Form of the Earth. This important question has raised such a clamor in recent times, learned men – above all in France – have so occupied themselves with it, that we believe we must make it the subject of a specific article, without turning to the word Earth, which will otherwise provide us with plenty of material for other topics.

We will not go into any detail about the extravagant opinions that the ancients held, or that are attributed to them about the form of the Earth . One may inform oneself about these in the Almagest of Riccioli [1] and elsewhere. Anaximander, it is said, believed the earth to be like a column, Leucippus a cylinder, Cleanthes a cone, Heraclitus a skiff, Democritus a hollow disc, Anaximenes and Empedocles a flat disc, and finally Xenophanes of Colophon imagined it had an infinite root on which it carried itself. This last opinion reminds one of that of the peoples of India, who believe the earth to be borne on four elephants. But we will allow ourselves to doubt whether the great majority of the philosophers we have just named had such absurd ideas. Astronomy had by their time already made great progress, as Thales, who preceded them, had predicted eclipses. Hence it is not plausible, it seems to me, that in a time in which astronomy was already so advanced, one was still so ignorant of the form of the Earth ; for one will see that the earliest astronomical observations would have led one to know that it is round in every sense. Also, Aristotle, who was a contemporary, or even a predecessor of several of the philosophers named above, established and proved the roundness of the earth in his second book on the sky, chapter xiv , [2] by means of reasoning that was very solid, and very similar to what we are about to provide.

First of all one noticed that among the stars that one saw turning round the earth, there were some that always stayed in the same place, or nearly so, and that consequently the entire celestial sphere turned around a fixed point in the sky; one called this point the pole ; one noted very soon after, that while the sun found itself each day at its greatest elevation above our head, it was then constantly on the plane that passed through the pole and a plumb line; one called this plane the meridian : one then observed that when one travelled in the direction of the meridian, the stars towards which one was travelling appeared to approach from above the head, and that the others by contrast appeared to move away from it; and that furthermore these latter stars, in declining, entirely disappeared, and that others began to appear from the opposite side. From that it was easy to conclude that the plumb line, that is the line that is perpendicular to the surface of the Earth, and passing through the top of our head, changed direction to the same extent that one advanced along the meridian, and did not always remain parallel to it; as a consequence, the surface of the earth was not a plane, but curved in the direction of the meridian. Hence with the planes of all the meridians converging at the pole, as one has just noted, it needs only a little reflection (even without the least tint of Geometry), to see that the earth could not be curved in the direction of the meridian, without it also being curved in the direction that is perpendicular to the meridian, and that consequently is it curved in all senses. Moreover, other astronomical observations, such as those of the rising and the setting of stars, and the difference in times at which these occur according to the where on the Earth one is standing, confirm the curvature of the Earth in the sense of the perpendicular to the meridian. Finally, the observation of eclipses of the Moon, during which one sees the shadow of the Earth advance across the disc of the Moon, lets one know that this shadow is not only curved, but detectably circular; from that one concludes with reason that the Earth also has more or less a spherical form; I say more or less , because there were some in antiquity who believed that the Earth did not have exactly this form; see the Memoires de l’Académie des Belles-Lettres, t. XVIII p. 97 . But notwithstanding this opinion from antiquity, the non-sphericity of the Earth should be regarded as a discovery that belongs absolutely and uniquely to modern philosophy, for the reasons laid out in the article Erudition, volume V page 918 column 1 . Whatever the case may be, it is at least certain that in general the ancient philosophers attributed to the Earth a perfect spherical form; and it was natural to believe in this until one was disabused of this observation.

If the roundness of the Earth has need of yet another proof in the eyes of the whole world, those who have often made a voyage around the Earth also assure us of its roundness. The first time anyone made the trip was in 1519. It was Ferdinand Magellan who undertook it, and he took 1124 days to make the whole trip; Francis Drake, and Englishman, did as much from the year 1577 in 1056 days; Thomas Cavendish in 1586 made the same voyage in 777 days; Simon Cordes of Rotterdam did it in 1590; Olivier Hoort, Dutchman, in 1077 days. Guillaume Corn. Van Schout, in the year 1615, in 749 days. Jacques Heremites and Jean Huyghens, the year 1653, in 802 days. Most recently, this voyage has been made by Admiral Anson, of whose trip such an interesting and curious account has been published. All these navigators went from east to west, to return at length to Europe whence they had left, and the phenomena, whether celestial or terrestrial, that they observed during their voyages proved to them that the Earth is round.

The sphericity of the Earth being admitted, it was easy enough to know the value of one degree of the meridian, and consequently the circumference and the diameter of the Earth. We have explained in general at Degree how one measures a degree of the meridian; we refer to this, and that will suffice for the present, saving a greater detail for the remainder of this article; a degree of the meridian is found by this method to be around 25 of our leagues, [3] and as there are 360 degrees, one therefore concludes that the circumference of the Earth is 9000 leagues, and the radius or half-diameter of the Earth is 14 or 15 hundred leagues, all in round numbers, for there is still no need here for an exact and rigorous measure. [4]

The physics → of the time joined itself with observations to prove the sphericity of the Earth; one supposed that gravity makes all bodies tend towards the same center; one additionally believed almost generally that the earth is immobile. Hence with that stated, the surface of the seas should be spherical, so that the waters rest there in equilibrium; and this conclusion, and thus the principle that it has produced, were regarded as incontestable, even after one had discovered the movement of the Earth around its axis. See Copernicus, etc. Let us now see how one was disabused of this sphericity, and what is the actual state of our knowledge on this point; let us begin with some general reflections.

The nature of the learned, in this little different than that of other men, leads them first of all to seek neither uniformity nor law in the phenomena they observe; let them begin to notice that, or even to suspect some regular process, and straight away they imagine one that is the most perfect and the simplest; soon an observation followed further disabuses them, and often even leads them back to their initial opinion in a rush, and in disappointment; finally a lengthy study, assiduous, removed from prejudice and from any system, brings them back within the limits of the truth, and teaches them that ordinarily the law of phenomena is not so ordered that it may be perceived at a stroke, nor so simple as one might think it; that each effect coming almost always from the coming together of several causes, the manner in which each one acts is simple, but the result of their mingled action is complicated, while regular, and that everything comes down to breaking down this result to disentangle the different parts. Among an infinity of examples that one could produce from what we put forth here, the orbits of the planets furnish a striking one: scarcely had one suspected that the planets move in a circular manner, than one has them describing perfect circles, and with a uniform movement, first around the Earth, and then around the Sun as centers. Observation having soon after shown that the planets are sometimes more, and sometimes less, distant from the Sun, this star was displaced from the center of the orbits, but without at all changing either the circular form , or the uniformity of the motion that had been supposed; it was then perceived the orbits were neither circular nor uniformly laid out, so they were made into ovals and were given elliptical forms , the simplest oval we know. Finally it was seen that this form still did not apply to all, and that several planets, among others Saturn, Jupiter, the Earth itself, and above all the Moon, do not adjust themselves exactly to it during their passages. Many have tried to discover the law of their inequalities, and this is the great topic that today occupies the learned. See Earth, Moon, Jupiter, Saturn, etc.

It has been more or less the same with the form of the Earth : scarcely was it recognized that it was curved, than one began to suppose it spherical; at length during the past centuries, for the reasons that we will state in a moment, one recognized that it was not perfectly round; one supposed it to be elliptical, because after the spherical form , this was the simplest that one could it. Today, observations and multiple experiments have begun to cast doubt on this form , and some scientists claim even that the Earth is absolutely irregular. We discuss all these different claims, and examine in detail the reasons on which they are based; but first of all let us see in detail how one comes to know the length of a degree of the Earth.

Everything is reduced to two operations; the measure of the amplitude of the celestial arc, included between two places set on the same meridian at different latitudes, and the measure of the terrestrial distance between these two places. In effect, if one knows in degrees, minutes and seconds the amplitude of the celestial arc included between these two places, and if separately one knows their terrestrial distance, one will make this proportion: just as the number of degrees, minutes and seconds that the amplitude contains, is to a degree, thus the known terrestrial distance between the two places, is to a degree of the Earth.

To measure the amplitude of the celestial arc, one observes at one of the two places the meridional altitude of a star, and at the other place, one observes the meridional altitude of the same star; the difference between the two altitudes gives the amplitude of the arc, that is to say the number of degrees in the sky that corresponds to the distance between the two terrestrial places. See the article Degree, where one explains the reasoning of this. It is pointless to add that one must correct the observed altitudes for refractions. See Refraction. What is more, so that the error caused by refraction is the least possible, one must be careful to take, to the extent that one can, a star near the zenith, because refraction at the zenith is nil, and almost undetectable at 4 or 5 degrees from the zenith. It is also good that the observations of the star from the two places be simultaneous , that is to say that they be made at the same time , as much as possible, by two different observers each set up at the same time in each of the two places; by this method one avoids all the reductions and corrections to be made because of the apparent movements of the stars, such as precession, aberration and nutation. See the months . However, if it is not possible to make simultaneous observations, then one must be aware of the corrections that these movements produce. We add that when the places are not set exactly on the same meridian, which almost infallibly happens, the observation of the amplitude, made with the precautions we have just noted, gives the amplitude of the arc between the parallels of these two places, and that suffices to know the degree one seeks, at least on the supposition that the parallels are circles; this supposition has always been held up to now in all the operations that have been undertaken to determine the form of the Earth ; it is true that in recent times one has sought to shake this; this is what we will examine below; we will content ourselves with saying that as present, that this supposition of circular parallels is absolutely necessary to be able to conclude anything in the operations by which one measures degrees, because if the parallels are not circles, it is absolutely impossible, as one will also see below, to know by this measure the form of the Earth , nor even to be assured that what one has measured is a degree of latitude.

