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This webpage reproduces an article in
Vol. 12 No. 3 (Dec. 1929), pp482‑514.

The text is in the public domain.

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 p482  History of determinations
of the heights of mountains​1

Thayer's Note: In the text as printed, the figures and their captions are all placed together on pp488‑491. There being no particular virtue in that arrangement, in this Web transcription I distributed them closer to where they are referred to in the text. (The original placement of the figures accounts for what appears to be a jump from p487 to p492.)

There have been in vogue three methods of measuring elevations of mountains above adjacent plains or above the level of the sea: The geodetic method, the barometric method, and the boiling point method. Of these, the geodetic method is the oldest and at the same time the most accurate of recent time. It may be considered under three heads involving, respectively, three more or less distinct processes: Determination of heights, (1) by leveling, (2) by measurement of triangles, (3) by determining the distance a mountain is visible at sea.

The Geodetic Method.

References to heights of mountains found in Greek authors. Omitting references to extravagant and uncritical statements in ancient writings, relating to heights of mountains, we note that the earliest known scientific measurements of mountain heights were made by Dicaearchus, Eratosthenes and Xenagoras. The only available information concerning Xenagoras is given by Plutarch. Dicaearchus and Eratosthenes described their geodetic determinations in books which are now lost. Dicaearchus (about 350‑290 B.C.), a philosopher about 30 years younger than Aristotle, was born in Messina, but lived in Greece. He was contemporary with Euclid who wrote his Elements between 330 and 320 B.C. Eratosthenes flourished about a century later and was a contemporary of Archimedes. He is celebrated for his measurement of the earth, and for his determination of the distances of the moon and sun from the earth. He is reported to have estimated the altitude of the highest mountains as 10 stades  p483 [1850 meters], but no particular mountains are cited, the height of which he determined.2

The principal direct sources of information about measurements made by Dicaearchus are Pliny, Theon of Smyrna, Geminus and Strabo. Speaking of the "earth" Pliny states:​3 "It is indeed wonderful that with such extensive ocean and land surface it [the earth] forms a sphere. This view is defended also by Dicaearchus, an exceedingly learned man, who on the command of the kings has measured the mountains and given the perpendicular height of the Pelion, which he considers the highest, to be 1250 paces [1905 m; modern estimate, 1585 m], but this height is vanishingly small in comparison with the circumference of the earth. To me this view seems untenable, for I know that some Alpine peaks, by gradual ascent, rise to not less than fifty thousand paces." This last figure, if not an error of some copyist, must represent, not the vertical height, but the slanting approach to a mountain and its summit. The mountains which Dicaearchus measured are supposed to be those of Macedonia.

Theon of Smyrna says​4 that "the difference in altitude of the highest mountains and the lowest places of the earth amounts, along a vertical, to 10 stadia [1850 m], according to statements of Eratosthenes and Dicaearchus, and such considerable distances are ascertained by appliances, with the aid of dioptra for measuring distances."

Strabo5 refers to Polybius as authority for the statement that Dicaearchus and Eratosthenes had treated the subject in a scientific manner, and prepared a treatise with maps which was still extant in the time of Cicero.

Geminus6 when arguing that clouds do not reach the top  p484 of high mountains, makes the following statement: "The height of the Cyllene mountains amounts, according to the measurements of Dicaearchus, to less than 15 stadia [2775 meters], that of Atabyrius to less than 10" [1850 meters; modern estimate 1390 meters].

These are all the extant direct references to Dicaearchus. As to Xenagoras, Plutarch7 states: "At this place Olympus is, ten furlongs and ninety-six feet in height [1879 m; modern estimate, 2930 m], as it is signified in the inscription made by Xenagoras, the son of Eumelus, the man that measured it. The geometricians, indeed, affirm, that there is no mountain in the world more than ten furlongs high, nor sea above that depth; yet it appears that Xenagoras did not take the height in a careless manner, but regularly, and with proper instruments."

Without naming the men who made the measurements, some further altitudes of Greek mountains are given by ancient writers. Strabo8 writes: "That which is called the Acrocorinthus is a lofty mountain, perpendicular, and about three stadia and a‑half [647 m] in height." According to Euripides, this is "the sacred hill and habitation of Venus."

Strabo9 says also: "Arcadia is situated in the middle of Peloponnesus, and contains the greatest portion of the mountainous tract in that country. Its largest mountain is Cyllene. Its perpendicular height, according to some writers, is 20 stadia [3700 m], according to others, about 15 stadia [2775 m; modern estimate 2374 m]." The wide divergence of 15 and 20 stadia would indicate that at least one of these figures is a mere guess. Wide from the truth is also the estimate for Cyllene of 80 feet less than 9 stadia [1641 m] which Stephanus Byzantinus10 (geographer, sixth century A.D.) takes from the writings of the Roman architect Apollodorus. Evidently greater uncertainty existed regarding  p485 the height of Cyllene than of any other prominent Greek Mountain. Of interest is the following passage in Cleomedes11 (second century A.D.): "Those who say that the earth cannot be spherical, because of the depressions occurring in the ocean and the elevations speak without reason. The height of a mountain does not exceed 15 stadia and the depth of the ocean is not more. But 30 stadia have a vanishingly small ratio to 80,000 stadia, (the earth's diameter). It is like dust upon a ball."

Accuracy of Greek determinations. It is to be noted that the particular mountain altitudes which are explicitly ascribed to Dicaearchus and to Xenagoras are more accurate than those not attributed to any one. Dicaearchus estimated Pelion about 20% too high; Atabyrius less than 33% too high, Cyllene less than 17% too high. Xenagoras estimated Olympus 36% too low. The anonymous estimates for Cyllene vary from 56% in excess to 31% in defect.

We have noted a discrepancy between the statements of Theon of Smyrna on the one hand, and Geminus, Pliny and Cleomedes on the other. Theon of Smyrna claims that Dicaearchus and Eratosthenes set 10 stadia as representing the maximum height of mountains (in Greece). This statement appears to be in conflict with Geminus who asserted that Dicaearchus estimated Cyllene to be less than 15 stadia. It is in conflict with Pliny who attributed to Dicaearchus the height of Pelion as being 1250 paces, which is more than 10 stadia.​a It is in conflict with Cleomedes who sets 15 stadia as the recognized maximum mountain height. The motive for setting an upper limit to the heights of mountains was, as previously indicated, the desire to meet the argument that the great height of mountains made it impossible to entertain the theory that the earth was spherical.

The Theory involved in Greek measurements of heights. The theory of the procedure which Dicaearchus, Eratosthenes and Xenagoras used in the determination of the altitudes of mountains is nowhere specifically and fully set forth, but it must have been essentially the same as that used by Thales in measuring the heights of the pyramids and the distances of ships at sea: namely, the comparison of similar triangles, usually right triangles. The  p486 theory of problems in surveying is explained by Euclid in his Optics, chapters 19‑22. Some centuries later, this subject was set forth more fully by Heron of Alexandria, in his Dioptra. A modern surveyor might measure an acute angle and a side of one right triangle and then compute the required side. It is not probable that Dicaearchus proceeded in this manner, for there is no evidence that there existed in his day tables which would supply the necessary trigonometric ratios corresponding to the modern sine, cosine or tangent. The earliest table of this sort is the table of chords ascribed to Hipparchus, who flourished about 150 years later. By using two similar triangles, geometers performed their computations without trigonometry, by using only the lengthsº of sides: they did not find it necessary to measure angles in degrees. The theoretical explanations given by Euclid and Heron, referred to above, involve similar triangles and the measurements only of lengths.​b

We have no direct knowledge of the instruments used by Dicaearchus, Eratosthenes and Xenagoras in finding the elevations of mountains. Theon of Smyrna mentions the dioptra in a passage already quoted, but it is not clear whether he meant to say that the dioptra was used at the time of Dicaearchus, or only in his own day. A line of inquiry is supplied by Polybius (about 210‑122 B.C.) who says​12 that a strategist in war must utilize the sciences of astronomy and geometry for the purpose of finding distances and the heights of walls which cannot be measured directly in the face of the enemy.​c Polybius does not explain the process. Recently Friedrich Hultsch13 pointed out the obvious fact that the height of walls could be ascertained by performing the following two triangulations explained by Heron: (1) To measure an unapproachable distance, (2) To measure the height of a wall or mountain when its distance from the observer is known. Heron's process of measuring the distance from Α to an unapproachable point Β is shown by Fig. 1, where the distanceº Α Γ, Α Δ, Γ Ε are measured by rods and the distance Α Β is computed by similar triangles. As in computing the answer, it is necessary to divide by the difference between Γ Ε  p487 and Α Δ, accurate measurements are here called for; a small error in that difference causes a large error in the answer.

