We are now to proceed to give some account of the double measurement with the chain and glass rods; wherein it must be remembered, as also in continuing the operation with the glass rods alone, that in referring to the map for the daily progress in the work, we are going from the forty-sixth towards the first station; and in having recourse to the general table of the base, for altitude, temperature, or correction for expansion, we are ascending from the bottom towards the top, contrarily to the order in which the operation with the deal rods was conducted. On the morning of the 18th of August, the stands with the various parts of the apparatus being placed in the manner just now described, the operation was begun by bringing the first end of the chain to coincide with the intersection on the tripod, answering to the end of the 1370th deal rod, and 4.31 feet distant from the centre of the pipe terminating the north-west extremity of the base. The chain being stretched along its five coffers by the fourteen pounds weight suspended over the pulley at the farther end, and the temperatures of the five thermometers being registered in a book kept for that purpose, a fine trace was made on a piece of card, fastened under the feather-edge at the farther handle, denoting the end of the first hundred feet. The chain being then moved on into the next five coffers, those that had been thus vacated were carried forward, to prepare for the third chain length, and thereby permit the first set of stands to be elevated for the reception of the glass rods; and so in succession with the others. In this manner we proceeded, and in the course of the day were only able to measure the length of ten chains, or 1000 feet, being the forty-sixth and forty-fifth hypothenuses of the base, the first of 400 and the last of 600 feet. Being arrived at this point, it was found, that the fine line on the brass slide, marking the extremity of the tenth chain, fell short of another fine line on the same slide, denoting the end of the fiftieth glass rod, just two-tenths of an inch. Now it will appear hereafter, when we come to shew, by the experiments with the pyrometer, what the real contractions of the chain and glass rods were, for the degrees of difference of temperature* below that in which their respective lengths were laid off, that this small apparent difference of two-tenths of an inch, between the two modes of measuring the thousand feet, should have been 0.17938 in. to have made the two results exactly agree, which is a real difference of only 0.02062 of an inch. Supposing then every thousand feet of the base to have been measured by the chain with the same attention, and consequently with the same, or nearly the same success (and there surely cannot be any reason to doubt of the practicability), we shall have 27.404 × 0.02062 in. = 0.565 in. or a defect of something more than half an inch on the whole length of the base. When the length of the chain was laid off, the heat was 66°, and that of the glass rods 68°. They will, therefore, only agree with each other accurately in these respective temperatures. The mean of twenty thermometers for the four chain lengths of the fortysixth hypothenuse gave a heat of 61°.6; and for the six chain lengths of the forty-fifth, the mean of thirty thermometers gave 59°.75. The temperature of the 400 feet of glass, by the mean of forty thermometers, was 65°.3; and of the 600 feet, by the mean of sixty thermometers, it was 60°.8. Now, from these data, and the expansions of steel and glass, as determined by the pyrometer, the computation will stand as follows: Steel S 400 { Glass{ In. 66.5-61.6 = 4.9 × 0.03052 = 0.14955}=0.45856 400 68.0-65.3 2.7 x 0.02068 0.05584 600 68.0-60.8 7.2 × 0.03102 = 0.22334 }=0.27918{ The 1000 feet of steel should have contracted more than the 1000 feet of glass, But the difference was found to be Therefore the error of the chain in defect was in. or little more than half an inch on the whole base. }=0.17938 0.20000 contract. of 1000 feet. contract. of 1000 feet. 0.02062 x 27.4040.565 So nice an agreement between two results, with instruments so very different, could not fail to be considered as astonishing; and as it rarely happens, that the graduation of thermometers will so nearly correspond with each other, as not to occasion a much greater error, all were very desirous that it could have been farther confirmed by continuing the operation in the same way through a more considerable proportion of the whole length. But besides the tedious nature of the double measurement, owing to the multiplicity of stands, platforins, coffers, and other articles, that were now successively to be moved forward, and for which purpose it had been found necessary to reinforce the party of soldiers with six additional men; the operation had already trained out to a much more