21 6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1758. 



to blame. The particular methods of description proposed or used by geogra- 

 phers 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 qua- 

 drilateral 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. 



The only maps Mr. M. remembers to have seen, in which the last of these 

 blemishes is removed, and the other lessened, are some of P. Schenk's of Am- 

 sterdam, a map of the Russian empire, the Germania Critica of the famous pro- 

 fessor Meyer, and a few more, several of which were drawn by Senex. In these 

 the meridians are straight lines converging to a point ; from which, as a centre, 

 the parallels of latitude are described: and a rule has been published for the 

 drawing of such maps, mentioned in the preface to the small Berlin Atlas. 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 ex- 

 hibit the superficial and linear measures in their truest proportions ? In order to 

 which, 



Let e/lp, fig. 1, pi. 8, be the quadrant of a meridian of a given sphere, its 

 centre c, and its pole p ; el, e/, the latitudes of two places in that meridian, em 

 their middle latitude. Draw ln. In, 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 ex = —. , and through .r draw xr, xr, equal each to 



Xil X MF X MT 



half the arc l/, and perpendicutar 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 sur- 

 face of the zone described in the same time by the arc l/ ; as will easily appear 

 by comparing that conical surface with the zone, as measured by Archimedes. 

 And lastly, if from the point t, in which Rr produced meets the axis, we take 

 the angle c/v in proportion to the longitude of the proposed map, as mp the 

 sine of the middle latitude is to radius, and draw the parallels and meridians as in 

 the figure, the whole space soav 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. 



This construction is illustrated by a calculation in an example, having the 

 breadth of tne zone 50° lying between 10° and 60*^ north latitude; its longitude 

 110°, from 20" east of the Canaries to the centre 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 then 

 it 18 remarked that a map drawn by this rule will have the following properties: 1. 

 The intersections of the meridians and parallels will be rectangular. 2. The distances 



