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by the efforts of the two men at the other end, in stretching it to its true position, for the front arrow.

On Monday, the 21st of June, the operations were resumed, by measuring twice with the chain (forwards and again backwards) the thirteen hypothenuses comprehended in the second section of the base, between Hanworth Summerhouse and the north-west bank of the great Road (an old Roman way) leading from Staines to London. This being the smoothest part of the Heath, and the holdfast being now applied, the two measurements differed only one inch and a half in the distance of 7800 feet. This instance of accuracy is alone sufficient to prove the great excellence of the chain, although another will be given hereafter still more surprising.

On the same day that the second section of the base was measured, the levels of that and the first were taken. The operation of levelling is so universally known, as to render any detail of it unnecessary. It will be sufficient to say, that the spirit-level made use of on this occasion was a very good one, about eighteen inches in length, and could at all times be very readily and accurately adjusted, by inversion in its Y's. The tops of the pickets, marking the hypothenusal distances, were the points on which the levelling rods were placed on each side of the level; which being inverted at the intermediate picket, points equidistant from the centre of the earth were thereby obtained at the cross vanes of the levelling rods, and no correction for curvature or refraction necessary. It will be readily understood, that the relative heights of the pickets were found by measuring their distances from the centres of the cross vanes and axis of the telescope respectively.

The six first columns towards the left hand of the first or general table subjoined to this Paper, shew distinctly every thing relating to the levels of the whole base, those of the third section having been determined on the 22d of June. By examining the table it

will be seen, that the ascent on the first section is

on the second

and on the third

10.555 feet, 8.580

12.130

Total

31.265 feet,

between the lower extremity at Hampton Poorhouse, and the higher near King's Arbour.

The computed numbers in the seventh column are the reductions* depending on the aforesaid heights, or the differences between the hypothenusal distances of 600 feet each and the reduced base distances. With regard to the remaining columns of the table, or those towards the right hand, they will be severally spoken to hereafter, in taking into consideration the expansion of metals, as determined with great accuracy by the experiments with the py

rometer.

Hitherto no use had been made of the transit instrument: for, in order that it might be applied to advantage, there was a necessity for laying the wheel into the ground at the lower end of the

E

* The reduction in the seventh column, I have computed by the difference between the square of the hypothenuse, actually measured, and the square of the height found by the level; and Lieut. Col. Calderwood has done the same thing by a much shorter method. Thus, in the annexed figure, CE being the hypothenuse of 600 feet, DE the perpendicular height obtained by levelling, DB the reduction required, or the difference between the hypothenuse and true base; then, substituting the chord BE instead of DE, the following analogy is A obtained; AB BE:: BE: DB; consequently, = DB: that is, the

BE2
AB

C

square

DB

of

the perpendicular height being divided by double the distance, or 1200 feet, the quotient is equal to DB the reduction, without sensible error. For if DE were four feet, the greatest perpendicular height in the base, BE the chord would only exceed it zooooo, which would not be more than ' part of an inch. The difference between the results, by the two modes of computation, is so trifling as not to deserve notice.

I

F

base, and so to modify the St. George's flag-staff that, being placed in the pipe, it might be steadily supported by braces in a true vertical position; which we found, from experience, could not be effected by ropes only.

