Enclosure
Report to the Academy of Sciences
on a Unit of Measure

The idea of founding the whole Science of measure upon an unit of length taken from nature, presented itself to mathematicians from the moment that they knew the existence of such an unit, and the possibility of determining it: they saw that this was the only means of excluding every thing arbitrary from the system of measures, and of being sure of preserving it at all times the same, without it’s being subjected to uncertainty from any event short of a revolution in the present system of nature. They judged also that such a system not belonging exclusively to any particular nation, might eventually be adopted by all. Indeed were we to adopt for an unit the measure of any particular country, it would be difficult to assign to the others reasons of preference sufficient to over-weigh that degree of repugnance not philosophical, perhaps, but at least very natural, which all nations have against an imitation which implies an acknowledgment of a certain inferiority. There would then be as many measures as there are great nations. Besides, were nearly the whole of them to adopt any one of these arbitrary standards, numberless accidents easy to be foreseen would produce doubts of the true admeasurement of the standard. And as there would be no exact means of testing it, there would, in time, be a disagreement in these measures. The diversity which exists at this day among those in use in different Countries is not so much owing to an original diversity existing at the period of their first establishment, as to changes occasioned in process of time. In short, little would be gained even in a single nation, by preserving any one of the units of length therein used, since it would be still necessary to correct the other imperfections of the system of measures, an operation which would be nearly as inconvenient for the greater part of them.

The units which appear most proper for serving as the basis of measure, may be reduced to three, the length of the pendulum; a quadrant of the Equatorial circle; and lastly a quadrant of a terrestrial meridian.

The length of the pendulum has appeared in general to merit preference, as it possesses the advantage of being more easily determined, and consequently more easily tested or verified, whenever accidents to the Standard should render it necessary. Add to this, that those who should adopt this measure already in use with another nation, or, who after having adopted it should have occasion to verify it, would not be obliged to send observers to the place where the operation would have been first made.

In fact the law of the lengths of the pendulum is sufficiently certain and sufficiently confirmed by experience to be employed in operations without danger of sensible error; and even if we were to disregard this law, still a comparison of the difference in the lengths of pendulums, once executed could always be verified, and thus the unit of measure become invariable for all the places where the comparison should have been made; and thus we might immediately rectify any accidental change in the Standards, or find the same unit of measure at any moment when we should resolve to adopt it. But we shall see hereafter that this last mentioned advantage can be made common to all natural measures, and that we may employ the observations on the pendulum to verify them, although such observations have not served as the basis for their determination.

In employing the length of the Pendulum it seems natural to prefer that of the simple pendulum, which vibrates seconds at the forty fifth degree of Latitude. The law of the lengths of simple pendulums from the Equator to the Pole, performing equal vibrations, is such that that of the pendulum at the forty fifth degree is precisely the mean of all these lengths, that is to say, it is equal to their sum divided by their number: it is equally a mean between the two extremes taken the one at the pole, the other at the equator, and between any two lengths whatever corresponding at equal distances, the one to the north, the other to the south of the same parallel. It would not then be the length of the pendulum under any determinate parallel which would in this case be the Unit of measure, but the mean length of the pendulums, unequal among themselves, which vibrate seconds in the different Latitudes.

We must observe, however, than an unit of measure thus determined has something arbitrary in it: the second of time is the eighty six thousand four hundredth part of a day, and consequently an arbitrary division of this natural unit; so that to fix the unit of length we not only employ a heterogeneous element (time) but an arbitrary portion of it.

We might, indeed, avoid this latter inconvenience by taking for unit the hypothetical pendulum which should make but a single vibration in a day, a length which divided into ten thousand millions of parts would give an unit for common measure, of about twenty seven inches; and this unit would correspond with a pendulum which should make one hundred thousand vibrations in a day: but still the inconvenience would remain of admitting a heterogeneous element, and of employing time to determine an unit of length, or which is the same in this case, the intensity of the force of gravity at the surface of the earth.

Now if it be possible to have an unit of length, which depends on no other quantity, it would seem natural to give it the preference. Besides, an unit of measure taken from the earth itself has the advantage of being perfectly analogous to all the actual measures which for the common uses of life we take on the earth, such as the distances between certain points of it’s surface, or the extent of portions of that surface. It is much more natural in fact to refer the distance of one place from another to a quarter of a circle of the earth, than to refer it to the length of the pendulum.

