
earth. Owing to the latter cause, if two spectators should at the same
instant of time, but in different places, measure the distance of the
moon in the East, from a star still more to the eastward, it would
appear greater to the more easterly spectator than to the other
observer, who as seen from the star would be standing more directly
behind the moon. The mode of allowing for these alterations is taught by
trigonometry and astronomy.
The success of this method depends altogether upon the exact knowledge
which we now have of the moon's course, and till that knowledge was
perfected it would have been found altogether illusory. Such in fact was
the judgment which Galileo pronounced upon it. "As to Morin's book on
the method of finding the longitude by means of the moon's motion, I say
freely that I conceive this idea to be as accurate in theory, as
fallacious and impossible in practice. I am sure that neither you nor
any one of the other four gentlemen can doubt the possibility of finding
the difference of longitude between two meridians by means of the moon's
motion, provided we are sure of the following requisites: First, an
Ephemeris of the moon's motion exactly calculated for the first meridian
from which the others are to be reckoned; secondly, exact instruments,
and convenient to handle, in taking the distance between the moon and a
fixed star; thirdly, great practical skill in the observer; fourthly,
not less accuracy in the scientific calculations, and astronomical
computations; fifthly, very perfect clocks to number the hours, or other
means of knowing them exactly, &c. Supposing, I say, all these elements
free from error, the longitude will be accurately found; but I reckon it
more easy and likely to err in all of these together, than to be
practically right in one alone. Morin ought to require his judges to
assign, at their pleasure, eight or ten moments of different nights
during four or six months to come, and pledge himself to predict and
assign by his calculations the distances of the moon at those determined
instants from some star which would then be near her. If it is found
that the distances assigned by him agree with those which the quadrant
or sextant[154] will actually show, the judges would be satisfied of his
success, or rather of the truth of the matter, and nothing would remain
but to show that his operations were such as could be performed by men
of moderate skill, and also practicable at sea as well as on land. I
incline much to think that an experiment of this kind would do much
towards abating the opinion and conceit which Morin has of himself,
which appears to me so lofty, that I should consider myself the eighth
sage, if I knew the half of what Morin presumes to know."
It is probable that Galileo was biassed by a predilection for his own
method, on which he had expended so much time and labour; but the
objections which he raises against Morin's proposal in the foregoing
letter are no other than those to which at that period it was
undoubtedly open. With regard to his own, he had already, in 1612, given
a rough prediction of the course of Jupiter's satellites, which had been
found to agree tolerably well with subsequent observations; and since
that time, amid all his other employments, he had almost
unintermittingly during twenty-four years continued his observations,
for the sake of bringing the tables of their motions to as high a state
of perfection as possible. This was the point to which the inquiries of
the States in their answer to Galileo's frank proposal were principally
directed. They immediately appointed commissioners to communicate with
him, and report the various points on which they required information.
They also sent him a golden chain, and assured him that in the case of
the design proving successful, he should have no cause to complain of
their want of gratitude and generosity. The commissioners immediately
commenced an active correspondence with him, in the course of which he
entered into more minute details with regard to the methods by which he
proposed to obviate the practical difficulties of the necessary
observations.
It is worth noticing that the secretary to the Prince of Orange, who was
mainly instrumental in forming this commission, was Constantine
Huyghens, father of the celebrated mathematician of that name, of whom
it has been said that he seemed destined to complete the discoveries of
Galileo; and it is not a little remarkable, that Huyghens nowhere in his
published works makes any allusion to this connexion between his father
and Galileo, not even during the discussion that arose some years later
on the subject of the pendulum clock, which must necessarily have forced
it upon his recollection.
