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that it is not repugnant to the nature posed to be*. He then proceeds with of things that there should be a vacuum, ihese remarkable words :-“ If we raise but merely that it is difficult to produce. the plane higher and higher, one of these To explain myself more clearly : if we areas terminates in the circumference of allow that the air has weight, there is no a circle, and the other in a point, for difference between air and water except such are the upper rim of the basin and in degree. At the bottom of the sea the top of the cone. Now since in the the weight of the water above me com diminution of the two areas they to the presses everything round my body, and very last maintain their equality to one it strikes me that the same thing must another, it is in my thoughts proper to happen in the air, we being placed at say that the highest and ultimate terms of the bottom of its immensity ; we do not of such diminutions are equal, and not feel its weight, nor the compression one infinitely bigger than the other. It round us, because our bodies are made seems therefore that the circumference capable of supporting it. But if we of a large circle may be said to be equal were in a vacuum, then the weight of to one single point. And why may not the air above our heads would be felt. these be called equal if they be the last It would be felt very great, but not infi- remainders and vestiges left by equal nite, and therefore determinable, and it magnitudes $?" might be overcome by a force propor We think no one can refuse to adtioned to it. In fact I estimate it to be mit the probability, that Newton may such that, to make a vacuum, I believe have found in such passages as these we require a force greater than that of the first germ of the idea of his prime a column of water thirty feet high*." and ultimate ratios, which afterwards

This subject is introduced by some ob- became in his hands an instrument servations on the force of cohesion, Ga- of such power. As to the paradoxilileo seeming to be of opinion that, al- cal result, Descartes undoubtedly has though it cannot be adequately ac- given the true answer to it in saying counted for by “the great and principal that it only proves that the line is not a resistance to vacuum, yet that per- greater area than the point is. Whilst haps a sufficient cause may be found by on this subject, it may not be uninconsidering every body as composed of teresting to remark that something very minute particles, between every similar to the doctrine of fluxions seems two of which is exerted a similar resist- to have been lying dormant in the minds ance." This remark serves to lead to a of the mathematicians of Galileo's era, discussion on indivisibles and infinite for Inchoffer illustrates his argument in quantities, of which we shall merely ex- the treatise we have already mentioned, tract what Galileo gives as a curious that the Copernicans may deduce some paradox suggested in the course of it. true results from what he terms their He supposes a basin to be formed by absurd hypothesis, by, observing, that scooping a hemisphere out of a cylinder, mathematicians may deduce the truth and a cone to be taken of the same that a line is length without breadth, depth and base as the hemisphere. from the false and physically impossible It is easy to show, if the cone and supposition that a point flows, and that scooped cylinder be both supposed a line is the fluxion of a point s. to be cut by the same plane, parallel to A suggestion that perhaps fire dis

solves 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 the one on which both stand, that the proves the transmission of light to be area of the ring CDEF thus discovered in the cylinder is equal to the area of the equality of the solids standing on the cutting plane,

Galileo also reasons in the same way on the corresponding circular section AB of the but one is sufficient for our present purpose. cone wherever the cutting plane is sup

+ Gli altissimi e nitimi termini.

Le ultime reliquie e vestigie lasejate da grandezze eguali.

s Punctum fuere, et lineam esse Auxam puactia • Venturi, vol. ii.

Tract. Syllept, Romæ, 1633,

sage:

instantaneous, to which Sagredo cau- undulations which will be seen regutiously replies, that nothing can be ga. larly spreading round the glass, will thered from that experiment except that suddenly split into two, proving that light traveis more swiftly than sound; the vibrations that occasion the octave nor can we draw any decisive conclusion are double those belonging to the simfrom the rising of the sun. Who can ple note.” Galileo then describes a assure us that he is not in the horizon method he discovered by accident of before his rays reach our sight?" Sal- measuring the length of these waves more viati then mentions an experiment by accurately than can be done in the agiwhich he endeavoured to examine this tated water. He was scraping a brass question. Two observers are each to be plate with an iron chisel, to take out furnished with a lantern: as soon as some spots, and moving the tool rapidly the first shades his light, the second is to upon the plate, he occasionally heard a discover his, and this is to be repeated hissing and whistling sound, very shrill at a short distance till the observers are and audible, and whenever this occurperfect in the practice. The same thing red, and then only, he observed the is to be tried at the distance of several light dust on the plate to arrange itself miles, and if the first observer perceive in a long row of small parallel streaks any delay between shading his own light equidistant from each other.