The amplitude of the celestial arc being known, it is a question of measuring the terrestrial distance between the two points, or if they are not set on the same meridian, the distance between the parallels. For this one chooses on some high mountains different points, that form with the two places in question a set of triangles for which one observes the angles as exactly as possible. As the sum of the angles of each triangle is equal to 180 degrees ( see Triangle), one will be certain of the exactitude of the observation, if the sum of the observed angles equals 180 degrees or does not differ noticeably from that. It must also be noted that the different points that form the triangles are never ordinarily placed on the same plane, or on the same level, so it is necessary to reduce them there, by observing the heights of these different point above the level of a surface that is concentric to that of the Earth, that one imagines passing through one of these two points. That done, one measures out on some part on the land surface a base of some extent, such as 6 or 7000 toises; one observes the angles of a triangle formed by the two extremities of this base, and by one of the points of the set of triangles. Thus one has (including here the two extremities of the base) a set of triangles for which one knows all the angles and one side, that is the measured base; then by trigonometric calculation one will know the sides of each of these triangles: in addition, one knows the height of each point above the level; thus one knows the sides of each triangle brought down to the same level; finally one knows once more by observation the angles the verticals make where the sides of the triangles are set with the meridian that one imagines to pass through one of the two places, and consequently one knows by the reductions taught by Geometry what the angles are that the sides of the triangles brought down to the same level make with the meridian passing through that place. Then using trigonometric calculation, and having regard, if one judges it necessary, for the slight curvature of the meridian contained between these two places, one will know the length of the arc of meridian contained between the parallels of the two places. Finally, one makes a slight reduction in this length, having regard for the amount by which above sea level rises that of the two places from which one made the meridian begin. With this reduction made, one has the length of the arc, reduced to sea level. To verify this length, one ordinarily measures a second base in a place other than the first, and by this second base tied to the triangles, one calculates afresh one or several sides of these triangles; if the second result agrees with the first, one is assured of the soundness of the operation. The length of the terrestrial arc and the amplitude of the celestial arc thus being known, one deduces from them the length of a degree, as explained above.

One can see in the different works that have been published on the form of the Earth , and that we will note at the end of this article, the precautions that one must take to measure the celestial arc and the terrestrial arc with the greatest possible exactitude. These precautions are so necessary, and must be taken to such an extent, that according to M. Bouguer, one can only answer for 5 seconds in the measurement of the celestial arc by being as precise as possible there. Hence, an error of one second in the measurement of the celestial arc produces an error of about 16 toises in the terrestrial degree, because one second of terrestrial degree is around 16 toises; then, according to M. Bouguer, one can be sure of only 80 toises in the degree, if one has measured only one degree. If one were to measure 3 degrees, as was done at the equator, [5] then the error for each would be only around a third of 80 toises, that is to say around 27 toises. However, it should be added that if the instrument one uses to measure the celestial arc is made with extreme care, such as the sextant used in the operations in the north, [6] one can then count on a much greater exactitude, above all when this instrument is put into use as it was by the most skilled observers.

I do not speak at all of various other methods the ancients used to know the form of the Earth ; they are too inexact to be mentioned here, and the process we have just given deserves in every respect to be preferred. Furthermore, I will not speak at all, or rather, I will say only a word about another method that one can use to determine this figure, that of the measure of degrees of longitude at different latitudes. Regardless of the exactitude to which one can take this last measure, it will always be much more liable to error than the measurement of degrees of latitude. M. Bouguer estimates the that the error can be one 240 ^th part of the measure of an arc of two degrees of longitude, and six or seven times greater than the measure of an arc of latitude of two degrees.

Here now are the different values of a degree on the Earth, derived up to M. Picard inclusively, under the hypothesis of a spherical Earth. We have no need to say that the measurements of the ancients should be regarded as very inaccurate, the imperfection of the methods and the instruments they used being understood, but we believe that the reader will note with pleasure the progress in human knowledge of this subject.

According to Aristotle the circumference of the Earth was 400000 stades, [7] which gives a degree of 1111 stades in dividing it by 360.

According to Eratosthenes, this circumference was 250000 stades, or 252000 when taking 700 stades to the degree.

According to Hipparchus, the circumference of the Earth is 2520 stades greater than 252000, however, he held to the latter measure of Eratosthenes.

According to Posidonius, the circumference of the Earth is 240000 stades. Strabo, in correcting the calculation of Posidonius, gave only 180000 stades as the circumference of the Earth. This last observation was adopted by Ptolemy. See the work of M. Cassini, entitled De la grandeur & la figure de la Terre , 1718. [8]

The mathematicians of the caliph al-Ma’mūn in the 9 ^th century determined the degree on the plains of Mesopotamia at 56 miles, and estimated it at only 10 thousand toises less than what Ptolemy had given.

The geography of Nubia in the 12 ^th century gave 25 leagues to the degree.

Fernel, physician to Henry II, determined the degree at 56746 toises, but by a very inexact measurement discussed at the word Degree. Snellius at 57000 toises (this measurement has since been corrected by M. Musschenbroek, and set at 57033); Riccioli, at 62650 (that is to say 5650 toises greater than Snellius, which gives a difference of ¹ ⁄ ₁₀ for the circumference of the Earth), Norwood, in 1633, at 57300.

Finally in 1670, M. Picard, having measured the distance between Paris and Amiens by the method laid out above, found the degree in France to be 57060 toises at the latitude of 49° 23’, half-way between these two towns; but still one did not think the Earth could have any form other than a sphere.

In 1672, M. Richer, having traveled to the island of Cayenne, about 5° from the equator, to make astronomical observations there, found that his pendulum clock, which he had regulated in Paris, fell behind by 2’ 28” a day. [9] From that one concludes, all deduction having been made of the amount by which the pendulum would be lengthened at Cayenne by the heat, see Pendulum, etc., that the same pendulum moves more slowly at Cayenne than at Paris; that consequently the action of the weight would be less at the equator than in our regions. The academy had already suspected this fact (as M. le Monnier remarks in the Histoire celeste published in 1741 [10]) after some experiments made in various places in Europe; but it seems, to mention in passing, that one would have been able to doubt this without the help of experiments, because the body at the equator being further from the axis of the earth, the centrifugal force produced by the rotation is greater there, and consequently, everything else being equal, more than counterbalances the weight; see Centrifugal force, etc. It is thus that through a kind of fate linked to the advancement of science, certain facts that are only the simple and immediate consequences of known principles, nevertheless often remain ignored before observation uncovers them. Whatever it may be, once it was recognized that weight was less at the equator than at the pole, the following reasoning was made: the earth is to a great extent fluid on its surface, and one can suppose without much error that it has the more or less the same form as if it were entirely fluid. Hence, in this case weight being less at the equator than at the pole, and the column of liquid that would go from one of the points on the equator to the center of the earth, having necessarily to counterbalance the column that would go from the pole to the same center, the first of these columns would have to be longer than the second; then the earth must be higher at the equator than at the poles, and hence, the Earth is a sphere flattened at the poles.

This reasoning has been confirmed by an observation. One has discovered that Jupiter turns very quickly around its axis ( see Jupiter); this rapid rotation must impose a considerable centrifugal force on parts of this planet, and consequently flatten it noticeably; thus in measuring the diameters of Jupiter, one has discovered these to be very unequal, a new proof in favor of the flattened Earth.

Some have gone to the effort of attempting to determine the amount of its flattening, but in truth the results differ amongst themselves, according to the nature of the hypotheses on which they were founded. M. Huyghens, by supposing that the primitive gravity, that is not altered by the centrifugal force, was directed towards the center, had discovered that the Earth was an elliptical spheroid, of which the axis was to the diameter at the equator in the ratio of 577 to 578. [11] See Earth, Hydrostatic and Spheroid. M. Newton held to another principle, that he supposed that the primitive gravity came from the attraction of all parts of the globe, and found that the Earth was yet an elliptical spheroid, but of which the axes were between them in a ratio of 229 to 230, [12] a flattening more than double that of M. Huyghens.

These two theories, while most ingenious, do not sufficiently resolve the question of the form of the Earth : primarily one must decide which result is the more in conformity with the truth, and the system of M. Newton, then at its birth, has not yet made enough progress that it can lead to the exclusion of the hypothesis of M. Huyghens; in the second place, in each of these two theories, one supposes that the Earth were absolutely the same form as if it were entirely fluid and homogenous, that is to say equally dense in all its parts; hence one feels that this spurious supposition could contain much that is arbitrary, and that if it strays a little from the truth (which is not impossible), the real form of the Earth could be very different from that which theory gives to it.

From this one concludes with reason, that the surest means by which to know the true form of the Earth , is the actual measure of degrees.

In effect, if the Earth were spherical, all degrees would be equal, and consequently, as has been proven at the word Degree, one would have to travel the same distance at every point along the meridian, so that the elevation of the same given star would increase or diminish by one degree. But if the Earth is not spherical, then its degrees will not be equal, and it will be necessary to travel a greater or lesser distance along the meridian, according to the place on Earth at which one will be, so that the elevation of a star one is observing increases or diminishes by one degree. Now, to determine in just what way the degrees should be believed or disbelieved in this hypothesis, let us first of all suppose the Earth to be spherical and malleable, and let us imagine that a double force applied to the ends of the axis, compresses the Earth from outside to within, following the direction of the axis: what will happen? Certainly the axis will decrease in length, and the equator will rise: but furthermore the Earth will be less curved at the extremities of the axis than it was before, it will be more flattened towards the axis, and by contrast it will be more curved at the equator. Hence, the more the Earth is curved in the direction of the meridian, the less it will be necessary to travel in that same direction, so that the observed elevation of a star increases or diminishes by one degree; consequently if the Earth is flattened towards the poles, one would need to travel a shorter distance along the meridian near the equator than near the pole to gain or to lose one degree of latitude; consequently if the Earth is flattened, the degrees should diminish from the equator towards the pole and reciprocally; [13] the reasoning that has just been given is sufficient for those who are not geometers; here is a rigorous version for those who are.

