275.4 Colonel Herschel] John Herschel (1837-1921), English astronomer who came to the U.S. to connect the English gravity survey with that going on in the U.S. Herschel oscillated Kater pendulums of the Royal Society, first at the Stevens Institute in Hoboken, N.J., and later at the Smithsonian. Herschel participated in the Conference on Gravity Determinations which was held in Washington D.C. in May 1882. The report of this conference, of which Peirce was the editor, includes work by Herschel (see W4:349-77).

275.9 Washington Observatory] The U.S. Naval Observatory in Washington, D. C., which was then located on the hill north of where the Lincoln Memorial now stands (i.e. Foggy Bottom). The observatory was established in 1830 as the Depot of Charts and Instruments. In 1844 its mission was expanded, and the observatory became an important center for scientific research.

275.13 Professor Baird] Spencer Fullerton Baird (1823-1887), American zoologist.

275.16 Cornell.] On the day Peirce completed his work at Cornell, the Cornell Daily Sun (5 February 1886) described his work as follows (p. 1):

Professor Peirce employed himself in observing the vibration of two pendulums which were kept swinging, one a cold pendulum, in the equatorial room of the Engineering building and the other, a warm one, in a closely walled room in the basement of the Physical laboratory, allowing only a small aperture through which to observe the vibrations by means of a telescope.

275.23-24 Professor E. A. Fuertes] Estevan Antonio Fuertes (1838-1903), American engineer, Dean of the Department of Civil Engineering, Cornell University, 1873-90. On 8 January 1886, Peirce wrote to Edward S. Holden a glowing letter of recommendation expressing his hope that Fuertes could become the next Coast Survey Superintendent.

276.25 White] Andrew Dickson White (1832-1918), founder and first president of Cornell University.

276.25 Adams] Charles Kendall Adams (1835-1902), president of Cornell University 1885-92.

276.26 Anthony] William Arnold Anthony (1835-1908), professor of physics at Cornell University 1872-87.

276.31 Villarceau régulateur] Instrument made by Antoine Joseph Francois Yvon Villarceau (1813-1883), French engineer who constructed an equatorial meridian-instrument and an isochronometric regulator for the Paris Observatory.

276.33 Professor Newberry] John Strong Newberry (1822-1892), American paleontologist and geologist.

277.2 Professor Schaeberle] John Martin Schaeberle (1853-1924), American astronomer.

277.7 President Angell] James Burrill Angell (1829-1916), President of the University of Michigan, Ann Arbor, 1871-1909.

277.8 Langley] Samuel Pierpont Langley (1834-1906), director of the Allegheny Observatory (1867-1887). He supervised the construction of the observatory's first telescope and clock. From 1887 until his death he was secretary of the Smithsonian Institution.

277.28 Professor Holden] Edward Singleton Holden (1846-1914), director of the Washburn observatory from 1881 until 1885.

277.29 Washburn Observatory] The observatory at the University of Wisconsin, named after Wisconsin governor Cadwallader C. Washburn and completed in 1881.

277.29 Mr. M. Updegraff] Milton Updegraff, an assistant at the Observatory from 15 August 1884 to about 1888, who worked primarily on observations and reductions of the positions of stars using a meridian circle. Circa 1890, after spending two years at the national observatory in Argentina, he became director of the observatory at the University of Missouri in Columbia, Missouri.

279.26 Krille] Peirce seems to have had first hand experience with the Krille clock, whose maker could not be further identified. On 25 August 1885 he sent B. A. Colonna an inventory of Coast Survey property in his possession at Ann Arbor, which listed a Krille clock with accessories.

280.4 Allegheny Observatory] The Allegheny Observatory is located ten miles north of Pittsburgh, Pennsylvania, in Allegheny City. It was constructed in the 1860s and affiliated with the Western University of Pennsylvania in Pittsburgh.

280.30 the Criterion] See Benjamin Peirce, "Criterion for the Rejection of Doubtful Observations," Astronomical Journal 2 (1852): 161-63. Though cumbersome to apply, Peirce's Criterion received some renown. There is no sign that Charles Peirce used it in the present report. For a discussion and statement of the Criterion see W3:132 and the corresponding annotation in W3. In CD Charles Peirce wrote simply that it was a "certain rule for preventing observations from being rejected without sufficient reason." The criticism of R. M. Stewart is now regarded as having revealed a fatal flaw in the procedure ("Peirce's Criterion," Popular Astronomy 28 [1920]: 2-3). Further references are given in Raymond Clare Archibald, Benjamin Peirce, 1809-1880: Biographical Sketch and Bibliography (Oberlin, Ohio: The Mathematical Association of America, 1925).

284.37 No. 1 was abandoned] Greely brought back Pendulum No. 1 with the expedition records at the time of their 1884 rescue (see 218.6-7 with ann.). Commander Robert E. Peary retrieved Pendulum Head No. 1 several years later.

