The promises and pitfalls of precision: random and systematic error in physical geodesy, c. 1800–1910

ABSTRACT This article discusses the ways in which nineteenth-century geodesists reflected on precision as an epistemic virtue in their measurement practice. Physical geodesy is often understood as a quintessential nineteenth-century precision science, stimulating advances in instrument making and statistics, and generating incredible quantities of data. Throughout most of the nineteenth century, geodesists indeed pursued their most prestigious research problem – the exact determination of the earth’s polar flattening – along those lines. Treating measurement errors as random, they assumed that remaining discordances could be overcome by manufacturing better instruments and extending statistical analysis to a larger amount of data. In the second half of the nineteenth century, however, several German geodesists developed sophisticated methodological critiques of their discipline, in which they diagnosed a too-narrow focus on precision among their peers. On their account, geodesists urgently needed to identify and anticipate the causes of the remaining measurement errors that arose from the earth’s little understood interior constitution. While mostly overlooked in the literature, these critiques paved the way for many empirical successes in late nineteenth- and early twentieth-century geodesy, including the first convergent measurements of the earth’s polar flattening.


Introduction
Few scientific disciplines occupy such a central position in the modern history of precision measurement as physical geodesy.The design and refinement of the international metric system, statistical error analysis, the construction of numerous astronomical observatories, and the development of modern gravimetry and geography were all, at least partially, motivated by the desire to empirically determine the earth's figure and gravity field.While the importance of geodesy to the broader history of precision science has been explored from various angles, 1 relatively little has been written on the methodological significance that precision had for geodesists.My concern here is to fill that gap, analysing how the role of precision as an epistemic virtue in physical geodesy changed throughout time.
My historical analysis of the role of precision in geodetic practice is composed of two parts.In the first part, I make use of some canonical sources and secondary literature to establish the epistemic 'promises' that geodesists associated with precision measurement.I show that geodetic research began to overwhelmingly focus on precision measurement between the late eighteenth century and the second half of the nineteenth century, driven by decisive advances along three vectors: instrument making, statistical error analysis, and quantity of available data.As geodesists secured access to fine-grained instruments, adopted a shared statistical approach, and acquired funding for a growing number of surveys, many of them became convinced that further progress along those lines would pave the way for accurately determining the earth's physical parameters.The second part of my analysis is both more innovative and more historiographically ambitious.I show how some geodesists eventually began to criticize what they perceived as a toonarrow focus on precision measurement among their peers, whom they accused of misunderstanding the systematic nature of geodetic measurement errors.I trace back this critical tradition to two works published by German geodesists Philipp Fischer and Johann Benedict Listing in 1868 and 1872.Heinrich Bruns, then professor of mathematics in Berlin, later publicized these critiques among geodesists, ultimately leading to a thorough reorientation of geodesists' research focus.Fischer, in particular, has been a marginal figure in extant historiographical scholarship, while Listing is generally only remembered as a topologist and inventor of the term 'geoid'.Focussing on the changing perspectives on precision in geodetic practice allows us to reappreciate their seminal importance to the history of geodesy and physical geoscience at large.
My overall argument is that a historical study of precision's role as an epistemic virtue among geodesistsits practical interpretations, implications, and criticismsignificantly improves our understanding of geodesy's methodological development as a discipline, shedding light on overlooked contributions and enriching existing explanations of its empirical advances.Put more analytically, the 'promise of precision' that I discuss consisted of the methodological conception that increasing precision is the best strategy to improve the accuracy of measurements.Historicizing this idea and its problems might also prove insightful beyond historiography, providing new insights into the autonomous importance of systematic error analysis in physical measurement.This puts us in a position to appreciate the promises, pitfalls, and internal debates in the development of geodetic measurement as contributions to an ongoing methodological discourse. 2

The aims and problems of geodetic measurement
Before the twentieth century, determining the earth's polar flattening was regarded as the most general and prestigious problem in geodetic measurementhere understood as encompassing both the measurement of terrestrial distances and terrestrial gravity.Starting with Isaac Newton's and Christian Huygens's influential work in 1687 and 1690, geodesists had modelled the earth as an oblate ellipsoid of revolutiona hydrostatic equilibrium figure of rotating, self-gravitating fluid bodiesand defined its polar flattening as the ellipticity of that model.Consistently measuring the earth's polar flattening promised to offer evidence for Newtonian gravitation, which implied that its magnitude was bound between 1/230 and 1/579depending on the earth's internal density distribution.When performing empirical measurements, geodesists used latitudinal variations in either (i) effective surface gravity or (ii) the length of meridional arcs with latitude as indicator quantities, whose relationship to earth's polar flattening was fixed by the ellipsoid model.This broad epistemic framework was firmly established by the mideighteenth century after Newtonian gravitation had become widely accepted in continental Europe.Alexis Clairaut proved that the earth's rotation and Newtonian gravitation indeed implied the abovementioned limits for the model's ellipticitygiven an idealized internal composition of homogenous ellipsoidal shells with inwardly increasing density, hydrostatic equilibrium, and uniform rotation-, and several French arc measurements had shown that the earth is indeed oblate in shape.Although most histories have focussed on these early developments in physical geodesy, the international heydays of geodetic measurement must be located in the subsequent development of the discipline.Early geodesists had indeed established a rough framework for empirical measurement, but the available data was scarce and indicated conflicting results (Figure 1), leaving major questions unanswered: What is the exact magnitude of the earth's polar flattening, and is it compatible with Newtonian gravitation?To which extent does the earth's actual surface even resemble a smooth ellipsoid? 3During the eighteenth and nineteenth centuries, geodesists followed several pathways in pursuing these open questions, all of which centred around improving the precision of their measurements.Of course, several theoreticians also continuously tried to improve the derivation of equilibrium figure models, with Pierre Simon Laplace proving that one could further relax the symmetry requirements on the earth's internal density distribution,4 and George Biddell Airy introducing an additional parameter to indicate possible deviations of the earth from an ellipsoid at 45°latitude.5That being said, the standard ellipsoid model continued to be the basis of most measurements for long into the twentieth century.Consequently, the majority of geodetic research concentrated on improving the inferences to the model's parameters, not calling into question the model's capacity to represent the physical earth.Expressed in contemporary vocabulary, this means that geodesists understood themselves to tackle a problem of precisionare our measurements good enough to produce consistent outcomes at the required level of detail?rather than a problem of accuracy or validitydo our measurements produce meaningful data about their intended physical target?Geodesists pursued their inquiry along three separate but interlinked dimensions of precision measurement: the quality of instruments, the statistical interpretation of raw data, and the  Of course, geodesists also faced a host of practical challenges relating to the uses of their measurement results in regional cartography and navigation.My deliberate focus on the very general physical problems in geodetic measurement should not be taken to imply that these problems exhausted geodetic practice.If anything, focussing on these general contexts is as a first step towards a richer understanding of the development of a geodetic culture of precision, which included but was not exhausted by the project of measuring the earth's global physical parameters.
quantity of raw data.Indeed, Kathryn Olesko has shown that the very notion of 'precision' co-evolved with the advances in instrument making and the statistical analysis of measurement error discussed in what follows. 6Hence, I do not merely reconstruct how geodesists applied a prefigured concept of 'precision' to their scientific work.Rather, I trace how a relatively autonomous practice of precision measurement came into being, wherein the concept of 'precision' played a central role.