Let [14] (figure 12 Geography [15]) C be the center of the Earth; CP the axis, EC the radius at the equator, EHP a portion of the meridian; from any point H , the perpendicular HO is taken to the meridian EHP , with the line HO touching tangent GOF at O . See Tangent. HO will be the osculating radius of H . See Osculator; let then the point h be taken such that the osculating radius ho makes an angle of one degree with HO ; it is easy to see that Hh will represent a degree of meridian; that is to say, as was proved at the word Degree, that an observer who moves from H to h , will find at h one degree more or less than at H in the elevation of all the stars set on the meridian. Hence, Hh being very near to the arc of a circle depicted with radius HO (or ho which is roughly the same) it is clear that if the degrees Hh increase from the equator E towards the pole P , the osculatory radii HO will also increase; thus the radius of a circle is greater than the degree or the 360 ^th part of a circle further extended. Thus the tangent GOF will always be entirely within the angle ECF . Hence, by the property of the tangent, see Tangent, one has EGOF = FCP , and it is clear from the axioms of Geometry that EGOF is «i> EC + CF ; then EC + CF » CP + CF ; thus EC » CP ; thus the Earth is flattened if degrees continue to increase from the equator towards the pole . Those who, following M. Picard, measured the first degrees of meridian in France to know whether or not the Earth was spherical, did not reach this conclusion; whether by inattention, or by a lack of sufficient geometric understanding, they believed on the contrary that if the Earth was flattened, degrees would have to diminish when going from the equator towards the pole. Here, according to all the evidence, is the reasoning they applied: let a line be drawn from the center that makes with EC an angle of one degree, and from the same center C let a line be drawn that makes with PC an angle of one degree, it is certain that EC being supposed larger than PC , the portion of the Earth delimited in E between the two lines that make an angle of one degree, will be larger than that in P ; thus (perhaps they concluded) a degree near the equator will be larger than a degree at the pole. The fallacious nature of this reasoning lies in the fact that a degree of the earth is not determined by two lines that go to the center, and that make an angle of one degree; but by two lines that are perpendicular to the surface of the Earth, and that make an angle of one degree. It is in relation to these perpendiculars (determined by the placement of a plumb bob) that one measures the distance of stars to the zenith, and consequently their elevation; for these perpendiculars will not pass through the center of the Earth , when the Earth is not spherical. See Tangent, Osculator, etc.

Regardless of this conjecture, those who were the first to measure degrees within the extent of France, perhaps preoccupied with the idea that a flattened Earth would produce degrees towards the north smaller than those towards the south, in effect discovered that in the entire extent of France in terms of latitude, degrees diminished towards the north. But scarcely had they shared this result with the learned men of Europe, that it was demonstrated to them that in consequence the Earth had to be elongated. It was necessary for this to happen; for how does one go back over measurements that one was assured were very exact? In France, one then remains convinced enough of the elongation of the Earth, notwithstanding the contrary results derived from theory.

This conclusion was confirmed in the book De la grandeur & de la figure de la Terre , published in 1718 by M. Cassini, whom the Academy of Sciences of Paris has just lost. [16] In this work M. Cassini gave the results of all the operations carried out by him and by his father M. Dominique Cassini, to determine the length of degrees. He concluded that the average degree in France was 57061 toises, [17] and that the degrees diminished across the entire extent of France from the south to the north, from Collioure as far as Dunkerque. See Degree. Other operations carried out since then in 1733, 1734, 1736, confirmed this conclusion; thus all measurements were in agreement with each other, in spite of the theory, to make the Earth elongated.

But the partisans of Newton, both in England and the rest of Europe, and the principal geometers in France itself, judged that these measurements did not invincibly overturn the theory; they dared to believe that they were perhaps not exact enough. However, in supposing they had been taken with care, it was possible, they said, that through errors of observation, the difference between degrees that were immediately beside one another, or not far apart (a very small difference in itself) could not be determined with such assurance. Hence with regard to this it was determined to measure two degrees that were very far apart, so that their difference would be great enough that it could not be imputed to an error of observation. It was proposed to measure the first degree on the meridian at the equator, and a degree as close to the pole as possible. MM. Godin, Bouguer and de la Condamine departed on the first voyage in 1735; and in 1736 MM. de Maupertuis, Caliraut, Camus and le Monnier, departed for Lapland. The latter returned in 1737. They had measured the degree of latitude that passes through the polar circle, at around 23 ^d ½ from the pole, and they had found it to be considerably larger than the average degree in France; from which they concluded that the Earth is flattened.

The degree in Lapland, at 66 ^d 20’, was found by the skilled observers to be 57438 toises, 378 toises longer than the 57060 toises of M. Picard, measured at 49 ^d 23’; but before concluding from that the form of the Earth , they considered it appropriate to correct M. Picard’s degree, having regard for the aberration of the stars, that M. Picard had not known, as well as the precession and the refraction, that this astronomer had neglected. By this means the degree of 57060 toises, measured by M. Picard, was reduced to 56925 toises, shorter by 513 toises than that of Lapland. [18]

In supposing that the meridian of the Earth be an ellipse little different from a circle, one knows from Geometry the increase in degrees, in going from the equator towards the pole, should be noticeably proportional to the squares of the sine of latitude. Furthermore the same geometry demonstrates that if one has on an elliptical meridian the value of two degrees at known latitudes, one will have the relationship of the axes of the Earth by leans of a very simple formula. In effect, if one names E , F the lengths of two degrees measured at latitudes whose sines are ∫ and s , one will have by the difference the axes ^{E - F} ⁄ _{3( E ∫∫ - F ss)} . M. de Maupertuis provided this formula in the Memoirs of the Academy for 1737, and in his book La figure de la Terre, déterminée , and it is very easy to find it by different methods. If the degree F is at the equator, one has s=0 , and the formula become simpler, reducing to ^{E - F} ⁄ _{3 E ∫∫ ☐} . The academicians of the North, by applying to this formula the measurements of the degree in Lapland and in France, found that the relationship of the axis of the Earth to the diameter of the equator was 173 to 174; which does not stray greatly from the relationship of 229 to 230 given by M. Newton, always supposing the inevitable errors in measuring a degree. It is not without use to mention that the academicians of the North had neglected about 1” from the refraction in the amplitude of their celestial arc . This small correction having been made, the Lapland degree should be shortened by 16 toises, and is reduced to 57422; but the relationship of the axis to the diameter at the pole always remains notably the same, that of 173 to 174. Following M. Cassini’s measurements, the Earth would be an elongated spheroid, whose axis would exceed the diameter of the equator by about ¹ ⁄ ₁₀₀ . The degree in Lapland should be, by this hypothesis, around 1000 toises smaller than was determined by the academicians of the North; an error into which one cannot suppose them to have fallen.

The partisans of the elongation of the Earth first of all made all the objections it was possible to imagine against the operations upon which were based the measurement in the North. One believed, said one modern author, that it involved the honor of the nation not to allow the Earth to be given a foreign form, a form imagined by an Englishman and a Dutchman, very like it had long been believed that the honor of the nation was concerned with defending vortices and subtle matter, [19] and to proscribe Newtonian gravitation. Paris, and even the Academy, split into two parts: in the end the measurement from the North was victorious, and its adversaries were so convinced of it, that they demanded a second measurement of all the degrees of the meridian within the entire extent of France. The operation was performed more exactly than the first time, astronomy having been greatly perfected during the interval between the two measurements: all were assured in 1740 that the degrees increased from south to north, and consequently the Earth once more found itself flattened. This is what one can see in the book entitled La méridienne vérifiée dans toute l'étendue du royaume , etc., by M. Cassini de Thury, son of M. Cassini, and today pensionary and astronomer of the Academy of Sciences. Paris 1744 . [20] One must however remark that, for greater exactitude in this account, that the degrees in France did not all and without exception diminish from north to south, but that was true of the greatest number, and in the degrees that strayed from this law the difference was so exceedingly small, that one could, and should, wholly attribute them to inevitable observation errors.

It is necessary to add that the academicians of the North, once returned to Paris, believed in 1739 that it was necessary to make some corrections to M. Picard’s degree, which they had already reduced to 56925 toises. Here is their reasoning. In general the measurement of this degree depends, as has already been said, on two observations, that of the difference between the elevations of a star observed at the two extremities of the degree, and that of the geographical distance between the parallels traced at the two extremities of the degree. One has no doubt whatever that this latter distance had not been measured very exactly by M. Picard, but one was also not sure of the celestial observation: however exact that astronomer was, he ignored, as we have already stated, certain movements since observed in the fixed stars; he neglected some other factors such as refraction; furthermore, modern astronomical instruments have been brought to a degree of precision that they did not have in his time. Therefore scientists restarted the observation of the amplitude of the celestial arc contained between the two extremities of the degree from Paris to Amiens, and in consequence in place of 57060 toises for the degree, found 57183: this new degree, larger than that determined by M. Picard, was still much smaller than that from the North, and the flattening of the Earth remained; but this flattening was a little less than 173 to 174; it was 177 to 178, although always within the hypothesis of an elliptical Earth.

In 1740, those who had first upheld the elongation of the Earth, having had occasion to verify the base that had been used for the measurement of M. Picard, claimed that this base was nearly six toises shorter than that which M. Picard had found, and as a consequence admitting the correction made to the amplitude of the arc of M. Picard by the academicians of the North, they set M. Picard’s degree at 57074 toises ½, within 14 toises of the length that M. Picard had given it; thus M. Picard’s two errors in the measurement of the base and of that of the celestial arc formed, according to them, a kind of offset.

However, several academicians still doubted whether M. Picard had been mistaken on his base. M. de la Condamine appears to us to have treated this matter very well in his Mesure des trois premiers degrés du méridien , art. 24 page 246ff.  [21] He did not at all believe that M. Picard’s error, if in effect there was one, came, as M. Bouguer thought it did, from that astronomer having made his toise ¹ ⁄ ₁₀₀ too short: his reasoning is that the length of the pendulum at Paris, determined by M. Picard, differs by scarcely a line [22] from that which M. de Mairan had recently determined. With that suggested, one can only doubt that the toise of the two observers was not exactly the same; for the toise of M. de Mairan is also the same that served for the measurement of degrees at the equator and at the polar circle, and the same that was used in 1740 to verify M. Picard’s base. But from another angle M. Cassini has verified this base up to five times, and on different occasions, and has always found it to be 6 toises shorter than that of M. Picard. Many other methods both direct and indirect, of which M. de la Condamine makes mention, have been used to verify this base, and it has always been found shorter by 6 toises. M. de la Condamine suspects that M. Picard’s error, if there is one, could arise, 1º. from the length of the wooden perches [23] he used, and into which could slip various errors against which one was less on guard than one is today; 2º. from the manner in which they were laid on the ground. This is a detail that must be viewed in his book, and to which we will return, taking no stance at all on the true or false error of M. Picard, as far as this error be observed or fully justified, as it will be soon.