286.3 New Stand] Peirce described the development of the second pendulum stand in a letter of 1 Sept. 1885 from Ann Arbor:

I have had an expensive oaken pendulum support constructed, costing about $80. This was necessary for two reasons; because the old stand was not sufficiently stiff, so that the pendulum made 4 oscillations too few per diem upon it, and also because I desired to economize by oscillating two pendulums at once and so abridging the time consumed at each station. The new support is a great success, and has been much admired by scientific men.

286.20 6 feet long.] R 1096 (1889.8) contains two large typed sheets. One lists the instruments that are needed for fieldwork (R 1096.4), and the other describes how to work with the pendulum (R 1096.5).

§ 26. Rules for the use of these pendulums.1st.

The following instruments will be needed in the field:

2 Peirce pendulums,
1 paper double pointed tacks,
2 heads,
2 vulcanite observing-keys with cords,
Bolts and leaden washers,
2 fine glass scales for reading chrono-graph sheets,
2 wooden pendulum stands,
1 mercurial barometer,
1 five-inch lens mounted on castors,
7 thermometers graduated to tenths of a Centrigrade degree,
1 reading lens (convenient with handle),
1 large Steinheil reading telescope,
1 microscope with eye-piece micrometer and 2/3 inch objective
2 condensing lenses,
3 Rochester lamps,
1 stage microscope with fine lines, divided into tenths and hundredths of a millimetre,
1 oil can with pump,
1 engine divided scale, 40 lines to the inch,
1 holder for the micrometre,
1 Atwood machine pulley, strongly made,
1 army prescription scales, with weights down to a centigramme,
3 salmon lines, down to a centigramme,
1 weight of 10 kilograms and 2 of 5 kilo-grams,
2 pendulum rests for laying pendulum down,
1 spring balance,
1 rest for end of pendulum,
1 small spirit level for leveling knife-edge plane, the divisions about 1'each,
1 large transit instrument,
1 current American Ephemeris,
1 box of tools, including monkey wrenches, wood clamps, etc.
1 Crelle's Rechentafeln,
1 four place table of natural trigonometric functions
1 high support for microscope,
1 astronomical clock (if possible),
1 Barlow's tables,
1 box-chronometer,
Stationary, record books, etc.
1 accurate pocket time-piece with second hand,
1 step ladder,
1 chronograph table,
1 Fauth chronometer,
1 computing table,
Chronometer sheets,
1 observing table,
Tin boxes for chronograph sheets,
4 chairs,
8 large cells gravity battery with blue and white vitriol,
1 cot bed and blankets,
2 foot rugs,
1 reel paraffin-coated wire,
1 petroleum stove,
1 spool silk covered wire,
2 shunts,
1 levelling instrument and staves.

The pendulum should first be compared in Washington, by means of the vertical comparator, a metre pendulum with Metre B, and the Yard pendulum with Yard and Metre Bar No. 1.

The knife-edges should first be carefully scrutinized; but they should not be removed unnecessarily. The thumb screws should be made very tight, and scratches made on their heads to detect any subsequent loosening. The positions of these should be noted in the record.

The values of the screw-revolutions of both microscopes of the vertical comparator should first be ascertained by measuring spaces on Metre B. State whether the screw head reading increases or decreases as the wire descends. The sum of the distances of the lines on the abutting pieces should be measured. A perfect illumination should be arranged with special precautions against the shifting of the lines. The comparator should be adjusted. 1st, its length should be intermediate between that of the pendulum, and the most nearly equal length on the standard. 2nd, it must be adjusted so that when its axis of rotation is vertical a plumb-line falls on the vertical lines of both microscopes at once, and is in focus of both microscopes at once. 3rd, the microscopes must be level, when the axis is vertical. 4th, the whole staff carrying the microscopes must be put in such a position, that with its axis of rotation vertical the defining line of the piece on which the pendulum rests is in focus. The lower piece should then also be in focus. 5th, the standard is brought into the vertical, and into focus. Thermometers should have been bound with tin-foil to the middles of the pendulum and standard. Sufficient time should be allowed for the temperature to settle. Two persons should not enter the comparing room at once; but the recorder should stand just without the door. The observer should work as quickly as possible, in the following order. 1st, lamps lighted to illuminate thermometers. 2nd, readings of thermometers by telescope, first on standard, second, on pendulum. Lights turned down. 3rd, one reading of micrometer on standard below, one on standard above. Comparator carefully turned on pendulum. One reading on pendulum above, two on pendulum below, one on pendulum above. Comparator turned back onto standard. One reading on standard above, one on standard below. 4th, lights turned up on thermometers. Thermometer-readings, first on the pendulum, second on the standard. The observer now goes out and reduces his observations. Two comparisons should be made in a day, at intervals of two hours. At least one complete readjustment of the comparator should be made before concluding the measurements of each pendulum.