The Dutch commissioners had chosen one of their number to go into Italy
for the purpose of communicating personally with Galileo, but he
discouraged this scheme, from a fear of its giving umbrage at Rome. The
correspondence being carried on at so great a distance necessarily
experienced many tedious delays, till in the very midst of Galileo's
labours to complete his tables, he was seized with the blindness which
we have already mentioned. He then resolved to place all the papers
containing his observations and calculations for this purpose in the
hands of Renieri, a former pupil of his, and then professor of
mathematics at Pisa, who undertook to finish and to forward them into
Holland. Before this was done, a new delay was occasioned by the deaths
which speedily followed each other of every one of the four
commissioners; and for two or three years the correspondence with
Holland was entirely interrupted. Constantine Huyghens, who was capable
of appreciating the value of the scheme, succeeded after some trouble in
renewing it, but only just before the death of Galileo himself, by which
of course it was a second time broken off; and to complete the singular
series of obstacles by which the trial of this method was impeded, just
as Renieri, by order of the Duke of Tuscany, was about to publish the
ephemeris and tables which Galileo had entrusted to him, and which the
Duke told Viviani he had seen in his possession, he also was attacked
with a mortal malady; and upon his death the manuscripts were nowhere to
be found, nor has it since been discovered what became of them. Montucla
has intimated his suspicions that Renieri himself destroyed them, from a
consciousness that they were insufficient for the purpose to which it
was intended to apply them; a bold conjecture, and one which ought to
rest upon something more than mere surmise: for although it may be
considered certain, that the practical value of these tables would be
very inconsiderable in the present advanced state of knowledge, yet it
is nearly as sure that they were unique at that time, and Renieri was
aware of the value which Galileo himself had set upon them, and should
not be lightly accused of betraying his trust in so gross a manner. In
1665, Borelli calculated the places of the satellites for every day in
the ensuing year, which he professed to have deduced (by desire of the
Grand Duke) from Galileo's tables;[155] but he does not say whether or
not these tables were the same that had been in Renieri's possession.
We have delayed till this opportunity to examine how far the invention
of the pendulum clock belongs to Galileo. It has been asserted that the
isochronism of the pendulum had been noticed by Leonardo da Vinci, but
the passage on which this assertion is founded (as translated from his
manuscripts by Venturi) scarcely warrants this conclusion. "A rod which
engages itself in the opposite teeth of a spur-wheel can act like the
arm of the balance in clocks, that is to say, it will act alternately,
first on one side of the wheel, then on the opposite one, without
interruption." If Da Vinci had constructed a clock on this principle,
and recognized the superiority of the pendulum over the old balance, he
would surely have done more than merely mention it as affording an
unintermitted motion "like the arm of the balance." The use of the
balance is supposed to have been introduced at least as early as the
fourteenth century. Venturi mentions the drawing and description of a
clock in one of the manuscripts of the King's Library at Paris, dated
about the middle of the fifteenth century, which as he says nearly
resembles a modern watch. The balance is there called "The circle
fastened to the stem of the pallets, and moved by the force with
it."[156] In that singularly wild and extravagant book, entitled "A
History of both Worlds," by Robert Flud, are given two drawings of the
wheel-work of the clocks and watches in use before the application of
the pendulum. An inspection of them will show how little remained to be
done when the isochronism of the pendulum was discovered. _Fig. 1._
represents "the large clocks moved by a weight, such as are put up in
churches and turrets; _fig. 2._ the small ones moved by a spring, such
as are worn round the neck, or placed on a shelf or table. The use of
the chain is to equalize the spring, which is strongest at the beginning
of its motion."[157] This contrivance of the chain is mentioned by
Cardan, in 1570, and is probably still older. In both figures the name
given to the cross bar, with the weight attached to it, is "the time or
balance (_tempus seu libratio_) by which the motion is equalized." The
manner in which Huyghens first applied the pendulum is shown in _fig.
3._[158] The action in the old clocks of the balance, or _rake_, as it
was also called, was by checking the motion of the descending weight
till its inertia was overcome; it was then forced round till the
opposite pallet engaged in the toothed wheel. The balance was thus
suddenly and forcibly reduced to a state of rest, and again set in
motion in the opposite direction. It will be observed that these
balances wanted the spiral spring introduced in all modern watches,
which has a property of isochronism similar to that of the pendulum.
Hooke is generally named as the discoverer of this property of springs,
and as the author of its application to the improvement of watches, but
the invention is disputed with him by Huyghens. Lahire asserts[159] that
the isochronism of springs was communicated to Huyghens at Paris by
Hautefeuille, and that this was the reason why Huyghens failed to obtain
the patent he solicited for the construction of spring watches. A great
number of curious contrivances at this early period in the history of
Horology, may be seen in Schott's Magia Naturæ, published at Nuremberg
in 1664.
[Illustration: _Fig. 2. Fig. 1. Fig. 3._]
Galileo was early convinced of the importance of his pendulum to the
accuracy of astronomical observations; but the progress of invention is
such that the steps which on looking back seem the easiest to make, are
often those which are the longest delayed. Galileo recognized the
principle of the isochronism of the pendulum, and recommended it as a
measurer of time in 1583; yet fifty years later, although constantly
using it, he had not devised a more convenient method of doing so, than
is contained in the following description taken from his "Astronomical
Operations."
