I received your favour of the 30th Ulto, in reply to my communication, on the difference, between lines due East and West, and parallels of Latitude, commencing from the same point. The apprehension of intruding upon your quiet & Retirement, made me hesitate, in sending you the demonstration, which I forwarded; and the same apprehension, confirmed, by the tenor of a part of your letter, determined me, to offer nothing farther on the subject; But upon reflection, I have thought it might be proper, to make a few remarks, by way of reply, knowing, that if I trespass in so doing, it will be without [. . .]ing to you, and with impunity to myself.—You think the question, depends on definitions, and that it is rather philological than Mathematical.—Grant, that it does depend on definitions, and that it may be philological as well as mathematical.—The result is the same. The Cardinal points, of East and West, have each a precise definition, in which Astronomers, Geographers, and surveyors essentially concur,—admitting it to be perfect. According to these definitions, the lines due East and due West, (whether curve or direct) of any place, or rather point, on the Earth, between the Equator and the poles, will leave the Meridian of that point, (which you appropriately call the Meridian of departure) at right angles. This point is necessarily supposed to be in the Zenith, or to speak more precisely, immediately under it, on the top of the upper hemisphere of the Earth. The parallel of Latitude passing through this point, will leave the same meridian, of acute angles towards the nearest pole, and hence it cannot coincide with the lines due East and West, commencing from that point. Therefore Mathematically and philologically, we should not say, in the words of the Treaty,—“Thence (from the middle of Connecticut River) due west on the parallel of 45°–North,” but, Thence, westerly on the said parallel.—

Mr. Campbell, professor of Mathematics in Wm & Mary, College, in a friendly letter, says, that he concurs with me, in every thing, except that part of the theory, relating to sailing, and the interpretation of the words of the Treaty, quoted above. He has calculated the deviation of the parallel of 45° from the due East or West line, for the distance of one mile, and adopted a method of describing a parallel on Land, and of cor[. . .]ting the deviation of the parallels of Latitude from [. . .] lines due East and West, which he proposes shortly to publish. But, from what he admits, I know, that he can offer nothing in opposition, except that which is supported, either by prescription or authority, founded upon a want of precision, and if allowed at all, to be allowed only through courtesy.

If you think it will answer an useful or agreeable purpose, and if it should be consistent with your sense of propriety, I should be pleased for you to offer this curious and interesting subject, to the consideration of the Emperor Alexander, and the Grand Turk, in order that we may learn their opinions, or the result of their investigations of it.

However, I leave this, as all things else, entirely with you, and will offer no other apology for this communication, than, that the freedom with which it is written, is connected with the highest respect for your Taste, Judgment, & Character. And were I to particularize on these points, I might imitate a part of the dedication, written a hundred years ago, to the Duke of Chandos by Heill: Who says, “You, my Lord, are the publick and standing mark of all men’s admiration, the beautiful pattern, which all desire to imitate, tho’ few can hope to equal. In publick affairs, what statesman more able? In domestical management, what private man move expert? In the constant stating and [. . .]ing of accompts, no Body more provident[. . .] Body more frugal. In expenses, no Body more liberal: In Largesses, no Body so magnificent. To which, on the present occasion, he would add, In taste, no Body more exquisite.