TO DOCTOR ROBERT PATTERSON

      Dear Sir,—Your favor of September 23d came to hand in due time, and I thank you for the nautical almanac it covered for the year 1813. I learn with pleasure that the Philosophical Society has concluded to take into consideration the subject of a fixed standard of measures, weights and coins, and you ask my ideas on it; insulated as my situation is, I am sure I can offer nothing but what will occur to the committee engaged on it, with the advantage on their part of correction by an interchange of sentiments and observations among themselves. I will, however, hazard some general ideas because you desire it, and if a single one be useful, the labor will not be lost.

      The subject to be referred to as a standard, whether it be matter or motion, should be fixed by nature, invariable and accessible to all nations, independently of others, and with a convenience not disproportioned to its utility. What subject in nature fulfils best these conditions? What system shall we propose on this, embracing measures, weights and coins? and in what form shall we present it to the world? These are the questions before the committee.

      Some other subjects have, at different times, been proposed as standards, but two only have divided the opinions of men: first, a direct admeasurement of a line on the earth's surface, or second, a measure derived from its motion on its axis. To measure directly such a portion of the earth as would furnish an element of measure, which might be found again with certainty in all future times, would be too far beyond the competence of our means to be taken into consideration. I am free, at the same time, to say that if these were within our power in the most ample degree, this element would not meet my preference. The admeasurement would of course be of a portion of some great circle of the earth. If of the equator, the countries over which that passes, their character and remoteness, render the undertaking arduous, and we may say impracticable for most nations. If of some meridian, the varying measures of its degrees from the equator to the pole, require a mean to be sought, of which some aliquot part may furnish what is desired. For this purpose the 45th degree has been recurred to, and such a length of line on both sides of it terminating at each end in the ocean, as may furnish a satisfactory law for a deduction of the unmeasured part of the quadrant. The portion resorted to by the French philosophers, (and there is no other on the globe under circumstances equally satisfactory,) is the meridian passing through their country and a portion of Spain, from Dunkirk to Barcelona. The objections to such an admeasurement as an element of measure, are the labor, the time, the number of highly-qualified agents, and the great expense required. All this, too, is to be repeated whenever any accident shall have destroyed the standard derived from it, or impaired its dimensions. This portion of that particular meridian is accessible of right to no one nation on earth. France, indeed, availing herself of a moment of peculiar relation between Spain and herself, has executed such an admeasurement. But how would it be at this moment, as to either France or Spain? and how is it at all times as to other nations, in point either of right or of practice? Must these go through the same operation, or take their measures from the standard prepared by France? Neither case bears that character of independence which the problem requires, and which neither the equality nor convenience of nations can dispense with. How would it now be, were England the deposit of a standard for the world? At war with all the world, the standard would be inaccessible to all other nations. Against this, too, are the inaccuracies of admeasurements over hills and valleys, mountains and waters, inaccuracies often unobserved by the agent himself, and always unknown to the world. The various results of the different measures heretofore attempted, sufficiently prove the inadequacy of human means to make such an admeasurement with the exactness requisite.

      Let us now see under what circumstances the pendulum offers itself as an element of measure. The motion of the earth on its axis from noon to noon of a mean solar day, has been divided from time immemorial, and by very general consent, into 86,400 portions of time called seconds. The length of a pendulum vibrating in one of these portions, is determined by the laws of nature, is invariable under the same parallel, and accessible independently to all men. Like a degree of the meridian, indeed, it varies in its length from the equator to the pole, and like it, too, requires to be reduced to a mean. In seeking a mean in the first case, the 45th degree occurs with unrivalled preferences. It is the mid-way of the celestial ark from the equator to the pole. It is a mean between the two extreme degrees of the terrestrial ark, or between any two equi-distant from it, and it is also a mean value of all its degrees. In like manner, when seeking a mean for the pendulum, the same 45th degree offers itself on the same grounds, its increments being governed by the same laws which determine those of the different degrees of the meridian.

      In a pendulum loaded with a Bob, some difficulty occurs in finding the centre of oscillation; and consequently the distance between that and the point of suspension. To lessen this, it has been proposed to substitute for the pendulum, a cylindrical rod of small diameter, in which the displacement of the centre of oscillation would be lessened. It has also been proposed to prolong the suspending wire of the pendulum below the Bob, until their centres of oscillation shall coincide. But these propositions not appearing to have received general approbation, we recur to the pendulum, suspended and charged as has been usual. And the rather as the laws which determine the centre of oscillation leave no room for error in finding it, other than that minimum in practice to which all operations are subject in their execution. The other sources of inaccuracy in the length of the pendulum need not be mentioned, because easily guarded against. But the great and decisive superiority of the pendulum, as a standard of measure, is in its accessibility to all men, at all times and in all places. To obtain the second pendulum for 45° it is not necessary to go actually to that latitude. Having ascertained its length in our own parallel, both theory and observation give us a law for ascertaining the difference between that and the pendulum of any other. To make a new measure therefore, or verify an old one, nothing is necessary in any place but a well-regulated time-piece, or a good meridian, and such a knowledge of the subject as is common in all civilized nations.

      Those indeed who have preferred the other element, do justice to the certainty, as well as superior facilities of the pendulum, by proposing to recur to one of the length of their standard, and to ascertain its number of vibrations in a day. These being once known, if any accident impair their standard it is to be recovered by means of a pendulum which shall make the requisite number of vibrations in a day. And among the several commissions established by the Academy of Sciences for the execution of the several branches of their work on measures and weights, that respecting the pendulum was assigned to Messrs. Borda, Coulomb & Cassini, the result of whose labors, however, I have not learned.

      Let our unit of measures then be a pendulum of such length as in the latitude of 45°, in the level of the ocean, and in a given temperature, shall perform its vibrations, in small and equal arcs, in one second of mean time.

      What ratio shall we adopt for the parts and multiples of this unit? The decimal without a doubt. Our arithmetic being founded in a decimal numeration, the same numeration in a system of measures, weights and coins, tallies at once with that. On this question, I believe, there has been no difference of opinion.

      In measures of length, then, the pendulum is our unit. It is a little more than our yard, and less than the ell. Its tenth or dime, will not be quite 4 inches. Its hundredth, or cent, not quite .4 of an inch; its thousandth, or mill, not quite .04 of an inch, and so on. The traveller will count his road by a longer measure. 1,000 units, or a kiliad, will not be quite two-thirds of our present mile, and more nearly a thousand paces than that.

      For measures of surface, the square unit, equal to about ten square feet, or one-ninth more than a square yard, will be generally convenient. But for those of lands a larger measure will be wanted. A kiliad would be not quite a rood, or quarter of an acre; a myriad