The promises of precision in geodesy
3.1.Advances in instrument making and standardization The instruments that geodesists used to measure the earth's figure differed for each of the two indicator quantities.Latitudinal variations in surface gravity were generally measured by observing changes in the length of isochronous pendulums with a fixed period.The first of such measurements were conducted by Jean Richer and Christian Huygens, with pendulum clocks constructed by the latter. 7Huygens originally intended these clocks to measure the supposedly constant gravitational acceleration and act as reliable timekeepers, exploiting the celebrated observation that the period of a sufficiently constrained pendulum seemed only to depend on the strength of terrestrial gravity and the length of the pendulum.Richer first noted that the length required to realize the same period seemed to vary with latitude, leading to a widespread recognition that such variations offered insights into the latitudinal changes in terrestrial gravity and, ipso facto, the physical figure of the earth that could explain these changes (Figure 2).Over the subsequent two centuries, several geodesists took on the challenges of practical gravimetry and successfully designed pendulums that measured local surface gravity and its variation with increasing precision.
After the initial observations with Huygens's clocks, later devices soon involved more complex constructions, beginning with a coincidence pendulum constructed by Jean Charles de Borda and Jacques-Dominique Cassini de Thury in 1792. 8The period of their pendulum was now assessed based on the 'coincidence' of a simple pendulum's period and time intervals simultaneously measured by a pendulum clock.Moreover, their results were corrected for the confounding effects of the buoyancy of the air, thermal expansion of the instrument, and the rule used for determining the wire length, as well as changes in the length of the pendulum due to the weight of 6 Kathryn M. the bob and its swings. 9Further necessary corrections were implemented by the Prussian astronomer and geodesist Friedrich Wilhelm Bessel in 1828, who corrected a simple free-swinging pendulum for a lag of rigidity in the connection between bob and wire. 10These early measurements invariably faced problems arising from the uneven density in the imperfect metal pendulums, which caused discrepancies between the pendulum's pivot point and its actual centre of oscillation.Instead of further employing thin wires to circumvent the issue, the British geodesist Henry Kater solved this problem by successfully designing a 'convertible' pendulum with a pivot point at each of its ends.The centre of oscillation could now be practically determined by moving a weight along the length of the pendulum and alternating the two pivot points until both of them lead to equal periods. 11The instrument makers Georg and Adolf Repsold from Hamburg, finally, achieved a breakthrough in the second half of the nineteenth century.Using a design from Bessel, they constructed a class of reversion pendulums that were quickly adopted all across central Europe, Switzerland, Russia, India, and the United States and eventually allowed for measurements of gravitational force up to a precision of 981.274 ± 0.003 cm s −2 . 12By the mid-nineteenth century, geodesists thus had access to incredibly precise and standardized instruments for measuring one of the two major indicator quantities of the earth's figure, making gravimetry a highly successful enterprise in precision measurement.A similar development took place in terrestrial distance measurement, which geodesists used to determine the latitudinal variations in the length of meridional arcs.Such measurements required instruments for (i) optically determining the angular position of different observation stations and their altitude relative to sea level and (ii) physically determining one or multiple 'baselines'.In an arc measurement, the baseline is integrated into a network of triangles, whose sides are then computed using trigonometry with additional corrections for the surface's assumed curvature.The different altitudes of stations in the network are further 'reduced' to an ellipsoidal surface coinciding with the sea level base on angular or barometric measurements of altitude.In the early triangulations of the late seventeenth and early eighteenth century, angles were still measured with horizontally oriented sextants, which gradually gave way to a specialized repetition circle designed by Borda in 1785 and so-called theodolites designed by English instrument firms from the 1730s onwards (most notably, Ramsden, Dollond, and Troughton & Simms) (Figure 3).German instrument making followed suit in the early nineteenth century, with Repsold in Hamburg, Breithaupt in Kassel, and Reichenbach & Utzschneider in Munich taking on leading roles. 13arl Friedrich Gauss, finally, designed the first heliotrope, an instrument used to send light signals from one observation point to another. 14Heliotropes soon replaced conventional observation points (such as church towers or similar landmarks) and were soon produced in standardized form by most major instrumentmaking firms.By the mid-nineteenth century, angular measurements were based on a class of standardized and increasingly precise instruments, leading to altitude reduction with mean errors of only 5 mm/km and relative coordinate determination for entire federal states that depart from twentieth-century values by less than ± 0,7 metre. 15he two main problems in the physical measurement of baselines concerned the choice of local physical standard, the so-called étalon, and its calibration relative to a general physical unit of length.The first of these issues was originally dealt with by directly measuring a baseline with a physical copy of the étalon, corrected for various unavoidable degradations and the thermal expansion of the metal.Various strategies were used to improve these measurements, eventually culminating in optical, microscopic measurements of the alignments between étalon and baseline from the 1820s onwards.The second issue had been debated since the early sixteenth century, with various physical units being proposed to define the toise, metre, yard, or inch.These units were systematically intercompared with increasing scrutiny (Figure 4).Eventually the Central European Arc Measurement organization successfully pushed for the 1875 international metre convention, which included a new set of physical standards.16