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Fig. 1.

From Heron's DioptraVIII, ed. H. Schöne, Leipzig, p220. This shows how the distance Α Β is obtained, after Α Δ, Α Γ and Γ Ε have been measured.

The second triangulation is evident from Fig. 2. Sighting the mountain top through the dioptra fixes the points Η and Κ on vertical rods. The water-level attached to the Dioptra supplies points for finding Κ Ξ and Η Ξ. By the first triangulation the distance Β Π to the center of the mountain is ascertained. Thus the mountain height Π Α may be computed. It will be observed that in the determination of the horizontal distance to the invisible center of a mountain, the surveyor would be obliged to point his instrument to the peak of the mountain, and to determine the points Δ and Γ in Fig. 1 by the use of vertical rods such as are shown in Fig. 2.

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Fig. 2.

From Heron's DioptraXII, ed. H. Schöne, p229. It shows how the height Π Α may be obtained, when Β Π and the triangle Η Ξ Κ are known.

Instruments used by the Greeks. In addition to a rod or a hodometer for the direct measuring of distances, there are five instruments known to Greek antiquity, any one of which might have been used for measuring the heights of mountains: namely the dioptra Fig. 3, the quadrant Fig. 4, the ancient astrolabe Fig. 5, the ancient cross-staff Fig. 6, and the gnomon Fig. 7. Which of these five is most likely to have been used? We are inclined to eliminate the ancient cross-staff. As far as known it was used only by Archimedes in certain astronomical determinations. The circular cross-piece was designed to measure the disc of the sun and moon, but was not particularly suitable for sighting the top of a mountain. We eliminate also the ancient astrolabe. It was designed more especially for use in astronomy to measure angles in degrees. We eliminate the quadrant; its degrees are superfluous; its radius was too small for marking the level accurately. As regards the gnomon, it is unsuitable, except for mountains favorably located, so that the shadow of the peak of the mountain could be conveniently marked. It demands a special location for the observer. There is no mountain which is known to have been measured by the Greeks, using the method of shadows. Pliny14 states that Mt. Athos at the summer solstice casts its shadow upon the market place of Myrina, which was 5000 paces from the island Thasos. That this fact was utilized for ascertaining the height of Athos has been conjectured

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Fig. 3.

Heron's DioptraIII, ed. H. Schöne, p193, showing the instrument called "dioptra", the forerunner of the theodolite. It contains no graduation into degrees.

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Fig. 4.

Quadrant, from Ptolemy's Composition Mathématique, ed. Halma, Vol. I, Paris, 1927, p48. Using the plumb-line, this could be made to serve as a level. Placing the eye at the center and looking at the mountain top, the points Η and Κ demanded in Fig. 2 could be located on the rods.

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Fig. 5.

Ancient astrolabe (armilles solsticiales), from Ptolemy's Composition Mathématique, ed. Halma, Vol. I, 1927, p64. Using a plumb-line, the inner circle, which turned in the plane of the graduated outer circle, could be used to fix a horizontal direction. Placing the eye near the center of the astrolabe, the points Η and Κ in Fig. 2 could be located.

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Fig. 6.

A forerunner of the cross-staff, used by Archimedes for determining the diameter of the sun and moon. If placed on a level platform, and if the position of the movable circular disc be adjusted, so that the eye at the end of the staff would see the top edge of the disc exactly with the mountain top, then without the use of surveying rods, data could be obtained corresponding to the triangle Κ Ξ Η in Fig. 2. This drawing is taken from Archiv. f. Geschichte d. Math., d. Naturw. u. d. Technik, Vol. 10, 1928, p464. The level platform could have been the chorobates in Fig. 9.

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Fig. 7.

The Gnomon, first used by the Babylonians. Simply a vertical rod, casting a shadow upon a horizontal plane. From the lengths of rod and shadow, and the length of the simultaneous shadow of a mountain, its height could be computed. This figure is from De quadrante geometrico, by Cornelius de Judeis, Nürnberg, 1594.

 p492 by Varenius,​15 but without satisfactory evidence. Varenius supplies additional but hypothetical data, and explains the method of computation. Reference to the shadow of Athos is made by more recent writers, for example, Mandeville16 and Kepler.​17 Robert Hues,​18 a sixteenth century classical scholar, comments as follows: "One of the most miraculous things which they have discovered of the mountain Athos is, that whereas it is situate in Macedony, it casts a shadow into the market-place of Myrrhina, a towne in the Island Lemnos, from whence Athos is distant 86 miles. But for as much as Athos lies westward from Lemnos, no marvaile that it casts so large a shadow, seeing that we may observe by daily experience, that as well when the Sun riseth as when it sets, the shadows are always extraordinarily long."

The dioptra of Heron, or some similar instrument which may be less complicated, is the only one remaining. Some such type was probably used by Dicaearchus and later surveyors, in finding the elevations of mountains. Heron says that his one instrument accomplished everything which several other older instruments did. Probably one of those older instruments was the water-level.

The height of hills could be ascertained by the process of leveling, as shown by Fig. 8, taken from Heron's Dioptra. Operations of leveling must have been practised ages before the time of Heron. Says a modern engineer: "There are many examples of large ancient engineering works which required precise leveling, such as the canal between the Nile and the Red Sea, open B.C. 250, after 600 years' construction; the canal through Mount Athos, six miles long, cut by Xerxes, B.C. 480, probably the most famous as well as the most ancient of ship canals. Traces of it are still visible. An interesting example is that of the canal at Corinth, projected by Periander, B.C. 600; 650 years later, in the time of Nero, his engineers came to the conclusion that the canal could not be dug, as the sea was higher on one side of the Isthmus  p493 than on the other, and consequently would wash away the island of Aegina."19

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Fig. 8.

Finding differences in altitude by leveling. Taken from Heron's DioptraVI, ed. H. Schöne, p207.

Leveling is the only operation in surveying which in ancient times had attained a precision which is comparable with the precision in modern leveling. The reason for this lies no doubt in the inexorable demands of nature; water would not run up hill.

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Fig. 9.

The chorobates of Vitruvius, an early instrument used in leveling, as restored by Newton. Taken from Transactions of the Newcomen Society, Vol. II, p51. It was a rod twenty cubits (9 m) long, having two legs and a water-level.

From Eratosthenes to Jean Picard (about 150 A.D. to 1672). For about 1800 years after the time of Eratosthenes, no noteworthy progress was made in the accuracy of geodetic measurement. To be sure, tables of the natural trigonometric functions and the general development of plane trigonometry made the solution of triangles easy. But practical measurement remained as crude as ever. A slight modification of the procedure described by Heron is found in the writings of Gerbert in the tenth century, who measures a base line directed to the center of the mountain. The general plan is clear from Fig. 10. Gerbert did not actually determine mountain heights by the process which he describes. Nor did Orontius Fine, who was professor in Paris and described this method by the use of a quadrant as shown in Fig. 11. We shall see that this same placing of the base line was used in 1724 by Father Feuillée at the Peak of Teneriffe of the Canary Islands.

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Fig. 10.

Gerbert's drawing to explain the determination of heights. Oeuvres de Gerbert, ed. A. Olleris, Paris, 1867, Figs. 56, 57.

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Fig. 11.

Use of the quadrant in determining elevations, as explained by Orontius Fine in his geometry, Paris, 1556.

