TO CAPT. A. PARTRIDGE.

Monticello, January 2d, 1816.

Sir,—I am but recently returned from my journey to the neighborhood of the Peaks of Otter, and find here your favors of November 23d and December 9th. I have therefore to thank you for your meteorological table and the corrections of Colonel Williams' altitudes of the mountains of Virginia, which I had not before seen; but especially for the very able extract on Barometrical measures. The precision of the calculations, and soundness of the principles on which they are founded, furnish, I am satisfied, a great approximation towards truth, and raise that method of estimating heights to a considerable degree of rivalship with the trigonometrical. The last is not without some sources of inaccuracy, as you have truly stated. The admeasurement of the base is liable to errors which can be rendered insensible only by such degrees of care as have been exhibited by the mathematicians who have been employed in measuring degrees on the surface of the earth. The measure of the angles by the wonderful perfection to which the graduation of instruments has been brought by a Bird, a Ramsden, a Troughton, removes nearly all distrust from that operation; and we may add that the effect of refraction, rarely worth notice in short distances, admits of correction by well-established laws; these sources of error once reduced to be insensible, their geometrical employment is certainty itself. No two men can differ on a principle of trigonometry. Not so as to the theories of Barometrical mensuration. On these have been great differences of opinion, and among characters of just celebrity.

Dr. Halley reckoned one-tenth inch of Mercury equal to 90 feet altitude of the atmosphere. Derham thought it equal to something less than 90 feet. Cassini's tables to 24° of the Barometer allowed 676 toises of altitudes.

Mariole's,to the same544toises.
Schruchzer's "559"

Nettleton's tables applied to a difference of .5975 of mercury, in a particular instance gave 512.17 feet of altitude, and Bonguor's and De Luc's rules, to the same difference gave 579.5 feet. Sir Isaac Newton had established that at heights in arithmetrical progression the ratio of rarity in the air would be geometrical, and this being the character of the natural numbers and their Logarithms, Bonguor adopted the ratio in his mensuration of the mountains of South America, and stating in French lignes the height of the mercury of different stations, took their Logarithms to five places only, including the index, and considered the resulting difference as expressing that of the altitudes in French toises. He then applied corrections required by the effect of the temperature of the moment on the air and mercury. His process, on the whole, agrees very exactly with that established in your excellent extract. In 1776 I observed the height of the mercury at the base and summit of the mountain I live on, and by Nettleton's tables, estimated the height at 512.17 feet, and called it about 500 feet in the Notes on Virginia. But calculating it since on the same observations, according to Bonguor's method with De Luc's improvements, the result was 579.5 feet; and lately I measured the same height trigonometrically, with the aid of a base of 1,175 feet in a vertical plane with the summit, and at the distance of about 1,500 yards from the axis of the mountain, and made it 599.35 feet. I consider this as testing the advance of the barometrical process towards truth by the adoption of the Logarithmic ratio of heights and densities; and continued observations and experiments will continue to advance it still more. But the first character of a common measure of things being that of invariability, I can never suppose that a substance so heterogeneous and variable as the atmospheric fluid, changing daily and hourly its weight and dimensions to the amount, sometimes, of one-tenth of the whole, can be applied as a standard of measure to anything, with as much mathematical exactness, as a trigonometrical process. It is still, however, a resource of great value for these purposes, because its use is so easy, in comparison with the other, and especially where the grounds are unfavorable for a base; and its results are so near the truth as to answer all the common purposes of information. Indeed, I should in all cases prefer the use of both, to warn us against gross error, and to put us, when that is suspected, on a repetition of our process. When lately measuring trigonometrically the height of the Peaks of Otter (as my letter of October 12th informed you I was about to do), I very much wished for a barometer, to try the height of that also. But it was too far and hazardous to carry my own, and there was not one in that neighborhood. On the subject of that admeasurement, I must premise that my object was only to gratify a common curiosity as to the height of those mountains, which we deem our highest, and to furnish an à peu près, sufficient to satisfy us in a comparison of them with the other mountains of our own, or of other countries. I therefore neither provided such instruments, nor aimed at such extraordinary accuracy in the measures of my base, as abler operators would have employed in the more important object of measuring a degree, or of ascertaining the relative position of different places for astronomical or geographical purposes. My instrument was a theodolite by Ramsden, whose horizontal and vertical circles were of 3½ inches radius, its graduation subdivided by noniuses to one-third, admitting however by its intervals, a further subdivision by the eye to a single minute, with two telescopes, the one fixed, the other movable, and a Gunter's chain of four poles, accurately adjusted in its length, and carefully attended on its application to the base line. The Sharp, or southern peak, was first measured by a base of 2806.32 feet in the vertical plane of the axis of the mountain. A base then nearly parallel with the two mountains of 6,589 feet was measured, and observations taken at each end, of the altitudes and horizontal angles of each apex, and such other auxiliary observations made as to the stations, inclination of the base, &c., as a good degree of correctness in the result would require. The ground of our bases was favorable, being an open plain of close grazed meadow on both sides of the Otter river, declining so uniformly with the descent of the river as to give no other trouble than an observation of its angle of inclination, in order to reduce the base to the plane of the horizon. From the summit of the Sharp peak I took also the angle of altitude of the flat or northern one above it, my other observations sufficing to give their distance from one another. The result was,

The mean height of the Sharp peak above the surface of Otter river2946.5 inches.
Mean height of the flat peak above the surface of Otter river3103.5 inches.
The distance between the two summits 9507.73 inches.

Their rhumb N. 33° 50´ E. the distance of the stations of observation from the points in the bases of the mountains vertically under their summits was, the shortest 19002.2 feet, the longest 24523.3 feet. These mountains are computed to be visible to fifteen counties of the State, without the advantage of counter-elevations, and to several more with that advantage. I must add that I have gone over my calculations but once, and nothing is more possible than the mistake of a figure, now and then, in calculating so many triangles, which may occasion some variation in the result. I mean, therefore, when I have leisure, to go again over the whole. The ridge of mountains of which Monticello is one, is generally low; there is one in it, however, called Peter's mountain, considerably higher than the general ridge. This being within a dozen miles of me, north-eastwardly, I think in the spring of the year to measure it by both processes, which may serve as another trial of the Logarithmic theory. Should I do this you shall know the result. In the meantime accept assurances of my great respect and esteem.