This uncertainty about the length of M. Picard’s degree necessarily rendered very uncertain the amount of the flattening of the Earth; for in supposing the Earth to be an elliptical spheroid, one has seen that one could determine, by measuring two degrees of latitude, the amount of the flattening, and one then had only two degrees of latitude, that of the North and that of France, of which the latter (a very singular thing) was much less well known than the former after 80 years of work, the difference between the two values that one assigned to it being nearly 110 toises.

Upon their return, the academicians of Peru made the question even more difficult to resolve. They had measured the first degree of latitude, and had found it to be 56753 toises, that is to say considerably shorter than the degree of France, whether one were to put the latter at 57074 toises, or at 57183. The comparison of the degrees from the equator and from Lapland gave, under the elliptical hypothesis, the ratio between the axes at 214 to 215, very close to that of M. Newton; hence by this hypothesis, and this flattening being supposed, the degree in France would necessarily have to have a certain value; that value would have to conform enough to the length of 57183 toises, assigned to the degree of France by the academicians of the North, and not at all that of 57074 toises that one had given it in the first place. It is not useless to add that in 1740, once one had found the diminution of degrees in France from north to south, just as one would in a flattened Earth, one had measured a degree of longitude, at the latitude of 43 ^d 32’; and this degree of longitude agreed very closely with what it should be under the hypothesis of an elliptical Earth and a flattening equivalent to ¹ ⁄ ₂₁₅ .

However, M. Bouguer, without regard for the four degrees that accorded with the elliptical hypothesis, and which gave the flattening of ¹ ⁄ ₂₁₅ , believed it better to prefer the French degree determined to be 57074 toises, to that same degree determined at 57183; he then removed the elliptical form from the Earth, and gave it that of a spheroid, in which the increases in the degrees followed the proportion, not of the squares of the sines of latitude, but of the fourth powers of these sines. He found that the degree of the North, that of Peru, that of France held to be 57074 toises, and the degree of longitude measured at 43 ^d 32’ of latitude, all agreed with one another under this hypothesis. He then concluded that the Earth is a non-elliptical spheroid, in which the ratio of the axes was 178 to 179, almost equal to that of 177 to 178, determined most recently by the academicians of the North, but in truth under the elliptical hypothesis; which would give two very different spheroids, even though almost equally flattened. One will see in an instant that measurements taken since in other places cannot be maintained under M. Bouguer’s hypothesis, which in truth could not then have been foreseen, and which believed everything was done for the best, in adjusting the facts one had chosen to the same hypothesis.

This is how things stood, when in 1752 M. l’Abbé de la Caille, one of those who had taken the largest part in the measurement of degrees in France in 1740, finding himself at the Cape of Good Hope at 33 ^d 18’ latitude, to which he had been sent by the academy to make some astronomical observations, related principally to the parallax of the moon, there measured a degree on the meridian, and found it to be 57037 toises. This degree accorded very well with the elliptical hypothesis and flattening of ¹ ⁄ ₂₁₅ , and, it should be noted, with the French degree held to be 57183 toises; but it was almost equal to the French degree held to be 57074 toises; and if that were true, it would result in not only that the Earth was not elliptical, but that the two hemispheres of the Earth were not identical, because the degrees were almost equal at such different latitudes as that of France at 49 ^d , and that of the Cape at 33 ^d . As for the rest it is clear that the degree at the Cape no longer accords with M. Bouguer’s hypothesis, because the French degree of 57074 toises, almost equal to a degree at the Cape, although at a very different latitude, is in conformity with this hypothesis.

Lastly, the measurement of a degree recently done in Italy by Fathers Maire and Boscovitch at 43 ^d 1’ latitude produced new difficulties. This degree was found to be 56979 toises, so not only did it differ a great deal from what it should be under the hypothesis of an elliptical Earth and the supposed flattening of ¹ ⁄ ₂₁₅ , but furthermore it was found to differ by more than 70 toises from one of the degrees measured in France in 1740, at almost the same latitude as the degree in Italy; for the degree of latitude in France, at 43 ^d 31’, had been determined at 57048 toises.

If this last difference were real, it would follow that the meridian that traverses Italy would not be similar to the meridian that traverses France, and thus the meridians not being the same, the Earth could no longer be regarded as perfectly or even noticeably circular at the equator, as one had always supposed up to this point. It would furthermore result in other very annoying consequences, as one will see in the remainder of this article. One can note at the same time that the degree of Italy squares well enough with M. Bouguer’s hypothesis, with which the Cape’s degree does not at all; thus whatever way one turns, no single hypothesis agrees with the length of all the degrees measured up to now. Nothing more is needed, as one can see, to make the form of the Earth as uncertain as Pyrrhonism [24] could wish it.

To place at a glance under the eye of the reader the degrees measured up to the present, we will assemble them in this table.

This table confirms what we noted above, that all the degrees in France do not diminish exactly from north to south; but the last degree towards the south in France is 36 toises shorter than the last degree towards the north; and that is enough to be sure that the degrees shorten from north to south across the extent of France.

To this table I will add the following, which M. l’Abbé de la Caille communicated to me.

Under the hypothesis of the length a degree of the meridian at the equator, of 56753 toises, as results from measurements taken at the equator, and that of 57422 toises at the parallel of 66 ^d 19’½ according to the measurement of the north, after having removed 16 toises to account for the effect of refraction, as have done all those who measured degrees, one has as the ratio of the axes 214 to 215, or 1 to 1.00467, supposing the Earth to be a regular elliptical spheroid. And in supposing that the increases in the degrees of the meridian are like the squares of the sines of the latitudes, one derives the following lengths:

One sees from this table that the degree at the Cape of Good Hope is shorter by only 44 toises that the measured degree; that the degree of France at 49 ^d 22’ is longer by only 29 toises than the French degree considered to be 57183, but longer by 138 toises than the degree considered to be 57074; and finally that the Italian degree is 152 toises longer than the measured degree. Thus there is properly only the Italian degree, and the French degree considered to be 57074 toises (degree still in dispute) that do not square with the elliptical hypothesis and the flattening of ¹ ⁄ ₂₁₅ ; for the differences among the others are too small not to be due to observation. I do not speak at all of the value of the other degrees of France; this is still uncertain, until one has verified the correction made to the base of M. Picard. It is not useless to add that the degree of longitude measured at 43 ^d 32’, and found to be 41618 toises, also differs very little in terms of toises from what it should be under the hypothesis of an elliptical Earth and the supposed flattening of ¹ ⁄ ₂₁₅ . In effect M. Bouguer found that this degree differed by only 11 toises from the length it should have, in supposing a flattening of ¹ ⁄ ₂₂₃ which differs little from ¹ ⁄ ₂₁₅ . Furthermore it is not useless to note that in making slight corrections to degrees that square with this last flattening of ¹ ⁄ ₂₁₅ one finds exactly the flattening of ¹ ⁄ ₂₃₀ , just as Newton gave it. M. de la Condamine, comparing two by two under the elliptical hypothesis the four following degrees, that of Peru, that of Lapland, that of France held to be 57183 toises, and the same held to be 57074, discovered that the ratio of the axes varies from ¹ ⁄ ₁₃₂ to ¹ ⁄ ₃₀₃ . See his work, page 261 . Finally we should add that the flattening of the Earth has always been found much greater than that of M. Huyghens, either by the measurement of degrees, or by the observation of the pendulum; from which it seems that one can conclude with plenty of basis that primitive gravity is not directed towards the center of the Earth, nor even towards a single center, as M. Huyghens supposed.

Before passing our judgment on the present state of this great question of the form of the Earth , and above all what has been done to resolve it, it is necessary that we speak of experiments on the lengthening and shortening of the pendulum, observed at different latitudes; for these experiments bear directly on the question of the form of the Earth . It is in general certain that if the Earth is flattened, gravity must be less at the equator than at the pole, as a consequence of which the seconds-pendulum should slow down while moving from the pole towards the equator, and by the same reasoning, the pendulum that beats the seconds at the equator, should be lengthened while moving from the equator towards the pole. Furthermore, if the flattening of ¹ ⁄ ₂₃₀ , given by M. Newton, is correct, it is demonstrated that the gravity at the equator would be ¹ ⁄ ₂₃₀ less than the gravity at the pole, and furthermore, that the increase in gravity, from the equator to the pole, should follow the ratio of the squares of the sines of latitude. Hence, by the observed law governing the lengthening of the pendulum, in moving from the equator to the pole, one would know the law of the increase in gravity in the same way, and this increase which is proportional to the lengthening of the pendulum ( see Pendulum) is found, by observation, to be almost exactly proportional to the squares of the sines of latitude.

In effect the lengths of the pendulum corrected by the barometer, and reduced to that of a pendulum that oscillates in a non-resistant medium, are

Hence, according to the calculations of Father Boscovitch, the differences that are proportional to the squares of the sines of latitude, or, what comes to the same, to half the sine over twice the latitude ( see Sine), are 7, 24, 138, 206, in truth a little smaller than those in the table, as I had already noted in my Recherches Sur le Système du Monde, II part pages 288 and 289  [28] in using a calculation less rigorous than the preceding; however, as the greatest spread between the observation and the theory here is ⁸ ⁄ ₁₀₀ of a line, [29] it seems that one could regard the proportion of the squares of the sines of latitudes as having been observed exactly enough in the lengthening of the pendulum. It should be noted here that in the preceding table, the pendulum lengths observed at Paris and Pello have been increased by ¹ ⁄ ₁₀ of a line (something I had not done in the place cited in my Recherches Sur le Système du Monde ); because the observed lengths 440, 57, and 441, 17 are those of the pendulum in air, and the lengths 440, 67, 441, 27, are those of the same pendulum in a non-resistant medium, as are the three that precede them.

