La Condamine argues for the toise de l'Équateur

Introductory note

The paragraphs have been numbered (in square brackets) to facilitate comparisons with the French original.

Readers not familiar with the subject may find one 17th century usage confusing. “la mesure des Degres de la Terre”(¶3). distance betwee lines of latitude The result of such a project is not an angle, in degrees, but a length, which, in the case of the French projects, was measured in toises.

The toise that La Condamine calls the toise de l'Équateur or “our” toise, is usually called the toise du Pérou.

Académie des sciences (France).
Histoire de l'Académie royale des sciences.
Année MDCCLXXII. Seconde Partie.
Avec les mémoires de mathématique & de physique, pour la meme année,
Tirez des registres de cette Académie.
Paris: L'Imprimerie Royale, 1776.
Pages 482-501

on the Toise-Standard of the Châtelet,
and on the diverse Toises employed in measuring Degrees of the Meridian
and on that of the Seconds Pendulum.
by M. de la Condamine,
[29 July] 1758

portrait of la Condamine

Charles Marie de La Condamine

Pastel by Maurice-Quentin de La Tour (1704-1788)

[1] It would be easy for me to prove that the toise of the Châtelet, if by that name one meant some particular fixed measure, has never existed; but what is sufficient for the present case, and what everyone tends to agree on, is that it no longer exists today. M. Mairan on 24 May last, warns the Academy that the iron bar fixed to the wall at the foot of the stairway of the Châtelet to serve as a standard for the toise, was altered and distorted, and that its length had changed. He added that the Magistrates by whom the inspection is made had agreed with him to rely on the Academy for restoration of this public measure.

location of old standard

The letter B marks the staircase where the standard for the toise of the Châtelet was fastened to the wall.

Fig. 52, page 115 of Book I, Des Elemens de Géometre. (1702)

[2] It must be confessed that a standard that seems intended only for legal verification of ordinary toises of masons, carpenters, or, if you will, architects, does not appear to require very scrupulous precautions. In measuring buildings lignes are neglected; sometimes even pouces are of no consequence. But since the Academy is consulted on fixing the new standard, doubtless one expects an accuracy worthy of it.

[3] To fulfill these views, you must first agree what is the toise used by the Academy in the measurement of the degrees of Earth, and clarification does not permit the slightest negligence. The slightest mistake on the length of the toise is multiplied nearly sixty thousand times in the length of the Degree of the Meridian. A hundredth of a ligne—if our senses, helped by the most perfect instruments, can reach that far—is not to be willingly neglected in the seconds pendulum, when it is possible to take it into account. It leads to some differences in the shape of the Earth and in the theory of gravitation, important problems in celestial physics.

[4] Astronomy, geography and navigation are therefore interested in fixing the length of the Academy's toise. It is above all a question of noting the differences, if any, between the different toises that have been used for the measurements of the various degrees, and to report the various degrees, and to relate everything to the same measure.

[5] If I seemed to want to delay the decision on this issue by asking to read a memoir, it is only to put the Academy in a position to pronounce with full knowledge of the facts on a subject with which I have been occupied for a long time, whose full importance I felt, and on which I already did my best to fix the attention of the Academy to leave no doubt in the future on this matter. In 1749, finding myself Director of the Academy, I had commissioners appointed, by a resolution of the Company duly recorded, to go to the Observatory to verify the precise length of the toise with which the French meridians had been measured. I have since requested, several times, the execution of this decision. Permit me to say that for more than twenty years I have anticipated the sort of embarrassment into which one might some day fall, which [now] begins to be felt, and whose consequences could over time become more dangerous. There is still time to prevent them. Before proposing remedies, let us go back to the origin of the uncertainty. I have collected some facts about that, whose oral tradition may fade away, and which deserve the attention of the Academy.

[6] Of the Toise of M. Picard

[7] Mr. Picard in his Latin treatise On Measures, inserted in Volume VI of the old memoirs of the Academy, says with his usual concision; “the old Masons toise was reformed and shortened by five lignes in 1668”, without informing us of any other circumstances. One learns only through oral tradition, that to give the new standard the true length it ought to have, they measured the width of the arcade or inner door of the large house that serves as the entrance to the old Louvre, the side on rue Fromenteau. According to the [builder's] plan this opening was to be twelve pieds wide. Half of that length [six pied] became the new standard of the toise, which proved shorter than the old by five lignes.