The emergence of statistical error analysis
Beyond the successful manufacture of increasingly standardized and finegrained instruments, geodetic precision measurement thrived on the back of statistical methods for inferring the best estimate from a large number of measurement results.Such methods were originally developed as practical heuristics in astronomy and geodesy and were only later connected to the mathematical theory of probability.17The Jesuit Roger Boscovich pioneered the first principled method for combining different measured values for the earth's ellipticity, determining a value from ten combinations of five measured arcs so that the magnitude of positive and negative corrections, as well as their sum, is minimized (Figure 5). 18While Boscovich had expressed his method geometrically, Laplace subsequently provided it with an algebraic formulation, in which he assigned arcs different evidential weights according to their amplitude. 19In 1776, the German physicist Johann Heinrich Lambert also applied principled error analysis to determine the flattening from pendulum measurements at different latitudes, by taking the bivariate mean between two equal-sized groups of the largest and smallest observations.20Finally, error analysis could also be applied to individual arc measurements, by determining the same distances in a triangulation based on different physical baselines and adjusting the total network (and corresponding arc length) according to some principle of error minimization.
The different methods for minimizing errors by determining the best estimate from a group of outcomes culminated in the method of least squares, according to which linear equations are adjusted such that the sum of the squares of residuals across the group of measurement outcomes is minimized.The French mathematician Adrien-Marie Legendre first discussed this method in print in 1805,21 while Gauss and the Irish mathematician Robert Adrian independently employed it in empirical problems.Laplace later used the theory of inverse probability to prove that least square is the optimal method for an infinitely large sample of observations, while Gauss proved that the method is optimal for any linear solution in which observational errors have a mean of zero, are uncorrelated, and have equal variances (Howarth 2001).Before, during, and after these theoretical results were established, least-squares was already being popularized among practitioners through the influential introductions written by the French geodesist Louis Puissant (1805), the German astronomer Johann Frank Encke (1834-36), and Gauss's student Christian Ludwig Gerling (1843). 22By the mid-nineteenth century, least squares error Figure 5. Boscovich's geometric representation of his method for determining the best fitting elliptic meridian l(θ) = l e +x sin 2 (θ), where θ is an arc's mean latitude, l its length, l e is the length of an arc at the equator and x is the excess length of an arc at the poles.In the graphic, Aa -Ee give the measured arc lengths and the horizontal axis AF gives the respective sin² of the arc's mean latitude θ.Boscovich's solution was to find A'H such that the corrections aA', bO, cK, dL, and eM fulfilled his two conditions.Taken from Roger Boscovich and Christopher Maire, Voyage astronomique et géographique dans l'État de l'Église, Paris 1770, plate 1.
analysis was routinely applied to angular and base-line determinations, as well as arc and pendulum measurements of the earth's polar flattening across Europe and many of its colonies.In 1858, two teams of computers of the British Ordnance Survey went as far as spending two and a half years to adjust their main triangulation, represented in a system of 1554 equations involving 920 unknowns. 23

Geodesy's geographic expansion
Statistical error analysis equipped geodesists with a piece of machinery for drawing measurement inferences in light of imperfect data.The utility of this machinery depends on the amount of data.In light of the ratio between the earth's surface area and the parts of it covered by pendulum stations and triangulation networks, many eighteenth-century geodesists had little confidence in the most likely outcomes they determined for the earth's polar flattening.A new age of large-scale surveying began in the 1790s with the foundation of the Ordnance Survey in Britain, the Great Trigonometrical Survey in India, and Pierre Méchain and Jean-Baptiste Delambre's arc measurement between Barcelona and Dunkirk.Partly due to the Napoleonic expansion, several German states subsequently established their own precision surveys, with Prussia, Bavaria, Denmark, Hessen, and Hannover soon conducting some of the most precise triangulations in the world. 24These networks expanded at an increasing pace, with Friedrich Georg Wilhelm von Struve spending nearly forty years coordinating the measurement of a 2820 km long arc between the Northern coast of Norway and the coast of the black sea in Ukraine, 25 and the Survey of India establishing multiple arcs across the entire subcontinent. 26In the mid-nineteenth century, the different Central and Western European triangulations were increasingly interconnected, a development culminating in Johann Baeyer initiating the European Arc Measurement, a pan-European organization with its central bureau located in Potsdam. 27he development of individual arc measurements into extensive and interconnected triangulation networks was enabled by the economic and political interests of increasingly centralized governments.High precision triangulations gradually became the basis of many regional and local topographical surveys, which served the military, administrative, and fiscal purposes. 28ince gravimetric measurements lacked similar practical applications, a larger network of pendulum measurements generally emerged later and more rarely.The first of such kind were constructed by Kater and the French physicist Jean-Baptiste Biot, who measured the swings of pendulums with fixed lengths across Western Europe and the Ordnance Survey's triangulation network in 1818 and 1819. 29Edward Sabine expanded on their work and conducted the first global gravimetric survey with a set of attached and detached invariable pendulums constructed by Kater. 30 The Russian Counter Admiral Friedrich Benjamin von Lütke followed Sabine's example in 1829, as did several other British, Russian, Spanish, Indian, and German surveys in the next decades.By 1830, geodesists had constructed at least 79 gravimetric measurement stations, supplying valuable raw data for measurement of the earth's figure.