But if on the one hand the law of the shortening of the pendulum conforms enough with the elliptical hypothesis, then on the other hand the amount of the shortening of the pendulum at the equator is not found to be what it should be, if the flattening of the earth were ¹ ⁄ ₂₃₀ ; it is greater than this fraction. Thus the experiences with the pendulum seem also to give some check to the Newtonian theory of the form of the Earth , under which one regards the planet as fluid and homogenous. This naturally leads us to speak of everything that has been done up to our time, to extend and perfect this theory.

M. Huyghens had determined the form of the Earth under the hypothesis that basic gravity was directed towards the center, and that gravity altered by centrifugal force was perpendicular to the surface. M. Newton had supposed that basic gravity resulted from the attraction of all parts of the Earth, and that the central columns were in equilibrium, with no regard for the perpendicularity to the surface. Furthermore, Messrs. Bouguer [30] and de Maupertuis [31] brought to notice in the Mémoires de l’académie de Sciences de 1734 that the Earth being supposed fluid as with Messrs. Huyghens and Newton, it would be necessary, for there to be equilibrium among the parts, under whichever hypothesis of gravity towards one or several centers, that the two hydrostatic principles of M. Huyghens and M. Newton to agree with one another, that is to say that the direction of the gravity would be perpendicular to the surface, and that furthermore the central columns could be in equilibrium. They have each demonstrated that there is an infinity of cases in which the central columns could be in equilibrium, without gravity having to be perpendicular to the surface, and in reverse; and that there is no equilibrium at all, at least in that the observation of these two principles does not agree in giving the same form . For the rest, these two able geometers have principally envisaged the question of the form of the Earth by the supposition that basic gravity had given directions towards one or several centers: the Newtonian hypothesis of the attraction of particles made the problem much more difficult.

It was all the more that the manner in which it had been solved by M. Newton could be regarded as not only indirect, but also as insufficient and imperfect in certain regards: in this solution, M. Newton first supposed that the Earth was elliptical, and on this hypothesis he determined the flattening it should have: for as much as this supposition of the elliptical Earth was legitimate under the hypothesis of a homogenous Earth, it had however to be proven; without that it was proper to suppose that this was in question. M. Sterling rigorously demonstrated the first in the Philosophical Transactions , that M. Newton’s supposition was in effect legitimate, in regarding the Earth like a homogenous fluid, and very little flattened. [32] Soon after that M. Clairaut , in the same Transactions, nº 449, [33] greatly extended this theory. He proved that the Earth should be an elliptical spheroid, in supposing not only that it was homogenous, but that it was composed of concentric layers, of which each particular one differed in density from the other layers; it is true that he then regarded these layers as being alike; for the similitude of the layers, as we shall see further below, and of which M. Clairaut had been assured, cannot be maintained under the hypothesis that the layers were fluid.

In 1740, M. Maclaurin, in his excellent piece on the flux and reflux of the sea, which shared the prize of the Academy of Sciences, was the first to demonstrate this elegant proposition, that if the Earth is supposed to be a homogenous fluid, whose parts draw one another, and are drawn apart from that by the Sun or by the Moon, following the ordinary laws of gravitation, this fluid, in turning around its axis at whatever speed, will necessarily assume the form of an elliptical spheroid, whatever its flattening may be, which is to say very little or not. Furthermore M. Maclaurin showed that in this spheroid, not only was gravity perpendicular to the surface, and the central columns in equilibrium, but also that any point whatever chosen at will inside the spheroid, was equally pressured on every side. This last condition was no less necessary than the other two, that there be equilibrium; however, none of those who had up to then worked on the form of the Earth had thought of it; one had limited oneself to gravity’s perpendicularity to the surface, and to the equilibrium of the central columns, and one did not dream that according to the laws of Hydrostatics ( see Fluid and Hydrostatics), any point whatever of the fluid must be equally pressured on every side; that is to say that the columns of fluid, directed at any point whatever, and not only towards the center, are in equilibrium among themselves.

M. Clairaut, having considered this last condition, deduced from it some profound and curious consequences, which he laid out in 1742 in his treatise entitled, Théorie de la figure de la terre, tirées des principes de l’Hydrostatique . [34] According to M. Clairaut, for a fluid to be in equilibrium, that the efforts of all the parts gathered in a channel of whatever form that one imagined traversing the entire mass, should mutually destroy themselves. This principle is apparently more general than that of M. Maclaurin; but I stated in my Essai sur la résistance des fluides, 1752, art. 18 . [35] that the equilibrium of curvilinear channels is only a corollary of M. Maclaurin’s simpler principle of the equilibrium of rectilinear channels; which, besides, in no way diminishes the merit of M. Clairaut, because he deduced from this principle a great number of important truths that M. Maclaurin had not drawn, and that he had even followed up so little as to fall into various errors; for example, in those of supposing alike among themselves the layers of a fluid spheroid, as one can see in the Traité des fluxions, art. 670 et seq .

M. Clairaut, in the work we have just cited, proves (which M. Maclaurin did not do directly) that there is an infinity of hypotheses, where the fluid would not be in equilibrium, even though the central columns counterbalance themselves, and the gravity is perpendicular to the surface. He provides a method by which to recognize hypotheses of gravity, in which a fluid mass can be in equilibrium, and by which to determine its form ; furthermore he demonstrates that in the system of the attraction of parts, provided that the gravity be perpendicular to the surface, all the points of the spheroid will be equally pressured on all sides, and that thus the equilibrium of the spheroid under the hypothesis of attraction reduces itself to the simple law of the perpendicularity at the surface. Based on this principle, he seeks the laws of the form of the Earth under the hypothesis that the parts attract one another, and that it is composed of heterogenous layers, whether solid, or fluid; he finds that the Earth should in all these cases have an elliptical form more or less flattened, according to the disposition and the density of the layers; he proves that the layers need not be similar, if they are fluid; that the increase in gravity from the equator to the pole should be proportional to the squares of the sines of latitude, as with the homogenous spheroid; a proportion that is very remarkable and very useful in the theory of the Earth; furthermore he proves that the earth would be flattened more only in the case of homogeneity, that is by ¹ ⁄ ₂₃₀ , but this proposition holds only by supposing that the layers of the Earth, if it is not homogenous, increase in density as one goes from the circumference towards the center; a condition that is not absolutely necessary, above all if the interior levels are supposed as solid; furthermore, in supposing even that the densest levels are the closest to the center, the flattening could be greater than ¹ ⁄ ₂₃₀ , if the Earth has a solid interior core flattened more than ¹ ⁄ ₂₃₀ . See part III of my Recherches sur le système du monde, p. 187 . Finally M. Clairaut demonstrates, by a very elegant theorem, that the diminution of gravity from the equator to the pole [36] is equal to twice ¹ ⁄ ₂₃₀ (the flattening of the homogenous Earth) less the real flattening of the Earth. This is but a very light sketch of what is excellent and remarkable in this work, quite superior to everything that has been done up to now on the same topic. See Hydrostatic, Capillary Tubes, etc.

After having reflected for a long time in this important topic and having read with attention all the studies it has led to, it appeared to me that one could still push them much further along.

Up to this point we have supposed that in a fluid made up of layers of different densities, the layers should all be on a level , that is to say that the force of gravity should be perpendicular to each of these layers. In my Reflexions sur la cause des vents 1746  [37] article 86 I had already proven that this condition was not absolutely necessary to equilibrium, and since then I have demonstrated this more directly and more generally, in my Essai sur la resistance des fluids 1752, articles 167 and 168 . [38] In the same work, from article 161 and up to and including article 166 , I have proven that the concentric and dissimilar layers of this same fluid need no longer be necessarily of the same density in their entire extent for the liquid to be in equilibrium; and I have presented, it seems to me, from a point of view that is broader than any up to now, and in a very simple and direct manner, the equations that express the law governing the equilibrium of fluids ( see the article Hydrostatic for much more detail on these different topics, and on some others that are related to the laws of the equilibrium of fluids, and for other remarks I have made with regard to these laws). Finally in article 169 of the same work, I determined the equation for the different layers of the spheroid, not only by supposing, as others had done before me, that the layers were fluid, that they attracted one another, and that they increased or diminished in density, following some sort of law, from the center to the circumference, but in furthermore supposing (as no-one had yet done) that the force of gravity was not perpendicular to the layers, except at the top layer; I derived from this hypothesis a general equation, of which those that had been given before me, were only a particular case; it should be noted that in all cases in which these limited and particular equations could be integrated, the much more general equations I have given could also be integrated; it is this that results from various particular investigations into integral calculus, that I have published in the Mémoires of the Academy of Sciences of Prussia of 1750 . [39]

Nevertheless, in these generalized formulas, I had always supposed the Earth to be elliptical , as had all who preceded me, not having until now found any means by which to determine the attraction of the Earth under other hypotheses; but having made new efforts to solve this problem, in 1754, at the end of my Recherches sur le système du monde , I finally gave a method that Geometers have, it seems to me, for a long time sought to determine the attraction of the terrestrial spheroid in an infinite number of other suppositions than that of the elliptical form . I then imagined that the equation of the spheroid could be represented like this, r´=r+a+bt+ct²+et³+ft4+gt5 , etc., r´ being the radius of the Earth at any point, r the half-axis of the Earth, t the sine of the latitude, a, b, c, etc. being the ordinary constant coefficients; and I determined the attraction of a similar spheroid. This equation is infinitely more general than what has been supposed to date; for in the case of the supposed elliptical Earth, one has only r´=r+a-at2.

In the third part of my Recherches sur le système du monde , which is in press as I write (May 1756) and which will probably have appeared before the publication of the sixth volume of this Enclyclopédie , I have deduced some very great consequences from this important problem. Furthermore I have shown that the problem would not be more difficult, but would just entail a longer calculation, under the hypothesis of attraction proportional not only to the inverse square of the distance, but to whatever sum of whatever powers of that distance; which can be very useful in research into the form of the Earth , while one takes note of the action that the sun and the moon exert on it, or (which comes back to the same thing) in research on the elevation of the waters of the sea by the action of these two heavenly bodies; see Flux and Reflux: finally I have noted that in supposing the spheroid to be fluid and heterogenous, and the layers on a level or not, it could very well be in equilibrium without having an elliptical form ; and I have given the equation that expresses the form of its different levels.

