**Robert Patterson to Thomas Jefferson**

Philadelphia March 15 1803^{1}

Sir

I have been honoured with your favor of the 2d and thank you for your confidence, which I will never abuse. I am preparing a set of astronomical formula for Mr. Lewis and will, with the greatest pleasure, render him every assistance in my power. I take the liberty of subjoining the formula which I commonly use for computing the longitude from the common lunar observation, illustrated by an example. The other formula for computing the time, altitudes, etc., are all expressed in the same manner, viz. by common algebraic signs; which renders the process extremely easy even to boys or common sailors of but moderate capacities

Example

Suppose the apparent angular distance of the sun and moon's nearest limbs (by taking the mean of a set of observations) to be 110°02'30", the apparent altitude of 's lower limb measuring 20°40' and that of 's lower limb 35°24', height of the eye 18 feet, estimated Greenwich time September 18th 1798 about 6 hours p.m., time at place of observation, allowing for error of watch, or computed from the sun's altitude and latitude of place 4h 20m 30s p.m. apparent time. Required the longitude of the place of observation, from the meridian of Greenwich.

Solution

From the apparent altitudes of the lower limbs of and find the apparent altitudes of their centers by subtracting the dip corresponding to the height of the eye, and adding the apparent semidiameters: Also from the apparent distance of limbs find the apparent distance of centers by adding the semidiameters. The longitude may then be computed by the following formula, in which the capital letters represent the corresponding arches in the adjoining column; and the small letters, the logarithmic functions of those arches. When the small letter is omitted, the arch is found from the log function. The logs need not be taken out to more than 4 decimal places, and to the nearest minute only of their corresponding arches except in the case of proportional logs. Where an ambiguous sign [. . .] as ± or (expressing the sum or difference) the one or the other is to be used as directed in the explanatory note to which the number in the margin refers

Explanatory notes

- Add when C is greater than B, otherwise subtract
- Subtract when C is greater than B, otherwise add
- Subtract when either H or I exceeds 90°, or when H is greater than I, otherwise add
- Add when either H or I exceeds 90°, or when H is less than I, otherwise subtract
- In Table 13 (requisite tables) under the nearest degree to Q at the top find two numbers, one opposite the nearest minute to 's correction of altitude found in Table 8, and the other opposite the nearest minute to the 1st correction (N) and the difference of those two numbers will be the 3rd correction This correction may generally be omitted.
- Add when Q is less than 90°, Otherwise sub.
- These are to be found in Nautical Almanac from page 8th to page 11th of the month, and the sun or star from which the moon's distance was observed, taking out the two differences which are the next greater, and next less than the true distance (S) calling that the
*preceding*distance which comes first in the order of time, and the other the*following dist*. - The Greenwich time and the time at place of observation must both be reckoned from the same no. [day?]
- When Y is greater than Z the longitude is West, otherwise it is East and when the longitude comes out more than 12 hours or 180° subtract it from 24h or 360° and change its name.

Note, the logarithmetical part of the operation may, with sufficient accuracy be wrought in Gunthers scale thus:

- Extend the compassers from Tangent E to Tangent D, and that extension will reach from Tangent F to Tangent G.
- Extend from Tangent I to sine B and that extent will reach (on the line of numbers) from K to L.
- Extend (on the line of numbers) from W to 180, and that extent will reach from v to X

I am, Sir, with the most perfect respect & esteem, your obedient Servant.

R. Patterson

1. Autograph letter signed, recipient's copy; Library of Congress; Jackson,Letters, 1:28.Funded in part by a grant from the NPS Challenge-Cost Share Program.