It is not surprising that even mathematically trained writers will err in making estimates of mountain heights which they did not base on actual determination of distances and angles. Their imagination, unchecked by geodetic measurement and computation, took flights which would do justice to a Jules Verne. Thus the Arabic physicist Alhazen, famous for his experiments on light, would have the tops of the highest mountains to eight Arabic miles [about 14800 meters]. Of Cardan, the sixteenth century algebraist, Robert Hues says: "Cardan and some others profest Mathematicians are bold to raise them up to 288 miles, but with no small staine of their name have they mixed those trifles with their other writings."​20 Nor were literary men any wiser. Scaliger, an opponent of mathematical science, gives credence to the stories that Peak Teneriffe rises to fifteen leagues  p494 [about 90,000 m; modern estimate 3711 m], while Francesco Patrizzi, an Italian philosopher and orator, when arguing against the sphericity of the earth, elevates this peak to seventy miles [about 105,000 m].

Three noteworthy determinations of mountain heights were made about the beginning of the seventeenth century. The cone-shaped Cerro diº Potosi in Peru, where in 1546 rich silver deposits were discovered, was found to tower 4872 Spanish feet [1377 m] above the adjoining plain.​21 According to modern estimates, the city of Potosi is 4146 m, and the mountain Cerro 4830 m, above sea-level. Accordingly, Cerro towers 684 m above the city, which is about half of the seventeenth century estimate.

In Italy the Jesuit Giuseppe Biancani22 (Latinized Blancanus), from elevations taken at Parma with dioptric instruments, determined the height of Monte Baldo above lake Garda, to be 804 passus of 5 Bologna feet each [1528 m]. The modern figure is 2095 m.

In England, Edward Wright23 in 1589, by measuring a short base line on a hill near Plymouth sound and also certain angles, showed by computation that this hill was 375 feet above the sea. From this and other data he computed the diameter of the earth, which is about 15% too small. This was a very creditable performance.

A method of computing the height of mountains visible from the sea came into vogue among many writers, which is theoretically extremely simple, but which practically is encumbered by difficulties of such grave nature that it may lead to extravagant error. It is the determination of height by the distance a mountain is visible at sea or by the distance at sea that a ship can be seen from the top of the mountain, or by the depression of the horizon of the sea below the artificial horizon at the mountain top. This method calls for a simple computation, the solution of a right triangle, the vertices of which are the top of the mountain, the extreme point of visibility at sea and the center of the earth. In solving this  p495 triangle, the data most commonly used were the radius of the earth and the distance of visibility. But often the radius of the earth was computed from two other data assumed to be known. Computations of this sort were made by many writers, including Maurolycus,​24 Kepler,​25 Huygens26 and Varenius.27

Practically the difficulties of this method were insuperable. Before the seventeenth century there prevailed extreme uncertainty relating to the size of the earth. To be sure, Eratosthenes had given a good value for the size of the earth, but there was in early days no way of telling that this value was superior to others. Edward Wright28 pointed out in 1610 that the earth's radius was estimated by different writers all the way from 8000 miles to 3200 miles. A second grave source of error arose from atmospheric refraction. Along the horizon, rays of light deviated largely from straight lines, and no reliable tables of correction existed at that time. Mirages are due to this curving. The bending of the rays may make a mountain look very much higher than it really is. To add to the defects of this method, there was no means of measuring with reasonable accuracy the distance from a ship at sea to the center of the mountain at sea level. Sailors reported the Peak Teneriffe to be visible at a distance of four degrees of the meridian. Varenius29 computes from this that this Peak must be eight Italian miles high [11840 meters] "quae incredibilis fere est." He then allows for refraction, replacing 4° by 3° and finally by 2½°, and obtains for the height 5 and 4 Italian miles! The difference between guess data and measured data is well illustrated in the case of Snell. From hear-say data he​30 computed the Peak of Teneriffe allowing 1° for refraction to be 27,000 feet high (an error in excess of 121%) and of Aetna 25,416 feet (an error in excess of 149%), but his estimate of the size of the earth, obtained in 1617 from his own very painstaking measurements, although taken at a time when  p496 the telescope with cross wires was still unavailable, was less than 5% in error.

W. Snell and J. Picard inaugurate modern methods. The realization dawned upon scientists that the finding of mountain heights and the radius of the earth must be approached much more seriously than had been the case before. In place of hours or days of work there was need of months or years. Measurements of base-lines and of angles must be repeated to afford better values and also to convey an idea of the amount of error in the measurements. Moreover, instruments of greater precision were needed. Picard was the first to use, in geodetic work, the telescope with cross-wires. The problem of finding mountain heights and the radius of the earth came to be intimately connected. Accurate triangulation over large areas made it necessary to erect signals on the crests of hills and mountainsº tops which could be seen many miles away. The accurate computation of a meridian line made it necessary to reduce distances to sea level. This could not be done unless the heights of the mountain stations above sea level were known.

Snell's measurements in Holland (1617) initiated an epoch of laborious effort and new technique: so did that of Picard, in France, 55 years later. Even then approximately reliable tables for correcting atmospheric refraction were lacking. More accurate surveys were made by Dominique Cassini, Maupertuis, and others, for the purpose of determining the shape of the earth. Celebrated are the meridian measurements of Bouguer, Godin, La Condamine, George Juan and de Ulloa in the plain north-east of Quito in Equador, which involved the finding of mountainsº heights.​d

The Teneriffe Peak. As already stated, the peak was very conspicuous to navigators. Early in the eighteenth century a party of merchants ascended the mountain and an account of that remarkable trip is described by Thomas Sprat in his History of the Royal Society of London (1722, p200). Its height became a matter of general interest. The first geodetic determination of its height, since the time of the inauguration of improved procedure, was made by the Franciscan Father Louis Feuillée in 1724. With a chain of 60 feet, he measured a base line pointing toward the center of the mountain, and 210 toises [409.3 m]  p497 in length. With a quadrant he measured the elevation of the Peak at each end of the base-line. From these data he computed the height of the Peak above sea level to be 2213 toises [4313 m]. His determination was published in 1751.​31 A foot note to the article, apparently by Abbé de la Caille, calls attention to the fact that the method followed is not accurate because of the shortness of the base-line, so that an error of only 20 seconds in measuring the angles would result in an error of 60 toises in the height of the mountain. In 1771 Borda and Pingré32 combined two base-lines in one base, and from its extremities took measurements on Teneriffe; they obtained for it a height above sea level of 1904 toises [3711 m] which is 309 toises [602.2 m] less than Feuillée's value. According to Humboldt,​33 1904 toises has a probable error of 6 toises or 1317th of the height.

Nineteenth Century. This century brought many refinements in the technique of geodetic measurements. The process of leveling, which was brought to remarkable perfection by the Greeks, continued to be used in favorable localities. Thus in the Report of the Superintendent of the U. S. Coast and Geodetic Survey for the year 1883, the height of Mount Diablo in California was determined by leveling. Proceeding one way it was found to be 3661.618 feet high; the opposite way, it was found to be 3661.864 feet. The mean was 3661.741 feet or 1116.09 m ± .03 m.

An idea of the accuracy of modern triangulation is conveyed by T. C. Mendenhall who in 1893, as Superintendent of the U. S. Coast and Geodetic Survey, stated that in primary triangulation, angular measurements are made with instruments of such precision, and the observations are repeated so many times that "the average probable error of a direction should not exceed 0″.1." Testing triangulation by comparing the computed length of a line obtained through a long chain of triangles with an actual measurement of the same line, it was found, says Mendenhall, that "in the triangulation between the Maryland and  p498 Georgia base lines, 602 miles apart, the discrepancy was scarcely perceptible, being little over half an inch in a thirty-mile line."