From random to systematic error
As we have seen, nineteenth-century geodesists could claim significant progress in the statistical analysis of errors, the quantity of available data, and the manufacture of measuring instruments for the indicator quantities of the earth's figure.These advances in all the dimensions of geodetic precision measurement also led to a closer agreement in the measured outcomes for the earth's polar flattening, which were continuously modelled as the ellipticity of a rotating ellipsoid composed of homogenous ellipsoidal shells with an inwardly increasing density.While Boscovich struggled with strong discrepancies among the five contemporary measurements, the most likely values inferred from dozens of arcs and pendulums stations now all fell within an interval between 1/283 and 28 While the number and interconnection of high-precision triangulations did gradually increase throughout the nineteenth century, this increase was not linear.Governments' willingness to afford the costly precision standards required by large geodetic networks was far from self-evident.Many military or fiscal purposes of surveying could also be realised by individual and less precise surveys.For the case of India, this is discussed in: Miguel 1/316. 32All was not well, however, since the remaining discrepancies were not evenly distributed.Rather, the existing outcomes showed a systematic conflict, with pendulum measurement concentrated at one end and arc measurements concentrated at the other end of the distribution.As soonto-be Astronomer Royal Airy had it in 1826: The ellipticity of the earth, deduced by Captain Sabine from a series of pendulum experiments the most extensive, and apparently the most deserving of confidence, that has ever been made, differs considerably from that which, as is generally believed, is indicated by geodetic [arc] measures. 33is was not the only emerging problem.Whichever global ellipticity value geodesists' chose, they had ascribed significant errors to individual pendulum and arc measurements, which far outstripped their assumed precision.In other words, the local raw data seemed to resist being accommodated in the global ellipsoid model that was assumed in the measurementsa problem that Laplace already discussed in 1796 and which was subsequently reiterated by the Austrian general and geodesist Anton von Zach (elder brother of astronomer Franz Xaver von Zach) and his more widely known colleagues Carl Friedrich Gauss and Friedrich Wilhelm Bessel. 34In principle, geodesists had a plausible epistemic explanation for this problem: they had no detailed models of the earth's interior density distribution and could not predict its physical effects on different measurement stations.Various more specific explanations had been attempted in ad hoc fashion, ranging from the predicted attraction of mountains 35 to unknown geological anomalies, 36 or peculiar subterranean masses below islands, 37 but none of them had an established theoretical or empirical basis.This situation was further complicated when measurements near the Himalayas first showed that some topographic mass surpluses were compensated by subterranean mass deficitsa phenomenon whose underlying physical mechanism, as well as its geographical frequency, were the subject of additional dispute among geodesists. 38lthough these problems were indeed considerable, few geodesists cast doubt upon the reliability of their measurement procedures.Rather, the dominant reaction throughout the first half of the nineteenth century was to count on the further extension of the existing successes in precision measurement.Thus, almost all geodesists treated these errors as inevitable but random, promising to be filtered out by the statistical analysis of large enough quantities of precise measurement data.As Bessel argued in a seminal paper in 1838: 'one cannot expect to find anything lawful in the irregularities of the earth's surface […].What new research can show is their effect on geodetic work 39 [original italics]'.Even if the earth's subterranean strata remained inaccessible and the earth's figure was not a smooth ellipsoidas Bessel believedthe only way forward he saw was to improve how geodesists determined the best-fitting ellipsoid model.
Bessel's opinion on the matter gained widespread traction because he also offered a sophisticated method to limit the distorting influence of such random anomalies, exploiting the advances in precision measurement made during the preceding decades.His methodlater dubbed astrogeodetic network adjustmentrequired geodesists to record the differences between (i) astronomical and (ii) triangulated coordinates and consequently treat them as variables in a statistical error minimization problem.Let us quickly spell out the rationale for this procedure before returning to its historical significance.Both telescopes used for determining (i) and theodolites used for determining (ii) measure angles and thus need to be oriented relative to the direction of gravity, i.e. the direction of a plumbline.All theodolite stations are projected on the same (ellipsoidal) reference surface, which is supposed to be perpendicular to the direction of the plumbline at every point.If the earth's surface covered by triangulation is not ellipsoidal, reducing multiple observation points to one ellipsoidal surface introduces hidden systematic errors corresponding to the total magnitude of deflections of the vertical throughout the triangulation network.Contrary to triangulation networks, astronomical coordinates can be determined with a telescope that is oriented towards the direction of gravity at a single point.For conducting an astrogeodetic network, geodesists needed to construct several such astronomical stations across a triangulation network and compare absolute and local astronomical coordinates with the network-relative ellipsoidal coordinates.One could then choose a best-fitting ellipsoidal reference surface in such a way that the total amount of deviations between the two coordinate types is statistically minimized. 40Bessel's procedure was soon adopted by The Ordnance Survey 39 Friedrich Wilhelm Bessel, 'Ueber den Einfluss der Unregelmässigkeiten der Figur der Erde auf geodätische Arbeiten und ihre Vergleichung mit den astronomischen Bestimmungen', Astronomische Nachrichten, 14.19-21 (1837), 269-312 (p.272). 40Bessel, 'Ueber den Einfluss der Unregelmässigkeiten der Figur der Erde auf geodätische Arbeiten und ihre Vergleichung mit den astronomischen Bestimmungen', pp.295-304.
of Great Britain and Ireland and The Great Trigonometrical Survey of India and became one of the central motivations for Johann Baeyer's proposal of a unified European Arc Measurement. 41Leading geodesists widely interpreted it as a method to locally control 'irregular deviations and separating them from the earth's regular figure', eventually fulfilling 'the hopeful expectation that the arc measurements converge toward the ratio [given by pendulum measurements]'. 42In 1878, Johann Benedict Listing, holder of a prestigious mathematics chair in Göttingen, went as far as defining the 'extensive value' of an arc measurement by the product of its amplitude and the number of astrogeodetic control stations in the triangulation network. 43y 1870, the global investment in geodetic measurement reached an all-time high, with large parts of Europe, North America, and South Asia covered in triangulation networks of hitherto inconceivable scale (Figure 6) .In 1828, Gauss had cautiously referred to the geodetic 'interconnection of all European observatories' as a 'perhaps unreachable ideal' (Gauss 1828, 50), but this very ideal was now on the verge of being realized.Yet, despite all these networks (i) being measured with increasingly precise and intercalibrated procedures, (ii) being adjusted based on the statistical analysis of plumb-line deflections, (iii) and different arcs being combined based on established statistical methods, the global discordance with gravimetric results persisted.In the 1860s and 1870s, geodesists reacted to these developments with increased scrutiny and formulated radical critiques of their discipline.In what follows, I highlight three such critiques that would have a lasting impact on geodetic practice, formulated by the German geodesists Philip Fischer, Johann Benedict Listing, and Heinrich Bruns.At their heart lies a deep concern about the manners in which geodesists had treated errors in the measurements of the earth's polar flattening and a clear conviction that such errors are of a systematic, not a random nature.
The first of such critiques was formulated by Philipp Fischer, who took over the chair in Geodesy at the Technische Hochschule Darmstadt in 1843.Most notably, he wrote the first complete 'higher geodesy' textbooks in German, synthesizing applied statistics, systematic arc measurement, and the physical theories of the earth's figure and gravity field.After himself advancing the cause of more precise and extensive triangulation for decades, his writings culminated in an extensive and systematic critique of geodetic measurement practice, the 300-pages-long Untersuchungen zur Figur der Erde ('Investigations of 41 J. J. Baeyer, Ueber die Grösse und Figur der Erde: Eine Denkschrift zur Begründung einer mitteleuropäischen Gradmessung (Berlin: Verlag Georg Reimer, 1861), pp.44-45. 42Ibid., p. 44. 43Johann Benedikt Listing, Über unsere jetztige Kenntniss der Gestalt und Grösse der Erde, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen (Göttingen: Dieterich, 1872), p. 12. Listing's proposal is even more striking since he calls into question the reliability of arc measurements of polar flattening towards the end of the book.He does mention that the 'extensive value' of an arc does not take into account 'other than extensive value relationships' (pp.12-13).This cryptic remark may be read as hinting toward systematic errors that were not yet accounted for in established methods of assessing the relative evidential weight of arc measurements.
the Earth's Figure ').Fischer's book opens with a personal reflection on the 'complete security' and 'admiration' that he and other geodesists had felt towards 'the recent arc measurements the calculation methods worked out by Schmidt 44 and the work conducted by Bessel' 45a feeling that he now criticized as unjustified.Before giving his detailed argument for why this was so, Fischer linked the flaws of geodesy to a habit of treating measurement errors as random and overestimating the power of least square error analyses: I was significantly aided by an experience I have made earlier, which remains unexplained.When writing a textbook on the method of least squares in 1849, a series of examples thoroughly convinced me that the method of least squares sometimes does not lead to the best but erroneous results in unsuspicious circumstances. 46stead of a problem of statistical and instrumental precision, Fischer argued that geodetic measurements faced a problem of unaccounted systematic errors, which he described as a rot infecting scientific knowledge: All seems to point towards a rotten element in the science that has to be identified somewhere, and that we have indeed not arrived at the actual solution to the problem of the earth's figure; that several confusions need to be uncovered before can again occupy a clear scientific standpoint. 47scher went on to give a historical and systematic analysis of arc and pendulum measurements, in which he linked their ongoing discordance to geodesists' inability to predict the impact of gravity anomalies on triangulation networks.While he discussed several physical causes of such anomalies (subterranean geology, general departure of the earth's figure from an ellipsoid, different mass distribution along coastlines and on islands), his chief practical conclusion was that 'an injustice has been done to the pendulum measurements'. 48As we saw, large triangulation networks could only be corrected for gravity anomalies indirectly through scattered astrogeodetic control stations, leaving room for unaccounted errors to accumulate.In a series of pendulum measurements, however, such anomalies would be directly apparent in outliers from a regular, elliptic variation of surface gravity with latitude and could simply be compensated or excluded from the calculation: As we have seen, we can identify irregularities in surface gravity from the observations themselves, whose magnitude provides a measure for irregular influences.Consequently, we compensate those which are too large and others which are too small.This is not the case with arc measurements. 49scher thus inferred that arc measurements did not offer a suited tool for studying the earth's figure and should be used to measure its size after its ellipticity had been determined independently, i.e. by pendulum measurements.The radical departure that this conclusion marked from previous work can hardly be overstated.Not only did geodesists conventionally use arc measurements to determine the earth's polar flattening, but they routinely used their importance for solving the prestigious problem of the earth's figure to secure funding for their enormous cost. 50n 1873, Johann Benedict Listing took up Fischer's critique in a short but influential book published as part of the Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, entitled Unsere jetzige Kenntniss der Gestalt und Größe der Erde (Our Current Knowledge of the Shape and Size of the Earth). 51A student of Gauss's, Listing spent most of his life working on problems in mathematical topology for which he is remembered most frequently.His geodetic work, likewise, is usually remembered for its conceptual, rather than empirical contributions.Listing took up an idea of Gauss, who had distinguished between the ellipsoid model and the earth's 'geometrical surface' and had defined the latter as 'the surface which stands perpendicular to the direction of gravity everywhere and is partially realized by the ocean […] whose form is, at every point, determined by the irregular density of the solid part of the earth'. 52Listing gave this equipotential surface model its now common name 'geoid' and redefined astrogeodetic network adjustments as a means for choosing an ellipsoid that approximates the geoid most closely. 53The objective of Listing's paper, however, was not simply proposing a new label for Gauss's ideas, but further exploring their empirical consequence for geodetic practice.Listing's conceptual discussions offer his reader a conceptual framework for physically studying the effects that the earth's gravity field and corresponding density distribution of its outer strata have on polar flattening measurements.He justified this line of argument by appealing to Fischer's 'important investigation', which he quoted repeatedly with great enthusiasm.Listing echoed the idea that pendulum measurements should be preferred for measuring polar flattening.Moreover, he closed his paper by forcefully repeating Fisher's call for a shift from a merely statistical approach to the physical analysis of systematic errors: The previously preferred method in geodetic investigations, the discrepancies between pole heights or amplitudes, as they are measured on the geoidal surface or projected on the ellipsoid, were merely treated like simple observation mistakes, and physical anomalies in the direction of gravity were reduced to insignificant magnitudes through [astrogeodetic] adjustments.Now that the importance of deflections of the plumbline has acquired new foundations through the works of Pratt and Fischer's critical analysis, it is time to treat these discordances as falling under the 51 Listing further contrasted these ideas with new empirical data in a second book in 1878, whichto the best of my knowledgehas not received any mention or discussion in standard histories of German or international geodesy so far: Johann Benedikt Listing, Neue Geometrische und Dynamische Constanten des Erdkörpers, Nachrichten Der Königlichen Gesellschaft Der Wissenschaften Zu Göttingen (Göttingen: Dieterich Verlagsbuchhandlung, 1878).Among other things, the book illustrates just how much of an empirical focus Listing's work had and how closely he engaged with contemporary measurement results. 52Carl Friedrich Gauss, 'Bestimmung des Breitenunterschiede zwischen den Sternwarten Göttingen und ona durch Beobachtungen am Ramsdenschen Zenithsector', in Werke, Volume 9, ed.by Carl Friedrich Gauss, (Cambridge: Cambridge University Press, 2011 [1828[), p. 49). 53Listing, Über unsere jetzige Kenntniss der Gestalt und Grösse der Erde, p. 9.It would take until the twentieth century before satellite measurements would allow geodesists to turn the geoid into an empirically parametrized, global model of an equipotential plane approximating the earth's surface.Cf.Irene Fischer, 'The category of constant errors, 54 which cannot be controlled through the method of least squares.The inquiry will reach new stages and will have to follow new paths. 55