After the measures are completed, the two thermometers should be immersed in water at the same temperature as the comparing-room, together with a third; and the three should be compared. The zero of the third should then be determined by immersing it for about three minutes in a pail of melting pounded ice, from which the water is not allowed to drain off.

3rd. The pendulums should be swung at the station in the Smithsonian building. The error of the barometer should be ascertained. In setting up the stand, pains should be taken to have the bearings of the parts, as well as the floor under the feet of the stand as clean as possible.

The stand having been set up, and wrenched up which such force as almost to endanger the perfect coherency of the wood, one of the brass heads is put on. The top of the stand should be already nearly level, and the head should be bolted on with the use of the leaden washers. By compressing these it should be leveled pretty accurately.

The next step is to measure the flexure. A weight of 10 kilograms should be hung from the middle of the knife-edge plane, during these measures. A flexible and inelastic cord must be bound to the middle of the knife-edge plane, and, passing off horizontally and at right angles to the direction of the knife-edge, pass over the Atwood machine pulley and carry a weight of 5 or 10 kilograms. No soft or yielding substances should be interposed between the cord and the steel. The centre of the cord must be exactly on the level of the plane. The stage micrometer must have its plane vertically and the line of readings coincident with the line of the cord produced on the other side of the knife-edge plane. The microscope should be supported in the finest possible way independently of the pendulum stand.

The observer should record his own observations while his assistant gently lets the weight bear and relieves it alternately. This may be done by a spring balance. Both should be sitting and neither must stir. The one who raises and lowers the weight may very well be seated on a shelf below the pulley. The observer must cautiously assure himself that there is no mistake about his measure of the flexure. The inclination of the knife-edge must now be measured. These measures having been concluded, the thermometers should be attached. One of these should have the centre of its bulk on the level of the lower knife-edge, or if the Yard and Metre pendulums are both used, half way between the levels of the two lower knife-edges. It must be in front of the stand and a little to one side. Another thermometer should be on the level of the upper knife-edge, towards the back of the stand and as far to one side as the lower thermometer is to the other.

The observer, wearing lined leather gloves, should then hang the pendulum upon the raised Y's, an operation requiring a certain dexterity. A weighed piece of engine-divided scale [divisions 40 to the inch] 1 11/20 inch long and 1/2 inch wide should already have been fixed to the pendulum with two weighed bits of wax. The observing telescope and illuminating lamp, mirror and lens should be a [djusted.]

286.23 W. B. Fairfield] Walter B. Fairfield, who was employed by the Coast Survey as "extra observer," assisted Peirce in his pendulum observations at Key West, on which occasion Peirce spoke very highly of him. Fairfield also assisted Peirce for the Greely Report (sel. 30).

286.25 W. B. Curtis] Unidentified.

286.26 E. D. Preston] Erasmus Darwin Preston (1851-1906), subassistant at the Coast Survey; after 1900, editor of publications of the Coast and Geodetic Survey.

286.28-29 D. C. Chapman] Coast Survey mechanician; see also sel. 30, 221.15-16.

289.8-11 + 275 . . . - 86 feet] These values are obtained by multiplying by 100, not 1,000 feet as Peirce has it. This implies that the denominator in the formula at 291.4 should be 0.0069 rather than 0.00069 logarithmic seconds.

289.13-14 "logarithmic seconds,"] The "logarithmic second" is Peirce's own invention. On 19 July 1890, Superintendent T. C. Mendenhall objected to Peirce's system for expressing gravity; Peirce's eight-page written response defended the use of the concept as a way of simplifying calculations involving values of relative gravity (Peirce to Mendenhall, 22 July 1890, NARG 23):

I apprehend that what you mean by saying my system is unnecessarily obscure, is that the C.G.S. system affords an unexceptionable mode of expressing gravity. In this, I am entirely with you. I have no idea of using any other system for expressing any quantity which the C.G.S. system expresses. But you will agree that there are innumerable cases in which we have to record physical determinations which have not yet been carried to the point at which we can compare them with the centimetre, the gramme, and the second. There is no more marked instance of this than determinations of the relative acceleration of gravity. These are much more accurate than absolute determinations. Their principal use, that of ascertaining the figure of the geoid, would be in no degree subserved by knowing the absolute value. The measurement of the absolute value and its calculation from observations is most advantageously separated from the relative determinations. Thus, we are often in the situation of having to express gravity relatively, with high precision, when we are quite unable to assign its absolute value with anything like that precision. In such a case, we plainly cannot make use of absolute units. . . . Values of relative gravity to be of any use must be carried to six significant figures. The only way they can be reduced, or used after they are reduced, is by multiplying and dividing them. If, therefore, they are expressed in a logarithmic form, that generally saves going twice or more to the table of seven place logarithms every time you have to do with one. . . . [Logarithmic seconds] have the effect of making all the operations of reduction and comparison additions & subtractions in place of multiplications and divisions.