Elevations of mountains have not attained such extreme precision by triangulation, because no such high degree of accuracy has been attempted, and also because of uncertainty in the proper corrections for atmospheric refraction. The amount of refraction varies constantly, as was shown at two elevations in California, Bodega Head and Ross Mountain. The most reliable result for this difference in height was obtained by the spirit-level, viz., 598.74 ± 0.06 m. Utilizing zenith distances, the computed differences for angles taken at Bodega Head were as follows: At 7 A.M., 600.365 m; at 11 A.M., 599.119 m; at 1 P.M., 599.489 m; at 5 P.M., 600.508 m. The corrections in atmospheric refraction were made in accordance with C. M. Bauernfeind's investigations​34 which were preferred on account of their completeness. The results indicate that the amount of refraction continually changes. Moreover, the angular determinations made at Bodega Head taken by themselves were found to give too much difference in height, and the angular determinations at Ross Mountain too little difference in height. The observations at Bodega Head, sighting Ross Mountain, yielded 599.473 m; the observations at Ross Mountain, sighting Bodega Head, yielded 598.533 m. The average of the two is very close to the truth. Similar results regarding atmospheric refraction were obtained at Ragged Mountain in Maine in 1874.35

In recent publications of the U. S. Coast and Geodetic Survey, stations are divided into three classes, with reference to elevations: "first, those fixed by direct connection with sea level, the elevations of which are subject to a probable error of ± 0.04 meter; second, the stations in the main scheme fixed by reciprocal measures of vertical angles and subject to probable errors varying from ± 0.1 to ± 1.2 meters; and third, the intersection stations the elevations of which are fixed by measurement of vertical angles which are not reciprocal, the stations not being occupied, and subject to probable errors which may be as great as ± 3 meters."36  p499 In other reports there is a slight variation in the probable error.

The Barometric Method

Basic qualitative experiments. The "Torricellian experiment" thought out by Torricelli, but first performed by his pupil Vincenzo Viviani in 1643, after he had made the necessary glass tubing, was repeated in 1644 by Torricelli and Viviani who by that time had prepared a larger supply of glass ware. Torricelli described the experiments in a letter to Angelo Ricci in Rome, who, in turn, wrote to Mersenne through whom the experiments became known to B. Pascal and others. Pascal reasoned that if the mercury column is held up simply by the pressure of the air, then the column ought to be shorter at a high altitude. Whether this idea was altogether original with him, or whether it was first suggested to him in a letter from René Descartes, we shall not stop to discuss.​37 Pascal tried that experiment at a high church steeple in Paris, but, finding the results uncertain, he induced his brother-in‑law, Florian Périer, to try it on the Puy de Dôme, a high mountain in Auvergne, near Clermont. The experiment was made on September 19, 1648. During the three preceding days Périer had "rectified" 16 pounds of mercury. He filled two equal glass tubes, each 4 feet long, and hermetically sealed at one end, then inverted them over mercury in a vessel. Both marked 26 inches and 3.5 lines as the height of the mercury column. On the day of the experiment, Périer gave one tube to Father Chastin, to observe at Clermont during the day, while he himself, in company with distinguished ecclesiastical gentlemen and laymen, carried the other tube to the top of the mountain, Puy de Dôme [then believed to be 3000 feet or 973 m high; modern estimate is 1465 m above sea-level], where it registered a height of only 23 inches and 2 lines — a drop of 3 inches and 1.5 lines. This result, says Périer, "ravished us with admiration and astonishment." The day brought varying atmospheric conditions. He  p500 repeated the experiment five times at different places of the summit, under cover of a small chapel and in the open, when in the wind and when sheltered from it, when the atmosphere was clear and when it was rainy or foggy. He says that in each experiment he first carefully freed the tube of air bubbles. We infer from this that the tube was emptied before the apparatus was transported from one place to another, and then refilled for each trial. Arriving at the foot of the mountain, he found his tube to register 26 inches, 3.5 lines, as before.

Pascal first wrote an account of the experiment or Récit de la grande expérience de l'équilibre des liqueurs,​38 in November or December, 1648, in which he suggested that the method could be used for determining whether two localities, even when far apart, had the same or different elevations. The Récit was printed, but, for reasons now not definitely known, was withdrawn from circulation. Only very few copies were issued. This action was due, apparently, to Pascal's determination to confine his activities to matters of religion. The Puy de Dôme experiment did not become generally known among scientific men until 1651 when the celebrated anatomist Jean Pecquet published at Paris his Experimenta nova anatomica, in which he described not only the experiments performed by Périer on the Puy de Dôme, but also some experiments by Auzout on the Torricellian vacuum which has, however, nothing to do with the effects of high altitude upon the length of the barometric column.​39 Pecquet's book was widely read; it was from this source that Sinclair in Scotland, and the Academicians in Florence learned of the Puy de Dôme effect.

Experiments in Florence. The members of the famous Accademia  p501 del Cimento in Florence were led to determine differences in altitude on a new principle, though still using mercury, namely the expansion of the air under diminished pressure, the temperature being invariable. They designed an instrument shown in Fig. 12.​40 At the foot of a steeple in Florence, 142 yards (braccia) in height, they awaited the moment when the temperature was the same as at the top of the tower. Then they poured mercury into the lower tubing, sealed the fine terminus of the glass tube, and carried the apparatus to the top of the tower. The diminished air pressure caused the air in the bulbs to expand so that the two mercury columns which before had been equally high, now showed a difference in height. From this, by Boyle's law, the difference in air pressure could be ascertained. This apparatus was frail and hardly portable, though some tests were made with it on neighboring hills — "sopra diverse colline di quelle, che la Città coronano." A modification of this design is shown in Fig. 13, where the tube at D is sealed after the mercury is introduced. When the external air pressure decreases or increases, the mercury in the tube rises or falls. A serious defect of this apparatus was that the amount of the expansion of the air in the receptacle was greatly affected by even slight changes in temperature. This method of determining altitudes never passed beyond the experimental stage. The academicians doubted the possibility of satisfactory altitude determinations by the barometer, for the reason that they had observed fluctuations of barometric readings at one and the same place.

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Fig. 12.

An instrument designed by the Academicians at Florence for measuring changes in atmospheric pressure.

Fig. 13.

A modified instrument designed by the Academicians at Florence for determining changes in atmospheric pressure.

Experiments in Scotland. George Sinclair, at one time professor of philosophy at the University of Glasgow, took observations in the years 1661‑1666.​41 From reading Pecquet's book he was familiar with the experiments of Pascal and Périer. He selected four suitable elevations in southern Scotland, Tinto Hill south east of the town of Lanark, a cliff north of the town of Moffat in the Annan basin, Arthur's Seat east of Edinburgh, and the Glasgow cathedral. He also made tests in mines. Some of the experiments were made in the presence of distinguished persons as witnesses.

 p502  Sinclair selected Tinto Hill because it could be seen from a distance, even from Glasgow. The ascent was made October 12, 1661. "At the summit I filled the baroscopic tube with mercury, observing the greatest care and diligence, foreign things being driven out by means of an iron wire, air bubbles in particular, which lie hid in the mercury. Then, the tube being inverted in the usual manner and the orifice immersed in a bath of mercury, the mercury in the tube did not descend to 29 inches, as was commonly expected, but nearly to 27, after it had come to rest . . . A long strip of paper was attached to the tube and graduated into divisions and subdivisions, for the purpose of measuring. Sinclair obtained the height of Tinto Hill by measuring at Glasgow its angle of elevation by means of a quadrant, and taking the usual estimate for its distance from Glasgow. He obtained for its height 500 paces or 2500 feet. The modern figure is 2335 feet or 711.7 m.

The vertical cliff north of Moffat indicated a drop of the mercury of ¼ of an inch. The height of the cliff was not accurately known, but was said to be 50 paces.

At the Glasgow cathedral the mercury tube and reservoir were attached to a rope and raised and lowered repeatedly through a vertical distance of 186 feet; the difference in height was found to be 532 of an inch.

At Arthur's Seat near Edinburgh, the column fell from 29 inches to 28¼ inches, but Sinclair gives no estimate of the height of the hill. Modern determinations are 822 feet above the sea.

Boyle established the law of gases known by his name, in 1662. Sinclair was not familiar with Boyle's results and proceeded to frame a rule for the variation of the length of the barometric column for different elevations of observation, which was altogether false. He assumed that, if ascending from sea level to a certain height resulted in a drop of the mercury of, say, two inches, then double that height would yield a drop of exactly four inches. Sinclair interested himself not only in finding a rule for the barometric determination of mountain heights, but even more in the determination of the actual height of the atmosphere. Taking his Tinto Hill data, he solved the proportion 2 inches is to 500 paces as 29 inches is to x paces. This gave  p503 x = 7250 paces or 36250 feet for the height of the atmosphere. He had more confidence ("plùs fido") in his cathedral data than in the data for the cliff near Moffat, because he had carefully measured the altitude at the cathedral. Taking his data obtained at the Glasgow cathedral, he obtained the proportion: 532 inches are to 186 feet, as 29 inches are to x feet. Here he gets approximately x = 34336 feet, a result for the height of the atmosphere somewhat smaller than the first.