[my italics]
Ten years after its first publication, Fischer's book would inspire a third critique of geodetic practice by Heinrich Bruns, professor of mathematics at the University of Berlin and subsequent head of Leipzig's observatory.His book Die Figur der Erde: Ein Beitrag zur Europäischen Gradmessung (The Figure of the Earth: A Contribution to the European Arc Measurement) was widely read internationally and found its way into the existing histories of nineteenth-century geodesy. 56Bruns enjoyed a high standing in German-speaking mathematics, astronomy, and geodesy while embracing an openly critical perspective toward institutional geodesy.Himself associated with the European Arc Measurement, he was known to have a somewhat uneasy relationship with its head, Johann Jacob Baeyer. 57his is not entirely surprising, since Bruns used his book to publicly criticize the gap between the goals and methods of the project, which still counted on more extensive and precise arc measurements to improve geodetic measurements of polar flattening.With one decade passed since Fischer's critique, geodesists across the world kept following this practice; a matter that seriously irritated Bruns: It is a vain enterprise to achieve anything but a merely apparent increase in the accuracy of the dimensions of the earth's body derived from arc measurements.Whichever models of earth's surface one adopts […], their only reasonable purpose can be to offer a first approximation.Any other attempt is merely a calculation exercise, useful only for the person performing it. 58ready at the beginning of his book, he linked this situation to a mistaken view of the nature of measurement errors in geodesy, lamenting a drastic lack of theoretical tools to break the empirical deadlock between their conflicting measurements: One previously did not further use deflections of the plumbline in the determinations of the earth's figure, but simply treated them as random errors, satisfied with noting their existence and smallness.The geodesist was approximately in the same position as an astronomer that is expected to discuss current planetary observations based on perturbation theories from the time before Laplace.The existing solutions to the problem of geodesy are insufficient in so far, as they do not exhaust the existing numerical material and leave contradictions between the [ellipsoid] hypothesis and experience untouched. 59fter the publication of the book, Bruns's arguments were intensely discussed at the 1878 general meeting of the European Arc Measurement, 60 and he subsequently provided personal commentary on their practical implications in 1880. 61The large amount of attention paid to Bruns's work significantly contributed to the diffusion of Fischer's and Listing's ideas among geodesists.With these critiques being read widely, other aspects of formerly established approaches to geodetic practice gradually crumbled.Most notably is their reception by the German geodesist Friedrich Robert Helmert, who soon became the head of the Royal Prussian Geodetic Institute and the Bureau of the European Arc Measurement (soon to be renamed Internationale Erdmessung or Association Géodésique Internationale).In his canonical 1880 and 1884 textbooks, Helmert extended earlier critiques to pendulum measurements and proposed a physical, rather than a statistical approach for correcting the associated measurement data.While Fischer had still proposed to anticipate the effect of gravity anomalies by sorting out pendulum measurements that significantly departed from a measurement series extending across several latitudes, Helmert now argued for a new theoretical approach.Not only did his 'terrain reduction' anticipate the potential subterranean compensation of topographic surplus masses, but he proposed to divide up the area surrounding a pendulum station 'into a series of concentric circles of increasing radii, each ring subdivided into a number of sectors, each defining a prism of [geologically analyzed] rock', so that 'the topographic contributions from each zone can be summed to obtain the final correction'. 62espite considerable disciplinary inertia, 63 the call for a shift from mere precision measurement to new methods of systematic error analysis was now widely embraced.How far the underlying critiques of geodetic practice had been diffused is apparent in the reaction to a paper that Hervé Auguste Faye, an established astronomer and Helmert's successor as the head of the Association Géodésique Internationale, read at the Sorbonne in 1886.In it, he defended the view that the measures of arcs of meridians already made have done away with all irregularities, which at the beginning of this century were supposed to exist; and that one 60 As Wilko Graf von Hardenberg thankfully pointed out to me, a TF-IDF analysis of the minutes of the European Arc Measurement from 1875 to 1912 gives 'Bruns' as the most relevant word in the 1878 meetings.TF-IDF is a measure of the relevance of terms to specific texts from a collection, defined as the product of their frequency in the specific text and their inverse frequency across the entire collection.can assign for the form of the surface of the sea an ellipsoid of revolution, having an eccentricity of 1: 292 (accurate to one unit in the denominator).
In response to this argument, the geologist Albert de Lapparent rushed to criticize him in a supplementary article in Science, which was published just a few months later: I do not feel able to say how the assertions of M. Faye can be reconciled with the diametrically opposite ideas which have been developed in recent German works, noticeably in the ' Lehrbuch der Geophysik' of Gunther, the works published in 1868 by P. H. Fischer, in 1873 and 1877 by Listing, and, above all, in the important memoir which Bruns published at Berlin in 1876; which last is not even mentioned by the learned French astronomer. 64 the 1880s, scientists in and beyond geodetic institutions had thus begun to think about geodetic measurement errors as systematic and deserving of detailed physical investigation, marking a significant break with nineteenthcentury geodetic practice.

