Part 3
During the years from 1827 to 1840, various types of pendulum were constructed and swung by Francis Baily, a member of the Royal Astronomical Society, who reported in 1832 on experiments in which no less than 41 different pendulums were swung in vacuo, and their characteristics determined.[35] In 1836, Baily undertook to advise the American Lt. Charles Wilkes, who was to head the United States Exploring Expedition of 1838-1842, on the procurement of pendulums for this voyage. Wilkes ordered from the London instrument maker, Thomas Jones, two unusual pendulums, which Wilkes described as "those considered the best form by Mr. Baily for traveling pendulums," and which Baily, himself, described as "precisely the same as the two invariable pendulums belonging to this [Royal Astronomical] Society," except for the location of the knife edges.
The unusual feature of these pendulums was in their symmetry of mass as well as of form. They were made of bars, of iron in one case, and of brass in the other, and each had two knife edges at opposite ends equidistant from the center. Thus, although they resembled reversible pendulums, their symmetry of mass prevented their use as such, and they were rather equivalent to four separate invariable pendulums.[36]
Wilkes was taught the use of the pendulum by Baily, and conducted experiments at Baily's house, where the latter had carried out the work reported on in 1832. The subsequent experiments made on the U.S. Exploring Expedition were under the charge of Wilkes, himself, who made observations on 11 separate occasions, beginning with that in London (1836) and followed by others in New York, Washington, D.C., Rio de Janeiro, Sydney, Honolulu, "Pendulum Peak" (Mauna Loa), Mount Kanoha, Nesqually (Oregon Territory), and, finally, two more times in Washington, D.C. (1841 and 1845).
Wilkes' results were communicated to Baily, who appears to have found the work defective because of insufficient attention to the maintenance of temperature constancy and to certain alterations made to the pendulums.[37] The results were also to have been included in the publications of the Expedition, but were part of the unpublished 24th volume. Fortunately they still exist, in what appears to be a printer's proof.[38]
The Kater invariable pendulums were used to investigate the internal constitution of the earth. Airy sought to determine the density of the earth by observing the times of swing of pendulums at the top and bottom of a mine. The first experiments were made in 1826 at the Dolcoath copper mine in Cornwall, and failed when the pendulum fell to the bottom. In 1854, the experiments were again undertaken in the Harton coalpit, near Sunderland.[39] Gravity at the surface was greater than below, because of the attraction of a shell equal to the depth of the pit. From the density of the shell as determined from specimens of rock, Airy found the density of the earth to be 6-1/2 times greater than that of water. T. C. Mendenhall, in 1880, used a Kater convertible pendulum in an invariable manner to compare values of gravity on Fujiyama and at Tokyo, Japan.[40] He used a "simple" pendulum of the Borda type to determine the absolute value of gravity at Tokyo. From the values of gravity on the mountain and at Tokyo, and an estimate of the volume of the mountain, he estimated the mean density of the earth as 5.77 times greater than that of water.
In 1879, Maj. J. Herschel, R.E., stated:
The years from 1840 to 1865 are a complete blank, if we except Airy's relative density experiments in 1854. This pause was broken simultaneously in three different ways. Two pendulums of the Kater pattern were sent to India; two after Bessel's design were set to work in Russia; and at Geneva, Plantamour's zealous experiments with a pendulum of the same kind mark the commencement of an era of renewed activity on the European continent.[41]
With the statement that Kater invariable pendulums nos. 4 and 6 (1821) were used in India between 1865 and 1873, we now consider the other events mentioned by Herschel.
Repsold-Bessel Reversible Pendulum
As we have noted, Bessel made determinations of gravity with a ball ("simple") pendulum in the period 1825-1827 and in 1835 at Koenigsberg and Berlin, respectively. In the memoir on his observations at Koenigsberg, he set forth the theory of the symmetrical compound pendulum with interchangeable knife edges.[42] Bessel demonstrated theoretically that if the pendulum were symmetrical with respect to its geometrical center, if the times of swing about each axis were the same, the effects of buoyancy and of air set in motion would be eliminated. Laplace had already shown that the knife edge must be regarded as a cylinder and not as a mere line of support. Bessel then showed that if the knife edges were equal cylinders, their effects were eliminated by inverting the pendulum; and if the knife edges were not equal cylinders, the difference in their effects was canceled by interchanging the knives and again determining the times of swing in the so-called erect and inverted positions. Bessel further showed that it is unnecessary to make the times of swing exactly equal for the two knife edges.