Etching of western facade of the Louvre, with entrance that was measured

Western facade of the Louvre

Etching by Adam (?) Perelle (16??).

[8] Note in passing that if this expedient was good enough to recognize an error of five lignes, it was surely not suitable for finding the true length of the toise with high precision. For how sure are we,

1. that the twelve pieds width of the door of the Louvre, probably measured during construction with a common pied, and indicated by two stakes in the ground, was determined more accurately than the other fundamental measures of a building, in which a few lignes more or less, if not a few pouces, are barely perceptible and count for nothing?

2. Assuming gratuitously, and against all probability, that the distance of twelve pieds between the jambs of the arcade was originally fixed with the greatest precision, how could one afterwards be sure, to a nearest ligne, of the distance between two walls built of porous stone, exposed to injuries of the air, covered with a coating of dust thinned by rain and a mud made from the same stone? If anyone thinks that I exaggerate the difficulty here, I invite him, on leaving the Academy, to lay eyes on the spot I speak of, and I will rely on him.

[9] Be that as it may, the old toise of the Châtelet was reformed in 1668. It is M. Picard who assures us of this, and says that he used the new toise for the measurement of his meridian¹, between the latitudes of Paris and Amiens. If the toise used by M. Picard had remained on deposit at the Academy or at the Observatory, where M. Picard formally says that it will be carefully preserved, one would not have failed to make it serve in all the measurements of Degrees subsequent to his; they would have all been related to this toise, and the doubts which have arisen since then on the true length of the base of M. Picard's toise would have been promptly cleared up. But Mr. Picard's toise no longer exists, and we have no authentic monument, from his time, other than an iron bar sealed in the wall at the foot of the staircase of the Grand Châtelet, ending in two projections or steps at right angles, and which served as a standard for public measurements. This standard had been roughly built; its angles had become dull, and the interior faces of the two steps which must fit a toise when one tests it there, were never polished nor filed square and parallel with one other. No wonder the toises calibrated at different times and by different people on this defective original, are not perfectly equal to each other.

[10] We know neither in what season of the year, nor by whom the new Châtelet standard was set. If Mr. Picard had presided over it, the circumstance of the season seems too important for him to have neglected to inform us. Everything seems to indicate that this care was left to some worker, or at least to some unintelligent subordinate. It is therefore possible that the standard was, from the time it was installed, longer than the toise that Mr. Picard took to measure his degree, or that it had lengthened since by striking with a hammer the nails which attached it to the wall; moreover the two projections must have been worn by rust, by the continual rubbing of the measures being gauged, and perhaps by polishing them; it is thus very apparent that the distance between them increased. It would not be astonishing in this case that the new toises, gauged to this standard for twenty-five or thirty years; are longer than that of Mr. Picard, and consequently that one would have found fewer toises than he in the measurement of the base between Villejuîve and Juvify (Mém. De l'Acad. year 1754, p. 172).

[11] It is fair to look for all that can excuse this famous Academician, who does not deserve to be condemned lightly; but it must be admitted that the previous conjecture cannot save the error recognized in the base of Villejuive, unless to attribute another error to M. Picard, since he himself has left us the means of verifying the length of his toise, by attaching it, these are his terms, to an original, which being drawn from Nature itself, must be invariable and universal. He found that the measure of the seconds pendulum, in Paris, was 36 pouceslignes of his toise, and this length, very different from the true one, is incompatible with the number of toises that he gives to his base. It is thus necessary to agree that M. Picard was mistaken, either by assigning two fifths of a ligne too much to his pendulum, if his toise was good, or by employing a toise which was too short by more than four fifths of a ligne, if the measure of his pendulum is exact.

[12] The other toises known to the Academy, and different from that of M. Picard, are
1. that which Messrs. Godin this link goes to another website, Bouguer this link goes to another website and I, carried to Peru in 1735, and which served us in measuring the three Degrees of the meridian nearest the Equator;
2. the toise with which Messrs. de Maupertuis this link goes to another website, Clairaut this link goes to another website, Camus this link goes to another website and le Monnier this link goes to another website measured, in 1736 and 1737, the Degree of the meridian which intersects the Arctic Circle in Lapland;
3. the toise that M. Cassini de Thury this link goes to another website and Abbé de Lacaille this link goes to another website made use in 1739 and 1740, for the verification of the Meridian of Paris;
4. the toise employed by Abbé de Lacaille in his measurement of the 34th and 35th degree of southern latitude, at the Cape of Good Hope in 1752;
5. M. de Mairan's toise, with which he made his experiments with the pendulum, in 1735.