This is not all. I supposed that in this spheroid the meridians would not have been similar, that not only did each layer differ from the others in density, but that all the points of the same layer differed in density between themselves; and that I indicated the method of finding the attraction of the parts of the spheroid in this very general hypothesis; a method that could be most useful in what follows, if the Earth were in effect found to have an irregular form . It remains only for us to examine this last opinion, and the reasons one could have for either supporting or opposing it.

M. de Buffon is the first (that I know) to have advanced the idea that the Earth probably has great irregularities in its form , and that its meridians are not identical. See Histoire naturelle, volume I page 165 et seq . [40] M. de La Condamine did not distance himself very far from this idea in the same work in which he gave an account of the measurement of a degree at the equator, page 262 . M. de Maupertuis, who first opposed it in his Elements de Géographie, [41] seems since to have adopted it in his Lettres sur le progrès des Sciences ; [42] finally Father Boscovitch, in the work he published last year on measuring a degree in Italy, not only inclines towards believing that the meridians of the Earth are not alike, but seems to be strongly convinced of it, because of the difference found between the degree of Italy and that of France at the same latitude.

First of all, it is certain that astronomical observations do not prove beyond doubt the regularity of the Earth and the similarity of its meridians. One takes for true in these observations that the line of the zenith or the plumb bob (which is the same thing) passes through the axis of the Earth; that it is perpendicular to the horizon; and that the meridian, that is to say the plane where the Sun is found at midday, and which passes through the line of the zenith, also passes through the axis of the Earth, but I have proved in the third part of my Recherches sur le système du monde (and I believe I am the first to make this remark), that none of these suppositions has been demonstrated rigorously, that it is as if impossible to assure oneself by observation of the truth of the first and the third, and that it is at the least very difficult to assure oneself of the truth of the second. However it must at the same time be affirmed that these three suppositions being natural enough, the sole difficulty or even impossibility of proving the truth of them is not a reason to ban them, above all if the observations are not notably contradictory. The question then comes down to whether the measurement of a degree recently done in Italy is a sufficient proof of the dissimilarity of the meridians. This dissimilarity having once been admitted, the Earth would no more be a revolving solid; and not only would it remain very uncertain whether the line of the zenith passes through the axis of the Earth, and if it is perpendicular to the horizon, but the contrary would be yet more probable. In this case the plumb bob would indicate nothing more than the perpendicular to the surface of the Earth, not that of the plane of the meridian; the observation of the distance of the stars from the zenith would no longer give the true measure of a degree, and all the operations carried out up to the present to determine the form of the Earth and the length of a degree at different latitudes would be to no purpose. This question, as one can see, merits a serious examination; let us first of all consider it from the physical side.

If the Earth had been particularly fluid and homogenous, the mutual gravitation of its parts, combined with the rotation around its axis, would certainly have given it the form of a flattened spheroid, of which all the meridians would have been alike: if the Earth had originally been formed of fluids of different densities, these fluids, in seeking to put themselves into mutual equilibrium, would also have been distributed in the same manner in each of the planes that would have passed through the axis of rotation of the spheroid, and consequently the meridians would again have been alike. But, as one will say, has it been firmly proven that the Earth was originally fluid? And were it to have been, had it assumed the form that this hypothesis requires, is it completely certain that it would have kept it? Neither to dissimulate nor to diminish the force of this objection, let us further support it before appraising its value, by the following reflection. The fluidity of the spheroid demands a certain regularity in the disposition of its parts, a regularity that we do not observe in the Earth that we inhabit. The surface of the fluid spheroid should be homogenous; that of the Earth is made up of fluid parts and solid parts, different by their density. The evident disruptions that the surface of the Earth has suffered, disruptions that are hidden only to those who do not wish to see them (and of whom we have only a faint, but sorry image among those who have come to know Quito, Portugal and Africa), the evident changing of land into seas and seas into land, the depression of the globe in certain places, its raising in others, has not all of this considerably altered the primitive form ? ( See Physical Geography, Earth, Attraction, etc ., the Géographie of Varenius, [43] and the first volume of M. Buffon’s Histoire naturelle ). For the primitive form of the Earth once being altered, and the greater part of the Earth being solid, what will assure us that it has kept any regularity in its form or in the distribution of its parts? Would it furthermore be more difficult to believe that this distribution seems, so to say, done by chance in the part that we can know of the interior and of the surface of the Earth? The evident circularity of the Earth’s shadow during lunar eclipses proves nothing other than that the meridians and the equator are near-circles; hence the equator would have to be exactly a circle for the meridians to be identical. The evident circularity of the shadow does not at all prove that the meridians are exact circles, because measurements have proven they are not so; why would they prove the perfect circularity of the equator? The same observed elevations of the pole, after having been surveyed from equal distances on different meridians, leaving from the same latitude, do not prove anything either, because one must be certain that no error at all has been committed, either in the terrestrial measurement or in the astronomical observation; as one knows that errors are inevitable during these measurements and these operations. Finally the rules of navigation that ever more surely direct a vessel, the better they are practiced, prove only that the Earth is nearly spherical, and not that the equator is a circle. For the most exact practice of these rules is itself subject to many errors.

These are the reasons for doubting the regularity of the Earth on which we live, and even for giving it an irregular form . But would there not be other inconveniences in admitting this irregularity? The uniform, constant rotation of the Earth about its axis, does not this seem to prove (as has already been noted by other scientists) that its parts are nearly equally distributed around its center? It is true that this phenomenon could absolutely have occurred under the hypothesis of the dissimilarity of the meridians, and of the irregular density of the different parts of our globe; but then the axis of rotation of the Earth would not pass through its formal center, and the relationship between the length of days and nights at each latitude would not be that which observations and calculation give it; or if one wished that the axis of rotation passed through the center of the Earth, as observations appear so to prove, it would be necessary to suppose among the irregular parts of the globe a particular arrangement, whose symmetry would be much more singular and more surprising, than the similarity of the meridians would allow it to be, above all if this similarity were only very approximate, as one supposes in astronomical operations, and not absolutely rigorous.

Moreover the phenomena of the precession of the equinoxes, so much in accord with the hypothesis that the meridians are alike, and that the arrangement of the parts of the Earth is regular, do they not in effect appear to prove that this hypothesis is legitimate? Equally, would these phenomena even occur, if the exterior parts of our globe were laid out without order and without laws? For the precession of the equinoxes coming uniquely from the non-sphericity of the of the Earth, these exterior parts would greatly influence the amount and the law of this movement of which they could then disarrange the uniformity. Finally the greatest part of the surface of the Earth is fluid, and consequently homogenous; the solid matter that covers the rest of this surface is almost everywhere little different in gravity from the common water: is it not then natural to suppose that this solid matter has nearly the same effect as fluid matter, and that the Earth is nearly in the same state, as if the its surface were everywhere fluid and homogenous; that thus the direction of gravity is noticeably perpendicular to this surface, and in the plane of the axis of the Earth, and that consequently all the meridians are alike, if not in rigor, at least perceptibly? The inequalities of the surface of the Earth, the mountains that cover it, are less considerable in relation to the diameter of the globe, than would be little bumps of a tenth of ligne in height, scattered here and there across the surface of a globe two feet in diameter. Moreover the slight attraction that the mountains exercise in relation to their mass ( see Attraction and Mountains), appears to prove that this mass is very small in relation to their volume. The attraction of the mountains of Peru that are elevated by more than a league, deviates the pendulum from its direction by only seven seconds: hence a hemispherical mountain one league in height should shift the pendulum by around a 3000 ^th part of the whole sine, that is to say one minute 18 seconds: the mountains then appear to contain very little of their own matter in relation to the rest of the terrestrial globe; and this conjecture is supported by other observations, that have revealed to us immense cavities in several of these mountains. This inequalities that to us appear to considerable, and that are so few, have been produced by the turmoil that the Earth has undergone, and of which the effect is very likely not extended much beyond the surface and the first levels.

Thus of all the reasons that one can bring to support that the meridians are dissimilar, the only one of any weight is the difference in a degree measured in Italy and a degree measured in France at a like latitude and on different meridians. But this difference that is only of 70 toises, that is to say of about 35 for each of the two degrees, is it big enough not to be attributed to observations, no matter how exact one supposes them to be? Two seconds of error in the sole measurement of the celestial arc give 32 toises of error in the degree; and what observer could answer for two seconds? Those who are at the same time the most exact and the most sincere, would they even dare to answer for 60 toises in the measurement of a degree, because do not 60 toises assume an error of four seconds in the measurement of the celestial arc, and none in the geographical operations?

Nothing then obliges us to believe the dissimilar meridians; fully to allow this opinion, it would be necessary to have measured two or more degrees at the same latitude, in places on the Earth that are very far apart, and there to have found too great a difference for it to be imputed to the observers: I say in places that are very far apart , for when the meridian of Italy, for example, and that of France, are in reality different, as these meridians are not that far distant from each other, one could always reject as observational errors the differences that one would find in the degrees corresponding to France and to Italy at the same latitude.

The would be another means by which to examine the truth of the opinion it concerns; this would be to observe the pendulum at the same latitude and at greatly removed distances: for if (taking into consideration the inevitable errors of observation) the length of the pendulum should be found to be different at these two places, [44] one would then conclude that (at least in all likelihood) the meridians were not alike. Here then are two important operations that are still to be done to decide the question, the measurement of the degree, and that of the pendulum, at the same latitude, at extremely different longitudes. It is to be desired that some exact and intelligent observer would be willing to charge himself with this enterprise, which deserves to be encouraged by sovereigns, and above all by the ministry of France, which has already done more than any other in the determination of the form of the Earth .