The basic relation between two values of gravity, g₁ and g₂, and the corresponding oscillations n ₁ and n₂, is g ₁ / g ₂ = (n ₁ / n ₂ )². Consequently, 100,000 log(g ₁ / g ₂) = 200,000 log(n ₁ / n ₂) = 200,000 log(e)ln(n ₁ / n ₂) which Peirce takes as approximately equal to 86859 x 2 x (n ₁ - n ₂) / 2 n₁ or, equivalently, (n ₁ - n ₂) (see Victor Lenzen, "An Unpublished Scientific Monograph by C. S. Peirce," Transactions of the Charles S. Peirce Society 5 [1969]: 5-24, esp. 13-14, 21-22).

290.22 g = . . .] See ann. 291.22-23.

291.15-16 unpublished investigation] This investigation has not been found.

291.22-23 Coast Survey Report for 1881, Appendix 15] "On the Deduction of the Ellipticity of the Earth from Pendulum Experiments," Report of the Superintendent of the United States Coast an Geodetic Survey . . . 1881 (Washington: Government Printing Office, 1883), 442-56 (W4:529-34). On p. 445 (W4:534) Peirce gives a value for the ellipticity of the earth "taking = .0052375." There is no explicit explanation of the calculation, of what represents, or of the formula used in this conclusion to his 1881 writing. The formula used in the present report for reduction for latitude is based on Clairaut's Theorem, and the coefficient, to which Peirce has assigned the value of 0.0052375, is equal to 5m /2 - e where m is the ratio of the centrifugal force at the equator to the force of gravity at the equator and e is the ellipticity of the earth. A standard theoretical derivation of this formula is given in A. R. Clarke, Geodesy (Oxford: Clarendon Press, 1880), 66-82.

292.7 8 forms of pendulum apparatus] The eight forms of apparatus which Peirce usually used were the following:

(1) Pendulum 2, heavy end down, Old Stand
(2) Pendulum 2, heavy end down, New Stand
(3) Pendulum 2, heavy end up, Old Stand
(4) Pendulum 2, heavy end up, New Stand
(5) Pendulum 3, heavy end down, Old Stand
(6) Pendulum 3, heavy end down, New Stand
(7) Pendulum 3, heavy end up, Old Stand
(8) Pendulum 3, heavy end up, New Stand

At the Smithsonian in 1884-85 the four forms were (1), (3), (5), and (7). The reduced station numbers are calculated on the basis of the data contained in the first set of "Results of Single Swingings" charts (R 1096a:49 and following). Richard Tursman determined that the corrected oscillations per diem for each of the eight forms of apparatus (when reduced to sea level and to the equator) are:

Form	Smithsonian 1884-85	Smithsonian 1886	Ann Arbor	Madison	Cornell
(1)	86010.44	86011.64	86011.27	86010.07	86009.81
(2)	--	86012.52	86011.96	86010.37	86010.22
(3)	86005.04	86004.59	86004.53	86003.22	86000.17
(4)	--	86006.13	86004.09	86003.07	86001.68
(5)	89912.13	89912.33	89911.65	89909.86	89911.28
(6)	--	89911.06	89913.83	89913.19	89908.85
(7)	89926.19	89926.93	89925.76	89925.12	89924.04
(8)	--	89927.21	89926.75	89926.07	89923.39

It is not clear in these concluding steps of his work how Peirce obtained precisely the values he gives for either of the two sets of station numbers he provides in Table II and in the footnote. It is possible to arrive at numbers close to Peirce's by the following procedure. Starting with the data in the above table, for each of the eight forms of apparatus (but excluding the incomplete Smithsonian 1884-85 for purposes of this example), subtract 86401.9 from the value given for Smithsonian 86 (in form (1) for example: 86011.64 - 86401.9 = -390.26) and add the difference to each value in the form (obtaining in (1): 86401.9, 86401.53, 86400.33, 86400.07). After completing these calculations for each of the eight forms, apply some best-fit scheme for each station; simply averaging the results across the forms would give 86401.9, 86401.58, 86400.53, and 86399.03 for Smithsonian, Ann Arbor, Madison, and Cornell respectively. Dropping some of the more anomalous values of forms and using a least-squares approximation it is possible to get values closer to those of Peirce. However, for the second set at least, Peirce apparently applied temperature corrections in addition to those already applied in the tables under "Results of Single Swingings" (324), and gave a reduced weight to the results at Cornell (290n.1-6).

296.4 The formula] In his "On the Flexure of Pendulum Supports," Report of the Superintendent . . . 1881, 359-441 (W4:515-28); see p. 430 (not printed in W4).