In reand the appearance of his companion's, peated experiments he produced differit is to be attributed to the time taken ent tones by scraping with greater or by the light in traversing twice the dis- less velocity, and remarked that the tance between them. He allows that he streaks produced by the acute sounds could discover no perceptible interval at stood closer together than those from the distance of a mile, at which he had the low notes. Among the sounds protried the experiment, but recommends duced were two, which by comparithat with the help of a telescope it should son with a viol he ascertained to differ be tried at much greater distances. Sir by an exact fifth; and measuring the Kenelm Digby remarks on this pas- spaces occupied by the streaks in both

It may be objected (if there be experiments, he found thirty of the some observable tardity in the motion one equal to forty-five of the other, of light) that the sunne would never be which is exactly the known proportion truly in that place in which unto our of the lengths of strings of the same eyes he appeareth to be; because that material which sound a fifth to each it being seene by means of the light other*. which issueth from it, if that light re

Salviati also remarks, that if the quired time to move in, the sunne (whose material be not the same, as for inmotion is so swifte) would be removed stance if it be required to sound an from the place where the light left it, octave to a note on catgut, on before it could be with us to give tidings wire of the same length, the weight of of him. To this I answer, allowing per- the wire must be made four times as adventure that it may be so, who great, and so for other intervals. “The knoweth the contrary ?

Or what in- immediate cause of the forms of musiconvenience would follow if it be ad- cal intervals is neither the length, the mitted * ?"

tension, nor the thickness, but the proThe principal thing remaining to be portion of the numbers of the undulanoticed is the application of the theory tions of the air which strike upon the of the pendulum to musical concords drum of the ear, and make it vibrate in and dissonances, which are explained, in the same intervals.

Hence we may the same manner as by Kepler in his gather a plausible reason of the differ“ Harmonices Mundi," to result from ent sensations occasioned to us by difthe concurrence or opposition of vibra- ferent couples of sounds, of which we tions in the air striking on the drum hear some with great pleasure, some of the ear. It suggested that these with less, and call them accordingly vibrations may be made manifest by concords, more or less perfect, whilst rubbing the finger round a glass set in some excite in us great dissatisfaction, a large vessel of water; "and if by pres- and are called discords. The disagreesure the note is suddenly made to rise able sensation belonging to the latter to the octave above, every one of the

• This beautiful experiment is more easily tried by

drawing the bow of a violin across the edge of glass Treatise of the Nature of Bodies. London, strewed with tine dry sand. Those who wish to see more 1663."

on the subject may consult Chladni's * Acoustique.'

a

probably arises from the disorderly being the farthest removed from the manner in which the vibrations strike wall. As an easy mode of describing the drum of the ear; so that for in- the parabolic curve for this purpose, he stance a most cruel discord would be recommends tracing the line in which a produced by sounding together two heavy flexible string hangs. This curve strings, of which the lengths are to each is not an accurate parabola: it is now other as the side and diagonal of a called a catenary ; but it is plain from square, which is the discord of the false the description of it in the fourth diafifth. On the contrary, agreeable con- logue, that Galileo was perfectly aware sonances will result from those strings that this construction is only approxiof which the numbers of vibrations made mately true. In the same place he makes in the same time are commensurable, the remark, which to many is so para“to the end that the cartilage of the doxical, that no force, however great, drum may not undergo the incessant exerted in a horizontal direction, can torture of a double inflexion from the stretch a heavy thread, however slender, disagreeing percussions." Something into an accurately straight line. similar may be exhibited to the eye by The fifth and sixth dialogues were left hanging up pendulums of different unfinished, and annexed to the former lengths : “if these be proportioned so ones by Viviani after Galileo's death: that the times of their vibrations cor- the fragment of the fifth, which is on the respond with those of the musical con- subject of Euclid's Definition of Ratio, cords, the eye will observe with pleasure was at first intended to have formed a their crossings and interweavings still part of the third, and followed the first recurring at appreciable intervals; but proposition on equable motion: the sixth if the times of vibration be incommen was intended to have embodied Galileo's surate, the eye will be wearied and worn researches on the nature and laws of out with following them.”