"A very exact time-measurer for minute intervals of time, is a heavy
pendulum of any size hanged by a fine thread, which, if removed from the
perpendicular and allowed to swing freely, always completes its
vibrations, be they great or small, in exactly the same time."[160]
The mode of finding exactly by means of this the quantity of any time
reduced to hours, minutes, seconds, &c., which are the divisions
commonly used among astronomers, is this:—"Fit up a pendulum of any
length, as for instance about a foot long, and count patiently (only for
once) the number of vibrations during a natural day. Our object will be
attained if we know the exact revolution of the natural day. The
observer must then fix a telescope in the direction of any star, and
continue to watch it till it disappears from the field of view. At that
instant he must begin to count the vibrations of the pendulum,
continuing all night and the following day till the return of the same
star within the field of view of the telescope, and its second
disappearance, as on the first night. Bearing in recollection the total
number of vibrations thus made in twenty-four hours, the time
corresponding to any other number of vibrations will be immediately
given by the Golden Rule."
A second extract out of Galileo's Dutch correspondence, in 1637, will
show the extent of his improvements at that time:—"I come now to the
second contrivance for increasing immensely the exactness of
astronomical observations. I allude to my time-measurer, the precision
of which is so great, and such, that it will give the exact quantity of
hours, minutes, seconds, and even thirds, if their recurrence could be
counted; and its constancy is such that two, four, or six such
instruments will go on together so equably that one will not differ from
another so much as the beat of a pulse, not only in an hour, but even in
a day or a month."—"I do not make use of a weight hanging by a thread,
but a heavy and solid pendulum, made for instance of brass or copper, in
the shape of a circular sector of twelve or fifteen degrees, the radius
of which may be two or three palms, and the greater it is the less
trouble will there be in attending it. This sector, such as I have
described, I make thickest in the middle radius, tapering gradually
towards the edges, where I terminate it in a tolerably sharp line, to
obviate as much as possible the resistance of the air, which is the sole
cause of its retardation."—[These last words deserve notice, because,
in a previous discussion, Galileo had observed that the parts of the
pendulum nearest the point of suspension have a tendency to vibrate
quicker than those at the other end, and seems to have thought
erroneously that the stoppage of the pendulum is partly to be attributed
to this cause.]—"This is pierced in the centre, through which is passed
an iron bar shaped like those on which steelyards hang, terminated below
in an angle, and placed on two bronze supports, that they may wear away
less during a long motion of the sector. If the sector (when accurately
balanced) be removed several degrees from its perpendicular position, it
will continue a reciprocal motion through a very great number of
vibrations before it will stop; and in order that it may continue its
motion as long as is wanted, the attendant must occasionally give it a
smart push, to carry it back to large vibrations." Galileo then
describes as before the method of counting the vibrations in the course
of a day, and gives the rule that the lengths of two similar pendulums
will have the same proportion as the squares of their times of
vibration. He then continues: "Now to save the fatigue of the assistant
in continually counting the vibrations, this is a convenient
contrivance: A very small and delicate needle extends out from the
middle of the circumference of the sector, which in passing strikes a
rod fixed at one end; this rod rests upon the teeth of a wheel as light
as paper, placed in a horizontal plane near the pendulum, having round
it teeth cut like those of a saw, that is to say, with one side of each
tooth perpendicular to the rim of the wheel and the other inclined
obliquely. The rod striking against the perpendicular side of the tooth
moves it, but as the same rod returns against the oblique side, it does
not move it the contrary way, but slips over it and falls at the foot of
the following tooth, so that the motion of the wheel will be always in
the same direction. And by counting the teeth you may see at will the
number of teeth passed, and consequently the number of vibrations and of
particles of time elapsed. You may also fit to the axis of this first
wheel a second, with a small number of teeth, touching another greater
toothed wheel, &c. But it is superfluous to point out this to you, who
have by you men very ingenious and well skilled in making clocks and
other admirable machines; and on this new principle, that the pendulum
makes its great and small vibrations in the same time exactly, they will
invent contrivances more subtle than any I can suggest; and as the error
of clocks consists principally in the disability of workmen hitherto to
adjust what we call the balance of the clock, so that it may vibrate
regularly, my very simple pendulum, which is not liable to any
alteration, affords a mean of maintaining the measures of time always
equal." The contrivance thus described would be somewhat similar to the
annexed representation, but it is almost certain that no such instrument
was actually constructed.
