I received yours of the 12th. 14th. and 17th. together with the several papers mentioned, to which I shall give as much attention, and as soon as my health will permit. In the mean time I thought it not amiss to transmit to you such observations as occurred to me on first reading them.

I am not quite satisfied with the reasons given (page 1.) for having recourse to motion for a Standard of Measure. The true reason seems to be, not because all matter is variable in its dimensions, for that is a difficulty we have to contend with after recourse is had to the motion of Pendulums, but because a Standard rod of any given lenghth may be irrecoverably lost, and because no such rod has been preserved for us from ancient times, nor can we undertake to transmit them to posterity with sufficient authenticity, or to different Countries for general use.

That the motion of the Earth, about its Axis, is Sufficiently uniform for every human purpose I have very little doubt. But there are good reasons for supposing that it is not perfectly Equable. The Unequal attractions of the Sun and Moon are sufficient to produce a sort of libratory motion in the Earth’s Axis, and when this motion is encreasing, the rotatory motion is probably decreasing, and the Contrary. There are other causes which may perhaps sensibly diminish the Earths motion on its Axis in a long course of time. Would it not therefore be best to avoid asserting that it is uniform and invariable, without any restriction?

Perhaps so great a preference is not due to the vibrating rod of an uniform Diameter. That preference seems to be founded on an Opinion that the radius of Oscillation is precisely ⅔ of the lenghth of the rod. But this is not true unless the thickness of the rod be infinitely little with respect to its lenghth. In all other cases a correction is necessary. A cube is but a short thick rod, and its radius of Oscillation is ⅚ of its lenghth. It is true that the Center of Oscillation of any uniform rod is easily found by Calculation. And so it is likewise when the pendulum has any given regular figure. Perhaps no other part of Mathematick affords so many theorems, beautiful for their simplicity, as the doctrine of Oscillations. The Globe, Cylinder, Cube, Pyramid &c. are figures which may all be executed with great Accuracy, and when used as pendulums their centers of Oscillation are easily found, nor does the weight of the rod by which they are suspended add much to the difficulty if its substance be uniform. Let the radius of a Globe be made = 1. and let the distance of its Center from the Center of Vibration in such measures be = a. Then a + 4/10a = radius of Oscillation.

If a rod, uniform in its thickness, be used, a Cylindrical rod will be preferable to a square one. I mean its Center of Oscillation will be nearer to the points of ⅔. Let the lenghth of the rod be made = a, and half its diameter = x. Then in the Square rod, whether it vibrate parallel to one of its sides, or to its diagonal, the radius of Oscillation will be = ⅔a + 2xx/3a. But in the Cylindrical rod it will be = ⅔a + xx/2a. And tho’ x must be greater in the round rod, that the weights may be equal, still its correction will be to that of the square rod as 5000. to 5236.

The Latitude of 45.° appears to me very proper for determining the Standard Measure, but I fear it will be difficult for us to make experiments in that Latitude. Wherever it is done, some little inaccuracies will no doubt arise in adapting the result to other Latitudes, as likewise from different elevations, which in some cases may lenghthen the pendulum and in others Shorten it. These errors will however, I think, be very inconsiderable.

Of the different relations between weights and measures, noticed by you, there is one which I wish may be preserved. I mean that a Cubic foot of Water be made 1000. ounces. So that if the foot be decimally divided a Cubic inch will be one ounce.

I much approve of your proposal of making the Bushell of Gallons equal to one cubic foot and a quarter.

The	Pint will be 2½ I. Square and 31/8 deep
	Quart 2½ I. Square and 6¼ deep
	Gallon 5. I. Square and 6¼ deep
	Peck. 5. I. Square and 12½ deep
	Bushell 1. foot square and 12½ deep

Then if the foot be divided into 10 Inches

Or if a Cylindrical Vessel should be prefered for common use let it [be] 12.4 in diameter and 10.3½ deep and it will contain but 1/10 of a cubic inch less than a Bushell of 1250 Inches.

It will be very convenient to connect the pound and Ounce Avoirdupois with the penny weights and grains of Troy weight as you propose.

I am, Dear Sir, with great respect Your sincere friend & servant,