Percussion, on which he was employed at The second dialogue is occupied en the time of his death. Considering these tirely with an investigation of the solely as fragments, we shall not here strength of beams, a subject which does make any extracts from them. not appear to have been examined by any one before Galileo beyond Aris

CHAPTER XVIII. totle's remark, that long beams are weaker, because they are at once the Correspondence on Longitudes.--Pen

dulum Clock weight, the lever, and the fulcrum; and it is in the development of this obser. In the spring of 1636, having finished vation that the whole theory consists. his Dialogues on Motion, Galileo reThe principle assumed by Galileo as sumed the plan of determining the lonthe basis of his inquiries is, that the gitude by means of Jupiter's satellites. force of cohesion with which a beam Perhaps he suspected something of the resists a cross fracture in any section private intrigue which thwarted his may all be considered as acting at the former expectations from the Spanish centre of gravity of the section, and that government, and this may

have induced it breaks always at the lowest point: him on the present occasion to negotiate from this he deduced that the effect of the matter without applying for Ferdithe weight of a prismatic beam in over- nand's assistance and recommendation. coming the resistance of one end by Accordingly he addressed himself to which it is fastened to a wall, varies Lorenz Real, who had been Governor directly as the square of the length, and General of the Dutch possessions in inversely as the side of the base. From India, freely and unconditionally offerthis it immediately follows, that if for ing the use of his theory to the States instance the bone of a large animal be General of Holland. Not long before, three times as long as the corresponding his opinion had been requested by the one in a smaller beast, it must be nine commissioners appointed at Paris to times as thick to have the same strength, examine and report on the practicability provided we suppose in both cases that of another method proposed by Morin, * the materials are of the same consist- which consisted in observing the dis

An elegant result which Galileo tance of the moon from a known star. also deduced from this theory, is that the Morin was a French philosopher, prin. form of such a beam, to be equally strong in every part, should be that of a para

One of the Commissioners was the father of bolical prism, the vertex of the parabola Blaise Pascal.

ence.

cipally known as an astrologer and zea- ridians by means of the moon's motion, lous Anti-Copernican ; but his name de- provided we are sure of the following serves to be recorded as undoubtedly one requisites: First, an Ephemeris of the of the first to recommend a method, moon's motion exactly calculated for which, under the name of a Lunar dis- the first meridian from which the others tance, is now in universal practice. are to be reckoned; secondly, exact in

The monthly motion of the moon is so struments, and convenient to handle, in rapid, that her distance from a given star taking the distance between the moon sensibly varies in a few minutes even to and a fixed star; thirdly, great practhe unassisted eye; and with the aid of tical skill in the observer; fourthly, not the telescope, we can of course appre- less accuracy in the scientific calculaciate the change more accurately. Morin tions, and astronomical computations ; proposed that the distances of the moon fifthly, very perfect clocks to number from a number of fixed stars lying near the hours, or other means of knowing her path in the heavens should be be- them exactly, &c. Supposing, I say, forehand calculated and registered for all these elements free from error, the every day in the year, at a certain hour, longitude will be accurately found; but in the place from which the longitudes I reckon it more easy and likely to err were to be reckoned, as for instance at in all of these together, than to be pracParis. Just as in the case of the eclipses tically right in one alone. Morin ought of Jupiter's satellites, the observer, when to require his judges to assign, at their he saw that the moon had arrived at pleasure, eight or ten moments of difthe registered distance, would know the ferent nights during four or six months hour at Paris : he might also make al- to come, and pledge himself to predict lowance for intermediate distances. and assign by his calculations the disObserving at the same instant the hour tances of the moon at those determined on board his ship, the difference between instants from some star which would the two would show his position in re- then be near her. If it is found that gard of longitude. In using this the distances assigned by him agree method as it is now practised, several with those which the quadrant or sexmodifications are to be attended to, tant* will actually show, the judges without which it would be wholly use- would be satisfied of his success, or less, in consequence of the refraction rather of the truth of the matter, and of the atmosphere, and the proximity of nothing would remain but to show that the moon to the earth. Owing to the his operations were such as could be latter cause, if two spectators should at performed by men of moderate skill, and the same instant of time, but in different also practicable at sea as well as on places, measure the distance of the land. I incline much to think that an moon in the East, from a star still more experiment of this kind would do much to the eastward, it would appear greater towards abating the opinion and conto the more easterly spectator than to ceit which Morin has of himself, which the other observer, who as seen from appears to me so lofty, that I should the star would be standing_more di- consider myself the eighth sage, if I rectly behind the moun. The mode knew the half of what Morin

presumes of allowing for these alterations is taught to know." by trigonometry and astronomy.