The simplified discussion for infinitely small oscillations in a vacuum is as follows: If T_{1} and T_{2} are the times of swing about the knife edges, and if h_{1} and h_{2} are distances of the knife edges from the center of gravity, and if k is the radius of gyration about an axis through the center of gravity, then from the equation of motion of a rigid body oscillating about a fixed axis under gravity
(T_{1})^{2} = [pi]^{2}(k^{2} + (h_{1})^{2})/gh_{1},
(T_{2})^{2} = [pi]^{2}(k^{2} + (h_{2})^{2})/gh_{2}.
Then
(h_{1}(T_{1})^{2} - h_{2}(T_{2})^{2})/(h_{1} - h_{2})
= ([pi]^{2}/g)(h_{1} + h_{2})
= [tau]^{2}.
[tau] is then the time of swing of a simple pendulum of length h_{1} + h_{2}. If the difference T_{1} - T_{2} is sufficiently small,
[tau] = (h_{1}T_{1} - h_{2}T_{2})/(h_{1} - h_{2}).
Prior to its publication by Bessel in 1828, the formula for the time of swing of a simple pendulum of length h_{1} + h_{2} in terms of T_{1}, T_{2} had been given by C. F. Gauss in a letter to H. C. Schumacher dated November 28, 1824.[43]
The symmetrical compound pendulum with interchangeable knives, for which Bessel gave a posthumously published design and specifications,[44] has been called a reversible pendulum; it may thereby be distinguished from Kater's unsymmetrical convertible pendulum. In 1861, the Swiss Geodetic Commission was formed, and in one of its first sessions in 1862 it was decided to add determinations of gravity to the operations connected with the measurement--at different points in Switzerland--of the arc of the meridian traversing central Europe.[45] It was decided further to employ a reversible pendulum of Bessel's design and to have it constructed by the firm of A. Repsold and Sons, Hamburg. It was also decided to make the first observations with the pendulum in Geneva; accordingly, the Repsold-Bessel pendulum (fig. 16) was sent to Prof. E. Plantamour, director of the observatory at Geneva, in the autumn of 1864.[46]
The Swiss reversible pendulum was about 560 mm. in length (distance between the knife edges) and the time of swing was approximately 3/4 second. At the extremities of the stem of the pendulum were movable cylindrical disks, one of which was solid and heavy, the other hollow and light. It was intended by the mechanicians that equality of times of oscillation about the knife edges would be achieved by adjusting the position of a movable disk. The pendulum was hung by a knife edge on a plate supported by a tripod and having an attachment from which a measuring rod could be suspended so that the distance between the knife edges could be measured by a comparator. Plantamour found it impracticable to adjust a disk until the times of swing about each knife edge were equal. His colleague, Charles Cellerier,[47] then showed that if (T_{1} - T_{2})/T_{1} is sufficiently small so that one can neglect its square, one can determine the length of the seconds pendulum from the times of swing about the knife edges by a theory which uses the distances of the center of gravity from the respective knife edges. Thus, a role for the position of the center of gravity in the theory of the reversible pendulum, which had been set forth earlier by Bessel, was discovered independently by Cellerier for the Swiss observers of pendulums.
In 1866, Plantamour published an extensive memoir "Experiences faites a Geneve avec le pendule a reversion." Another memoir, published in 1872, presented further results of determinations of gravity in Switzerland. Plantamour was the first scientist in western Europe to use a Repsold-Bessel reversible pendulum and to work out methods for its employment.
The Russian Imperial Academy of Sciences acquired two Repsold-Bessel pendulums, and observations with them were begun in 1864 by Prof. Sawitsch, University of St. Petersburg, and others.[48] In 1869, the Russian pendulums were loaned to the India Survey in order to enable members of the Survey to supplement observations with the Kater invariable pendulums nos. 4 and 6 (1821). During the transport of the Russian apparatus to India, the knives became rusted and the apparatus had to be reconditioned. Capt. Heaviside of the India Survey observed with both pendulums at Kew Observatory, near London, in the spring of 1874, after which the Russian pendulums were sent to Pulkowa (Russia) and were used for observations there and in the Caucasus.
