Boundary Lines.

As we were the first to set the mathematicians by the ears, by giving place to a very brief and well written, though modest communication from “A Citizen of Richmond,” it gives us pleasure to afford that gentleman an opportunity of demonstrating; in a more explicit way, the correctness of his proposition.

Editor

To the Editor of the Enquirer

The National Intelligencer of the 11th. of April, contains remarks on the boundary line, between the northern states and Lower Canada, by Mr J. Wallace of Georgetown College, in which he grossly misrepresents the communication published upon that subject, in the Enquirer of the 25th. of February last, by a citizen of Richmond.

Mr Wallace blends what the citizen advanced—with what a writer in the Petersburg Courier has urged to support the contrary; falsely charges the former with changing the words of the Treaty—“a line due west,” for a “right line due west;” insinuates that he maintained the impracticability of delineating a parallel of latitude; resorts to ridicule both against the citizen and the writer in the Courier—and finally—from premises, which support the Citizen’s proposition, deduces a contrary and preposterous conclusion, viz. The coincidence of lines due east and west, with parallels of latitude.—

The writer in the Courier (who from the signature J.W. is understood to be Mr John Wood, President of the Petersburg Academy,) has replied to Mr Wallace, so far as Mr Wallace’s remarks applied to him, and to that part of the Citizen’s proposition, which he supported; and in his communication of the 25th. of April, he has very ingeniously and concisely shewn the absurdity of Mr Wallace’s Theory. But as neither of those gentlemen has explained the subject, according to the view of the citizen, it becomes his duty to demonstrate, what he advanced to the public.

Therefore in order to correct misrepresentation; to refute the charge of changing the words of the Treaty; and to establish the truth of what the citizen advanced, Mr Ritchie is requested to republish from the original communication, the two following passages, with the subsequent remarks and demonstration—

1. “In the 5th. article of the Treaty it is stated, that the above line shall proceed from a point, in the middle of Connecticut River, on the 45th. degree of north latitude, due west on the same latitude. Now it is impossible that the line should proceed, both due west and on the 45th. parallel of latitude.”—

2 “If our commissioners had admitted as the northern boundary of Louisiana, a line due west from the north west corner of the Lake of the Woods, instead of a parallel of latitude, passing through the same point, we should have lost land enough for a large state.”

It is evident from the above, that there was not a change of the words a “line due west,” for a “right line due west.” It is also evident, that what was advanced by the Citizen, on this subject, was one proposition, consisting of two parts, viz. 1st. That a line due west, could not be delineated on the 45th. parallel of latitude, from a given point in the same: but 2dly. that it would lie between the parallel and Equator.

The writer in the Courier (or Mr John Wood) admits the 1st. part of the prop. but denies the 2d. Mr Wallace admits niether, but in attempting to demonstrate the contrary, (viz. the coincidence of east and west lines, with parallels of latitude,) makes use of premises and reasoning, which support both—This difference of opinion between two eminent Mathematicians (for it appears from Mr Wallace’s references, that he is a man of great mathematical reading; and Mr Wood, whose profoundity in the Mathematics is unquestionable, may be considered the Descartes of the age;) This difference of opinion, between these great men, I say, will justify the Citizen, in giving the public a demonstration of what he advanced.

1st. According to the definition of east and west points, they are on the Equator, being made by the intersection of this circle with the horizon. Hence at the time of the Equinoxes, the sun rises in the east, & sets in the west, of every place between the Equator and poles, as well as of every place on the Equator.

2d. If two compasses be set in the plane of the same meridian, one on the Equator, the other at a place of north or south latitude, the right lines from their centres to that of the Earth, will meet there,1 at an angle equal to the latitude of the place, and being produced, will (by the 15th. prop. of the 1st. Book of Euclid) meet the surface of the lower Hemisphere, in the same meridian, the first on the Equator, the other in the nadir of the given place of north or south latitude.

3d. The east and west line of every place, being in the same plane, with the right line passing from that place through the centre of the Earth to the nadir, it is evident that the east or west line of every place of north or south latitude, will, (by the 1st. and 2d. of this demonstration, the 6th. def. of the 11th. Book of Euclid, and2 the 3d. prop. of Spherics) when produced both ways, recede from the nearest pole—and consequently, from the parallel of latitude, and converge towards3 the Equator, until it meets it, in the Horizon at the distance of 90° from the place, and at an angle equal to the latitude of the place; thence produced, it will recede from the Equator and incline towards the other pole, until it completes a great circle, in the nadir of the given place, bisecting the Equator, (by the 1st. of Spherics) in the east and west points of the said place.