We have thought it our duty then to decide in favour of this species of unit, and then again to prefer a quarter of a meridian to a quarter of the equator. The operations necessary to determine this last element cannot be executed but in countries so distant from us that they occasion expenses and difficulties far above the advantages which could be hoped from them; the verifications of them, whenever they should be deemed necessary would be more difficult for all nations, at least till the time when the progress of civilization shall reach the inhabitants under the equator, a time unfortunately very distant from us. The regularity of this circle is not more certain than the similitude, or the regularity of the meridians.

The length of the celestial arc corresponding with the space measured, is less susceptible of being determined with precision: in short we may say that every nation belongs to one of the meridians of the earth, but that only part of them are situated under the equator.

The quarter of a meridian of the earth then would be a real unit of measure, and the ten millionth part of it would be the unit for common use. It will be seen here that we renounce the ordinary division of the quarter of the meridian into ninety degrees, of the degree into minutes, of the minute into seconds; because we could not retain this ancient division without breaking in on the unity of the system of measures, since the decimal division answering to the arithmetical scale ought to be preferred for the measures in common use, and thus we should have, for those of length only, two systems of division, the one of which would be adapted to the great measures, and the other to the small ones. For example, the league could not be at the same time a simple division of a degree, and a multiple of the toise in round numbers. The inconveniencies of this double system would be perpetual, whereas those of changing it would be temporary, and they would fall principally on a small number of persons accustomed to calculation; and we have imagined that the perfection of the operation ought not to be sacrificed to an interest which in many respects we may consider as personal.

In adopting these principles we introduce nothing arbitrary into the system of measures, except the arithmetical scale by which their divisions must necessarily be regulated. Also in that of weights, there will be nothing arbitrary but the choice of a substance homogeneous and easy to be always obtained in the same degree of purity and density to which we must refer the weight of all other substances; as for instance, if we should for a basis chuse distilled water weighed in vacuo, or reduced to the weight it would be of in vacuo, and taken at the degree of temperature at which it passes from a solid to a fluid form. To this same degree of temperature all the real measures employed in the operations should have relation; so that in the whole system there would be nothing arbitrary but that which is so of necessity and from the nature of things. And even the choice of this substance and of this degree of temperature is founded on physical reason, and the retaining the arithmetical scale, is prescribed by a fear of the danger to which this change in addition to all the others, would expose the success of the whole operation.

The immediate mensuration of a quarter of a meridian of the earth would be impracticable, but we may obtain a determination of it’s length by measuring an arc of a certain length, and inferring from thence the length of the whole, either directly, or by deducing from this mensuration the length of an arc of the meridian corresponding to the hundredth part of the celestial arc of ninety degrees, and so taken as that one half of this arc should be to the south and the other to the north of the forty fifth parallel. In fact, as this arc is the mean of those which from the equator to the pole answer to equal parts of the celestial arc, or which is the same thing, to equal distances of latitude, by multiplying this measure by a hundred, we shall find the length of the quarter of the meridian.

The increase of length in these terrestrial arcs follow the same law as those of the pendulum, and the arc which answers to this parallel is a mean of all the others, in like manner as the pendulum of the forty fifth degree is the mean of all the other pendulums.

It may be objected here that the law of the increments of length of the degrees from the equator towards the pole is not so well ascertained as that of the increments of the pendulum; although both are founded on the same hypothesis of the ellipticity of the meridians. It might be said that it has not been equally confirmed by observations; but 1st. there exists no other method of finding the length of a quarter of a circle of the earth. Secondly, there results from it no real inexactitude, since we have the immediate length of the arc measured with which that deduced from it will always have a known relation. Thirdly, the error which might be committed in determining the hundredth part of a quadrant of the meridian, would not be sensible. The hypothesis of the ellipticity cannot be far from reality in the arc whose length should be actually measured: it will represent necessarily, with sufficient exactness, the small portion of the curve almost circular and a little flatened, which forms this arc. And fourthly, if this error could be sensible, it might of necessary consequence be corrected by the same observations. There could be no error but such as could not be appreciated by observations.

The larger the measured arc is, the more exact would be the determinations resulting therefrom. In fact the errors committed in the determination of the celestial arc, or even in the terrestrial measurements and that of the hypothesis will have the less sensible influence on the results in proportion as this arc is of greater extent. In fine there is an advantage in the circumstance that the two extremities happen to be, the one to the South, the other to the North of the parallel of forty five degrees, at distances, which without being equal, are not too disproportionate.

