This Gentleman illustrates his theory with examples justly intended to point out the quantity of error, that will happen in a large extent. For instance; Between latitudes 10° and 60° N. and containing 110 degrees difference of longitude, Mr. Murdoch computes the distance at 5594 miles; which, upon the arc of a great circle, is found to be 5477, or by other methods 5462; so that the difference is only 117, or at most 132 miles in so great an extent, and to an high latitude; and the higher the latitude the greater the error is like to be, where-ever middle latitude is concerned. His courses also agree very nearly with computations made from the tables of meridional parts. In example the first they are the very same: In example the 2d they agree to half a minute: In example the 3d they vary 1° 4´, on account of the high latitudes, which extend from 56° to 80° N. However, I do not esteem this method so simple, easy, and concise, in the practice of navigation, as Mr. Wright’s construction, especially in determining the bearings or courses from place to place: nor will it (I presume) admit of a zone containing both north and south latitude. Of these inconveniences Mr. Murdoch seems to be extremely well acquainted, when he expresses himself in the following very candid and ingenuous terms, _viz._ “As to Wright’s or Mercator’s nautical chart, it does not here fall under our consideration: it is perfect in its kind; and will always be reckoned among the chief inventions of the last age. If it has been misunderstood or misapplied by geographers, they only are to blame.”--And again, at the end of his nautical examples, he concludes thus, _viz._ “It is not meant, however, that it ought to take place of the easier and better computation by a table of meridional parts.” I have the honour to be, with the greatest respect, SIR, The ROYAL SOCIETY’S, and _Your_ most obedient Servant, William Mountaine. ADDENDA _to Mr._ Murdoch_’s Paper_, Nº. LXXIII. IF it is required “to draw a map, in which the superficies of a given zone shall be equal to the zone on the sphere, while at the same time the projection from the center is strictly geometrical;” _Take_ Cx _to_ CM _as a geometrical mean between_ CM _and_ Nn, _is to the like mean between the cosine of the middle latitude, and twice the tangent of the semidifference of latitudes_; and project on the conic surface generated by _xt_. But here the degrees of latitude towards the middle will fall short of their just quantity, and at the extremities exceed it: which hurts the eye. Artists may use either rule: or, in most cases, they need only make C_x_ to CM as the arc ML is to its tangent, and finish the map; either by a projection, or, as in the first method, by dividing that part of _xt_ which is intercepted by the secants thro’ L and _l_, into equal degrees of latitude.