Some days since, I had the honor to transmit to you a table for computing the Moon’s longitude, latitude, &c. at any intermediate time between 0 and 12 hours, which may be called Table I; and I now beg leave to request your acceptance of the inclosed table for ascertaining the Moon’s hourly motion or velocity, as Table II, connected with, and essential in practice, to the former. I have never seen any tables of this kind formed with such minute exactness either in relation to the extension of the decimal fractions, or to the small portion of intermediate time; and to render them still more complete, it is my intention to annex the fractions for fourth differences, before the calculations are made to determine the longitude of places in this country by means of the approaching Solar Eclipse on the 16th. June next. The value of fourth differences in longitude, latitude, &c. is thus ascertained.

1 hour = 1/12 X 11/24 X 23/36 X 1/48 = 253./497664. or, in decimals, 0050837. the value of—d.

To obtain the value of d. (the fourth difference) in the Moon’s hourly velocity, we must have recourse to a complicated series indeed;—for let the intermediate time be 1 hour, then,

1/48 less 11 X 1 X 1/1728, less 1 X 1 X 1/2304, added to 1 X 1 X 1 X 1/165888, becomes the value of—d, at 1 hour, which collected and reduced, is = 2329/165888, or, 0140335, the value of—d. at 1 hour, in velocity.

The Moon’s place referred to the ecliptic, in longitude, latitude &c: as stated in the Nautical Almanac, may be supposed to be inaccurately given to us, when fourth differences exceed 4 seconds; otherwise, the motions must be uncommonly irregular, when such cases happen;—the greatest value of—d, in longitude, &c. is about the 6th. hour, when it becomes ½ X ¼ X 3/6 X ⅛ = 1/128, and supposing the fourth difference to be 4 seconds, 1/128 part will be = 1/32 part of a second.

The last letter I had the honor to address to you may require an apology for the freedom of some expressions which it contained:—having been then, and at this time, an object of persecution myself, I felt the subject on which I wrote, when applied to a character who deservedly commands my high respect and esteem: The person to whom the inclosed circular letter relates, has treated me while I was in his office in a very different manner than you did when I was in your office as Secretary of State; As I do not wish to behave with injustice or ingratitude to others, neither will I suffer myself to be misrepresented or oppressed with impunity by them. I am far from wishing or expecting that you would take any part in relation to this object; it rests with him to admit, or me to prove the truth of the facts I have stated, if denied. I have been compelled against my inclination, to the step I have taken; but I will not, if I can possibly help it, forfeit the good opinion of men who are an honor to this country, because some may wish to see me crushed and degraded, as standing in their way.

I am, Sir, with perfect respect, Your most obedient servant,