II. The particular methods of description proposed or used by geographers are so various, that we might, on that very account, suspect them to be faulty; but in most of their works we actually find these two blemishes, _the linear distances visibly false_, and _the intersections of the circles oblique_: so that a quadrilateral rectangular space shall often be represented by an oblique-angled rhomboid figure, whose diagonals are very far from equal; and yet, by a strange contradiction, you shall see a fixed scale of distances inserted in such a map. III. The only maps I remember to have seen, in which the last of these blemishes is removed, and the other lessened, are some of P. Schenk’s of Amsterdam, a map of the Russian empire, the Germania Critica of the famous Professor Meyer, and a few more[27]. In these the meridians are straight lines converging to a point; from which, as a center, the parallels of latitude are described: and a rule has been published for the drawing of such maps[28]. But as that rule appears to be only an easy and convenient approximation, it remains still to be inquired, _What is the construction of a particular map, that shall exhibit the superficial and linear measures in their truest proportions?_ In order to which, IV. Let E_l_LP, in this figure (_See_ TAB. XXI.) be the quadrant of a meridian of a given sphere, whose center is C, and its pole P; EL, E_l_, the latitudes of two places in that meridian, EM their middle latitude. Draw LN, _ln_, cosines of the latitudes, the sine of the middle latitude MF, and its cotangent MT. Then writing unity for the radius, if in CM we take C_x_ = N_n_ ⁄ (L_l_ × MF × MT), and thro’ _x_ we draw _x_R, _xr_, equal each to half the arc L_l_, and perpendicular to CM; the conical surface generated by the line R_r_, while the figure revolves on the axis of the sphere, will be equal to the surface of the zone that is to be described in the same time by the arc L_l_; as will easily appear by comparing that conical surface with the zone, as measured by _Archimedes_. [Illustration: _Philos. Trans. Vol. L._ TAB. XXI. _p. 554_. _J. Mynde sc._] And, lastly, If from the point _t_, in which _r_R produced meets the axis, we take the angle C_t_V in proportion to the longitude of the proposed map, as MF the sine of the middle latitude is to radius, and draw the parallels and meridians as in the figure, the whole space SOQV will be the proposed part of the conical surface expanded into a plane; in which the places may now be inserted according to their known longitudes and latitudes. EXAMPLE. V. Let L_l_, the breadth of the zone, be 50°, lying between 10° and 60° north latitude; its longitude 110°, from 20° east of the Canaries to the center of the western hemisphere; comprehending the western parts of Europe and Africa, the more known parts of North America, and the ocean that separates it from the old continent. And because C_x_ = N_n_ ⁄ (L_l_ × MF × MT), add these three logarithms.