In Germany, Otto von Guericke42 proceeded to repeat the Torricellian experiment on the Brocksberg, the highest point in the Harz mountains, southwest of Magdeburg, but he met with an accident. He says: "At the foot of the mountain everything went well, but when I gave the glass tube, which was packed in a tin box, to a servant to carry, he broke it in a fall and thereby the experiment miscarried."

Early attempts to establish a correlation between barometric height and mountain height. The apparent theoretic simplicity of the barometric method vanished on closer examination. The simultaneous accurate measurements were difficult to make. When one of the measurements was easy, the other was hard. For a steeple, the difference in the mercury column was too small for accuracy. The exact height of any mountain remained unknown before about the middle of the eighteenth century. But these were not the only obstacles. Indeed, a perfect correlation does not exist, for the reason that there are slight fluctuations in the height of the mercury column at one and the same location due to changing weather conditions. Nor is the difference in the simultaneous barometric readings of the two places always the same. Technical matters had to be considered in the construction of barometers, to avoid capillary action, to remove air bubbles in the mercury and to attain more accurate methods of measuring the length of the mercury column, as well as to secure portability of the instruments. It took a century to develop instruments and to determine accurately the constants which enter into the determination of the altitudes by the barometer. Even then the method did not reach the accuracy which is possible by the modern refined geodetic methods, but it enjoys the advantage  p504 over the latter of being more expeditious. Most of the mountain heights given in geographic tables were determined by the barometric method. In 1807 Humboldt knew of only 122 measured tops. In 1829 Gehler gave about 4600 heights, most of which were determined barometrically.43

The earliest explanation of a process by which a mountain altitude may be computed from the drop of the barometric reading below that at sea level, was given by the French physicist Edme Mariotte.​44 He did not develop a formula, but explained roughly the procedure, on the assumption (1) of "Mariotte's" ("Boyle's") law of gases and (2) the distance above sea level which causes a drop of, say, one line (one-twelfth of an inch) in the barometric reading. From his experiments at the Paris observatory, he concluded that a rise of 84 feet caused a drop of 19 of an inch, or 43 of a line. This amounts to about 63 Paris feet for 1 line. Being told that at Orleans a rise of 300 feet caused a drop in the barometric reading of 5 lines, which amounts to about 60 feet per line, he took at first, "pour la facilité du calcul", 60 feet per line, but later 63 feet per line. We see here that no great care was exercised to determine accurately the observational constant entering his calculation. He took 63 or 60 feet, whichever best suited his fancy. As a matter of fact, both values were much too small. He applied two checks to his procedure, one being the data obtained by Périer at the Puy de Dôme whose height, as we stated, was not known accurately, the other being D. Cassini's data for a mountain in Provence, 1070 feet high, the mercury dropping 16⅓ lines. As an explanation of Mariotte's method we repeat this last calculation. He computed how far one has to rise at each step to secure a barometric drop of 1 line. The atmospheric pressure at sea level is equal to that of 28 inches or 336 lines of mercury. At an altitude where the pressure is only half, or 168 lines, the altitude of the air column is doubled, that is twice 63 or 126 feet. In the interval between 336 and 168 lines the increase per line is nearly 63168. Thus, for 16⅓ lines the total increase would be 63168 (1 + 2 + 3 + . . .  p505 + 16) = 51. Adding to this the fundamental 63 feet per step, or 63 × 16⅓ = 1029, yields 1029 + 51 or 1080 feet, as the computed height of the mountain; this is only 10 feet in excess of the measured 1070 feet.

A more finished analysis was published ten years later by Edmund Halley.​45 He showed the great advantage of the use of logarithms, and deducted a formula which yielded the height H in English feet, viz. ,

H =

900 (log 30 - log h)


where 30 inches is the height of the mercury column at sea level, and h inches is its height at the place of observation. Halley also took the experimental data in round numbers, so that the constants had to be redetermined later. His formula indicates that a drop in the barometric reading from 30 to 29 inches corresponds to a difference in altitude of 915 English feet, which is about 26 feet too much. In deriving his formula he assumed that mercury is 13½ times heavier than an equal bulk of water and that water is 800 times heavier than air at sea-level, so that mercury is 10800 times heavier than air. The modern figures for sea-level are more nearly 10500. A cylinder of mercury one inch high weighs therefore, according to Halley, as much as a cylinder of air of equal base at sea-level, and 10800 inches or 900 feet high. By Boyle's law, pv = 30 × 900. Using modern symbols, the cylinder of air reaching from sea-level to the place where the barometric reading is h, is

vdp = 30 × 900 (dp/p) = 30 × 900 log p ] 30 = 30 × 900 × (log 30 - log h).

Changing from natural to common logarithms, by dividing by the modulus .434295, and simplifying, Halley's expression is obtained.

Halley's formula, involving logarithms, has the basic form which was eventually adopted, but for many years it received little attention. Jacques Cassini,​46 Daniel Bernoulli,47  p506 Johann Jakob Scheuchzer,​48 Peter Horrebow49 and others set up different rules, none of which proved altogether successful. Halley's rule, that altitudes increase as the differences of the logarithms of the atmospheric pressures, was not generally accepted. Thus Daniel Bernoulli rejected it as contrary to the observations available to him.

Earliest barometric height measurements. It is noteworthy that during the seventeenth century the barometer was not actually used in the determination of the height of mountains. Observation was confined to the comparison of the height of the barometric column with geodetically determined mountain heights. The first observer to compute previously unknown heights from barometric readings was the Swiss Johann Jakob Scheuchzer, in the years 1705‑1707. His son, J. K. Scheuchzer, in 1727 made the following statement:​50 "My Father, Dr. J. J. Scheuchzer, in his Journies over the Mountains of Swisserland, as they were more particularly calculated for the Improvement of Natural Philosophy in it's several Branches, neglected no Opportunity, along with his other Observations, to make such Experiments with the Barometer as might serve to illustrate the Qualities of the Air, to settle the respective Heights of Places, and particularly to shew, how much our Mountains rise, as well above the Level of the Sea, as above other neighboring Mountains in France, Italy, Spain, etc. Many of these Observations are scattered up and down in his Writings, particularly his Itinera Alpina, and the several parts of his Natural History of Swisserland, which last work was published in High German." During these alpine travels, barometric readings were taken also at Zurich. In his early computations, J. J. Scheuchzer took 1 line (i.e. 112 of an inch) to correspond to 80 feet of elevation; in other words, he took the height of a mountain to be 80 × 12 × 12 or 11520 times the difference of the barometric readings.51

It will be observed that this rule is theoretically incorrect.  p507 Practically the rule yields values that are too high for moderate distances above sea-level. But a few years later a new rule was deduced from new data obtained by J. J. Scheuchzer, at the watering place, Pfäffers, where a vertical cliff was measured to be 714 Paris feet high, "as appeared by letting a Line drop down perpendicularly from a Tree at Top, full to the Bottom." The barometric difference was found "by repeated experiments" to be 10 lines. This determination, says the son, was "the most considerable that ever was made, and which enabled him [J. J. Scheuchzer] more particularly to examine the two Tables made by Cassini the Younger, according to the Rule of M. Mariotte, and the Observations made by him [Cassini] and others, when the Meridian Line was perfected in 1703." The height of the cliff corresponding to the difference of 10 lines was found by Mariotte's rule to be 696 feet, by Jacques Cassini's rule 921 feet, the true height being 714 feet. Johann Scheuchzer, a brother of Johann Jakob, undertook to calculate a new table from the data obtained at Pfäffers and from the theoretical principles used by Halley. If the barometric heights at sea-level and at another locality (expressed in thirds of a line) are designated, respectively, by H and h, then the Scheuchzer new rule​52 may be stated in form of a proportion thus:

.0142717 : 714 ≂ log H - log h : x

where x is the required height in Paris feet. Johann Scheuchzer calculated by this rule that Piz Stella, in the Canton of Grisons, was 9585 Paris feet over the sea. By Mariotte's rule it would be 9441 feet, by Jacques Cassini's 12,196 feet.53

Peak Teneriffe. This conspicuous mountain of the Canary islands, rising from the ocean to a great height, made, as we have seen, a wonderful appeal to the imagination. Is its height 70 miles, as claimed by the philosopher Patrizzi, or only 7 miles, as guessed by the more restrained, yet exuberant imagination of Robert Boyle? What are the prosaic facts? In 1667 Boyle wrote:​54 "Perhaps I have told your Lordship already by word of mouth, that I have been solicitously endeavoring to get the  p508 Torricellian experiment tried upon the pic of Teneriff, but hitherto I have had no account of the success of my endeavors." The experiment was first performed more than half a century later. It was in 1724 that Father Louis Feuillée determined its height trigonometrically to be 13278 French feet. He also ascended the mountain with a party, taking barometric observations at several places during the ascent. At the top the barometric reading was 17 inches and 5 lines, indicating a rope from the mark at sea-level of 10 inches and 7 lines. No barometric estimate of the Peak's height was published by Feuillée. If we compare the height by the rule of Mariotte, taking 63 feet to correspond to a barometric line at sea-level, we obtain 10,424 Paris feet — the trigonometric value of Borda and Pingré being 11,424 Paris feet. Stern reality forced Boyle's imaginative 7 miles down to a paltry 2 miles.

Discordant results. In the first half of the eighteenth century, many discordant results were obtained in barometric heights. The French were engaged in meridian measurements, and in the region of the Pyrénées determined mountain heights both geodetically and barometrically. Geodetic measurements, involving checks by the comparison of the measured and the computed values of base-lines, were beginning to indicate such a high degree of accuracy before the mild of the eighteenth century, that the very delicate empirical determination of the flattening of the earth at the poles was triumphantly demonstrated. Careful geodetic measurements of heights were therefore available after 1740, as standards for testing the accuracy of barometric determinations. De Luc55 instituted a comparison of results obtained by the different early formulas for a drop of the barometric reading of 5 inches below the height of 28 inches at sea-level. The computed altitudes expressed in Paris feet were according to Mariotte, 4170 feet 9 inches; Halley, 4998 feet 8 inches; Maraldi, 5430; J. Scheuchzer, 4274 feet; Jacques Cassini, 4554 feet; Daniel Bernoulli, 4782 feet; Horrebow, 4954 feet 3 inches;º Bouguer, 4954 feet 3 inches.º The highest of these values, differed from the lowest by 1260 feet. The correct value is about 4980 Paris feet. The formulas were based partly on the number of  p509 feet of altitude corresponding to a drop of 1 line in barometric readings; this is exceedingly difficult to measure with accuracy. That altitude varied with different early observers from 60 to 80 feet. The true value at sea-level is about 75 Paris feet, slight variations being due to changes of temperature, geographic latitude, aqueous vapor, and other weather conditions.

To secure another glimpse of the discordant results obtained in barometric measurements in the early part of the eighteenth century, we quote data on trench mountains in the Pyrénées region of France given by Jacques Cassini in 1733:

Mont de St. Barthelemy 1190 toise 1012 toise 1427 toise 1192 toise
Mont du Mousset 1289 toise 1035 toise 1467 toise 1236 toise
Canigou 1453 toise 1183 toise 1728 toise 1440 toise

Mont Blanc. De Luc states that Mont Blanc had been measured trigonometrically above Lake Geneva,​56 but that, for accurate work, the base was too short and too far removed from the mountain. De Luc himself, about 1765, determined the height of Mont Blanc in sections, the lower sections being computed barometrically and the uppermost section trigonometrically, from data which he recognized to be of doubtful accuracy. He found the height of Mont Blanc over the sea-level to be 2391 toises which is 4660 meters, the modern estimate being 4810 meters. It was in 1786 that a mountain guide discovered a path to the top of Mont Blanc. The following year the physicist, H. B. Saussure57 of Geneva climbed to the top and there found the barometric reading 16 inches and .22 lines, the temperature of the mercury being 1.2°R., that of the air — 2.3°R. The simultaneous barometric reading at the astronomical observatory in Geneva was 27 inches, 3.12 lines. By de Luc's formula with temperature  p510 correction, the height of Mont Blanc above Lake Geneva was 13,333 Paris feet; this is 251 feet short of the true value.

Boiling the mercury. Corrections for capillary action and temperature. Differences in the readings of barometers, observed at the same time and place, pointed to the need of greater care in construction. The French physicist Du Fay in 1723 made the suggestion that the mercury in the barometric tube be boiled, in order to expel air bubbles and other impurities. Cassini de Thury and L. G. Monnier found that boiling helped, but Abbé de la Caille found that boiled and unboiled thermometers which he observed agreed within ¼ of a line. Hence, for some time, De Fay's suggestion was not widely followed. Nor were the advantages resulting from boiling at first fully understood. Only later was it recognized, that bubbles and aqueous vapor in the mercury may escape into the Torricellian vacuum and by their elastic force depress the mercury column and distort the barometric readings.

It was found also that insufficient attention had been given to capillary action which causes greater depression of the mercury in a narrow glass tube. De Plantade58 was one of the first to give this matter serious consideration; he suggested that in all observations the diameter of the glass tube should be recorded.

In the latter part of the eighteenth century there was a small army of scientific workers bending their efforts towards the improvement of the barometer. Outstanding achievements in the experimental and observational side of this work are those of J. A. De Luc of Geneva in Switzerland. He initiated careful temperature corrections for boiled barometers, and determined the coefficient of expansion of mercury. For unboiled barometers the expansion due to heat was found to be quite irregular. The astronomer Laplace59 endeavored to give due consideration to all reliable experimental data, especially those gathered in the numerous observations in the region of the Pyrénées in France, and taken by L. Ramond. Laplace developed a formula for the barometric determination of altitudes which made allowance also for variation of gravity with the altitude and with the latitude.  p511 Additional improvements were made later. To expedite computation, K. F. Gauss computed tables giving corrections for the temperature of the air at both stations, and also for the diminished gravity in high altitudes. F. W. Bessel in 1838 introduced corrections for the humidity of the air.​60 In an altitude of 879.63 toises, computed for a relative humidity of 50%, Bessel added 3.24 toises for the case of saturation, and deducted the same amount for the case of perfectly dry air. Since the time of Mariotte, many different formulas have been deduced for computing barometric heights. In 1870 Rühlmann61 collected twentythree of them as representing the most important.

Example of early nineteenth century accuracy. An idea of the accuracy and reliability of the barometer as used in 1818, is afforded by the record of Thomas Greatorex62 in measuring Skiddaw in the north of England. The height of Skiddaw above Derwent lake was, by leveling, 936.097 yards; the barometric height was, by Maskelyne's formula, 936.1685 yards; by Hutton's formula 935.285 yards. Tests were made 150 yards lower down which resulted as follows: By leveling 786 yards, by Maskelyne's barometric formula, 782.3 yards; by Hutton's formula 781.4 yards. At still lower points the discrepancy was even greater. Greatorex remarked: "From the near agreement of the measured and barometric heights on the summit of Skiddaw, I had formed sanguine hopes that the barometer would prove a most exact determinator of altitudes, . . . but the subsequent observations lead me to fear that the state of the atmosphere has an effect which we cannot yet account for, and to which we cannot apply a correction."