The wheel being accordingly laid in its place, and the other precautions taken for securing the flag-staff, which was likewise painted white, that it might be more distinctly seen from the farther extremity; on the 22d of June, the transit instrument was adjusted over the thirteenth picket at Hanworth Summerhouse, while directed upon the flag-staff. But it being now found, that the vertical plane passing through the flag-staff fell to the eastward of the centre of Bansted Spire, therefore the transit was gradually moved to the eastward, until by repeated trials the three points were perceived to be in the same vertical plane, when the picket was moved, and replaced exactly under the axis of the telescope, a few inches from its first position. The same operation was repeated at the twentysixth station, on the farther bank of the Staines Road; and, lastly, at the forty-sixth, forming the north-west extremity of the base; where a pit was immediately dug for the wheel, which was placed therein, without however filling in the earth for the present, that being deferred till near the completion of the measurement with the deal rods. Thus the two extremities, and two intermediate points of the base, being accurately placed, by the help of the transit instrument, in the same vertical plane with Bansted Spire, it was easily seen, that by arranging camp colours in the intervals at any time, all the other points might be brought so nearly to coincide with these first, as not to occasion, by deviation, any sensible error in the measurement afterwards to be made. This application of the transit shewed us, however, that some labour had been lost by not using it sooner: for at the Staines Road, the tract cleared by the soldiers deviated about two feet and a half too much to the westward for the true line; and at King's Arbour it was twice as

much so that we were now obliged to widen the cleared tract, by adding to the eastern side of it.

On the same day that the chief points in the base were fixed by means of the transit, and the levels of the third section taken as before mentioned, the rough measurement of that section with the chain was completed, and found to contain nineteen hypothenusal distances of 600 feet each, and one of 404.55, making in the whole 11804.55 feet, between the twenty-sixth station at the Staines Road and the centre of the pipe near King's Arbour, the mean temperature being 62°. Here it is to be observed, that this last section was only measured once with the chain, the tract not being yet sufficiently cleared to admit of its being done to the best advantage; and, when completed, it was judged to be better to proceed directly in the operation with the rods, than to lose time in the usual repetition, since the merits of the chain, in this way of applying it, were already sufficiently well established; and any future tests to which it was to be put were proposed to be of a more rigid nature.

When the length of the chain, in its original state, was ascertained by the dots on the brass pins in the New England plank, it was found, in the then temperature of 74°, to exceed the 100 feet by near one quarter of an inch, or 0.245 inch. Therefore, in the temperature of 63°, being that in which the lengths of the deal rods were laid off, and differing very little from what was likewise the meant heat of the air, when applied upon the Heath, the chain, according to the experiments on the expansion of the very same steel, would exceed the 100 feet by 0.161 inch, or 0.0134 foot. Hence the sum of the three sections of the base, 274 chains, being multiplied by 0.0134 foot, we shall have 3.67 feet for the equation of the chain + 4.55 feet, to be added to its length, which will then become 27408.22 feet from the centre of one pipe to the centre of the other and this would have been the true length of the base, as given by the rough measurement with the chain, if the surface had

been one uniform inclined plane throughout its whole extent. But, although the ascent of Hounslow Heath is so small, and so gradual, as to occasion little more than half an inch of reduction, from the 46 hypothenusal to the 46 base distances, into which it is divided, as may be seen by referring to the table; yet each of these hypothenuses containing again many other small irregularities, all of which affect the measurement by the chain, in proportion to their number and height, in every space of 600 feet, their united effects, including the lateral deviations from the true line in measuring, do somewhat more than compensate for the extra-length of the chain, as will be seen hereafter in comparing the length of the base just now obtained with that given by the rods.

The weather, which during the greater part of June had been wet, became still worse towards the end of the month and first week of July; so much so, that even if the deal rods had been ready, they could not have been used with advantage. The soldiers, nevertheless, were not idle, being, when the weather would permit, partly employed in clearing the Heath, and partly in assisting Mr. Reynolds in the survey, towards the perfecting of which many chief points were fixed by means of my astronomical quadrant, placed for that purpose at several different stations of the base. At this time too (July 8th) I levelled from the lower end of the base to the surface of the Thames at Hampton, and found the descent to be 36.1 feet.

Measurement of the Base with the Deal Rods. Pl. I. and III.

11. Such extraordinary care and pains had been bestowed in the construction of the deal rods, in order to render them the best which had ever been made, that, although begun early in June, they were not completely finished before the 15th of July. They were brought that afternoon by Mr. Ramsden, together with the

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