All these toises originally had for their model that of the Châtelet, which no longer exists: one supposed them all equal. If their equality were perfect, there would be no choice to be made, or this choice would be indifferent. But in the last comparison, which was made in 1756 on the occasion of the new measurement of the base of Villejuive, some differences between these various toises were recognized, and the precision that we seek here does not allow us to ignore them.

[13] Let us therefore examine which of these five toises deserves preference and must be considered the original toise to which it is appropriate to relate the others. I am not speaking of a sixth toise, compared to the same standard by the late M. du Fay, and which I acquired, because it was only used for particular experiments which do not deserve to be admitted to the competition.

On the Toise de l'Équateur.

[14] “We had taken with us in 1735²: a polished iron rule, seventeen lignes wide, by four and a half lignes thick. Mr. Godin, assisted by a skilled artist (Mr. Langlois), had put all his attention into adjusting the length of this ruler to that of the standard toise, which had been fixed in 1668, at the foot of the staircase of the Grand Châtelet of Paris. I foresaw that this old standard, made rather roughly, and moreover exposed to the shocks, the insults of the air, the rust, the contact of all the measures which are presented there, and to the malignity even of any ill-intentioned person, would be hardly suitable to verify in the future the toise which was going to be used for the measurement of the Earth, nor to become the original to which the other toises were to be compared. It therefore seemed to me very necessary, while taking a well verified toise, to leave in Paris another one of the same material and of the same shape, to which one could have recourse if some accident happened to ours during such a long journey. I took upon myself the care of having such a one made. This second toise was built by the same workman, and with the same precautions as the first. The two toises were compared together in one of our Assemblies, and one of the two remained in deposit at the Academy. It is the same one which has since been carried to Lappland by M. de Maupertuis, and which has been used for all the operations of the Academicians sent to the Arctic Circle.”

[15] When I wrote the above in 1748, and which I transcribe literally from my Measure of the Meridian, I believed that this iron rule, which I call our toise, was still in Quito in the hands of Mr. Godin. I did not know that it had been in Paris for several months. M. le Comte de Maurepas, then Minister of the Academy, had written in 1747 to M. Joseph de Jussieu, our traveling companion who remained in Quito, and had charged him, at my request, to retrieve from M. Godin a copy of his observations, our toise and the Academy's other instruments, and to bring everything back to France, since M. Godin was called to Lima in 1747 by the Viceroy of Peru, and seemed to have already settled in the service of the King of Spain. The letter of the Minister reached M. de Jussieu at the beginning of 1748, when he was fifty lieues from Quito, in the province of Canelos, ready to embark to return to France by the Amazon River. He retraced his steps to Quito, which he left as soon as possible to go to Lima, with the intention of carrying out the orders of the Minister ³. He found Mr. Godin preparing to return to Europe. Both took the road to Buénos Aires by land, after having placed on the ship le Condé, ready to sail for France, a portable geodetic quadrant and our toise enclosed in its case. These instruments arrived at the end of 1748, or towards the beginning of 1749, at the address of M. le Comte de Maurepas, and were carried without my knowledge to the Academy's cabinet of machines, and transported some years later to the Royal Botanical Garden. M. Bouguer, having known about it, withdrew only the portable geodetic quadrant that was part of the shipment, which was the one he had always used during the trip. Three years and more passed, from that time on, without my hearing anything about our toise, until in 1752 I eagerly asked Mr. Godin for news of it on his arrival from America. I was surprised to learn that the toise had been in France for nearly four years. I took all the necessary care to search for it; finally it was found in the storage room of the Royal Garden, in good condition, enclosed in a wooden case, solid and lined with serge, where it had always been kept. I did not want to withdraw it; I only repeated my request for the verification of the toise of Messrs. Cassini, preserved at the Observatory, and for its comparison with ours.*

* This toise of Peru is in the cabinet of the Academy (April 1776). Mémoires de l'Académie, 1772. Part II.

Reasons for preferring the toise of the Equator.

