[ 557 ] 
2. The distances north and fouth will be exadtj 
and any meridian will ferve as a fcale. 
3. 1'iie parallels thro’ z and y, where the line 
Rr cuts the arc L/, or any final 1 diftances of places 
that lie in thofe parallels, will be of their juft quantity. 
At the extreme latitudes they will exceed, and in 
mean latitudes, from x towards ^ or y, they will fall 
fhort of it. But unlefs the zone is very broad, nei- 
ther the excefs nor the defedt will be any-where con- 
liderable. 
4. The latitudes and the fuperficies of the map 
being; exadt, by the conftrudtion, it follows, that the 
exceftes and defedts of diftance, now mentioned, 
compenfate each other j and are, in general, of the 
leaft quantity they can have in the map defigned . 
5. If a thread is extended on a plane, and fixed 
to it at its two extremities, and afterwards the plane 
is formed into a pyramidal or conical furface, it may 
be eafily fhewn, that the thread will pafs thro’ the 
fame points of the furface as before ; and that, con- 
verfely, the fhorteft diftance between two points in a 
conical furface is the right line which joins them, 
when that furface is expanded into a plane. Now, 
in the prefent cafe, the fhorteft diftances on the coni- 
cal furface will be, if not equal, always nearly equal, 
to the correfpondent diftances on the lphere : and 
therefore, all redtilinear diftances on the map, ap- 
plied to the meridian as a fcale, will, nearly at leaft, 
fhew the true diftances of the places reprefented. 
6 . In maps, whofe breadth exceeds not io° or 
15 0 , the redtilinear diftances may be taken for fufti- 
ciently exadt. But we have chofen our example of 
a greater breadth than can often be required, on pur- 