Solutions to the pitfalls of precision
Historians and geodesists routinely recall the period from 1880 to 1914 as a heyday of geodesy, leading to a radical expansion in our physical knowledge of the earth's gravity field and stimulating advances in geology and geophysics. 65After the end of World War I, the robust convergence achieved between different polar flattening measures in that period finally motivated geodesists to adopt the first standard models of the earth's figure and its gravity field. 66These developments are usually explained by better international cooperation among scientists and a corresponding expansion of measurement data. 67However, they also presumed a drastic change in the ways that geodesists conceptualized their measurement errors, mirroring a recognition that their remaining empirical problems were no artifacts of lagging precisionbe it in form of imperfect instruments, lagging statistical power, or insufficiently large samples of measurement data.Rather, the methodological critiques by Fischer, Listing, and Bruns had now convinced many geodesists that their strong focus on increasing measurement precision had led them to unduly neglect the identification and physical analysis of systematic errors.In the remainder of this article, I will briefly show how different responses to these critiques contributed to the empirical successes in international geodesy after 1880.
In his influential 1878 book, Bruns offered a sophisticated proposal for restructuring geodetic practice.Most notably, he argued that 'the general task of geodesy' ('die Allgemeine Aufgabe der Geodäsie') is the determination of the earth's gravity field, while a more immediate 'special task' ('die speciellere Aufgabe der Geodäsie') should be the construction of a model of the geoid as an equipotential reference surface in that field. 68Focussing on the special task, Bruns proposed to use astrogeodetic network adjustments and pendulum measurements of surface gravity to map out the regional structure of the geoid as precisely as possible. 69While such a radical departure from geodesists' classical orientation did not gain traction among his contemporaries, various researchers indeed picked up the call for more detailed studies of the local geoid behaviour.In 1905, German geodesist Ludwig Haasemann followed Bruns's proposal and constructed a geoid map for the Harz Mountain from 66 gravimetric stations. 70ther responses were decisively more optimistic than Bruns's, not requiring geodesists to give up their attempts to measure the earth's global ellipticity.Helmert, well aware of recent critiques, argued 'that the current practice of geodesists, who treat the geoid as an ellipsoid of revolution appears to be justified'. 71In light of Bruns's, Listing's, and Fischer's arguments, he now justified this claim by showing the extent to which systematic errors affected different measurements and how they could be sufficiently anticipated theoretically.In his 1884 textbook, Helmert took on this task by invoking additional measures of the earth's polar flattening, obtainable from two independent astronomical quantities: the earth's precession and a particular pair of perturbations in the moon's orbit.He showed that only his new terrain reduction could make the gravimetric measurement outcomes cohere with their astronomical counterparts, as well as the mean value of earlier arc measurements. 72Helmert and several other geodesists, astronomers, and geophysicists pursued this strategy repeatedly over the next decades, investigating the ability of different physical corrections to anticipate measurement errors in particular measure by assessing the resulting coherence with its gravimetric, geodetic, or astronomical alternatives. 73Until 1901, these gradual, comparative assessments of different systematic corrections allowed geodesists to 'successfully reduce the maximum divergence between measured inverse ellipticities from about 15 across two measurement procedures to about 1.7 across four measurement procedures'. 74 final approach to systematic errors in arc measurementsthe key target of Fischer's, Listing's, and Bruns's critiquewas pioneered by the American geodesist John F. Hayford in the early twentieth century.Hayford tried to find a method to quantitatively predict how far topographic surplus masses above the ellipsoid are compensated by specific subterranean density distributions.That is, he tried to offer a clearer empirical picture of the elusive phenomenon that the American geophysicist Charles Dutton had previously dubbed 'isostatic compensation'.75 Studying a total of 509 astrogeodetic control stations across the Coast and Geodetic Survey's North American triangulation network, he showed that detected deflections of the plumbline almost completely vanish when the network was oriented relative to an ellipsoidal surface at a depth of 113.8 km.Thus, most non-ellipsoidal gravity anomalies cancel each other out if a sufficiently large layer of subterranean strata is considered, implying a gradual strive towards equilibrium in the earth's crust.Hayford immediately realized that geodesists might use this insight to avoid systematic errors when amalgamating data from different triangulation networks across the world, which could now be oriented relative to a common subterranean reference surface.He illustrated the power of this idea by inferring a new polar flattening value from the North American network, which immediately fell into the very narrow convergence interval previously achieved by Helmert and others.76 Not only did corrections for isostasy soon become widely accepted among geodesists.When the International Association of Geodesy first accepted a global standard model of the earth's figure in 1918, it used Hayford's data to define the model's polar flattening parameter.77