For the rest, while we wait for the direct observation of the pendulum, or the immediate measurement of the degrees to give us the knowledge that we are missing; analogy, sometimes so useful in Physics → , could up to a certain point clarify for us the question we are examining, by applying to it the observations of the moons of Jupiter. The flattening of that planet that had been observed since the year 1666 by M. Picard, had already led one to suspect that of the Earth a long time before it was invincibly proven by the comparison of degrees in the North and in France. Repeated observations of this planet would easily show us whether its equator is circular. By that it would be enough to observe the flattening of Jupiter at different periods. As its axis is more or less perpendicular to its orbit, and consequently its ecliptic forms an angle of only one degree to the orbit of Jupiter, [45] it is evident that if the equator of Jupiter is a circle, the meridian of that planet, perpendicular to a visual ray drawn from the Earth, should always be the same, and thus Jupiter should equally always appear flattened, at whatever time one observes it. The contrary would occur if the meridians of Jupiter were dissimilar. I know that this observation will not be demonstrable with regard to the similarity or dissimilarity of the Earth’s meridians. But in the end if the meridians of Jupiter are found to be similar, as I have occasion to suspect from the questions I put above to a very skilled astronomer, it would be, it seems to me, well enough founded to believe, failing more rigorous proofs, that the Earth would also have similar meridians. For observations prove to us that the surface of Jupiter is subject to alterations that are without comparison greater and more frequent than those of the Earth, see Bands, etc ., so if these alterations have no influence over the form of the equator of Jupiter, why would the form of the Earth’s equator be altered by much lesser movements?

But even when one would be assured by the methods that we have just noted that the meridians are noticeably alike, it would still remain to examine if these meridians have the form of an ellipse. Up to this point the theory has not at all formally provided for the exclusion of other forms ; it is limited to showing that the elliptical form of the earth agrees well with the laws of Hydrostatics: furthermore, I have noted, I repeat, in the third part of my Recherches sur le système du monde , that there is an infinity of other forms that agree with these laws, above all if one does not suppose the Earth to be homogenous. Thus in imagining that the meridian of the Earth is not elliptical, I have given in the same third part of my research , a method as simple as one could desire, to determine geographically and astronomically without any hypothesis, the form of the Earth , by measurement of as many degrees of latitude and longitude as one could wish. This method is even more necessary to practice, as not only the theory, but even the actual measurements, do not force us to assign to the Earth the form of an elliptical spheroid; for the five degrees of the North, of Peru, of France, of Italy and of the Cape, do not agree at all with this form : on the other hand the experiments with the pendulum agree well enough to assign an elliptical form to the Earth, but they make it more flattened than ¹ ⁄ ₂₃₀ : finally this flattening agrees well enough with the five following degrees, that of the North, that of Peru, the degree of France supposed to be 57183 toises, and the degree of longitude at measured 43 ^d 22’ of latitude; but the degree of France supposed to be 57074 toises, as we see today, and the degree of Italy, throw everything into disarray.

M. le Monnier, seeking to lift some of these doubts, has undertaken to verify anew the base of M. Picard, irrevocably to either proscribe or reestablish the degree of France, fixed by the academicians of the North at 57183 toise.

If this degree is reestablished, it will then be up to the Astronomers to decide up to what point the elliptical hypothesis would be disturbed by the degree of Italy, the only one that extends itself beyond this hypothesis, and even the supposed flattening of ¹ ⁄ ₂₃₀ . (Might not one believe that in a country as full of high mountains as Italy, the attraction of these mountains would influence the direction of the plumb bob, and that consequently the measurement of a degree there would be less exact and less sure? This is a little conjecture that I only venture here). It should be further examined up to what point the observations of the pendulum deviate from this same flattening of ¹ ⁄ ₂₃₀ , a deduction made from the errors that one can commit during observations.

But if the degree of 57183 toises is proscribed, one must in this case carefully discuss the errors one can commit in observations, whether of the pendulum or of degrees; and if these errors would be supposed too large to accommodate the elliptical hypothesis in the observations, one would be forced to abandon this hypothesis, and to make use of the new methods that I have proposed, to determine the form of the Earth by theory and by observation.

The observation of the flattening of Jupiter could here still be useful to us up to a certain point. It is easy to find by theory what the relationship between the axes of this planet should be, in regarding it as homogenous. If this relationship is noticeably the same as the observed relationship, one could conclude from that with much likelihood that the Earth would also be a like case, and that its flattening would be ¹ ⁄ ₂₃₀ , the same as in the case of homogeneity; but if the observed relationship of the axes of Jupiter is different from that which theory yields, then one will be able to conclude by the same reasoning that the Earth is not homogenous, and perhaps even that it does not have an elliptical form . This last conclusion could also be confirmed or disproven by observation of the form of Jupiter; for it would be easy to determine if the meridian of that planet is an ellipse, or not. for that, it would suffice to measure the parallel at the equator of Jupiter that is removed from it by 60 degrees; if this parallel is found to be noticeably equal or unequal to half the equator, the meridian of Jupiter would be elliptical, or it would not be.

I say nothing of the method of determining the form of the Earth by the parallaxes of the Moon: this method, first laid out by M. Manfredi in the Mémoires de l’Académie des Sciences of 1734, [46] is subject to too many errors to yield any certainty. It is indubitable that parallaxes would be different on a sphere and a spheroid; but the difference is so small that a few seconds of error in the observation void all the precision that one could desire here. It is much more certain to determine the difference in parallaxes with the form of the Earth supposedly being known, than the form of the Earth by the difference in parallaxes; and for this reason I am attached to the first of these two topics, in the third part of my Recherches sur le système du monde , already cited. See Parallax.

It remains only for us to say a word on the utility of this question on the form of the Earth . One must avow in good faith that having regard to the present state of navigation, and the imperfection of the methods by which one may measure the path of a vessel at sea, and know as a consequence the point of the Earth at which one finds oneself, it is pretty much a matter of indifference to us to know whether the Earth is exactly spherical or not. The errors in navigational estimates are a lot larger than those that could result from the non-sphericity of the Earth. But one day navigational methods will perhaps improve themselves enough that it would then be important for a pilot to know upon which spheroid he is setting his route. Nevertheless, is it not a fitting subject for our curiosity, that of the form of the globe that we inhabit? And this research, apart from that, is it not highly important for the perfection of astronomical observations? See Parallax, etc .

However that may be, this is the exact history of the progress made up to today on the form of the Earth . One sees how the full solution of this great question still demands discussion, observation, and investigation. Aided by the work of my predecessors, I have attempted in my most recent work to prepare the substance of what remains to be done, and to facilitate the methods. How much effort will it take until time brings us new insights? We know to wait and be skeptical.

It is time to finish this article, for which I fear I will be reproached for its length, even though I have abridged it as much as was possible; I fear even more that it will only act as a kind of reproach, however ill-founded, to learned men for the uncertainty they still show about the form of the Earth after more than 80 years of efforts undertaken to determine it. That which should however reassure me is that I have intended this article that one has just read principally for those who truly interest themselves in the progress of Science; who know that the true means to hasten that is to disentangle everything that could suspend it; who know, finally, the limits of our spirit and our efforts, and the obstacles that Nature imposes upon our researches; the sort of readers to whom alone learned men should pay attention, and not to that part of the public, indifferent and curious, who, more avid for the new than the true, try everything while being content to skim the surface.

Those who would want to inform themselves more deeply, or in more detail, on the subject of this article, should read: La mesure du degré du méridien entre Paris & Amiens , by M. Picard, corrected by the Academicians of the North, Paris , 1740; Le traité de la grandeur & de la figure de la Terre , by M. Cassini, Paris, 1718; Le discours of M. de Maupertuis sur la figure des astres , Paris, 1732; La mesure du degré au cercle polaire, by the Academicians of the North, 1738; La théorie de la figure de la Terre , by M. Clairault, 1742; La méridienne de Paris vérifiée dans toute l’étendue de France , by M. Cassini de Thury, 1744; La figure de la terre , by M. Bouguer, 1749; La mesure des trois premiers degrés du méridien , by M. de La Condamine, 1751; The work of Fathers Maire and Boscovitch, which is entitled, De litteraria expeditione per pontificiam ditionem, &c. , Rome, 1755; my Réflexions sur la cause des vents , 1746. The second and the third part of my Recherches sur le système du monde , 1754 and 1756; and several learned memoirs by Messrs. Euler, Clairault, Bouguer, de Maupertuis, etc. laid out in the Proceedings of the Academies of Sciences of Paris, of St. Petersburg, of Berlin, etc.

1. Giovanni Baptista Ricciolio (1651) Almagestum novum astronomiam veterem novamque complectens observationibus aliorum et propriis nouisque theorematibus, problematibus, ac tabulis promotam, in tres tomos distributam quorum argumentum ... Bononiae: ex typographia Haeredis Victorii Benatij.

2. See fol. 39, Nicolas de Grouchy and Joachim Périon (1560)Aristotelis De cælo libri IIII, Parisiis, Ex officina Gabrielis Buon... A modern English translation of this specific comment is in Aristotle, On the Heavens , Book II, Chapter 14, translated by J.L. Stocks (1922), Oxford: Oxford University Press; pp. 297-298.

3. The common, Paris or postal league was 2,000 toises. The toise was a pre-metric French measure of length of six French feet, most generally accepted to be equivalent to 1.949 meters. Hence, in terms of the metric system a Paris league was 3,898 meters or 3.898 kilometers, equivalent to about 2.4 statute miles.

4. The problem of determining the circumference and diameter of the earth had occupied investigators for centuries. In 1681 Jean Picard published Mesure de la terre, Paris, Imprimerie royale, in which on Page 47 he detailed his observations (summarized here by d’Alembert) that set the circumference of the earth at 9000 “leagues of 25 to the degree” or 7200 “naval leagues”, each total being equivalent to 20541600 “toises de Paris”. Picard’s statement indicates that by his measurement, the circumference of the earth would be equivalent in modern terms to ^{20541600 x 1.949} ⁄ ₁₀₀₀ km, or 40,036 km, which is about 0.10% smaller than the currently accepted 40,075 km.

