Appendix, containing illustrations and developments of some passages of the preceding report. (1.) In the second pendulum with a spherical bob, call the distance between the centres of suspension and of the bob, 2x 19.575, or 2d, and the radius of the bob-r; then 2d :r::r: and of this last proportional expresses the displacement of the centre of oscillation, to wit: Two inches have been 1 5x24 = ". 5d proposed as a proper diameter for such a bob. In that case r " will express the In the cylindrical second rod, call the length of the rod, 3x 19.575. or 3d, and its radius=r and displacement of the centre of oscillation. 1 2x d It is 6d thought the rod will be sufficiently inflexible if it be of an inch in diameter. Then will be=.1 inch, and inches, which is but the 120th part of the displacement in the case of the pendulum with a spherical bob, and but the 689,710th part of the whole length of the rod. If the rod be even of half an inch diameter, the displacement will be but of an inch, or of the length of the rod. 1 1879 110356 (2.) Sir Isaac Newton computes the pendulum for 45° to be 36 pouces 8.428 lignes. Picard made the English foot 11 pouces 2.6 lignes, and Dr. Maskelyne 11 pouces 3.11 lignes. D'Alembert states it at 11 pouces 3 lignes, which has been used in these calculations as a middle term, and gives us 36 pouces 8.428 lignes =39.1491 inches. This length for the pendulum of 45° had been adopted in this report before the Bishop of Autun's proposition was known here. He relies on Mairan's ratio for the length of the pendulum in the latitude of Paris, to wit: 504: 257::72 pouces to a 4th proportional, which will be 36.71428 pouces 39.1619 inches, the length of the pendulum for latitude 48° 50'. The difference between this and the pendulum for 45' is .0113 of an inch; so that the pendulum for 45° would be estimated, according to Mairan, at 39.1619-0113-39.1506 inches, almost precisely the same with Newton's computation herein adopted. (3.) Sir Isaac Newton's computations for the different degrees of latitude, from 30° to 45°, are as follows: (4.) Or, more exactly, 144: 175::224: 272.2. (5.) Or, more exactly, 62.5: 1728::77.7: 2150.39. (6.) The merchant's weight. (7.) The Eng. rood contains 10,890 sq. feet=104.355 feet sq. (8.) The Measures, Weights, and Coins of the Decimal System, estimated in those of England, now used in the United States. 1. MEASURES OF LENGTH. 10000. 10000. The hundredth, The tenth, The rood, The double acre, Equivalent in English measure. . .011 inch. . .117 1.174, about >. 11.744736 more than the Eng. inch. about less than the English foot. . 9.787, about less than the 10 feet rod 48 an English square rood. 978.728, about more than the Eng. fur. . 9787.28, about 14 English mile, nearly the Scotch and Irish mile, and the German mile. 2. SUPERFICIAL MEASURE. The square furlong, . 100. . 957.9 . 9579.085 . 2.199, or say 2.2 acres English. The bushel, 1. about The quarter, 16.2, about less than the English half-pint. . 162.005, about more than the Eng lish pottle. S 1620.05506862 .937531868414884352 cub feet. S less than the middle sized English bushel. . 10. . 9.375, about less that the Eng. qr. . 100.. 93.753, about 4. WEIGHTS. more than the Eng. last. It is scarcely necessary to observe that the measures, weights, and coins, proposed in the preceding report, will be derived altogether from mechanical operations, viz.: A rod, vibrating seconds, divided into five equal parts, one of these subdivided, and multiplied decimally, for every measure of length, surface, and capacity, and these last filled with water, to determine the weights and coins. The arithmetical estimates in the report were intended only to give an idea of what the new measures, weights, and coins, would be nearly, when compared with the old. The length of the standard or second rod, therefore, was assumed from that of the pendulum; and as there has been small differences in the estimates of the pendulum by different persons, that of Sir Isaac Newton was taken, the highest authority the world has yet known. But, if even he has erred, the measures, weights, and coins proposed, will not be an atom the more or less. In cubing the new foot, which was estimated at .978728 of an English foot, or 11.744736 English inches, an arithmetical error of an unit happened in the fourth column of decimals, and was repeated in another line in the sixth column, so as to make the result one ten thousandth and one millionth of a foot too much. The thousandth part of this error (about one ten millionth of a foot) consequently fell on the metre of measure, the ounce weight, and the unit of money. In the last it made a difference of about the twenty-fifth part of a grain Troy, in weight, or the ninety-third of a cent in value. As it happened, this error was on the favorable side, so that the detection of it approximates our estimate of the new unit exactly that much nearer to the old, and reduces the difference between them to 34, instead of 38 hundredths of a grain Troy; that is to say, the money unit instead of 375.64 Troy grains of pure silver, as established heretofore, will now be 375.98934306 grains, as far as our knowledge of the length of the second pendulum enables us to judge; and the current of authorities since Sir Isaac Newton's time, gives reason to believe that his estimate is more probably above than below the truth, consequently future corrections of it will bring the estimate of the new unit still nearer to the old. The numbers in which the arithmetical error before mentioned showed itself in the table, at the end of the report, have been rectified, and the table re-printed. The head of superficial measures in the last part of the report, it thought to be not sufficiently developed. It is proposed that the rood of land, being 100 feet square, (and nearly a quarter of the present acre,) shall be the unit of land measure. This will naturally be divided into tenths and hundredths, the latter of which will be a square decad. Its multiples will also, of course, be tens, which may be called double acres, and hundreds, which will be equal to a square furlong each. The surveyor's chain should be composed of 100 links of one foot each. VIII-Opinion upon the question whether the President should veto the Bill, declaring that the seat of government shall be transferred to the Potomac, in the year 1790. July 15, 1790. A bill having passed both houses of Congress, and being now before the President, declaring that the seat of the federal government shall be transferred to the Potomac in the year 1790, that the session of Congress next ensuing the present shall be held in Philadelphia, to which place the offices shall be transferred before the 1st of December next, a writer in a public paper of July 13, has urged on the consideration of the President, that the constitution has given to the two houses of Congress the exclusive right to adjourn themselves; that the will of |