296.4-5 C. S. Report . . . 430] "On the Flexure of Pendulum Supports," partially printed in W4:515-28 (1883). The page here referred to is not included in W4.

296.9 43429] This is half the coefficient used at 290.32 in what appears to be the same relationship. In the following line a conversion of units probably explains the appearance of the decimal point in 0.43429. Another value of this coefficient, 0.0434, is given a few lines later and is used in Table VIII.

296 TABLE VIII] The heading of the penultimate column was left blank by Peirce. The column gives the values of the coefficient of S in the formula preceding the table. In the heading of the last column Peirce or his typist appears to have failed to put in the delta and the solidus by hand (items which were normally added by hand) and these have been supplied.

300.20 "Measurements . . . p. 74] "Measurements of Gravity at Initial Stations in America and Europe," Coast Survey Report for 1876, 202-337, 410-16 (W4:79-144). Peirce's page comes from a separate, later printing and corresponds to p. 273 of the published report (or W4:122).

302.8 Breteuil] The Pavillon de Breteuil, located in the Parc de Saint-Cloud, at Sèvres, in the suburbs south-west of Paris, is since 1875 the site of the International Bureau of Weights and Measures.

302.8 Regnault's] Henri Victor Regnault (1810-1878), French chemist.

302.9 Wüllner] Adolph Wüllner (1839-1908).

302.18 "Measurements . . . stations"] See ann. 300.20. Peirce's reference is probably to his discussion of the effect of viscosity in W4:104-5.

302.18-19 The a priori . . . made, also.] This sentence was added in ink by Peirce to the typescript. Despite Peirce's explanation, it is not made clear in the following pages just what formulas he is using in Table XVIII. In his 1890 review of the typescript, Professor Ferrel stated his belief that Peirce's method might be "a better way" of determining the atmospheric effects but that Peirce had not yet published on this and that he, Ferrel, could make little sense of how the calculations presented in "Atmospheric Effects" and in "Descent of the Arc" were made or how they fitted together. He gave considerable attention, however, to Peirce's key expression in the "Descent of the Arc" (310), for the rate of change of the amplitude of oscillation. He wrote that "it at once seemed to me that there must be something wrong, and after mature consideration, and fruitless efforts to understand some parts of the methods, I am constrained to come to the conclusion that the whole matter is not only unnecessarily complicated, but the first part even erroneous." "I know of no such expression as that of D _n" he asserted. Peirce had published the same expression in "Measurements of Gravity" (264; W4:93), but Ferrel did not have access to any literature while composing his report. In any case, Peirce is not much more helpful in justifying this expression in his earlier report. Peirce once stated that to find out what the matter was with Stokes's theory was "one of the most difficult mathematical problems conceivable" (Peirce to Thorn, 29 Aug. 1887, NARG 23/22). Ferrel pointed out the failure of Peirce's efforts to make the formula fit the data and proceded to derive an expression that differed only in the constant term, -a , from Peirce's. In the course of doing this he showed how the second term, -b, came from Stokes's theory, which takes atmospheric viscosity only into account, and that Peirce's contribution was essentially the addition of the second-order term, -c², which attempts to account for additional atmospheric effects. The motivation for this term could come, as Ferrel described it, by first obtaining the decrement of arc through an application of Stokes's theory for the oscillations given in Table XIX for Pendulum No. 4, and then comparing those values with the decrement of arc calculated from the observed amplitudes given in the table. Ferrel found that the former were on the whole one-fifth less than the latter. After completing his report Ferrel added a postscript stating that he had overlooked Peirce's tables for the "A Priori Calculation of the Second Effect" and "A Posteriori Calculation of the Second Effect" where the final values of the former are one-fifth those of the latter. He stated that as he was "not acquainted with the formulae and notation I am not quite certain as to what the final results refer to," but the one-fifth difference was evidently sufficient cause for Ferrel to conclude that "Peirce's method is therefore the better, as I at first surmised, since it takes into account other resistances than those from viscosity, if there happen to be such." (Ferrel to Mendenhall, 19 Oct. 1890, NARG 23, Vol. 657, p. 20.)

302.19 Stokes's] Sir George Gabriel Stokes (1819-1903), British physicist and mathematician, and a pioneer in geodesy.

305(c8).11 9663] The mass of Pendulum No. 3 is given earlier as 9676 (see 285.32).

307(c8).5 3.3390] Instead of log 2083, this value is that of log 2183. It is used to calculate the logarithm in the next row but does not affect subsequent calculations.

309(t2,c2).10 2.2427] This and the corresponding value for Pendulum No. 3, 2.2452, appear to be twice the correct values. The corresponding logarithms are computed using the correct values.

310.9-11 Green . . . perfect fluid;] George Green (1793-1841), British mathematician. Peirce is probably referring to Green's "Researches on the Vibrations of Pendulums in Fluid Media," Transactions of the Royal Society of Edinburgh 13 (1836): 54-62.

