Variability of Pendulum Vibrations

Zetetic Astronomy, by 'Parallax' (pseud. Samuel Birley Rowbotham), [1881], at sacred-texts.com

VARIABILITY OF PENDULUM VIBRATIONS.

Many contend that because a pendulum vibrates more rapidly in the northern region than "at the equator, the earth is thereby proved not only to be a globe, but to have axial motion, and because the variation in the velocity is that of gradual increase as the north pole is approached, it is concluded that the earth's true shape is that of an oblate spheroid--the diameter through the poles being less than that through the equator. The difference was calculated by Newton to be the 235th part of the whole diameter; or that the polar was to the equatorial diameter as 680 to 692. Huygens gave the proportion as 577 to 875, or a difference of about one-third of the whole diameter. Others have given still different proportions; but recently the difference of opinion, each the result of calculation, has become so great that many have concluded that the earth is really instead of oblate, an oblong spheroid.

It is argued that as the length of a pendulum vibrating

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seconds at the equator is 39,027 inches, and at the north pole 39,197 inches, that the earth, like an orange, has a globular form, but somewhat flattened at the "poles." But this so-called argument proceeds and depends upon the assumption that the earth is a globe having a "centre of attraction of gravitation," towards which all bodies gravitate or fall, and as a pendulum is essentially a falling body under certain restraint, the fact that when of the same length it oscillates or falls more rapidly at the north than at the equator is a proof that the northern surface is nearer to the "centre of attraction," or centre of the earth, than the equatorial surface: and of course if nearer the radius must be shorter, and therefore the "earth is a spheroid flattened at the poles."

The above is very ingenious and very plausible, but unfortunately for its character as an argument, the evidence is wanting that the earth is a globe at all; and until proof of convexity is given, all questions as to its being oblate, oblong, or entirely spherical, are logically out of place.

It is the duty of those who, from the behaviour of a pendulum at different latitudes, contend that the earth is spherical, to first prove that no other cause could operate besides greater proximity to a centre of gravity in producing the known differences in its oscillations. This not being done, nor attempted, the whole matter must be condemned as logically insufficient, irregular, and worthless for its intended purpose.

M. M. Picart and De la Hire, two celebrated French savans, as well as many other scientific men, have attributed

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the variations of the pendulum to differences of temperature at different latitudes. It is certain that the average changes of temperature are more than sufficient to bring about the variations which have been observed. The following quotation will show the practical results of these changes:--

"All the solid bodies with which we are surrounded are constantly undergoing changes of bulk, corresponding to the variations of temperature. . . . The expansion and con-traction of metals by heat and cold form subjects of serious and careful attention to chronometer makers, as will appear by the following statements:--The length of the pendulum vibrating seconds, in vacuo, in the latitude of London (51° 31' 8" north) at the level of the sea, and at the temperature of 62° Fahrenheit, has been ascertained with the greatest precision to be 39.13929 inches; now, as the metal of which it is composed is constantly subject to variation of temperature, it cannot but happen that its length is constantly varying, and when it is further stated that if the 'bob' be let down 1-100th of an inch, the clock will lose ten seconds in twenty-four hours; that the elongation of 1-1000th of an inch will cause it to lose one second per day; and that a change of temperature equal to 30° Fahrenheit will alter its length 1-5000th part, and occasion an error in the rate of going of eight seconds per day, it will appear evident that some plan must be devised for obviating so serious an inconvenience." ¹

"The mean annual temperature of the whole earth at the level of the sea is 50° Fahrenheit. For different latitudes it is as under:--

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From the above table it is seen that the temperature gradually decreases from the equator towards the pole, which would of necessity contract the substance of the pendulum, or starting it and cause it to vibrate more rapidly.