The introduction of the Repsold-Bessel reversible pendulum for the determination of gravity was accompanied by the creation of the first international scientific association, one for geodesy. In 1861, Lt. Gen. J. J. Baeyer, director of the Prussian Geodetic Survey, sent a memorandum to the Prussian minister of war in which he proposed that the independent geodetic surveys of the states of central Europe be coordinated by the creation of an international organization.[49] In 1862, invitations were sent to the various German states and to other states of central Europe. The first General Conference of the association, initially called _Die Mittel-Europaeische Gradmessung_, also _L'Association Geodesique Internationale_, was held from the 15th to the 22d of October 1864 in Berlin.[50] The Conference decided upon questions of organization: a general conference was to be held ordinarily every three years; a permanent commission initially consisting of seven members was to be the scientific organ of the association and to meet annually; a central bureau was to be established for the reception, publication, and distribution of reports from the member states.
Under the topic "Astronomical Questions," the General Conference of 1864 resolved that there should be determinations of the intensity of gravity at the greatest possible number of points of the geodetic network, and recommended the reversible pendulum as the instrument of observation.[51] At the second General Conference, in Berlin in 1867, on the basis of favorable reports by Dr. Hirsch, director of the observatory at Neuchatel, of Swiss practice with the Repsold-Bessel reversible pendulum, this instrument was specifically recommended for determinations of gravity.[52] The title of the association was changed to _Die Europaeische Gradmessung_; in 1886, it became _Die Internationale Erdmessung_, under which title it continued until World War I.
On April 1, 1866, the Central Bureau of _Die Europaeische Gradmessung_ was opened in Berlin under the presidency of Baeyer, and in 1868 there was founded at Berlin, also under his presidency, the Royal Prussian Geodetic Institute, which obtained regular budgetary status on January 1, 1870. A reversible pendulum for the Institute was ordered from A. Repsold and Sons, and it was delivered in the spring of 1869. The Prussian instrument was symmetrical geometrically, as specified by Bessel, but different in form from the Swiss and Russian pendulums. The distance between the knife edges was 1 meter, and the time of swing approximately 1 second. The Prussian Repsold-Bessel pendulum was swung at Leipzig and other stations in central Europe during the years 1869-1870 by Dr. Albrecht under the direction of Dr. Bruhns, director of the observatory at Leipzig and chief of the astronomical section of the Geodetic Institute. The results of these first observations appeared in a publication of the Royal Prussian Geodetic Institute in 1871.[53]
Results of observations with the Russian Repsold-Bessel pendulums were published by the Imperial Academy of Sciences. In 1872, Prof. Sawitsch reported the work for western Europeans in "Les variations de la pesanteur dans les provinces occidentales de l'Empire russe."[48] In November 1873, the Austrian Geodetic Commission received a Repsold-Bessel reversible pendulum and on September 24, 1874, Prof. Theodor von Oppolzer reported on observations at Vienna and other stations to the Fourth General Conference of _Die Europaeische Gradmessung_ in Dresden.[54] At the fourth session of the Conference, on September 28, 1874, a Special Commission, consisting of Baeyer, as chairman, and Bruhns, Hirsch, von Oppolzer, Peters, and Albrecht, was appointed to consider (under Topic 3 of the program): "Observations for the determination of the intensity of gravity," the question, "Which Pendulum-apparatuses are preferable for the determination of many points?"
After the adoption of the Repsold-Bessel reversible pendulum for gravity determinations in Europe, work in the field was begun by the U.S. Coast Survey under the superintendency of Prof. Benjamin Peirce. There is mention in reports of observations with pendulums prior to Peirce's direction to his son Charles on November 30, 1872, "to take charge of the Pendulum Experiments of the Coast Survey and to direct and inspect all parties engaged in such experiments and as often as circumstances will permit, to take the field with a party...."[55] Systematic and important gravity work by the Survey was begun by Charles Sanders Peirce. Upon receiving notice of his appointment, the latter promptly ordered from the Repsolds a pendulum similar to the Prussian instrument. Since the firm of mechanicians was engaged in making instruments for observations of the transit of Venus in 1874, the pendulum for the Coast Survey could not be constructed immediately. Meanwhile, during the years 1873-1874, Charles Peirce conducted a party which made observations of gravity in the Hoosac Tunnel near North Adams, and at Northampton and Cambridge, Massachusetts. The pendulums used were nonreversible, invariable pendulums with conical bobs. Among them was a silver pendulum, but similar pendulums of brass were used also.[56]
In 1874, Charles Peirce expressed the desire to be sent to Europe for at least a year, beginning about March 1, 1875, "to learn the use of the new convertible pendulum and to compare it with those of the European measure of a Degree and the Swiss and to compare" his "invariable pendulums in the manner which has been used by swinging them in London and Paris."[57]