310.13-17 Stokes . . . space.] See George Gabriel Stokes, "On the Effect of the Internal Friction of Fluids on the Motion of Pendulums," Transactions of the Cambridge Philosophical Society 9 (1856): 8-106, esp. 23.

310.17 Meyer] Oskar Emil Meyer (1834-1909), physicist and professor and director of the Department of Physics and the Mathematical Institute at the University of Breslau.

310.21 the old resistential formula] Peirce's equation was inspired by Coulomb's formula for the electrostatic force between two charged particles. In his "Gravity at Initial Stations" (W4:93), Peirce cites Benjamin Peirce's Analytic Mechanics where it occurs on p. 46. Peirce seems not to have succeeded here in determining appropriate values of a, b, and c. See ann. 302.18-19.

316.10-11 Their comparison . . . given.] The calculated values in the second column of the table are 1.2 less than those the formula gives. This indicates that one or both of the two values given for b and c is wrong, probably through mistyping.

317.9-10 I now calculate . . . and find] It is not clear what formula is being used; the first three values are identical to those before, as expected.

317.14-15 The calculation is as follows.] In the second table: instead of the stated value of Q as 0.47 -500 q, Q is taken as 0.47.

319 TABLE XX] How the values in the second column are obtained is not clear. In the fourth column, the fifth data value should be +2 based on the corresponding values in the previous two columns.

320.3-4 rule of false,] See ann. 252.8.

322.3-4 For different . . . will be] Since the present edition only includes the descent of the arc data for one of the meter pendulums, the constants for the descent of the arc for the yard pendulum, No. 3, are given below (from R 1096a:36):

log 1/P = 1.201
b'= 0.0000575 [H. e. down]
0.000169 [H. e. up]
Q = 0.5782 [H. e. down]
0.5792 [H. e. up]

322.18-9 "Measurements . . . p. 66] See ann. 300.20. The page number corresponds to p. 265 of the published report (or W4:105-6).

322.27-323.1 Pendulum No. 1 . . . Greely] See sel. 30.

323.16 [Table XXIII omitted.] ] Peirce next includes Tables XXIII-XXIX, which give "smoothed out" mean values for curves showing the descent of the arc--that is, the descending amplitude of the swinging arc--for each pendulum as it was oscillated at the Smithsonian station. Tables XXX-XXXIX, which give the descent of the arc for individual swingings, provide the observed mean values used to calculate the curves. Only representative samples are needed to illustrate Peirce's calculation of the decrement of the arc; the present edition abridges these sections by including only Table XXIV, which gives the smoothed curve for Pendulum 2 (swung heavy end down), and Table XXXI, which provides the observed decrement for each of the 13 heavy-end-down swingings of Pendulum 2 as well as the uncorrected mean values.

323 TABLE XXIV] A formula similar to that used in Table XXI appears to have been used here but what the corresponding constants are is not clear.

326.5-327.6 The first tables . . . comprises.] The first set of the two sets of tables to which Peirce refers is here presented in its entirety (Tables XL-LI). Each table in this set summarizes all swingings for a given pendulum at a given site after initial corrections for arc, rate, pressure, temperature, and inclination of the axis of rotation. Peirce uses the data to determine a mean value which is then further corrected for atmosphere, flexure, and expansion; at the end of each chart in this series, he projects the number of oscillations per diem for a given pendulum at a given site. These calculations will reduce to the logarithmic station number for each site as shown in Table II (see 291.10).

The second set (Tables LII-XCII) shows the individual swingings in detail, and provides the initial corrections summarized in the first set. The second set records the results of individual swingings for the eight forms of the gravity apparatus. This adds up to thirty-two tables for the four locations (Smithsonian 1886, Ann Arbor, Madison, Cornell). In addition Peirce swung nine forms of apparatus (all on the Old Stand and five involving Pendulums 1 and 4) at the Smithsonian in 1884-85 (see ann. 292.7). This gives a total of forty-one forms of apparatus, which corresponds to the data recorded in Tables LII-XCII. Each of these forty-one tables gives the same kind of information; only one (Table LXI) has been included in the present edition to illustrate the way that Peirce determined the number of oscillations and corrected them for use in Tables XL-LI.

Table LXI gives the results of eight swingings of Pendulum 2 heavy end down on the Old Stand at the Smithsonian in 1886. Each swinging in this case lasted a little over four hours (Peirce usually swung the invariables with heavy end down for 4 hours 11 minutes and with heavy end up for 1 hour 23 minutes). The exact time for the first swinging recorded in Table LXI, after initial corrections, is 4 hours 11 minutes and 34.073 seconds, or 15,094.073 seconds. Peirce repeats such results of the times for the individual swingings in the third column of each table in the first set (Tables XL-LI); for example, the exact time for the first swinging of Table LXI is repeated in Table XLIV, first entry, where it then receives final corrections before the number of oscillations per day is projected. See also ann. 331.