[16] I assume the same care and the same skill in all those who calibrated the different toises between which it is a question of deciding today. If the Châtelet standard still existed, it would be so crudely manufactured and so badly treated, that it would not be possible to recognize which of the alleged copies was originally the most conformable to the defective original. There is therefore no other reason for preference between these different toises than that of priority of date. Ours was presented to the Academy in an assembly in April 1735, compared and recognized as equal to the one that I left there on deposit. It was then adopted by the Academy, and no one protested against this adoption. M. Godin, in adjusting it to the Châtelet standard, had noticed the temperature indicated by M. de Reaumur's thermometer; it was 13 above freezing, the average temperature, and the same one to which the last measurements for the verification of Villejuive's base in 1756 were related.

[17] The decision of the Academy, which has been in effect for twenty-three years, must be revoked only for good reasons. Will it be said that our toise, in such a long journey, may have suffered some shock and changed length? I would answer that independently of the precautions with which it has always been preserved, the rule which forms it is about two pouces longer than the six pieds; that it is cut to the measure of a toise only on half of its width, and that the two heels exceeding about one pouce at each end, protected it from any shock; that its edges are still sharp, and have never been rusted, as it is easy to convince oneself of at the first viewing. Another proof that it has not been shortened is that it is longer than that of M. de Mairan. Will one say that the great heat of the Torrid Zone can on the contrary have lengthened it? This conjecture, which does not have the slightest plausibility, is destroyed by constant facts. It is no longer unknown that the thermometer does not usually rise higher under the Line than in the hottest weather in France; I can assure you that in the ten years of our trip, I have not seen the thermometer of M. de Reaumur, in America, pass 29 degrees above freezing; moreover, I have direct proof that an iron rule exposed to considerable heat regains its ordinary length, in the experiments I made on the expansion of iron by heat, and which I reported in my measurement of the Meridian, page 78. In making these experiments in Paris in 1749, I had exposed the toise of the North, and not that of the Equator, which I did not know had returned to France, to the degree of heat indicated by M. de Reaumur's thermometer, by 55 degrees above freezing; a heat more considerable than all those to which the toise from the Equator was exposed; this toise from the North has regained its ordinary length: it is even a little shorter today than ours, to which it was equal, a difference whose cause we shall examine.

[18] Finally, our toise today fits effortlessly into its standard, which did not leave Paris, and fills it exactly. There is therefore no reason to suppose that it has changed in length.

[19] This same toise was used for the actual measurement of the four fundamental bases of our triangles, at the two extremities of our arc (see the Journal du voyage à l'Equateur). We have, Messrs. Godin, Bouguer and I, determined by these measurements the length of an arc of more than three degrees from the Meridian, and that of the pendulum at Quito, by a very grand number of experiments, the result of which agrees almost to the hundredth of a ligne. All these measurements have been recorded in several works for ten years, both by the Spanish mathematicians, our travelling companions, and by M. Bouguet and me. They have been engraved on marble and on stone in various monuments, and if some of them no longer exist, the printed books and journals have preserved their memory.

[20] If the toise equal to ours, adjusted to the same standard and by the same artist, had remained on deposit at the Academy, where I had left it with this intention, it would undoubtedly have served as a model for all the toises which have since been used for the measurement of meridians in France and Africa. But M. de Maupertuis, while leaving for the Arctic Circle, one year after our departure from Paris, wrote to me that he did not hesitate to use our toise, so that we would have a common standard of measurement. He kept his word. Not having had time to make a new toise equal to the deposited toise, he took the one [on deposit] to Tornio with the permission of the Academy; new proof that our toise was regarded by the Academy, as the one to which all the measures must be related.

Toise du Nord.

[21] Thus, the toise itself deposited at the Academy, and intended to find the right length of ours, if during transport some accidents had happened to it, is the one which the Academicians of the North made use of in all their operations. We are thus all the more assured, that the length of the Degrees measured under the Equator, and under the Arctic Circle, is determined by a common measure.