Besides the temperature of a given latitude the pressure and density of the air must be taken into account. In numbers 294 and 480 of the "Philosophical Transactions," Dr. Derham records a number of experiments with pendulums in the open air, and in the receiver of an air-pump, which he summarises as follows:--

"The arches of vibration in vacuo were larger than in the open air, or in the receiver before it was exhausted; the enlargement or diminution of the arches of vibration were constantly proportional to the quantity of air, or rarity, or density of it, which was left in the receiver of the air-pump. And as the vibrations were longer or shorter, so the time were accordingly; viz., two seconds in an hour when the vibrations were longest, and less and less as the air was re-admitted, and the vibrations shortened."

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Thus it is evident that two distinct and tangible causes necessarily operate to produce variability in the oscillations of a pendulum at different latitudes, without having recourse to a flattening at the poles of an imaginary globe. First the gradual diminution of temperature as the pendulum is carried from the equator to the polar region, tends to shorten its length, and thus to increase its number of vibrations per hour or day; and secondly, as the polar centre is approached the air is colder, therefore denser, and therefore the "arches of vibration" shorter, and the times of oscillation less, or in other words the number of vibrations greater in a given period. It has also been ascertained that the pendulum is influenced--other conditions being the same, by electric and magnetic states of the atmosphere. When intense electric conditions exist the arches and times of vibration are less than during the existence of opposite conditions. Hence if in different latitudes pendulum experiments are made in vacuo, at the same temperature, and always at the level of the sea, different electric and magnetic conditions prevailing, will induce variable results. The attention of some of the most accurate and patient observers has been directed to this mode of proving the oblate spheroidal form of the earth, but the results have never been satisfactory, nor such as were expected, or that the theory of rotundity should produce. The following remarks upon this subject are interesting:

"Newton was the first person who made a calculation of the figure of the earth on the theory of gravitation. He took the following supposition as the only one to which his theory could be applied. He assumed the earth to be fluid. This fluid

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matter he assumed to be equally dense in every part. . . . For trial of his theory he supposed the fluid earth to be a spheroid. In this manner he inferred that the form of the earth would be a spheroid in which the length of the shorter is to the length of the longer or equatorial diameter, in the proportion of 229 to 230." ¹

"The following table comprises the results of the most reliable pendulum experiments which have thus far been made, and. among which the extensive series of observations by General Sabine holds the first place."

(Particulars are here given of sixty-seven experiments made in every latitude north of the equator, from 0° 1´ 49″ north to 79° 49´ 58″ north; and of twenty-nine experiments in the south from latitude 0° 1' 34″ south, to Cape Horn, 55° 51´ 20″ south, and South Shetland. 62° 56´ 11″ south.) We have here before us the results of fifty-five observations of the seconds pendulum, and of seventy-six observations of the invariable pendulum; in all 131 experiments; which number, however, includes eight of the former and fifteen of the latter kind, differing to a remarkable extent, as compared with the results generally from the computed values. General Sabine observes of these discrepancies that 'they are due in a far greater degree to local peculiarities than to what may be more strictly called errors of observation.' And already Mr. Bailey (in Memoirs of the Royal Astronomical Society, vol. 7), had expressed the opinion 'that the vibrations of a pendulum are powerfully affected, in many places, by the local attraction of the substratum on which it is swung, or by some other direct influence at present unknown to us, and the effect of which far exceeds the errors of observation."

"General Sabine himself relates:--'Captain Foster was furnished

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with two invariable pendulums of precisely the same form and construction as those which had been employed by Captain Kayter and myself. Both pendulums were vibrated at all the stations, but from some cause, which Mr. Bailey was unable to explain, the observations with one of them were so discordant at South Shetland as to require their rejection'" 1

From the foregoing remarks and quotations it is obvious that the assumption of Sir Isaac Newton that the earth is an oblate spheroid, is not confirmed by experiments made with the pendulum.

Footnotes

237:1 Noad's "Lectures on Chemistry," p. 41.

238:1 "Million of Facts," by Sir Richard Phillips, p. 475.

240:1 Professor Airey's "Six Lectures on Astronomy." Edit. 4, p. 194.