327 TABLE XL] The summary tables for Pendulums 2 and 3 (XLI-XLII, XLIV-LI) are fully reduced to the number of oscillations per diem. But Pendulums 1 and 4 (Tables XL and XLIII), which were only oscillated at the base station, are not reduced beyond the corrected time of swing (Table XLIII contains no reductions at all). Although the reductions are missing, the number of oscillations per diem for Pendulums 1 and 4 appear in Peirce's calculations for the absolute value of gravity at the close of the report. See 349(t2,c8-11).2-3.

331 TABLE XLIV] In this, as in all the tables of this type, the values in the fourth column of each sub-table give the corrected time for a fixed number of oscillations, usually 15000. It is implied that these are taken from the subsequent tables of the report of which one, Table LXI for the eight values for heavy end down, Old Stand, is printed here on p. 339. Comparison of the two shows that most of the eight values match (where, for example, 4 hours, 11 minutes, and 34.073 seconds corresponds to 15094.073 seconds). However, three of the values do not match exactly and this lack of a complete correspondence is typical for all of the tables for all of the pendulums.

332 TABLE XLV] For heavy end down, New Stand, only if the first value of nT is taken as 14431.484 instead of 14431.185 are the given values for error and mean correct. The source, Table LXVI (not printed here), gives this value as 14431.485, but in transferring these values to the present table all but the eighth value from Table XLVI have, without explanation, been similarly changed by 0.001.

333 TABLE XLVI] For heavy end up, New Stand, three values of nT have been altered by hand: .960 from .968, .928 from .941, .937 from .953. These changes result from changes made to Table LXXI (not printed here); Peirce has not, however, changed the corresponding differences from the mean.

334 TABLE XLVII] For heavy end down: log T ² should be -3366.17 for Old Stand and -3368.34 for New Stand based on the previous data values. Subsequent calculations, however, use the given values.

334 TABLE XLIX] For heavy end up, Old Stand, the final value should be 90027.70 based on previous values, some of which have been altered by hand.

336 TABLE L] Problems with the Ithaca oscillations led Peirce to reject or even cancel a number of swingings at this site. He describes these circumstances on the tables for individual swingings at Ithaca (Tables LXXXV-XCII), which are not included in the critical edition. Swingings missing from the final corrections at Ithaca (Tables L-LI) of the present edition were omitted by Peirce for various reasons, including random vibrations, rapid changes in temperature, and disturbance of the air around the pendulum.

337 TABLE L] For heavy end down, New Stand, the last two digits of the oscillations per diem, 86110.33, have been written by hand over typed numerals; the value should be 86110.35 based on the previous entries.

339 TABLE LXI] In the seventh swinging the time for 15000 oscillations should be 4 11 34.233 based on the previous data; this would change the final corrected time to 4 11 34.111. In any case, as noted above (ann. 331), Peirce uses a different corrected time in tables that draw on this table.

340.1 CORRECTION FOR INCLINATION] This correction appears not to have been made by Peirce in his previous gravity work nor by others in the field before or since.

340.10-11 but . . . proceeding.] Peirce provides six sample tables (XCIII-XCVIII) of the correction for inclination: Smithsonian (1886) Old and New Stands; Madison Old and New Stands; and Ithaca Old and New Stands (no corrections for Ann Arbor are given). The initial table in this sequence (XCIII) is included as an example of this work.

341.2 CORRECTION . . . TEMPERATURE] The values are those derived by Peirce in "On the Effect of Unequal Temperature upon a Reversible Pendulum" (W5:319-22).

342.15 Tiede clock] Sidereal clock made by M. Tiede of Berlin and housed in the Detroit Observatory since 1854.

343.1-3 The following . . . exhibited.] Peirce included nine tables (XCIX-CVII) of time signals and clock corrections covering the periods of time that he swung pendulums at the various stations in 1884, 1885, and 1886. These are abridged in the present edition; only the initial entries of Table XCIX, listing Peirce's first clock signals from the Naval Observatory (23-31 December 1884) and the recorded loss/gain of his own instruments, are included as samples of this calculation.

344.14 station errors] Peirce previously referred to these as the excesses of the reduced station-numbers over 86400 logarithmic seconds.

344.27-29 "Determinations . . . [Formula]] See W4:96.

345.5 Fauth chronometer] Most likely a chronometer produced by Fauth & Co., Washington D.C. (1874-1900). The company was founded by George N. Saegmuller (1847-1934), then a Coast Survey employee, and his two brother-in-laws Henry Lockwood (1834-1897) and Camill Fauth (1847-1925), the latter having started a business making surveying and astronomical instruments.