Hourly and annual barometric fluctuations. L. Ramond noticed about 1810 that barometrically determined heights varied with the time of day when the measures were taken. His experimental data exhibit also an annual variation. Among those who later occupied themselves with this subject are Emile Plantamour, C. M. Bauernfeind, Richard Rühlmann and Major Indicates a West Point graduate and gives his Class.Williamson of the United States army. George Davidson of the U. S. Coast Survey, in 1871, gave the following summary:​63 "Among the  p512 conclusions reached are the following: Differences of heights, barometrically determined, appear to attain their maximum value shortly before the time of greatest heat of the day; they decrease rapidly during the afternoon, and slowly during the night, reaching their minimum about one or two hour before sunrise . . . Resulting heights, determined from daily or monthly means, also show an annual period; they are found too small in winter and too great in summer . . . Heights determined from annual means generally give results differing little from the truth." Davidson gave a table showing the barometric differences of height between Bodega Head and Ross mountain, in California, for different hours of the day. By his operation of leveling, the difference was 598.72 m. By the barometric method, the difference was at 7 A.M. 598.80, at 1 P.M. 610.34 m, at 5 P.M. 604.32 m. These hourly variations were supposed to be due to a variable difference in the observed mean temperature at the two stations and the actual average temperature of the intervening stratum of air.

Charles A. Schott, of the U. S. Coast and Geodetic Survey, expressed himself in 1876 as follows: "It is evident that further observations are desirable, and that barometric hypsometry cannot be made the basis of exact determinations of heights."64

Thermometric Method

In 1724 G. D. Fahrenheit65 observed the rise of the boiling point of water with the increase of the atmospheric pressure. He suggested the determination of the weight of the atmosphere by means of the thermometer alone. A few years later L. G. Monnier and Jacques Cassini showed that this change in the boiling temperature is very considerable;​66 it is 7.18 Reaumur degrees for the height of Canigou; this drop is equivalent to 16.15° Fahrenheit. But it was the physicist of Geneva J. A. de Luc, who in 1762, when on scientific expeditions in the Alps made numerous experiments of the boiling temperature at different elevations. He found that the boiling temperature depends somewhat on the  p513 depth of the water in the receptacle, and that the boiling temperature reaches its maximum when the water boils with violence. Certain discordant results were laid to failure to make the tests under uniform conditions. He found that thermometric changes are not proportional to the corresponding barometric changes. The rate of change of the thermometric readings is less. But he was not able to deduce a workable formula. In 1775 Sir George Schuckburgh67 repeated de Luc's experiments and found that the boiling temperature does not drop quite as rapidly as de Luc indicated. A theoretical study of this question was made by Andrew Ure68 who measured the elastic force of water vapor. Supposing that the water boils at 212° F, when the barometer reads 30 inches, the water boils at 202° and 192° when the barometer reads 30/1.23 and 30/(1.23 × 1.24) inches respectively. The study of vapor pressure was taken up by F. J. H. Wollaston,​69 who introduced by interpolation much smaller intervals, making nine between 212° and 202° degrees.º In a second table he gives boiling points and the corresponding elevations. Using a thermometer with a large tube specially designed for altitude determinations, in which the scale extended only a few degrees and a degree marked nearly 4 inches, so that, with a vernier, small fractions of a degree could be read, Wollaston determined the height of Snowdon Hill to be 3546.25 feet by his thermometer, 3548.9 feet by the barometer, while General Roy's trigonometrical value was 3555.4 feet. The agreement was not so close on Moel Elio, which by his thermometer rose 2350.55 feet, by the barometer 2391.8 feet, by General Roy's trigonometric survey 2371 feet. Why the agreement should be so very much closer on Snowdonº than on Moel Elio could not be discovered with certainty. Wollaston remarked that Snowdon terminates in a point, while Moel Elio had a large base summit; moreover, the test on Moel Elio was made in a very high wind.

In more recent time Henri Victor Regnault produced more exact tables of vapor tensions. After noting the vapor tension, the computations for fixing altitudes are the same as when a  p514 barometer is employed. Regnault constructed a small instrument for thermometric height determinations which he called the hypsometer. For approximate results, a formula due to Soret may be used, h = 295 (100 - t), where h is expressed in meters, t in degrees centigrade.

Later results. An idea of the accuracy due to later improvements is afforded by measurements at Mount Washington in New Hampshire, the height of which had been determined by the U. S. Coast Survey in its primary triangulation, to be 6288 feet (1917 m). For the purpose of comparing methods, this height was divided, in 1854, into steps of about 500 feet difference in elevation, thus forming 12 stations, at which Indicates a West Point graduate and gives his Class.T. J. Cram,​70 who was in charge of secondary triangulation, took simultaneous observations ("in the same minute") by the barometric and the boiling point method. The difference in elevation of two of the stations which was 6280 feet by the process of leveling, was found to be 6206 feet by the barometer, and 6116 feet by the boiling point. Cram states that "the results given by the barometer, however erratic they may seem, wander less from those by the leveling instrument than do the results given by the thermo-barometric boiling-point apparatus." He states that whatever error in the barometric method may be attributed to the constants in the barometric formula, will likewise be found in the results of the thermometric method, as this formula is used in both instruments.

Florian Cajori.

University of California.

The Author's Notes:

1 This article has been prepared in response to Question 7 in Isis, 9, p425, 1927.

2 See Eratosthenica, ed. Bernhardy, 1822, frag. 39; Hugo Berger, Die geographischen Fragmente des Eratosthenes, Leipzig, 1880, p56, 80, 129, 173.

3 C. Plini Secundi Naturalis historiae libri XXXVII, Liber II, Chap. 65; see in particular § 162: "Dicaearchus, Vir in primis eruditus, regum cura permensus montes, ex quibus altissimum prodidit Pelion MCCL passuum ratione perpendiculi."

4 Theon Smyrnaeus (ed. Hiller, Leipzig, 1878, p124, 125).

5 The Geography of Strabo, transl. by H. C. Hamilton and W. Falconer, Vol. I, London, 1854, p156 (Book II, Chap. IV, § 1).

6 Gemini Elementa astronomiae, ed. C. Manitius, Leipzig, 1898, Chap. XVII, pp182, 183. There appears to be doubt concerning Dicaearchus' estimate of the height of Atabyrius; it is given in earlier editions of fragments of his writings, not as less than 10 stadia, but as less than 14 stadia. See O. Peschel's Geschichte der Erdkunde, München, 1877, p63.

7 Plutarch's Lives, translated by John Langhorne and William Langhorne, London, 1774, Vol. II, p169.

Thayer's Note: See the passage, Life of Aemilius, 15.10 (the edition onsite being a more recent one).

8 The Geography of Strabo, transl. by H. C. Hamilton and W. Falconer, London, (Casaub. 379 Book VIII, Chap. VI, § 21)º 1856, Vol. II, p31.

9 Op. cit., (Casaub. 387. Book VIII, Chap. VIII, § 1) Vol. II, p75.

10 Eustathii Commentarii ad Homeri Odysseam, Tome II, Lipsiae, 1826, p311 (1951, 15 Rom.)

11 Cleomedes, De Motu circulari corporum coelestium. German transl. in Ostwald's Klassiker, No. 220, 1927, p36.

12 Polybii Lycortae . . . Historiarum libri qui supersunt, Amstelodami, 1670, Vol. I, p776 (Liber IX, § 19, 5‑9).

13 Jahrbücher für Classische Philologie, Leipzig, 1897, p49 ff.

14 Pliny, Naturalis historiae, Book IV, Chap. 23, 8.

15 B. Varenius, Geographia generalis, edition by I. Newton, Cambridge, 1672, p62.

16 The Travels of Sir John Mandeville, London, 100, p12.

17 J. Kepler, Opera omnia (ed. Ch. Frisch), Vol. VIII, 1866, p93.

18 Robert Hues, Tractatus de Globis, edited by C. R. Markham, London, 1889, p8.

19 R. C. Skyring Walters, B. Sc., Assoc. M. Inst. C. E., "Greek and Roman Engineering Institutions", Transactions of the Newcomen Society for the Study of the history of Engineering and Technology, Vol. II, London, 1921‑1922, p51, 52.

20 Robert Hues, Tractatus De Globis, ed. 1889, p10.

21 Giuseppe d'Acosta,º Historia natural y moral de las Indias, lib. IV, Cap. 6. Reference taken from O. Peschel, Geschichte der Erdkunde, München, 1877, p426.

22 Blancanus, Sphaera mundi. Bonon. 1620, Pars III, p95; O. Peschel, op, p426.

23 Edward Wright, Certain Errors in Navigation detected and corrected, 3rd. ed., 1657, p90.

24 F. Maurolycus, Cosmographia, Venice, 1543, f. 73. Reference taken from G. Libri, Histoire des sciences mathématiques en Italie, Vol. III, Paris, 1840, p113.