Conclusion
In his 1861 proposal for a European arc measurement, Johann Baeyer stressed that 'contemporary metrology, practical astronomy, nautics, […] and all other sciences involving measurements are more or less indebted to geodetic measurement'. 78This statement did not just describe a particular vision of the history, but underlined geodesists' claim to exemplify an ideal, if not the ideal precision science. 79I have argued in this paper that ideas like these had long been constitutive of geodetic measurement practice, but would soon face severe criticism, ultimately leading to re-orientation of geodetic research at large.In particular, I have shown that Phillip Fischer, Johann Benedict Listing, and Heinrich Bruns argued that earlier successes in instrumental and statistical precision had lured geodesists into treating their systematic measurement errors as random, avoiding necessary investigations into their physical causes.Both Fischer's and Listing's important contributions to these problems have so far been overlooked in extant scholarship.
The role that geodesists initially assigned to precision as an epistemic virtue, as well as its subsequent criticism not only offer a new perspective into the discipline's development in the mid and late nineteenth century but are crucial for understanding its subsequent empirical successes at the turn of the twentieth century.While existing historical accounts more exclusively focussed on internationalization as a driver of scientific progress in the late nineteenth century, my aim here has not been to refute or challenge them.Rather, I hope that a focus on the epistemic role of precision and the rise of systematic error analysis offers a valuable complementary perspective.Fischer's, Listing's, and Bruns's critiques motivated new and successful empirical research into systematic errors and the earth's physical constitution.Among other things, such research finally allowed them to overcome the unexplained conflict between pendulum and arc measurements of ellipticity, construct detailed models of the regional gravity field, and empirically test hypotheses about subterranean isostatic compensation.Studying these developments is not merely of historiographical value but might offer transferable methodological lessons on investigating systematic errors in measurements of large and partially inaccessible physical systems. 80

Disclosure statement
No potential conflict of interest was reported by the author(s).

