Note, A geographical mile is the ¹/₆₀th part of a degree; whereof if you multiply the number of degrees by 60, the product will be the number of geographical miles of distance sought; but to reduce the same into English miles, you must multiply by 70, because about 70 English miles make a degree of a great circle upon the superficies of the Earth.
Thus, the distance betwixt London and Rome will be found to be about 13 degrees, which is 780 geographical miles.
If you rectify the globe for the latitude and zenith of any given place, and bring the said place to the meridian; then turning the quadrant of altitude about, all those places that are cut by the same point of it, are at the same distance from the given place.
Prob. VIII. To find the angle of position of Places, or the angle formed by the meridian of one Place, and a great circle passing through both the Places.
Having rectified the globe for the latitude and zenith of one of the given places, bring the said place to the meridian, then turn the quadrant of altitude about, until the fiducial edge thereof cuts the other place, and the number of degrees upon the horizon, contained between the said edge and the meridian, will be the angle of position sought.
Thus, the angle of position at the Lizard, between the meridian of the Lizard and the great circle, passing from thence to Barbadoes is 69 degrees South-Westerly; but the angle of position between the same places at Barbadoes, is but 38 degrees North-Easterly.
SCHOLIUM
The angle of position between two places is a different thing from what is meant by the bearings of places; the Bearings of two places is determined by a sort of spiral line, called a Rhumb Line, passing between them in such a manner, as to make the same or equal angles with all the meridians through which it passeth; but the angle or position is the very same thing with what we call the azimuth in astronomy, both being formed by the meridian and a great circle passing thro’ the zenith of a given place in the heavens, then called the azimuth, or upon the Earth, then called the angle of position.
From hence may be shewed the error of that geographical paradox, viz. If a place A bears from another B due West, B shall not bear from A due East. I find this paradox vindicated by an author, who at the same time gives a true definition of a rhumb line: But his arguments are ungeometrical; for if it be admitted that the East and West lines make the same angles with all the meridians through which they pass, it will follow that these lines are the parallels of latitude: For any parallel of latitude is the continuation of the surface of a Cone, whose sides are the radii of the sphere, and circumference of its base the said parallel; and it is evident, that all the meridians cut the said surface at right (and therefore at equal) angles; whence it follows, that the rhumbs of East and West are the parallels of latitude, though the case may seem different, when we draw inclining lines (like meridians) upon paper, without carrying our ideas any farther.