[22] It is true that since the two toises returned to France, one seems to find, by a new comparison, a slight difference between them, which one judged to be a twentieth or a thirtieth of ligne (by which the North toise is shorter), pending a more precise determination . But it is more than probable that this difference has arisen only since the measurement of the Degree which cuts the Arctic Circle, and that it arises from the shortening of the toise carried to the North: and here is how I prove it. It is known that the ship on which it was embarked, on its return, was shipwrecked in the Gulf of Bothnia. The toise was wet with seawater; it is especially at the ends and edges of a filed iron that rust attaches. This rust could not be removed without the toise losing some of its length; it must therefore necessarily be a little shorter today than it was in 1735, when it was compared to ours. It is indeed, and this is a new proof of their original equality; but the change which occurred to the North toise, occurred after all the operations made in Lapland; the base of 7000 toises which served as their foundation, was thus measured with a toise equal to ours; the length of the North Degree, and that of our three Degrees close to the Equator, were thus determined by a common measure. The experiments on gravity made by the Academicians of the North, their measurements of the seconds pendulum, and all ours were thus related to the length of our toise, to which theirs was then equal. All these results have been published for twenty years in the books on the Figure of the Earth, by M. de Maupertuis and M. Clairaut, in the relation of Abbé [R.] Outhier, in the measure of the Degree between Paris and Amiens, in the works of the mathematicians of Europe who have treated of this matter, and in the literary journals of all nations.

Toise de l'Observatoire or the Degrés de France.

[23] The degrees of the meridian, measured in France, exceed in number the sum of the degrees measured elsewhere. The toise with which they have been measured in France, has thus played a larger part than all the others in the measurement of the Earth, and consequently deserves the greatest attention.

[24] In 1739 and 1740, M. Cassini de Thury and M. l'abbe de la Caille verified again the Meridian which crosses France, and which includes 8 and a half degrees from Dunkerque to Collioure, passing by the Royal Observatory. They left without measuring a base, and calculated all their triangles according to the old base from Villejuive to Juvisi, measured by M. Picard. When they arrived in Bourges, they measured a new base on the ground to verify their operations. They found it, by their actual measurement, notably shorter than by the calculation based on the length they attributed, according to M. Picard, to the base of Villejuive. Then, for the first time, they began to suspect this base of error. This suspicion, communicated by them to the late M. Cassini, made no impression on him. Biased like everyone else in favor of the exactitude of M. Picard, and personally interested in believing a base that he had taken for foundation of all the measurements of the old Meridian, had been exactly measured, he at first rejected all the error on the operations of M. his son and M. l'abbe de la Caille. He exhorted them both to revise their calculations, and to repeat the measurement of their angles, and that of their base of Bourges. All this was done: they obtained the same result. M. Cassini was then obliged to verify in his turn the base of Villejuivè. He determined to do it only at the last extremity. Only after having measured it three times, and having always found it shorter than M. Picard by about one toise per thousand, did he finally take it upon himself to speak about it to the Academy, and to ask for Commissioners to establish the measure by trial. And he did so a fourth time in particular, before the fifth measurement of which the commissioners appointed by the Academy were witnesses.

[25] We know what happened afterwards. The last verifications were made in 1756. If there is still any doubt in this regard, it can only be about the plus or the minus; it is agreed that M. Picard counted too many toises in his base. The four Commissioners find, like Messrs. de Thury and l'abbe de la Caille, that that M. Picard has counted almost one toise per thousand or 56 toises on the Degree, too many. They find, however, the difference less than M. de Thury, of a tenth of a toise on the base; namely, 5748 toises 7 pouces and a half, instead of exactly 5748 toises , which makes one toise less on the degree. M. le Monnier finds that M. Picard's error is only ¾ of a toise per thousand and a little less, which would make only 42 toises on the Degree for the error in M. Picard's geodetic measurement, instead of 56 which the four Commissioners found.

[26] The three 20-pied iron rulers which M. Cassini used to verify the base of Villejuive in 1740, had been adjusted with two iron toises preserved at the Observatory, of which one with four equal faces had belonged to the late M. de la Hire. The measurement of the same base by the four Commissioners, whose report is printed , was made with the North toise, in 1756. This measure agrees to within half a pied with the averaged measure of M. Cassini, who had repeated it five times in 1740, and this serves as the basis for all the distances concluded in the Book of the Meridian of Paris, verified. This comparison takes the place of the verification ordered by the deliberation of the Academy, of which I spoke above, and of which I solicited several times the execution, to know the ratio of the toise of the Observatoiré, or of M. Cassini, to ours; since by the new measurement of l756, executed with the North toise, one found the same number of toises and pieds at the base of Villejuive, that the late M. Cassini and M. l'abbe de la Caille had found in 1740, with their toise. It follows that the toise of the Observatory, with which the Degrees of France were measured, does not differ appreciably from the North toise, and that one, in spite of its shortening, differs today from ours only of a 25th of ligne (the difference can be verified even more exactly, as well as that of the one and the other of these two toises to the one which was used for the measurement of the Degrees of France), which would produce only two and half toises of difference on the length of the Degree, if one would neglect to take it into account. The toise used for the measurements of the Degrees of France, can thus be taken for ours.