4th. The parallel of latitude, passing through any place north or south of the Equator, being a less4 circle, produced from the same poin[t] from which the east and west line,5 of the place is produced in a different direction, (which is evident from the definition of a parallel, and the 3d. Art. of this demonstration) it follows that the parallel of latitude and the east and west line are two circles lying out of each other, and touching in one and the given point; and that they can touch each other at no other point, is demonstrated by Euclid in the 13th. prop. of the 3d. Book.—

And since no two points in the east and west line; can coincide with any two points in the parallel of latitude; and since a line must lie between two points, it is evident, that a line due east or west cannot coincide with one parallel to the Equator. Therefore, the Citizen was perfectly correct, in asserting that “a line due west, from a point on the 45th. parallel of north latitude, could not be delineated on that parallel.” And by the 3d. Art. of this demonstration, it is evident that “if the northern boundary of Louisiana, should be established due west, instead of, on the parallel of latitude of the same place, we should lose land enough for a large state.”

Cor. 1st.—By the 3d. art. of this demonstration it is also evident, that the east or west line of any place, north or south of the Equator, cuts none of the meridians, except the one from which it proceeds6 at right angles; but meets them at obtuse, angles, towards the Equator, until it arrives there. Therefore the Surveyor in attempting to deliniate a line due east or west with his compass, from a place in north or south latitude, will depart from his original course every time he changes his place and sets his compass for a new object; and every departure will proceed at right angles to the meridian of the station, but meeting the meridians in advance, at obtuse angles on the side towards the Equator; therefore he will incline (from the parallel of latitude) towards the Equator, although he is declineing from the east or west line of the given place.—

Hence if one place be due7 east of another, the latter is not due8 west of the former; and hence when a line is run due east or west from a place, north or south of the Equator, it is impossible for the line forwards, through the sights of the compass, to coincide with the one delineated behind.

Cor. 2d.—If, in running with a compass, a line due east or west, from a place north or south of the Equator, the distance between the several stations, be considered very small and equal, the line thus discribed will represent the Spiral described by a ship, sailing due east or west from a similar place. And since the Equator cuts all meridians at right angles; and a parallel of latitude cuts them at acute angles towards the nearer pole, in consequence of their convergency, and a ship sailing from a place in a parallel of latitude, meets all the meridians in advance at obtuse angles towards the Equator; Therefore, it is evident, that the Spiral or Loxodromic described by a ship sailing due east or west from a place of north or south latitude will lie between the parallel and the Equator, inclining to the latter, if pursued in the direction, in which the vessel sails. And if the ship in sailing, meets the meridians at obtuse angles towards the Equator, and leaves them at right angles; it is evident that the Spiral which she discribes will be denticulated on the sides,9 or zig-zag.

This Theory of sailing is also supported, by the consideration that the angle, between the east and west line and touching parallel of Lat,10 is not evanescent; since it is not the point, from which the vessel starts, & its great circle, (in which the east and west lines lie)11 which move around the globe; but the vessel itself which begins its motion in the great circle, and yielding to the first impulse, must move therein, the first moment of its motion, and consequently recede from the parallel and incline towards the Equator.

It appears from the above demonstration, that the 16th. prop. of the 3d Book of Euclid, is perfectly consistent with it; since there is no occasion for the object moving due east or west, to divide the angle contained by this course, and the parallel of latitude passing through the place from which the object moves. Therefore Mr Wallace has misapplied this proposition.* Again if the idea of delineating a parallel of latitude, through sights on a staff, inclining (from the staff of the compass) towards the Equator, at an angle equal to the latitude of the place, be a new one to Mr Wallace; is it not probable that he derived it from the proposition of the Citizen, since it is evidently deduceible therefrom, by the 6th. def. of the 11th. Book, and the 29th. prop. of the 1st. Book of Eucli[d?]

In dismissing this subject, the Citizen declares that he never considered the ambiguity as operating against the reputation of the very respectable personages, who were concerned in making the Treaty. He supposes that the commissioners, being occupied in settling the important interest of the two nations, adopted the common opinion on the subject in question, without discussion, and that the Prince Regent, President and Senate, afterwards admitted, in the same manner, what the Comissioners had adopted.—

He did at first suppose, that the ambiguity might arise from an error in the press, but finding that several gentlemen, whom he consulted on the subject, believed the expression to be both original and correct, he thought it a favourable opportunity to bring the subject before the public for discussion, as the attending circumstances, would probably cause more attention to be paid to it, than would be, if it were abstractly proposed. He therefore conceives,13 that henceforward, all parties and classes will admit both the truth and propriety of what he advanced, and has demonstrated.