considerable length than had been expected; the summer was now far advanced, and the continuance of good weather uncertain; the coffers likewise for the chain, having been constructed in a hurry, were found to be defective in short, all these reasons contributed to induce us to give up, for the present, any farther experiment with the chain, and to proceed with the glass rods alone in the completion of the measure ment. Accordingly, on Thursday the 19th of August, the operation with the glass rods was continued for the five hypothenuses, from the forty-fourth to the fortieth inclusive. It will be remembered, that in proceeding with the deal rods, double pickets had been placed in the ground, at the middle of the forty-first hypothenuse, or that point which terminated the 1215th rod, reckoning from the south-east, or the 155th from the north-west end of the base. Now, in returning to this point with the glass rods, the extremity of the 155th fell short of the silk thread stretched from picket to picket, just one-tenth of an inch. The expansion of the brass standard scale, and that of glass being taken into the account, it appears, that the small expansion* of the deal rods from the humidity of the air, must, at this point, have exceeded what it was estimated at in the general table by 0.931 of an inch, supposing no error of any kind whatever to have arisen in the execution, from bringing the rods into contact, or otherwise. On Saturday the 21st of August, the measurement was resumed at the thirty-ninth station, and continued for five hypothenuses to the thirty-fifth inclusive. This day, about noon, his Majesty deigned to honour the operation by his presence, for the space of two hours, entering very minutely into the mode of conducting it, which met with his gracious approbation. On Monday the 23d, the mensuration was farther continued for five hypothenuses, that is, to the thirtieth inclusive. On Tuesday the 24th, we proceeded with the measurement for the space of seven hypothenuses, finishing the business of the day at the twenty-second station. It will be remembered, that in carrying on the operation with the deal rods, double pickets were left in the ground at the twentyseventh station, answering to the extremity of the 810th rod from the first, or the 560th from the last end of the base. on arrival at this point, the 560th glass rod overshot the silk thread, 155 deal rods =3100 feet 155 glass rods =3100 feet In. Now, +0.383 for 1° excess of temperature of the brass scale from 62° to 63°. +0.651 proportionable part of the estimated expansion from humidity. +1.034 equation of the deal rods on 3100 feet. +2.301 for 6° excess of the heat of the brass scale from 62° to 68°. -0.436 observed contraction of the glass from the 11th and 12th columns of the table. +0.100 by which the 155th rod fell short of the thread. +1.965 equation of the glass rods on 3100 feet. Difference of the two equations, under-rated in the ex0.931 pansion of the deal rods. K stretched from one picket to the other, 2.525 inches. Here again we find, that the lengthening* of the deal rods, from the moisture of the atmosphere, differs but little from what it has been estimated at by comparison with the standard, being over-rated only two-tenths of an inch on the 560 rods. In this day's operation, in passing the bridge laid over the old river, the measurement, instead of being made in the hypothenusal, was carried on in the level line, for the space of twenty rods, namely, fifteen rods of the twenty-seventh, and five of the twenty-sixth hypothenuse; which occasions the alteration in the reduction of these two spaces, marked with asterisks in the general table. As some trouble had been found to attend the crossing of the great road, in the first measurement, owing to the number of carriages that were continually passing, the depth of the ditches, and height of the banks of the old Roman way; therefore tressels, suited for the purpose, had been now prepared and lest any accident might have happened in conducting this part of the operation, so as to oblige us to a repetition, double pickets were placed in the usual manner in the ground, two rod lengths from the twenty-sixth station, to which we could have referred, without going back as far as the tripod left at the twenty-ninth station, the point from which we had departed in the morning. Bad weather prevented any progress being made on the 25th; In. 560 deal rods=+1.390 for 1° excess of heat of the brass scale from 62° to 63°. +6.648 equation of the 560 deal rods. +8.343 for 6° excess of heat of the brass scale from 62° to 68°. +1.821 observed expansion of glass 560 glass rods= 11200 feet from columns 11th -1.191 observed contraction of ditto and 12th. -2.525 over-shot the silk-thread. +6.448 equation of 560 glass rods. 0.200 difference over-rated in the expansion of the 560 deal rods. |