VOL. L.] PHILOSOPHICAL TRANSACTIONS. 217 



north and south will be exact; and any meridian will serve as a scale. 3. The 

 parallels through z and y, where the line Rr cuts the arc l/, or any small distances 

 of places that lie in those parallels, will be of their just quantity. At the ex- 

 treme latitudes they will exceed, and in mean latitudes, from x towards z or y, 

 they will fall short of it. But unless the zone is very broad, neither the excess 

 nor the defect will be any where considerable. 4. The latitudes and the superficies 

 of the map being exact, by the construction it follows, that the excesses and de- 

 fects of distance, now mentioned, compensate each other; and are, in general, 

 of the least quantity they can have in the map designed. 5. If a thread be ex- 

 tended on a plane, and fixed to it at its two extremities, and afterwards the planfr 

 be formed into a pyramidal or conical surface, it may be easily shown, that the 

 thread will pass through the same points of the surface as before: and that, con- 

 versely, the shortest distance between two points in a conical surface is the right 

 line which joins them, when that surface is expanded into a plane. Now, in the 

 present case, the shortest distances on the conical surface will be, if not equal, 

 always nearly equal, to the correspondent distances on the sphere; and therefore 

 all rectilinear distances on the map, applied to the meridian as a scale, will, 

 nearly at least, show the true distances of the places represented. 6. In maps, 

 whose breadth exceeds not 10° or 15°, the rectilinear distances may be taken for 

 sufficiently exact. But the above example is chosen of a greater breadth than 

 can often be required, on purpose to show how high the errors can ever arise; 

 and how they may, if needful, be nearly estimated and corrected, as follows: 

 Write down, in a vacant space at the bottom of the map, a table of the errors 

 of equidistant parallels, as from 5^ to 5° of the whole latitude ; and having taken 

 the mean errors, and diminished them in the ratio of radius to the sine of the 

 mean inclination of the line of distance to the meridian, you shall find the cor- 

 rection required: remembering only to distinguish the distance into its parts that 

 lie within and without the sphere, and taking the difference of the correspondent 

 errors, in defect and in excess. 



7. The errors on the parallels increasing fast towards the north, and the line 

 sa having at last, nearly the same direction, it is not to be wondered that the 

 errors in our example should amount to V^. Greater still would happen, if we 

 measured the distance from o to a by a straight line joining those points; for 

 that line on the conic surface, lying every where at a greater distance from the 

 sphere than the points o and q, must plainly be a very improper measure of the 

 distance of their correspondent points on the sphere. And therefore, to prevent 

 all errors of that kind, and confine the other errors in this part of our map to 

 narrower bounds, it will be best to terminate it towards the pole by a straight 

 line Ki touching the parallel oq in the middle point k, and on the east and west 

 by lines, as hi, parallel to the meridian through k, and meeting the tangent at 



VOL. XI. F F 