25 J. Kepler, Opera omnia (ed. Ch. Frisch), Vol. VI, 1866, p130.

26 Chr. Huygens, Oeuvres complètes, Tome XV, La Haye, 1925, p533.

27 Varenius, Geographia (ed. Isaac Newton), 1672, p61.

28 E. Wright, Certain Errors in Navigation detected and corrected, 2d.,º 1610.

29 Varenius, Geographia, 1672, p60.

30 W. Snell, Eratosthenes Batavus, Leiden, 1617, p257‑263.

31 Histoire de l'académie r. d. sciences, Année 1746, Paris, 1751, Mémoires p129.

32 Verdun, Borda et Pingré, Voyage fait par ordre du Roi, Paris, 1785, Tome I, p117, 378.

33 A. v. Humboldt, Voyages aux régions équinox., Paris, 1814, Tome I, p284.

34 Astronomische Nachrichten, 1866, Nos. 1478‑1487, and 1587‑1590.

35 U. S. Coast Survey, Report for 1876, Washington, 1879, p3623.

36 U. S. Coast and Geodetic Survey. Special Publication No. 74, Washington, 1921, p22.

37 This question is discussed minutely by Felix Mathieu in La Revue de Paris, Vol. 13, 1906, Part II, p565‑589, 772‑794; Part III, 179‑206; Vol. 14, 1907, Part II, p176‑224, 347‑378, 835‑876.

38 Printed in Oeuvres de Blaise Pascal (Ed. L. Brunschvicg and P. Boutroux) Vol. II, Paris, 1908, p365‑373. Later (probably in 1654) Pascal wrote his well known Traitez de l'équilibre des liqueurs et de la pesanteur de la masse de l'air, which was first printed in 1663, after his death. It is reproduced in the Oeuvres, Vol. III, 1908, p143 ff. A facsimile of the Récit was brought out in 1893 by G. Hellmann as No. 2 of his Neudrucke von Schriften und Karten über Meteorologie und Erdmagnetismus.

39 Of experiments on air pressure described by Pecquet, a part (but not the Puy de Dôme experiment) is reprinted in the Oeuvres de Blaise Pascal, Vol. III, Paris, 1908, p238. An account of Pecquet on air pressure is given by F. Mathieu in La Revue de Paris, Vol. 13, part II, 1906, p781, and Vol. 14, Part II, 1907, p203.

40 Saggi di naturali esperienze fatte nell' accademia del cimento. Seconda edizione, Firenze, 1691, p62, 64, 66‑72.

41 Georgii Sinclari Ars nova et magna gravitatis et levitatis. Sive Dialogorum philosophicorum Libri sex. Roterodami, 1669, p125‑149.

42 O. Guericke, Experimenta nova, Amsterdam, 1672, Lib. III, Cap. 30, p114. See Ostwald's Klassiker, No. 59.

43 J. S. T. Gehler, Physikalisches Wörterbuch, Vol. 5, Leipzig, 1829, pp339‑397.

44 E. Mariotte, Essai sur la nature de l'air, 1676; Oeuvres, nouvelle édition, La Haye, Tome I, 1740, p174, 175.

45 E. Halley, Philosophical Transactions, London 1686, p104‑116; Abridged Transactions, Vol. II, 1705, p14.

46 J. Cassini in Mém. de l'Académie r. d. Sciences, Paris, année 1705 and année 1733.

47 Daniel Bernoulli, Hydrodynamica, Sect. X.

48 J. J. Scheuchzer, Bergreise, Theil II, Zürich, 1746.

49 P. Horrebow, Elementa philosophiae naturalis, Chap. VIII.

50 J. K. Scheuchzer, Philosophical Transactions. Abridged, Vol. VI, Part II, London, 1734, p33.

51 J. J. Scheuchzer. Itinera alpina, Londini, 1708, Iter sec. p7. We have not seen this work and we take this reference from O. Peschel's Geschichte der Erdkunde, München, 1877, p690.

52 Philosophical Transactions. Abridged, Vol. VI, Part II, p35.

53 Op. cit., p42.

54 Robert Boyle's Works, Vol. III, 1772. A Continuation of New Experiments Physico-Mechanical, touching the Spring and Weight of the Air, p225‑228, Experiment XXIII.

55 J. A. De Luc, Recherches sur les modifications de l'atmosphère, n. ed., Vol. 1. Paris, 1784, p285.

56 "Remarque sur l'histoire naturelle des environs du Lac de Genève," à la fin du IIe Vol. de l'histoire de Genève par M. Spon.

57 H. B. Saussure, Voyages dans les Alpes, Tome VII, p304. Reference taken from O. Peschel, Geschichte der Erdkunde, 1877, p696. Applying formulas developed later, that of Laplace with Ramond's constant, yielded an altitude of Mont Blanc 56 feet in excess, that of Gauss yielded 38 feet in excess.

58 Magasin Français pour Décembre, 1750; J. A. De Luc, Recherches sur les modifications de l'atmosphère, n. éd., Vol. I, Paris, 1784, p94.

59 Laplace, Mécanique Céleste, tome IV, livre 10.

60 Astronomische Nachrichten, Vol. 15, No. 357, 1838, p360.

61 R. Rühlmann, Die barometrischen Höhenmessungen, Leipzig, 1870, p21‑32.

62 Philosophical Transactions, Vol. 108, for the year 1815, London, p398‑402.

63 U. S. Coast Survey, Report for 1871, Washington, 1874, p165.

64 U. S. Coast and Geodetic Survey, Report for 1876, Washington, 1879, p367.

65 Philosophical Transactions, Vol. 33, 1724, p179.

66 Mémoires de l'académie r. d. sciences, année 1740.

67 Philosophical Transactions, Vol. 69, for the year 1779, p363.

68 Philosophical Transactions, Vol. 108, 1818, p338.

69 Philosophical Transactions, Vol. 110, 1820, p295, 302.

70 U. S. Coast Survey, Report for 1854, p102*, 103*.

Thayer's Notes:

a Not really; a frequent rough-and‑ready conversion among the ancients was that one Roman mile was equal to eight stadia, and Pliny himself explicitly gives it in N. H. II.85: "one stadion is 125 of our paces". 1250 paces (1.25 Roman miles, 1850 meters) by this rule would be exactly 10 stadia, with the passage in Pliny thus confirming the 10‑stadia idea.

More exactly, however, there were several species of stadion — the main ones were the Attic, the Olympian, the Aeginetan: see this good page at Sizes.Com — and, as Sarah Pothecary writes in "Strabo, Polybius, and the Stade" (Phoenix, XLIX.1, p49), "The length of the stade [. . .] is one of the vexed issues of classical metrology". Although the commonest in extant written sources seems to have been the Attic, for which an exact conversion of 10 stadia gives 1775 meters and thus slightly less than 1250 paces, the other stadia were longer and converting ten of those stadia exactly we get figures slightly higher than 1250 Roman paces; and finally, there is some evidence to indicate that geographical writers, at least, have in mind a standard stadion that, apparently by coincidence, is in fact exactly one-eighth of a Roman mile, making 1250 paces not a rough-and‑ready but an exact conversion of 10 stadia.

No conclusion can be drawn as to whether Pliny confirms the 10‑stadia idea or not.

b Prof. Duncan Agnew of the University of California at San Diego kindly alerted me to a more recent treatment of the trigonometric method, providing also a more reasonable view, based on resurveys of aqueducts, of what the Romans could and couldn't do with leveling: M. J. T. Lewis, Surveying Instruments of Greece and Rome (Cambridge University Press, 2001).

c This is not quite exactly what Polybius says, as the gentle reader will see by following the link in footnote 12 above. The general needs astronomy and geometry (IX.20.5), but the former has nothing to do with measuring heights.

d The story of the celebrated Meridian Expedition is well told, and at some length, in "Antonio de Ulloa" (Hisp. Am. Hist. Review, Vol. 15 No. 2, pp157‑170), and more briefly in Gayarré, History of Louisiana, Vol. III, pp141‑146.

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