1
Volker Bialas, Erdgestalt, Kosmologie und Weltanschauung: Die Geschichte der Geodäsie als Teil der Kulturgeschichte der Menschheit (Stuttgart: Wittwer, 1982); Stephen M. Stigler, The History of Statistics: The Measurement of Uncertainty before 1900 (Cambridge, MA: Belknap Press of Harvard Univ.Press, 1993), chap.1; Ken Alder, 'A Revolution to Measure: The Political Economy of the Metric System in France', in The Values of Precision, ed. by M. Norton Wise (Princeton University Press, 1995), pp.39-71; Michael Kershaw, 'The "nec plus Ultra" of Precision Measurement: Geodesy and the Forgotten Purpose of the Metre Convention', Studies in History and Philosophy of Science Part A, 43.4 (2012), 563-76; Michael Kershaw, 'A Different Kind of Longitude: The Metrology of Location by Geodesy', in Navigational Enterprises in Europe and Its Empires 1730-1850, ed. by Richard Dunn and Rebekah Higgitt (Houndmills, Basingstoke, Hampshire: Palgrave Macmillan, 2015), pp.134-56.

Figure 1 .
Figure 1.Strongly varying magnitudes for polar flattening (ellipticité) and corresponding excess arc lengths that resulted from different pair-wise combinations of existing arc measurements in Peru, France, Lapland, Papal States, and South Africa.Taken from: Roger Boscovich and Christopher Maire, Voyage astronomique et géographique dans l'État de l'Église, Paris 1770, 483. 3

Figure 2 .
Figure 2. Engraving depicting Jean Richer during his astronomical and geodetic work in Cayenne (note the pendulum clock in the background), as imagined by Sébastien Leclerc.Taken from: Jean Richer, Observations astronomiques et physiques faites en l'isle de Caïenne, Paris 1679, 1.

Figure 3 .
Figure 3. Exemplary section of the Cassinis' arc measurement between 1680 and 1720 and the sketch of Thomas Sisson's theodolite in a later edition of Samuel Wyld's 1725 surveying manual.Sisson's theodolite had predecessors in name but pioneered the essential features of later theodolites by allowing for vertical and horizontal movement of the telescope.Taken from Jacques Cassini, De la grandeur et de la fure de la terre.Paris 1720, 232; Samuel Wyld, The Practical Surveyor, or, the Art of Land-measuring Made Easy, London 1765, appendix.

Figure 4 .
Figure 4. Alexander Ross Clarke's 1867 comparison of different étalons in terms of general units.Taken from Alexander Ross Clarke and Henry James, 'Abstract of the Results of the Comparisons of the Standards of Length of England, France, Belgium, Prussia, Russia, India, Australia, Made at the Ordnance Survey Office, Southampton'.Philosophical Transactions of the Royal Society of London 157 (1867), 180. 31

Figure 6 .
Figure 6.1870 chart of the GTSI triangulation network in South Asia, providing an exemplary illustration of the size and latitudinal and longitudinal integration of geodetic triangulation at the time.The chart only shows 'principal' triangulations that could be used for determinations of the earth's figure.Survey of India, Public domain, via Wikimedia Commons.
Ohnesorge, 'Theodolites at 20 000 Feet: Justifying Precision Measurement during the Trigonometrical Survey of Kashmir, 1855-1865', Notes and Records: The Royal Society Journal of the History of Science, Ahead of print edition (2021). 29Henry Kater, 'XXII.An Account of Experiments for Determining the Variation in the Length of the Pendulum Vibrating Seconds, at the Principal Stations of the Trigonometrical Survey of Great Britian', Philosophical Transactions of the Royal Society of London, 109 (1819), 337-508; Jean Baptiste Biot, Recueil d'observations Géodésiques, Astronomiques et Physiques, Exécutées Par Ordre Du Bureau Des Longitudes de France En Espagne, En France, En Angleterre et En Écosse, Pour Déterminer La Variation de La Pesanteur et Des Degrés Terrestres Sur Le Prolongement Du Méridien de Paris (Paris, 1821). 30Edward Sabine, An Account of Experiments to Determine the Figure of the Earth by Means of the Pendulum Vibrat- ing Seconds in Different Latitudes: As Well As on Various Other Subjects of Philosophical Inquiry (London: John Murray, 1825), p. 351. 31George Biddell Airy, 'Figure of the Earth', in Encyclopaedia Metropolitana, ed. by Edward Smedley, Hugh J. Rose, and Henry J. Rose (London: Fellowes & Rivington, 1845), 165-240 (p.230).
The relevant collection of annual minutes can be accessed through the 'Measuring the Earth' data base of the Max Planck Institute for the History of Science Berlin.61CentralbureauderEuropäischenGradmessung,Berichtüber die Verhandlungen der Sechsten Allgemeinen Conferenz der Europäischen Gradmessung, abgehalten in München Vom 13.Bis 16.September 1880 (Berlin: Georg Reimer, 1881), p. 22. 62 Richard Howarth, 'Gravity Surveying in Early Geophysics.II.From Mountains to Salt Domes', Earth Sciences History,26.2 (2008), 229-61 (pp.234-35).63Asexemplified by the contemporary results published by the British Ordnance Survey and the Survey of India.