[27] I confess that when I began this Memoir, I was only thinking of proving that the toise of the Equator, adopted from April 1735 by the Academy, and that of the North, which was perfectly equal to it when it was used to measure the Degree under the Arctic Circle, were the originals to which all the other toises used later should be related. , especially since the late M. Cassini, present when we presented our toise to the Academy when leaving for the Equator, and that we left a model of it, did not speak about his, which we would have taken gladly, not having then any reason to prefer ours. It is only by examining the thing in detail, that I recognized, as I have just proven, that the toise of the Observatory, with which the Degrees of France were measured, and from which I desired and feared the comparison with ours, differed so little that it was hardly allowed to hope for a similar agreement of two measures which were not built on the same standard; so that the reduction of the Degrees of France, to bring them to our length, could be neglected.

Toise of the Cape of Good Hope.

[28] M. l'Abbé de la Caille tells us, in the Memoirs of the Academy, for the year 1751, that the toise which he carried to the Cape of Good Hope, and which he used for the measurement of his two Degrees in 1752, was verified on the standard of Mr. Langlois, which was used to fix the precise length of the other toises carried to Peru and Lapland . This toise, brought by M. de la Caille to the Academy in 1756, to compare it with the others during the last verification of the base of Villejuive, is no longer to be found; but having been constructed by Mr. Langlois, then very practiced in this kind of work, and on the same standard as the toises of the Equator and of the North, which is extant, we can only suppose this toise equal to ours.

[29] Here are therefore four measurements of terrestrial degrees executed, in different climates, by nine academicians, with the same toise or equal toises. It is therefore no longer necessary to ask which of the four is the true one; the choice between the toise of the Equator, that of the North, that of the Observatory and that of the Cape, becomes indifferent, since their length is the same.

M. de Mairan's toise.

[30] It remains for us to speak about the toise of M. de Mairan, very well known to the Academy. It has not been used for the measurements of the Degrees by our Academicians, but it is famous for the experiments with the pendulum, which have been carried out with the greatest precision, and which M. de Mairan reported on to the Academy on November 19, 1735. We read there that his toise is an iron rule quite similar to the one which was taken to Peru, and whose model was left at the Academy, and that he had verified it on the standard of the Châtelet. But either this verification was made at an air temperature very different from 13 degrees Reaumur, which M. Godin observed when he compared the iron toise that we carried to the Equator, or, as is more likely, that the difference came solely from the coarseness of the standard, such that, as I have already noted, the same observer, comparing the same toise twice, would find different results. The toise of M. de Mairan is about a tenth of a ligne shorter than ours, by the immediate comparison which was made. It was judged in 1756 to be at least 1/15th of a ligne shorter than the toise of the North, which in its present state is shorter than that of the Equator by 1/25th of a ligne. The toise of the Equator is therefore longer than that of M. de Mairan by a 1/15th at least, plus 1/25, which makes 8/75 at least, or more than a 1/10th ligne, and consequently, so are all those toises that are equal, and which are consecrated by the measurements of the Degrees of the Meridian in Lapland, France and Africa, as ours by the three Degrees measured in America.


[31] After the facts which I have just presented, which have taken place before our eyes, and which everyone can remember, and of which the written proofs are in the hands of everyone, can we question which is the toise of the Academy? The toise of the Academy is undoubtedly the one used in their operations by the Academicians in charge of the measurement of terrestrial degrees. It is designated under this name in all the works of the mathematicians of Europe, and it is this same name that Messrs. Hellot this link goes to another website and Camus gave it twelve years ago, in their report on the verification of the ¹⁰ aune. It is true that they supposed that the toise of M. de Mairan was equal; I assumed it myself in my Mémoire sur la Mesure universelle ¹¹. I have since set the demi-toise that I carried in Italy, on the toise of M. de Mairan. As long as this one passed for being equal to ours, it could be taken for the toise of the Academy: today that it is made to differ from it, they can no longer be taken one for the other, when it will be a question of precision. There is less inconvenience in cutting three or four hundredths of a ligne from the length of the Paris pendulum alone, which M. de Mairan has measured using his particular toise, than in changing not only the results of all the pendulum experiments made in Santo Domingo, in Portobelo, in Panama, in Manta, in Quito, in Pitchincha, in Para, in Cayenne, in Tornio and in the Cape of Good Hope, but also all the lengths of the Degrees measured in various parts of the Earth; by nine Academicians charged with this work by order of the King.