Charles S. Peirce, assistant, U.S. Coast Survey, sailed for Europe on April 3, 1875, on his mission to obtain the Repsold-Bessel reversible pendulum ordered for the Survey and to learn the methods of using it for the determination of gravity. In England, he conferred with Maxwell, Stokes, and Airy concerning the theory and practice of research with pendulums. In May, he continued on to Hamburg and obtained delivery from the Repsolds of the pendulum for the Coast Survey (fig. 17). Peirce then went to Berlin and conferred with Gen. Baeyer, who expressed doubts of the stability of the Repsold stand for the pendulum. Peirce next went to Geneva, where, under arrangements with Prof. Plantamour, he swung the newly acquired pendulum at the observatory.[58]
In view of Baeyer's expressed doubts of the rigidity of the Repsold stand, Peirce performed experiments to measure the flexure of the stand caused by the oscillations of the pendulum. His method was to set up a micrometer in front of the pendulum stand and, with a microscope, to measure the displacement caused by a weight passing over a pulley, the friction of which had been determined. Peirce calculated the correction to be applied to the length of the seconds pendulum--on account of the swaying of the stand during the swings of the pendulum--to amount to over 0.2 mm. Although Peirce's measurements of flexure in Geneva were not as precise as his later measurements, he believed that failure to correct for flexure of the stand in determinations previously made with Repsold pendulums was responsible for appreciable errors in reported values of the length of the seconds pendulum.
The Permanent Commission of _Die Europaeische Gradmessung_ met in Paris, September 20-29, 1875. In conjunction with this meeting, there was held on September 21 a meeting of the Special Commission on the Pendulum. The basis of the discussion by the Special Commission was provided by reports which had been submitted in response to a circular sent out by the Central Bureau to the members on February 26, 1874.[59]
Gen. Baeyer stated that the distance of 1 meter between the knife edges of the Prussian Repsold-Bessel pendulum made it unwieldy and unsuited for transport. He declared that the instability of the stand also was a source of error. Accordingly, Gen. Baeyer expressed the opinion that absolute determinations of gravity should be made at a control station by a reversible pendulum hung on a permanent, and therefore stable stand, and he said that relative values of gravity with respect to the control station should be obtained in the field by means of a Bouguer invariable pendulum. Dr. Bruhns and Dr. Peters agreed with Gen. Baeyer; however, the Swiss investigators, Prof. Plantamour and Dr. Hirsch reported in defense of the reversible pendulum as a field instrument, as did Prof. von Oppolzer of Vienna. The circumstance that an invariable pendulum is subject to changes in length was offered as an argument in favor of the reversible pendulum as a field instrument.
Peirce was present during these discussions by the members of the Special Commission, and he reported that his experiments at Geneva demonstrated that the oscillations of the pendulum called forth a flexure of the support which hitherto had been neglected. The observers who used the Swiss and Austrian Repsold pendulums contended, in opposition to Peirce, that the Repsold stand was stable.
The outcome of these discussions was that the Special Commission reported to the Permanent Commission that the Repsold-Bessel reversible pendulum, except for some small changes, satisfied all requirements for the determination of gravity. The Special Commission proposed that the Repsold pendulums of the several states be swung at the Prussian Eichungsamt in Berlin where, as Peirce pointed out, Bessel had made his determination of the intensity of gravity with a ball pendulum in 1835. Peirce was encouraged to swing the Coast Survey reversible pendulum at the stations in France, England, and Germany where Borda and Cassini, Kater, and Bessel, respectively, had made historic determinations. The Permanent Commission, in whose sessions Peirce also participated, by resolutions adopted the report of the Special Commission on the Pendulum.[60]
During the months of January and February 1876, Peirce conducted observations in the Grande Salle du Meridien at the observatory in Paris where Borda, Biot, and Capt. Edward Sabine had swung pendulums early in the 19th century. He conducted observations in Berlin from April to June 1876 and, by experiment, determined the correction for flexure to be applied to the value of gravity previously obtained with the Prussian instrument. Subsequent observations were made at Kew. After his return to the United States on August 26, 1876, Peirce conducted experiments at the Stevens Institute in Hoboken, New Jersey, where he made careful measurements of the flexure of the stand by statical and dynamical methods. In Geneva, he had secured the construction of a vacuum chamber in which the pendulum could be swung on a support which he called the Geneva support. At the Stevens Institute, Peirce swung the Repsold-Bessel pendulum on the Geneva support and determined the effect of different pressures and temperatures on the period of oscillation of the pendulum. These experiments continued into 1878.[61]