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much altered and misrepresented. In the common projection of the two hemispheres, the meridians and parallels of latitude do indeed intersect at right angles, as on the globe; but the linear distances are every-where diminished, excepting only at the extremity of the projection: at the center they are but half their just quantity, and thence the superficial dimensions but one-fourth part: and in less general maps this inconvenience will always, in some degree, attend the [[underlined]] stereographic [[/underlined]] projection.
The [[underlined]] orthographic [[/underlined]], by parallel lines, would be still less exact, those lines falling altogether oblique on the extreme parts of the hemisphere. It is useful, however, in describing the circum-polar regions: and the rules of both projections, for their elegance, as well as for their uses in astronomy, ought to be retained, and carefully studied. 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.
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, [[underlined]] the linear distances visibly false, [[/underlined]] and [[underlined]] the intersections of the circles [[strikethrough]] circles oblique [[/underlined]]: 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 the 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 [[double dagger symbol]]. 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 *. But as that rule appears to be only an easy [[strikethrough]] approximation [[/strikethrough]] and convenient approximation, it remains still to be inquired, [[underlined]] What is the construction of a particular map, that shall exhibit the superficial and linear measures in their truest proportion [[/underlined]]? In order to which,
IV. Let ElLP, in fig. 1. be the quadrant of a meridian of a given sphere, whose center is C, and its pole P; EL, El, 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 Cx = [[above division line]] Nn [[\above division line]] / [[below division line]] Ll x MF x MT [[\below division line]], and thro' x we draw XR, xr, equal each to half the arc Ll, and perpendicular to CM; the conical surface generated by the line Rr, 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 Ll; as will easily appear by comparing that conical surface with the zone, as measured by [[underlined]] Archimedes [[/underlined]].
And, lastly, If from the point t, in which rR produced meets the axis, we take the angle CtV 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
V. [[underlined]] EXAMPLE. [[/underlined]] V. Let Ll, the breadth of the zone, be 50[[degrees]], lying between 40[[degrees]] and 60 [[degrees]] north latitude; its longitude 110[[degrees]] from 20[[degrees]] east of the Canaries to the center of the western hemispher; comprehending the western parts of Europe and Africa, the more known parts of North America, and the ocean that seperates it from the old continent.
And because Cx = [[above division line]] Nn [[\above division line]] / [[below division line]] Ll x MF x MT [[\below division line]], add these three logarithms.
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[[double dagger]] [[underlined]] Senex [[/underlined]] drew several of that form.
* See the preface to the small Berlin Atlas.
Log.
