Robert Patterson to Thomas Jefferson, March 15, 1803
Thomas Jefferson Papers, Library of Congress
Robert Patterson explains the mathematical formulas he will show Meriwether Lewis in preparing him for his journey. Please note, the complex nature of the formulas and examples makes it more feasible to leave them off of the transcribed document than include them; see the images for a complete rendition of Patterson's work.
Philadelphia March 15th 1803.
I have been honoured with your favour of the 2d and thank you for your confidence, which I will never abuse. I am preparing a set of astronomical formula for Mr. L. and will, with the greatest pleasure, render him every assistance in my power. I take the liberty of sub-joining 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, alts. &c are all expressed in the same manner, viz. by the common algebraic signs; which renders the process extremely easy even to boys or common sailors of but moderate capacities.
Suppose the apparent angular distance of the sun & moon's nearest limbs (by taking the mean of a set of observations) to be no° 2' 30" the app. alt of O's lower limb measuring 20° 40' and that of )'s lower limb 35° 24' height of the eye 18 feet, estimated Greenwich time Sept. 18th 1798 about 6 hours p.m. time at place of observation, allowing for error of watch, or computed from the sun's alt. & lat. of place 4° 20" 30s p.m. apparent time. Reqd. the longitude of the place of observation, from the mend. of Greenwich.
From the app. alts. of the lower limbs of 0 & ) find the app. alts. of their centers by subtracting the dip corresponding to the height of the eye, and adding the app. semidiameters: Also from the app. dist. of limbs find the app. dist. of centers by adding the semidiameters. The longitude may then be computed by the following
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formula, in which the capital letters represent the corresponding arches in the adjoining column; & the small letters, the logarithmic functions of those arches. When the small letter is omitted, the arch is found from the log. funct. 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 is [unclear] ± or xxxxx (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
[Ed Note: See images for complete mathematical notation of this concept]