It is probable that Galileo was The success of this method depends al- biassed by a predilection for his own together upon the exact knowledge which method, on which he had expended we now have of the moon's course, and so much time and labour; but the obtill that knowledge was perfected it jections which he raises against Morin's would have been found altogether il- proposal in the foregoing letter are no lusory. Such in fact was the judgment other than those to which at that period which Galileo pronounced upon it. “As it was undoubtedly open. With regard to Morin's book on the method of find- to his own, he had already, in 1612, ing the longitude by means of the moon's given a rough prediction of the course motion, I say freely that I conceive this of Jupiter's satellites, which had been idea to be as accurate in theory, as found to agree tolerably well with subfallacious and impossible in practice. I sequent observations; and since that am sure that neither you nor any one of the other four gentlemen can doubt the possibility of finding the dif

. These instruments were very inferior to those ference of longitude between two me Opt. Instrum."

now in use under the same name,

See " Treatise on

time, amid all his other employments, and for two or three years the corhe had almost unintermittingly during respondence with Holland was entirely twenty-four years continued his obser- interrupted. Constantine Huyghens, vations, for the sake of bringing the who was capable of appreciating the tables of their motions to as high a state value of the scheme, succeeded after of perfection as possible. This was the some trouble in renewing it, but only point to which the inquiries of the States just before the death of Galileo himself, in their answer to Galileo's frank pro- by which of course it was a second posal were principally directed. They time broken off; and to complete the immediately appointed commissioners to singular series of obstacles by which the communicate with him, and report the trial of this method was impeded, just various points on which they required as Renieri, by order of the Duke of Tusinformation. They also sent him a cany, was about to publish the ephegolden chain, and assured him that in meris and tables which Galileo had enthe case of the design proving success trusted to him, and which the Duke ful, he should have no cause to com- told Viviani he had seen in his posplain of their want of gratitude and ge- session, he also was attacked with a nerosity. The commissioners immedi- mortal malady ; and upon his death the ately commenced an active correspon- manuscripts were nowhere to be found, dence with him, in the course of which nor has it since been discovered what he entered into more minute details with became of them. Montucla has intiregard to the methods by which he mated his suspicions that Renieri himproposed to obviate the practical dif- self destroyed them, from a consciousficulties of the necessary observations. ness that they were insufficient for the

It is worth noticing that the secretary purpose to which it was intended to apto the Prince of Orange, who was mainly ply them; a bold conjecture, and one instrumental in forming this commis- which ought to rest upon something sion, was Constantine Huyghens, father more than mere surmise: for although it of the celebrated mathematician of that may be considered certain, that the name, of whom it has been said that he practical value of these tables would be seemed destined to complete the disco- very inconsiderable in the present adveries of Galileo ; and it is not a little vanced state of knowledge, yet it is remarkable, that Huyghens nowhere in nearly as sure that they were unique at his published works makes any allusion that time, and Renieri was aware of to this connexion between his father and the value which Galileo himself had set Galileo, not even during the diseussion upon them, and should not be lightly that arose some years later on the sub- accused of betraying his trust in so gross ject of the pendulum clock, which must

In 1665, Borelli calculated necessarily have forced it upon his re- the places of the satellites for every day collection.

in the ensuing year, which he professed The Dutch commissioners had chosen to have deduced (by desire of the Grand one of their number to go into Italy for Duke) from Galileo's tables; * but he the purpose of communicating person. does not say whether or not these tables ally with Galileo, but he discouraged were the same that had been in Renieris this scheme, from a fear of its giving possession. umbrage at Rome. The correspondence We have delayed till this opportunity being carried on at so great a distance to examine how far the invention of the necessarily experienced many tedious de- pendulum clock belongs to Galileo.. It lays, till in the very midst of Galileo's has been asserted that the isochronism labours to complete his tables, he was of the pendulum had been noticed by seized with the blindness which we have Leonardo da Vinci, but the passage on already mentioned. He then resolved which this assertion is founded (as transto place all the papers containing his lated from his manuscripts by Venturi) observations and calculations for this scarcely warrants this conclusion. “A purpose in the hands of Renieri, a for- rod which engages itself in the opposite mer pupil of his, and then professor teeth of a spur-wheel can act like the of mathematics at Pisa, who under arm of the balance in clocks, that is to took to finish and to forward them into say, it will act alternately, first on one Holland. Before this was done, a new side of the wheel, then on the opposite delay was occasioned by the deaths which speedily followed each other of

Theoricæ Mediceorum Planetarum, Florentiæ, every one of the four commissioners; 1666.

a manner.

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