345(c2).1 9h.58m] Peirce does not state which station is represented in this example, but the time of last transit given here is that of the seventh swing of the yard pendulum (No. 3) on the Old Stand (heavy end down) at Ann Arbor.

346.5 a régulateur Villarceau] See ann. 276.31.

347.12 Negus 1589] Chronometer made by T. S. & J. D. Negus, New York (1850-).

348.4 unpublished investigation . . . earth] Probably 1888.4; see also ann. 241.2-3.

348.7 Sabine's] Sir Edward Sabine (1788-1883), British soldier and astronomer who participated in expeditions to find the Northwest Passage (1818, 1819-20), and conducted pendulum experiments at many locations world-wide.

348.8-11 I cannot admit . . . my own Memoir.] Friedrich Robert Helmert (1843-1917) discussed Peirce's work in Die mathematischen und physikalischen Theorien der Höheren Geodäsie (1884), vol. II, 209-212. The memoir Helmert was referring to is Peirce's "De l'influence de la flexibilité du trépied sur l'oscillation du pendule à réversion" (Verhandlungen der allgemeinen Konferenz der europäischen Gradmessung zu Stuttgart 1877, Berlin: Georg Reimer, 1878, 171-87). Helmert referred in particular to page 173 of Peirce's introduction. Note that this introduction differs slighty from that in the English version that appeared in the Coast Survey Report of 1881 (W3:217-34). Helmert objected to Peirce's finding (210): "Nur beachtete Peirce nach E. Plantamour und Cellerier noch nicht genügend die Verschiedenheit des Fundaments." Helmert thus believed that Peirce's failure to take into account differences in the foundation was why Peirce's Paris measurement departed so radically from the measurements of others ("mit den älteren Bestimmungen ganz und gar nicht stimmt"), and he discarded Peirce's result on this ground (212). The Paris measurements listed by Helmert are 3860, 3860, 3860, 3859, 3899, and 3950 respectively, of which the last measurement is Peirce's (211). The value 3950 does not stem directly from Peirce, but from Förster's article in the American Journal of Science 20 (1880): 327. Helmert seems to have been unaware of Peirce's discussion of Förster's paper in a later addition to his "On the Value of Gravity at Paris" (published in 1881), and where Peirce gave a new measure of 3917.5 (W4:150-51).

348.31 unable to account] Peirce's inability to account for the discrepancy was a major issue in Simon Newcomb's analysis of the report. In a 28 April 1890 letter to Superintendent Mendenhall, Newcomb referred to the difference as an "extraordinary discrepancy," stating that more investigation was needed than Peirce had given it.

349(t2,c8,9).18 990.9540 . . . 9553] The final values for the equatorial seconds pendulum for No. 1 and No. 2 should be 990.9542 and 990.9556 respectively based on the given data. Since these values differ considerably from those given by Peirce, and since it is not clear how Peirce arrived at his figures, Peirce's original values are retained.

350.17 Mr. Blair] Henry Wayne Blair (1851-1884), Coast Survey assistant.

350.19-20 U.S.C.S. C.S.P. 1878B] See ann. 221.10.

350.36 Clarke] Alexander Ross Clarke (1828-1914), English geodesist and author of Geodesy. See also ann. 291.22-23.

351.1 Blair comparator] Comparator designed by Henry Blair. See ann. 350.17.

351 TABLE CXII] The value for No. 2 at 15° - Metre should be -132.8, instead of -134, based on the previous values. The value for No 4 at 15° - Metre should be 241.4, instead of 241.6, based on the previous values. The original values are retained because the above values differ considerably from Peirce's, and since it is not entirely clear how Peirce arrived at them.

353 TABLE CXV] See also R 1095a:68 (1892) "Length of Metre in Inches," and R 1095a:60 (1892) "Inch in Millimetres," where in the latter Peirce gives a figure of 25.4001, based upon Clarke's value for the number of inches per meter as corrected by Benoît. In his 1892 Nation review of Hussey's Logarithmic Tables, Peirce gives a more precise figure of 25.40003 millimeters in an inch. Today the relationship between the inch and millimeters is not a matter of measurement but of definition: one inch = 25.4 mm exactly. See American National Standard "Metric Practice," ANSI/IEEE Standard 268-1982, American National Standards Institute, New York.

353(t2,c2).1 Kater] See ann. 241.3-4.

353(t2,c2).2 Clarke] See ann. 350.36. Peirce is most likely referring to Clarke's 1867 paper in Philosophical Transactions.

353(t2,c2).3 Rogers] Fairman Rogers (1833-1900), American engineer and a founding member of the National Academy of Sciences.

353(t2,c2).5 Heaviside] William James Heaviside (1840-1915), British geodesist.

353(t2,c2).6 Comstock] General Cyrus Ballou Comstock (1831-1910), U.S. Army Corps of Engineers.