We will propose then an actual mensuration of an arc of the meridian from Dunkirk to Barcelona, comprising alike1 more than nine degrees and an half: this arc will be of quite sufficient extent, and there will be about six degrees of it to the north, and three and a half to the south of the mean parallel. To those advantages is added that of having it’s two extreme points equally in the same level of the sea: it is to satisfy this last condition which gives points in the same level, invariable and determined by nature, to increase the extent of the arc to be measured in order that it may be divided in a manner more equal; in fine to extend it beyond the Pyrennees, and free it from any inaccuracies which their effect on the instruments might produce, that we propose to prolong the measure to Barcelona. Neither in Europe nor in any other part of the world (without measuring an arc of much greater extent) can a portion of meridian be found which will satisfy the condition of having it’s two extreme points in the level of the sea, and at the same time that of traversing the forty fifth parallel, unless it be the one now proposed, or another westward of this and extending from the coast of France to that of Spain. This last arc would be more equally divided by the parallel of forty five degrees;2 but we have preferred that which extends from Barcelona to Dunkirk, because it is in the track of the meridian already traced in France, that there exists already an admeasurement of this arc from Dunkirk to Perpignan, and that it is some advantage to find in the work already done a verification of that now to be executed. In fact, if in the new operations we find in the distance from Perpignan to Dunkirk the same result in all it’s parts, we shall have a reason the more for counting on the certainty of these operations. Should there be found any variations, by examining what are the causes, and where the error is, we shall be sure of discovering those causes and of correcting the error. Besides in following this direction we cross the Pyrennees where they are more passable.

The operations necessary for this work will be first, to determine the difference of Latitude between Dunkirk and Barcelona, and in general to make on that line all the astronomical observations which shall be deemed useful. 2d. to measure the old bases which were used for the measure of a degree made at Paris, and for the purpose of the map of France. 3d. to verify by new observations the suite of the triangles which were used for measuring the meridian, and to extend them to Barcelona. 4th. to make at the forty fifth degree observations for determining the number of vibrations which a simple pendulum equal to the ten millionth part of the arc of the meridian will make in a day in vacuo at the sea-side, and in the temperature of ice beginning to melt, in order that, this number being once known, the measure may be found again at any time by observations on the pendulum. By these means we unite the advantages of the system which we have preferred, and of that which takes for it’s unit the length of the pendulum. These observations may be made before this ten millionth part is known: having in fact the number of vibrations of a pendulum of a given length, it will suffice to know afterwards the proportion of this length to this ten millionth part in order to deduce from thence with certainty the number required. 5th. to verify by new experiments, carefully made, the weight in vacuo of a given quantity of distilled water taken at the freezing point. And lastly, to reduce to the present measures of length the different longitudinal, superficial and solid measures used in commerce, and the different weights in use, to the end that we may be able afterwards, by the simple rule of three, to estimate the new measures when they shall be determined.

We see that these different operations require six separate commissions, each charged with one of these portions of the work. Those to whom the academy shall trust the work should be required at the same time, to explain to them the method which they propose to follow.

In this first Report we have confined ourselves to what relates to the unit of measure. We propose in another to present the plan of the general System to be established upon this unit. In fact, this first determination requires preliminary operations which will take time, and which should be previously ordered by the National Assembly. We have nevertheless sufficiently meditated on this plan and the results of the operations, as well for the measure of the arc of the meridian, as for the weight of a given quantity of water, are known so nearly, that we may assert at present, that in adopting the unit of measure which we have proposed, a general system may be formed, in which all the divisions may follow the arithmetical Scale, and no part of it embarras our habitual usages: we shall only say at present that this ten millionth part of a quadrant of the meridian which will constitute our common unit of measure will not differ from the simple pendulum but about a hundred and forty fifth part; and that thus the one and the other unit leads to systems of measure absolutely similar in their consequences.

We have not thought it necessary to wait for the concurrence of other nations either in deciding upon the choice of the unit of measure or in beginning the operations. In fact we have excluded from this choice every arbitrary determination: we have admitted no elements but those which belong equally to all nations. The choice of the forty fifth degree of Latitude was not determined by the position of France, it is not here considered as a fixed point of the meridian, but only as that to which the mean length of the pendulum, and the mean length of a given division of that circle correspond: in fine we have chosen the only meridian wherein an arc can be found terminating at both it’s extremities at the level of the ocean, and cut by the mean parallel without being of too great extent, which would render it’s actual mensuration too difficult. There is nothing here then which may give the smallest pretence for reproaching us with an affectation of pre-eminence.

We conclude therefore to present this Report to the National Assembly, praying it to order the proposed operations and the measures necessary for the execution of those which must be made on the territories of Spain.

Done at the Academy the 19th. of March 1791.