[32] If there were appreciable differences between the toises which the various Academicians used for the measurement of their Degrees, in Peru, in Lapland, in France and at the Cape, the toise which we had approved by the Academy in 1735, and of which M. de Mairan acknowledges that we left the model, would deserve the preference, as the most authentic; but I proved that of the four toises employed by our Academicians to measure the Earth, three are identically the same as that of the Equator, and that the fourth differs from it only of a quantity which can be neglected. We can therefore say that we have three witnesses, and even four, who testify in favor of the toise of the Equator; in addition to its right which I have just established, it has twenty-three years of possession: The Academy, after having adopted it without anyone objecting, cannot today adopt another for its own, without falling into self-contradiction. Retracting its first judgment, against all likelihood, after such a long time, could give rise to the fear that the second decision would not be more irrevocable than the first.

[33] I have no objection to the Châtelet's toise being calibrated on that of M. de Mairan; but in that case, it should be remembered that the Châtelet's toise will not be exactly that of the Academy; however, the reform of the public standard must not be done in the name of the Academy, unless the Academy uses its own toise, the one that served for the measurement of the meridians.

Means to preserve the length of the Academy's toise.

[34] It is not enough to establish, by a new deliberation, the length of the academic toise. It is still a question of preserving this toise in an invariable way. An iron or steel standard is not enough for that. Another means, which one would think more permanent, was not more usefully practiced by the late Mr. Cassini. Assisted by M. l'abbé de la Caille, he had, in 1740, marked on very fine stones forming the pavement of the large Salle of the Observatory, which the prime meridian passes through, the length of ten toises formed by the assembly of the three rules of twenty pieds with which he later verified the base of Mr. Picard. The line which ended the ten toises still remains on the tile; but the other end of the measure, a wall seven pied thick, which seemed unshakeable, has separated from the pavement, and leaves a gap of more than a ligne.

[35] To protect the new standard of the Academy's toise from the ravages of time and accidents, it must not only be of one piece, but of a material on which rust cannot take hold. One could make this of marble, but this stone is not hard enough; it would be subject to wear by the contact of the measures presented. Porphyry is too rare a material in France; but we have granite in Normandy, on which the file has no hold. The ancient Egyptian obelisks, transported to Rome, are of this material, and most of them have been preserved healthy and whole for nearly four thousand years.

[36] I therefore consider that, in the absence of porphyry, the new standard should be carved in a granite table or shelf. Two protrusions of square, whose interior faces would be parallel and polished, would include exactly the length the length of the toise of the Academy. The iron toises, or of other materials, which are requested from several places in Europe where it is planned to measure degrees, or those which are sent from now on to foreign academies, would be adjusted on this standard, without it being possible to believe that in several centuries, it could receive any alteration; An iron rule would be made to fit exactly, which would serve at the Châtelet as a preserving rod, for the new publicly exposed standard; without prejudice to another similar standard, which M. de Mairan proposes to enclose and leave in the custody of the magistrates, to be used only once a year.

[37] To make it easy to take the length of the toise between the points of a compass, it would be advisable to draw with the diamond on the granite tablet a thin line parallel to the length of the standard, finished by two points, and divided at least by the two extremities, into pouces and lignes. One could also, on this same line or on another nearby and parallel one, mark the length of the pendulum.

[38] On this occasion, I cannot help noticing that the present circumstance, where the Academy is consulted to reform the standard of public measurement, would be very favourable to propose the use of such a desirable universal measure, and of which the least attention can make the advantages¹² felt.

[39] M. Mouton, Canon of Lyon, was the first, as far as I know; who proposed this measure drawn from the pendulum; this was in 1670¹³. It is quite likely that if this happy idea, adopted by M. Picard, in 1672 (in his Mesure de la Terre); and by M. Huygens, in 1673 (de Horologio oscillatorio), had been known as early as 1668, at the time of the reform of the toise du Châtelet, instead of shortening the old toise by 5 lignes, as was done then, M. Picard, who was consulted on this toise, would have proposed on the contrary to lengthen it, to give it, by a hardly noticeable change, a double length of the seconds pendulum. At that time it would have been sufficient to add 9½ lignes to the old toise to make the demitoise equal to the equinoctial pendulum which beats the seconds, and whose length hardly exceeds 36 pouces 7 lignes. Today since the toise of the Châtelet is shortened by 5 lignes, it would be necessary to lengthen it by a little more than 14 lignes.

