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The Attraction of the Pyramids: Virtual Realization of Hutton's Suggestion To Hist. Geo Space Sci., 9, 1–7, 2018 https://doi.org/10.5194/hgss-9-1-2018 © Author(s) 2018. This work is distributed under the Creative Commons Attribution 4.0 License. The attraction of the pyramids: virtual realization of Hutton’s suggestion to improve Maskelyne’s 1774 Earth density estimate John R. Smallwood UCL Hazard Centre, Department of Earth Sciences, UCL, Gower Street, London WC1E 6BT, UK Correspondence: John R. Smallwood ([email protected]) Received: 11 September 2017 – Revised: 9 November 2017 – Accepted: 22 November 2017 – Published: 9 January 2018 Abstract. Charles Hutton suggested in 1821 that the pyramids of Egypt be used to site an experiment to measure the deflection of the vertical by a large mass. The suggestion arose as he had estimated the attraction of a Scottish mountain as part of Nevil Maskelyne’s (1774) “Schiehallion Experiment”, a demonstration of Isaac Newton’s law of gravitational attraction and the earliest reasonable quantitative estimate of Earth’s mean density. I present a virtual realization of an experiment at the Giza pyramids to investigate how Hutton’s concept might have emerged had it been undertaken as he suggested. The attraction of the Great Pyramid would have led to inward north–south deflections of the vertical totalling 1.8 arcsec (0.0005◦), and east–west deflections totalling 2.0 arcsec (0.0006◦), which although small, would have been within the contemporaneous detectable range, and potentially given, as Hutton wished, a more accurate Earth density measurement than he reported from the Schiehallion experiment. 1 Introduction 2 Background Following his analysis as part of Astronomer Royal Nevil In the late 17th century, Isaac Newton proposed two exper- Maskelyne’s experiment to measure Earth’s mean density on imental methods by which the mass of the Earth could be the Scottish mountain Schiehallion in 1774, Charles Hutton measured. One of his proposals was to measure the force suggested that another plumb-line experiment at a geometri- of attraction between two known masses in the laboratory, cally simpler large mass would be preferable to address some and the other was to measure the deflection of a weight on of the challenges he had encountered (Maskelyne, 1775a; a plumb-line towards a “hill”. Newton took as an exam- Hutton, 1821). His suggestion was that a similar experiment ple a hypothetical hemispherical mountain with a height of be conducted at the pyramids in Giza, Egypt (Fig. 1). A dig- 3 miles and a basal diameter of 6 miles. He calculated that, if ital realization of such an experiment at the Giza pyramids is the mountain had the same density as the Earth, the moun- presented here to investigate what could have resulted from tain would “by its attraction, draw the plumb-line [less than] Hutton’s concept. 2 min out of the perpendicular” (Newton, 1731), recognizing On a uniform spherical Earth, plumb-lines would hang so that the measurement of such a deflection presented a chal- as to point towards the Earth’s centre, its centre of mass. lenge for the experimentalists of his day. However, a hanging plumb-line is deflected if a nearby mass The first attempt to carry out the plumb-line deflection ex- attracts the plumb-bob (or plummet) (Fig. 2a). The deflection periment was that of Pierre Bouguer, who, encouraged by is very small, as outweighing the advantage of its proximity, the scale of the Andean mountains, undertook such an ex- the mass is small relative to the mass of the Earth. periment in 1738 as one of his contributions to the Paris Academy’s meridian arc expedition (Bouguer, 1749). Before attempting the experiment, Bouguer estimated that the An- dean mountain Chimborazo would give a plumb-line deflec- tion of 104300 (0.03◦). Stellar observations from a position on Published by Copernicus Publications. 2 J. R. Smallwood: Pyramid attraction: Hutton’s Earth density idea 1821 thermore, as Maskelyne appreciated, the shallow density of 40˚ crustal rocks is often less than half the Earth’s mean density, and that makes any plumb-line deflection an even smaller quantity. Maskelyne detected the deflection of plumb-lines towards Schiehallion by taking astronomical observations 36˚ at two temporary observatory stations on the northern and southern flanks of the mountain. The difference between as- tronomically determined latitude separation of the two obser- vatories and their surveyed locations yielded the gravitational Mediterranean Sea attraction of the mountain relative to that of the earth. Maske- 32˚ lyne (1775b) offered the preliminary finding that, “::: the mean density of the earth is at least double of that at the Giza surface :::”, and that “The whole quantity of matter in the earth ::: will be about four or five times as great as if it were 28˚ all composed of water”. m The precision of Maskelyne’s zenith sector was suffi- 2000 cient to yield a measurement of the mean sum of the two plumb-line deflections towards Schiehallion which he quoted 24˚ 0 as 11.6600, rounded down to 11.600 (0.00322◦) in his final statement of results. This sum of deflections, though small, −2000 was treated with confidence and taken forward to give an −4000 Earth mean density estimate. This calculation of Earth’s 20˚ density relied on the measurement of the mountain’s mass, 28˚ 32˚ 36˚ which required a detailed topographic survey, conducted be- tween 1774 and 1776, and extensive geometric modelling. Figure 1. Topography of the study area (Becker et al., 2009). Black Charles Hutton carried out the conversion of the survey mea- triangle: the location of the Giza pyramids, towards the north of the River Nile in Egypt. Red line: location of the profile in Fig. 2c. surements into the expected plumb-line deflections (Hutton, The presence of the Mediterranean Sea contributes to a southward 1778). After analysis of the topographic survey, Hutton was deflection of the vertical at Giza, to the south, but the difference in able to improve on the precision of Maskelyne’s preliminary − the effect between nearby stations is negligible. estimate and report a mean Earth density of 4500 kg m 3. In doing so, Hutton noted that the reduction of the topographic survey data was labour intensive, and though it had been se- the snowline on the southern flank of Chimborazo, and from lected for its favourable geometry, the mountain’s irregular- a lower second station to the west of the mountain, showed a ity led to a somewhat laborious and maybe inaccurate cal- much smaller observed deflection of the vertical than he pre- culation. Weather conditions had been challenging, and the dicted: just 7.5 arcsec (0.002◦). Bouguer concluded his report “inconveniences” of the Scottish mountainside contributed with the hope that a mountain of sufficient size in France or towards Maskelyne’s Royal Society Copley Medal award in England could be identified for a plumb-line experiment, so 1775 (Higgitt, 2014). that measurements could be made both in more benign con- Hutton subsequently reviewed the experimental analysis, ditions and with an improved experimental layout utilizing giving a final average Earth density estimate of 4950 kg m−3 stations to the north and south of a mountain, doubling the (Hutton, 1821), commendably accurate, but still some way deflection effect being measured. Although recent modelling short of the currently accepted average density value of has suggested his measured deflection was as would be ex- 5515 kg m−3 (Lowrie, 1997). pected from the topography (Smallwood, 2010), the appar- Hutton laid down a challenge to his successors: “let any ent discrepancy encouraged further plumb-line experiments person ::: look over and repeat the calculations ::: and try if to be undertaken. he can find any inaccuracy in them” (Hutton, 1821). Also, Such an attempt was proposed in 1772 by Revd. Nevil recognizing the challenges of the variable topography of Maskelyne, the British Astronomer Royal. In scoping this Schiehallion, Hutton suggested the Egyptian pyramids as of- proposed experiment, Maskelyne returned to Newton’s hemi- fering considerable advantages for a plumb-line experiment: spherical mountain and “by a very easy calculation [stated] “I cannot conclude this paper of enquiry, without express- that such a mountain would attract the plumb-line 101800 ing a hearty wish for the repetition of the large or mountain from the perpendicular” (Maskelyne, 1775a). However, iso- experiment, in some other favourable situation, and with im- lated real mountains on such a scale are unusual, and the proved means, if possible. For this purpose, I shall venture deflection of the vertical caused by real mountains is nor- just to mention an idea which has sometimes occurred to my mally much smaller than that in Newton’s example. Fur- mind, namely, that one of the large pyramids in Egypt might Hist. Geo Space Sci., 9, 1–7, 2018 www.hist-geo-space-sci.net/9/1/2018/ J. R. Smallwood: Pyramid attraction: Hutton’s Earth density idea 1821 3 (a) 250 ) 200 N S Plumblines (not to scale) 150 t (m, msl t (m, 100 Heigh 50 0 Northw 400300 200 1000 100200 300 400 Distance from pyramid crest (m) 0.5 ar d vertical deflection -0.5 Pyramid effect Ground effect -1.5 (b) ( ” 250 1 ) Ground+Pyramid effect 0.8 North 0.6 ) 200 0.4 wa 0.2 rd vertical deflection 150 t (m, msl t (m, 0 -0.2 Heigh 100 -0.4 -0.6 50 -0.8 ( -1 ” Distance from pyramid crest (km) ) 0 -1.2 54(c) 331012345 60 North Surface effect Water effect 40 wa Total 3 rd vertical deflection 20 2 0 y (km, msl) y (km, 1 800 600 400 200 0 200 400 600 800 aph Distance from pyramid crest (km) 0 -20 Mediterranean Sea opogr -1 T -40 -2 (” N S ) -3 -60 Figure 2.
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