Daily National Intelligencer

THE BOUNDARY LINE,

Messrs. Gales & Seaton,

Having lately accidentally cast my eye over a curious piece in your paper of April 4th copied from the Petersburg Courier, relative to the boundary line lately agreed on by the American and British commissioners at Ghent, where a judicious writer in the Richmond Enquirer remarks and pretends to prove, very correctly, that the line which commences at lat. 45 deg. cannot be extended due west, and in that parallel of lat. and further says, that in any parallel of latitude, north or south, an east and west line will approach one of the poles and recede from the equator: Now, I affirm, without fear of contradiction, because I affirm it on the evidence of mathematical demonstration, that both the above assertions of this learned mathematician, however judicious and correct he may be, are utterly false; and that those gentlemen who formed the treaty have made no mistake or blunder against the most rigorous mathematical principles, in the enunciation of these words in the 5th article—"thence by a line due west on said latitude (lat 45 deg) until it strikes the river Iroquois or Cataraguay," however great the astonishment of this writer may be. As this is a subject which not only concerns the commissioners of both nations, but the public in general, and the practical surveyor in particular, the following remarks may not be undeserving attention. That a due east or west line in any parallel of latitude is a true rhumb line, is evident to any one that has the slightest knowledge of the nature of a rhumb line, because it cuts all the meridians at right angles, and therefore at equal angles, as is clearly demonstrated in a periodical work, published by the late George Baron, New York, page 16, art 3, and is evident from prop 16. B. 3, of Euclid's Elements. The line thus traced out is certainly a curve line, as every line on the surface of the earth must be, but it is the curve of a lesser circle, the centre of which is different from the centre of the sphere (allowing the earth a sphere) and therefore not a spiral. It is true, that if any two points be taken in a parallel of latitude, and the surveyor according to common practice takes sight with his instrument (the plane of which is parallel to the plane of the horizon) from one of those points to a staff or object erected perpendicularly to the plane of the horizon in the other point, and then traces this line by a continuation of such perpendicular objects, the line thus traced out will be the curve of a great circle, whose centre will be the same as that of the sphere, as is evident; and this curve continued will cut the equator in two opposite points, but will not pass through either pole. If east or west bearings be taken to those perpendicular objects, set up at any convenient distances in the same parallel, the shortest lines connecting these objects will be arches of great circles. But if the staff or objects be set up in the plane of the lesser circle or parallel, so as to make angles with the horizontal plane equal the complement of the latitude, and declining towards the equator, and the sights of the instruments be placed, or the telescope by a single contrivance be made to play in the plane of those objects, which plane will be the same as that of the lesser circle or parallel, and then sights be taken to the respective objects so placed, other objects may be set up in the same plane, by working the telescope until the different parts of the objects coincide, & thus a series of objects so placed will trace out the parallel required. Other methods might be pointed out of tracing a parallel of latitude, but the dispute is not about the practicability but the possibility of its being traced: however, as the tracing of it is attended with some difficulty, the above new and simple method may be of no small use to practical surveyors, as it will answer whatever be the figure of the earth. Now the bearing of each of the objects so placed will, evidently, be east or west, as the whole continued parallel is evidently so; but to ascertain the bearing of any one from the other, the distances must be taken at very small intervals, otherwise it would be the angle of position, and not the bearing that would be measured by the instrument, and it is well known that the angle of position is essentially different from the bearing, as is clearly shewn in pages 121 136 &c. of a late Treatise on the Globes & Astronomy by J. Wallace The method above given of tracing a parallel of latitude by having two points in the parallel given by astronomical observation, is independent of the bearing of any of the objects and therefore liable to no error. The mistake of the judicious writer in the Enquirer, evidently arose from his changing the words in the treaty, 'a line due west,' for 'a right line due west.' Had the commissioners mentioned a right line, any person, however little versed in these matters, would immediately perceive the absurdity, independent of the writer's learned demonstration. But as the treaty is worded, the writer must display more ability to demonstrate to the public, who is now in possession of his judicious remarks, that the absurdity is not altogether on his own side. The judicious writer might possibly have in view the impossibility, in most cases, of a ship steering due east or west and still sail on the same parallel of latitude.
Attitude. If so, the case is in no respect applicable to that which he has reference to. But to shew that his principles, even in this case, are erroneous, I shall select the following proposition from the periodical work above quoted, the demonstration of which, by R. Adams, the curious may see in page 109. If a ship in any latitude steers due east with any velocity, or steers due west with a velocity greater than twice the velocity of a given point of that latitude easterly, by the earth's rotation on its axis, she will gradually deviate from that latitude, and approach towards the equator. If her velocity due west be equal to twice the velocity of a given point, in that latitude moving easterly by the rotation of the earth, she will continue in that latitude and describe the parallel, but if her velocity due west be less than in the case last mentioned, she will gradually deviate from that latitude to the nearest pole; and when her velocity west is just equal to the rotary velocity east of a point in that latitude, her deviation towards her nearest pole will ceteris paribus be a maximum. But what has the action of gravity, the unbalanced tendency of the vessel to the equator from the centrifugal forces, &c. to do with simply tracing a parallel of latitude? It is evident, then, that our judicious writer either mistook the matter altogether, or knew nothing of the subject on which he attempted to write.

J. WALLACE.

Those Editors who have inserted the judicious piece copied from the Courier are requested to give the above an insertion.