[40] Although this change in the toise may seem considerable, the advantages of the measure, which I have detailed elsewhere, would soon be felt. The reduction of all the measures of France to that which I propose had been approved by the Government, under the ministry of the late M. Orry, and if an untimely death had not taken M. du Fay away, the demitoise of France would be today the common measure of all the Academies of Europe, so that it becomes that of all the Nations.

Notes on this Memoir [by the editor of the Histoire de l'Académie royale des sciences, writing circa 1772]

[41] The proposal to adopt the toise of Peru, made by M. de la Condamine in 1758 in the preceding Memoir, was not carried out at first because of the opposition of M. de Mairan. However, on May 16, 1766, there was a Declaration of the King, rendered by M. Trudaine de Montigny, in execution of which M. Tillet, of the Academy of Sciences of France, had about eighty toises made, similar to the one that had been used under the Equator, which were sent, as well as the àune of Paris and the weight of the marc, to the Châtelet of Paris and to the Procurators General of the various Parliaments. Thus, the toise that M. de la Condamine describes in this Memoir, is multiplied at present and can no longer be lost. It was also sent to Guyana, Corsica, Vienna in Austria, Turin, Florence, and Mr. Maskelyne brought back the measure of the degree made in English America, (Philosophical Transactions. year 1768, p. 326). The original of all these toises is deposited in the cabinet of the Academy.

42. That of M. Mairan was acquired by M. de la Lande; it has been found to be about a twelfth of a ligne shorter than the toise of Peru, currently adopted; but it is proposed to soon make an authentic and accurate comparison of the two toises with that of the North, which was employed in the verification of 1756 and which is in the hands of M. Monnier.

1. Art. IV. pages 3-5. in Mesure de la Terre by M. Picard.

Jean Picard.
Mesure de la Terre.
Paris: Imprimerie Royale, 1671.
A digital facsimile is available through / Bibliothèque nationale de France.


2. Pages 75 and 76 of Mesure des trois premiers Degrés du Méridien.

Charles-Marie de la Condamine.
Mesure des trois premiers degrés du méridien dans l'hémisphère austral, tirée des observations de MM. de l'Académie royale des sciences, envoyés par le roi sous l'Équateur.
Paris: Imprimarie Royale, 1751.
A digital facsimile is available through / Bibliothèque nationale de France.


3. Journal du Voyage à l'Équateur, 1751, page 217.

Charles-Marie de la Condamine.
Journal du voyage fait par ordre du roi, a l'Équateur, servant d'introduction historique a la Mesure des trois premiers degrés du méridien.
Paris: Imprimarie Royale, 1751.
A digital facsimile is available through / Bibliothèque nationale de France.


4. lnscriptions deposited at Quito, at Tarqui and at Cotchesqui, etc. BACK

5. See le rapport des quatre Commissaires, inféré dans les Mémoires dê l'Académie de l'année 1754, p. 178; and le Journal des opérations de M. le Monnier, imprimé au Louvre en 1757, page 8, line 11. BACK

6. See Mérid. vérif. page 36, and le rapport des Commissaires, page 21. BACK

7. See le rapport des Commissaires, imprimé in-8. reimprimé dans les Mémoires de l'Académie, année 1754, page 172. BACK

8. Mémoires de l'Académie, year 1751, page 433.BACK

9. Mémoires de l'Académie, year 1735, page 157.BACK

10. Jean Hellot and C. E. L. Camus. Sur l'etalon de l'aune au bureau des merchands merciers de la ville de Paris. Mémoires de l'Académie, year 1746, page 610. BACK

11. See Mémoires de l'Académie, year 1747, page 499.
Mémoires de l'Académie, year 1772. Part II. BACK

12. See the Mémoire on this subject, in the Recueil de l'Académie, year 1747. BACK

13. Observationes diam. Sol Lun. Lyon, published in 1670. BACK

Thanks to

for suggesting a sensible anglicization of quatre-de-circle.

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