A gradual introduction would lessen the inconveniences which might attend too sudden a substitution, even of an easier for a more difficult system. After a given term, for instance, it might begin in the custom-houses, where the merchants would become familiarized to it. After a further term, it might be introduced into all legal proceedings, and merchants and traders in foreign commodities might be required to use it in their dealings with one another. After a still further term, all other descriptions of people might receive it into common use. Too long a postponement, on the other hand, would increase the difficulties of its reception with the increase of our population.
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, 2x19.575, or 2d, and the radius of the bob = r; then 2d:r::r: rr 2d and ⅖ of this last proportional expresses the displacement of the centre of oscillation, to wit: 2rr 5x2d = rr 5d . Two inches have been proposed as a proper diameter for such a bob. In that case r will be = 1. inch, and rr 5d = 1 9787 inches.
In the cylindrical second rod, call the length of the rod, 3 x 19.575. or 3d, and its radius = r and rr 2x3d = rr 6d will express the displacement of the centre of oscillation. It is thought the rod will be sufficiently inflexible if it be ⅕ of an inch in diameter. Then r will be = .1 inch, and rr 6d = 1 11745 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 1⁄1879 of an inch, or 1⁄110356 of the length of the rod.
[(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:
| Pieds. | Lignes. | |
| 30° | 3 | 7.948 |
| 35 | 3 | 8.099 |
| 40 | 3 | 8.261 |
| 41 | 3 | 8.294 |
| 42 | 3 | 8.327 |
| 43 | 3 | 8.361 |
| 44 | 3 | 8.394 |
| 45 | 3 | 8.428 |
[(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.