5. A reference to the French geodesic expedition to northern Peru (now Ecuador), to help determine the shape of the earth, that left France in 1735: Pierre Bouguer (1749) La figure de la terre, déterminée par les observations de Messieurs Bouguer, & de la Condamine, de l’Académie royale des sçiences, envoyés par ordre du roy au Pérou, pour observer aux environs de l’équateur. Avec une relation abregée de ce voyage qui contient la description du pays dans lequel les opérations ont été faites . Paris: Institut de France, Paris. Suite des Mémoires. VIII. Académie des sciences.

6. A reference to the French geodesic expedition to Lapland, to help determine the shape of the earth, that left France in 1736: Pierre de Maupertuis (1738) La figure de la terre, déterminée par les observations de Messieurs de Maupertuis, Clairaut, Camus, Le Monnier ... & de M. l’Abbé Outhier ... accompagnes de M. Celsius ... faites par ordre du roy au cercle polaire, par M. de Maupertuis . Paris: Imprimerie royale.

7. According to Liddle and Scott’s Greek Lexicon, s.v. , a στάδιον ( stadion ) was 600 Greek feet, or 606¾ English feet. This calculation produces a circumference of 45,966 statute miles, or just over 74,000 kilometers, almost double what is now accepted.

8. Jacques Cassini (1723) Traité de la Grandeur et de la Figure de la Terre par M. Cassini, De l’Academie Royale des Sciences. Amsterdam, Chez Pierre de Coup, Marchand Libraire dans le Kalverstraat. MDCCXXIII.

9. Jean Richer (1679) Observations Astronomiques Et Physiques Faites En L’Isle De Caïenne. Paris: Imprimerie Royale. MDCLXXIX.

10. Pierre-Charles Le Monnier (1741) Histoire céleste, ou Recueil de toutes les observations astronomiques faites par ordre du roy; avec un discours préliminaire sur le progrès de l’astronomie, où l’on compare les plus récentes observations à celles qui ont été faites immédiatement après la fondation de l’Observatoire royal . Paris: De l’Imprimerie de C.F. Simon, fils.

11. This discussion is found on Page 115 of an 1885 edition of Christiaan Huyghens’ Discours de la Cause de la Pesanteur, included with his Traité de la Lumière ... , first published in 1690 in Leiden: “C’est-à-dire que la diameter EA de la Terre, est à son axe PQ, comme 289 à 288½, ou comme 578 à 577; car la raison de p à n estoit comme 289 à 1”.

12. As noted on page 415 of the 1871 reprint of Isaac Newton’s Philosophiae Naturalis Principia Mathematica (1726), “Est igitur diameter terrae secundum aequatorem ad ipsius diametrum per polos ut 230 ad 229”.

13. That is, to maintain correct proportions on a terrestrial projection, higher degrees of latitude must be adjusted to match degrees closer to the equator, or the image produced will be distorted.

14. What follows is a discussion of the branch of mathematics that studies curves, which is now considered part of differential geometry.

15. This set of illustrations does not appear to have been published. The 2001 facsimile of Recueil des Planches sur les Sciences, les Arts Libéraux et les Arts Mécaniques avec leur explication: Astronomie (Paris: Inter-Livres) contains only three plates for “Géographie” and none of the three has more than eleven figures, with no figure being anything like as complex as that discussed here by d’Alembert.

16. Jacques Cassini died in April 1756.

17. Op. cit, page 302: “Partageant cette distance par 8 ^d 31’11” ⁵ ⁄ ₆ , arc du Meridien intercepté entre les paralleles de ces deux Villes; on aura la grandeur du degré d’un Meridien, l’un portant l’autre, de 57061 toises, ce qui approche si fort de celle qui a été déterminée par M. Picard, que nous avons crû devoir nous y conformer.”

18. These stated measurements differ very slightly (by one toise) from those first reported by de Maupertuis, in his report “Mesure de la Terre au cercle polaire”, page 138: “Ayant ainsi répété deux fois notre opération, on trouve par un milieu entre l’amplitude conclue par δ, & l’amplitude par α, que l’amplitude de l’arc du Méridien que nous avons mesuré entre Torneå & Kittis, est de 57’ 28”¼, qui, comparée à la longueur de cet arc de 55023½ toises, donne le degré qui coupe le Cercle Polaire de 57437 toises , plus grand de 377 toises que celui que M. Picard a déterminé entre Paris & Amiens , qu’il fait de 57060 toiles. Mais il faut remarquer que comme l’Aberration des Etoiles n’étoit pas connue du temps de M. Picard, il n’avoit fait aucune correction pour cette Aberration. Si l’on fait cette correction, & qu’on y joigne les corrections pour la Préceffion des Equinoxes & la Réfraction, que M. Picard avoit négligées, l’amplitude de son arc est 1° 23’6”½, qui, comparée à la longueur, 78850 toises, donne le degré de 56925 toiles, plus court que le nôtre de 512 toises.”

19. Le tourbillon and la matière subtile are terms from the philosophy of Réné Descartes that were fundamental elements of his description of the universe.

20. César-François Cassini and Louis Guillaume le Monnier (1744). La meridienne de l’Observatoire Royal de Paris, verifiée dans toute l’étendue du royaume par de nouvelles observations; ... Par M. Cassini de Thury, ... avec des observations d’histoire naturelle, faites dans les provinces traversées par la meridienne, par M. Le Monnier, de la même Académie, docteur en medicine. Paris: H.-L. Guerin & J. Guerin.

21. Charles-Marie de la Condamine (1751) Mesure des trois premiers degrés du méridien dans l’hémisphere austral, tirée des observations de Mrs de l’Académie Royale des Sciences, envoyés par le roi sous l’équateur par m. De La Condamine. Paris: Imprimerie Royale. Note that the article in question (“Article XXIV: Détermination de la longueur du degré du Méridien aux environs de l’Equateur”) is on page 227 of the linked edition.

22. The ligne was a pre-metric French measure of length, defined by the Petit Larousse of 1967 as “Ancienne mesure française de longuer, représantant la douzième partie du pouce, soit 2,25 mm environ”. This particular mention appears to be a printer’s glitch for “1/15” of a ligne, a measurement equivalent to about 0.15 mm.

23. A perche was a pre-metric French measure of length, equal to 3 toises (5.85 meters); this description is of surveyors’ wooden measuring rods that could be linked together. The equivalent English measure, the perch, is 5½ yards (5.0292 meters).

24. A reference to the work of the highly influential Huguenot French philosopher Pierre Bayle (1647-1706), whose views on skepticism (that came to be known as “Pyrrhonism”) were greatly informed by the thinking of the ancient Greek philosopher Pyrrho (c. 365-275 BC).

25. On the Caribbean coast of Panama; the current statement of its coordinates is 9°33’N 79°39’W.

26. In Haiti; the current statement of its coordinates is 18°26’N 72°52’W.

27. In present-day Finland, close to the Arctic Circle; the current statement of its coordinates is 66°45’N 23°58’E.

28. A reference to d’Alembert’s 1754-1756 publication Recherches Sur Différens Points Importans Du Système du Monde , issued in Paris in three volumes, chez David l’aîné.

29. Equivalent to about 0.18 mm.

30. Pierre Bouguer (1736) “Comparaison des deux Loix que la terre & les autres Planetes doivent observer dans la figure que la pesanteur leur fait prendre.”

31. Pierre Louis Moreau de Maupertuis (1736) “Sur les figures des Corps Celestes.”

32. James Sterling (1735) “Of the Figure of the Earth, and the Variation of Gravity on the Surface.” Philosophical Transactions 1735 vol. 39 no. 436-444: 98-105.

33. Alexis Clairaut and John Colson (1737) “An Inquiry concerning the Figure of Such Planets as revolve about an Axis, Supposing the Density Continually to Vary, from the Centre towards the Surface.” Philosophical Transactions 1737 vol. 40 no. 445-451 277-306.

34. Alexis Claude Clairaut (1743) Theorie de la Figure de la Terre : Tirée des principes de l’Hydrostatique. Paris: Chez Durand, Libraire, rue Saint-Jacques ...

35. Jean-Baptiste le Rond d’Alembert (1752) Essai d’une nouvelle théorie de la résistance des fluides. Paris: chez David l’aîné, libraire, rue Saint Jacques, à la Plume d’or.

36. This statement appears to be reversed, as d’Alembert has already shown that the force of gravity at the equator is less that the force of gravity at the pole, not more, given that any point on the equator is at greater distance from the center of the earth than is the pole.

37. Jean-Baptiste le Rond d’Alembert (1747) Reflexions sur la cause generale des vents. Piece qui a remporte le prix propose par l’Academie royale des Sciences de Berlin, pour l’annee 1746. Paris: chez David l’aîné, libraire, rue Saint Jacques, à la Plume d’or.

38. Jean-Baptiste le Rond d’Alembert (1752) Essai d’une Nouvelle Theorie de la Résistance des Fluides. Paris: chez David l’aîné, Libraire, rue S. Jacques, à la Plume d’or.

39. Jean-Baptiste le Rond d’Alembert (1750 [published in 1752]) “Additions aux Recherches sur le Calcul Integral.” Mémoires de l'Académie Royale des Sciences de Berlin 6: 361-378.

40. Georges Louis Leclerc, comte de Buffon’s Histoire naturelle, générale et particulière, volume 1 contained an extended discussion on “Histoire & théorie de la Terre” (link is to the edition of 1769).

41. Pierre Louis Moreau de Maupertuis (1742) Elements de geographie. Paris: rue S. Jacques, Chez G. Martin, J. B. Coignard [etc.] libraires. Art. VII p. 45, “Mesures faites pour determiner la figure de la Terre”.

42. Pierre Louis Moreau de Maupertuis (1752) Lettre sur le progrès des sciences. “Parallaxe de la Lune, & son usage pour connoître la figure de la Terre” (page 58).

43. Bernhard Varenius (1671) Geographia generalis, in qua affectiones generales Telluris explicantur. Amsterdam: Elzevir.

44. That is, the time taken for the pendulum to complete its swing, not the physical length of the instrument.

45. By contrast, the ecliptic of the earth is tilted by about 23.4° from the vertical.

46. Sur la Détermination de la Figure de la Terre par la Parallaxe de la Lune.