[Illustration]
It must be owned that Galileo greatly overrated the accuracy of his
timekeeper; and in asserting so positively that which he had certainly
not experienced, he seems to depart from his own principles of
philosophizing. It will be remarked that in this passage he still is of
the erroneous opinion, that all the vibrations great or small of the
same pendulum take exactly the same time; and we have not been able to
find any trace of his having ever held a different opinion, unless
perhaps in the Dialogues, where he says, "If the vibrations are not
exactly equal, they are at least insensibly different." This is very
much at variance with the statement in the Memoirs of the Academia del
Cimento, edited by their secretary Magalotti, on the credit of which
Galileo's claim to the pendulum-clock chiefly rests. It is there said
that experience shows that the smallest vibrations are rather the
quickest, "as Galileo announced after the observation, which in 1583 he
was the first to make of their approximate equality." It is not possible
immediately in connexion with so glaring a misstatement, to give
implicit credence to the assertion in the next sentence, that "_to
obviate this inconvenience_" Galileo was the first to contrive a clock,
constructed in 1649, by his son Vincenzo, in which, by the action of a
weight or spring, the pendulum was constrained to move always from the
same height. Indeed it appears as if Magalotti did not always tell this
story in the same manner, for he is referred to as the author of the
account given by Becher, "that Galileo himself made a pendulum-clock one
of which was sent to Holland," plainly insinuating that Huyghens was a
mere copyist.[161] These two accounts therefore serve to invalidate each
other's credibility. Tiraboschi[162] asserts that, at the time he wrote,
the mathematical professor at Pisa was in possession of the identical
clock constructed by Treffler under Vincenzo's directions; and quotes a
letter from Campani, to whom it was shown by Ferdinand, "old, rusty, and
unfinished as Galileo's son made it before 1649." Viviani on the other
hand says that Treffler constructed this same clock some time after
Vincenzo's death (which happened in 1649), on a different principle from
Vincenzo's ideas, although he says distinctly that he heard Galileo
describe an application of the pendulum to a clock similar to Huyghens'
contrivance. Campani did not actually see this clock till 1659, which
was three years after Huyghens' invention, so that perhaps Huyghens was
too easily satisfied when, on occasion of the answer which Ferdinand
sent to his complaints of the Memorie del Cimento he wrote to Bouillaud,
"I must however believe, since such a prince assures me, that Galileo
had this idea before me."
There is another circumstance almost amounting to a proof that it was an
afterthought to attribute the merit of constructing the pendulum-clock
to Galileo, for on the reverse of a medal struck by Viviani, and
inscribed "to the memory of his excellent instructor,"[163] is a rude
exhibition of the principal objects to which Galileo's attention was
directed. The pendulum is represented simply by a weight attached to a
string hanging on the face of a rock. It is probable that, in a design
expressly intended to commemorate Galileo's inventions, Viviani would
have introduced the timekeeper in the most perfect form to which it had
been brought by him. Riccioli,[164] whose industry was unwearied in
collecting every fact and argument which related in any way to the
astronomical and mechanical knowledge and opinions of his time,
expressly recommends swinging a pendulum, or perpendicular as it was
often called (only a few years before Huyghens' publication), as much
more accurate _than any clock_.[165] Join to all these arguments
Huyghens' positive assertion, that if Galileo had conceived any such
idea, he at least was entirely ignorant of it,[166] and no doubt can
remain that the merit of the original invention (such as it was) rests
entirely with Huyghens. The step indeed seems simple enough for a less
genius than his: for the property of the pendulum was known, and the
conversion of a rotatory into a reciprocating motion was known; but the
connexion of the one with the other having been so long delayed, we must
suppose that difficulties existed where we are not now able to perceive
them, for Huyghens' improvement was received with universal admiration.
There may be many who will consider the pendulum as undeserving so long
a discussion; who do not know or remember that the telescope itself has
hardly done more for the precision of astronomical observations than
this simple instrument, not to mention the invaluable convenience of an
uniform and accurate timekeeper in the daily intercourse of life. The
patience and industry of modern observers are often the theme of
well-merited praise, but we must look with a still higher degree of
wonder on such men as Tycho Brahe and his contemporaries, who were
driven by the want of any timekeeper on which they could depend to the
most laborious expedients, and who nevertheless persevered to the best
of their ability, undisgusted either by the tedium of such processes, or
by the discouraging consciousness of the necessary imperfection of their
most approved methods and instruments.
