VOL. XXXIX.] PHILOSOPHICAL TRANSACTIONS. GA 



grees of latitude on one side, and to 15 degrees of latitude on the other side of 

 that parallel ; and so of the succeeding zones. 



Now we may conceive the first of these portions, or zones, to e converted 

 from a spherical surface to a plane in this manner, without sensible error; let 

 the middle line of this zone, that is the equator, continue in its situation, and 

 let the segments of the meridians on each side be conceived to unbend them- 

 selves gradually, till they are extended into right lines perpendicular to the 

 equator; then will that which was before a zone, or portion of a spherical 

 surface, with a small alteration become a portion of a cylindrical surface, cir- 

 cumscribed about the sphere; whose breadth is every where equal to 10 degrees 

 of the sphere, and whose circumference is equal to the equator. And thus 

 every parallel to the equator, as far as that of 5 degrees of latitude on each 

 side, will be stretched and extended into a circle as large as the equator; but 

 they will all keep the same distance from one another, and from the equator, 

 that they had before. This extension, or alteration, will be every where regular 

 and uniform, and will be but very little, even where it is most; for the least of 

 these circles, which is the parallel of 5 degrees of latitude, has the same pro- 

 portion to the circle it is stretched to, or the equator, as the sine of 85 degrees 

 has to the radius, or as 996 1 947 to 10000000; which approaches very near to 

 a ratio of equality. And now it will be easily conceived, that without under- 

 going any other alteration, or distortion, this portion of a cylindrical surface 

 may be rectified, or extended into a plane parallelogram, whose length will be 

 equal to that of the equator, and whose breadth will be equal to an arch of 10 

 degrees of the same equator. 



Consequently, by an operation that will be just the reverse of this, if on a 

 plane we delineate such a parallelogram as this, we may then lay down all the 

 places that are contained in it very exactly, in their proper situation of longi- 

 tude and latitude, and then apply its middle line, or equator, to that of a globe 

 of a due magnitude, which will then become a portion of a cylindrical surface, 

 circumscribed about the globe. Then by pressing it close to the body of the 

 globe, we shall cause it to contract itself a very little, but regularly, which 

 contraction will be only according to longitude, and not at all according to lati- 

 tude; and then the cylindrical surface will be changed into that of a sphere, and 

 will become the first spherical zone before described, with all its delineations in 

 their due position, without sensible error. 



In like manner in the second spherical portion, or zone, comprehended be- 

 tween the parallels of 5 and 15 degrees, whose middle line is the parallel of 

 10 degrees, we may conceive the segments of the meridians to unbend gradu- 

 ally on each side, and to extend themselves into tangent right lines, which 