The invariable regularity of the pendulum's motion was soon made
subservient to ulterior purposes beyond that of merely registering time.
We have seen the important assistance it afforded in establishing the
laws of motion; and when the theory founded on those laws was extended
and improved, the pendulum was again instrumental, by a species of
approximate reasoning familiar to all who are acquainted with physical
inquiries, in pointing out by its minute irregularities in different
parts of the earth, a corresponding change in the weight of all bodies
in those different situations, supposed to be the consequence of a
greater distance from the axis of the earth's rotation; since that would
occasion the force of attraction to be counterbalanced by an increased
centrifugal force. The theory which kept pace with the constantly
increasing accuracy of such observations, proving consistent in all
trials of it, has left little room for future doubts; and in this manner
the pendulum in intelligent hands became the simplest instrument for
ascertaining the form of the globe which we inhabit. An English
astronomer, who corresponded with Kepler under the signature of Brutius
(whose real name perhaps might be Bruce), had already declared his
belief in 1603, that "the earth on which we tread is neither round nor
globular, but more nearly of an oval figure."[167] There is nothing to
guide us to the grounds on which he formed this opinion, which was
perhaps only a lucky guess. Kepler's note upon it is: "This is not
altogether to be contemned."
A farther use of the pendulum is in furnishing a general and unperishing
standard of measure. This application is suggested in the third volume
of the 'Reflections' of Mersenne, published in 1647, where he observes
that it may be best for the future not to divide time into hours,
minutes, and seconds, but to express its parts by the number of
vibrations of a pendulum of given length, swinging through a given arc.
It was soon seen that it would be more convenient to invert this
process, and to choose as an unit of length the pendulum which should
make a certain number of vibrations in the unit of time, naturally
determined by the revolution of the earth on its axis. Our Royal
Society took an active part in these experiments, which seem,
notwithstanding their utility, to have met from the first with much of
the same ridicule which was lavished upon them by the ignorant, when
recently repeated for the same purpose. "I contend," says Graunt[168] in
a dedication to the Royal Society, dated 1662, "against the envious
schismatics of your society (who think you do nothing unless you
presently transmute metals, make butter and cheese without milk, and, as
their own ballad hath it, make leather without hides), by asserting the
usefulness of even all your preparatory and luciferous experiments,
being not the ceremonies, but the substance and principles of useful
arts. For I find in trade the want of an universal measure, and have
heard musicians wrangle about the just and uniform keeping of time in
their consorts, and therefore cannot with patience hear that your
labours about vibrations, eminently conducing to both, should be
slighted, nor your pendula called swing-swangs with scorn."[169]
FOOTNOTES:
[153] One of the Commissioners was the father of Blaise Pascal.
[154] These instruments were very inferior to those now in use under the
same name. See "Treatise on Opt. Instrum."
[155] Theoricæ Mediceorum Planetarum, Florentiæ, 1666.
[156] Circulus affixus virgæ paletorum qui cum eâ de vi movetur.
[157] Utriusque Cosmi Historia. Oppenhemii, 1617.
[158] Huygenii Opera. Lugduni, 1724.
[159] Mémoires de l'Academie, 1717.
[160] See page 84.
[161] De nova Temporis dimetiendi ratione. Londini, 1680.
[162] Storia della Lett. Ital.
[163] Museum Mazuchellianum, vol. ii. Tab. cvii. p. 29.
[164] Almagestum Novum, vol. i.
[165] Quovis horologio accuratius.
[166] Clarorum Belgarum ad Ant. Magliabech. Epistolæ. Florence, 1745,
tom. i. p. 235.
[167] Kepleri Epistolæ.
[168] Natural and Political Observations. London, 1665.
[169] See also Hudibras, Part II. Cant. III.
They're guilty by their own confessions
Of felony, and at the Sessions
Upon the bench I will so handle 'em,
That the vibration of this pendulum
Shall make all taylors' yards of one
Unanimous opinion;
A thing he long has vaunted of,
But now shall make it out of proof.
Hudibras was certainly written before 1663: ten years later Huyghens
speaks of the idea of so employing the pendulum as a common one.