h 
ca 
1 not 
mint of . As the arch of the middle 
se» (Fig. 
many equal parts as the number of meridians wanted 
on each side of NS, and curves drawn through the 
corresponding points will be the metidians required. 
» In the above example, each of the arches be divided 
- into four equal parts, will give a meridian for every 
tenth degree. tt the parallels ‘are at such a distance 
(from one another as to render it difficult to describe 
the meridian curves with sufficient accuracy, interme- 
diate parallels may be described with a pencil point, 
and afterwards erased. 
It may be p to observe, that the preceding me- 
thod of determining the limits of the map on each pa- 
~ yallel, gives the extent somewhat too great, the chord 
’ of the arch Mm, instead of the arch itself, being assu- 
med equal to 1.285 in. In ordinary cases, indeed, this 
difference is too small to affect the accuracy of the 
map, and therefore an expeditious and convenient me- 
thod of construction is not to he abandoned, on ac- 
count of an error which is scarcely, if at all, sensible. 
The truth of this remark will be obvious from the fol- 
lowing method of determining the arch of the middle 
parallel, by which the length of the chord is obtained 
with perfect accuracy. 
parallel of latitude Mm, 
9.) is terminated on the sphere, and in the pro- 
ection by the same points, but has for a radius, in the 
sinc case the cosine, and in the latter'the cotangent 
of the latitude, the number of degrees which the arch 
contains in the projection, will be less than the num- 
ber which ‘tponinee on he goe: or which it re- 
presents in jection, in the same proportion as 
the cosine ofthe itude is less than cote t. 
Hence, if @ denote the amplitude of the arch of the 
middle parallel 7 on the sphere, or the number of de- 
grees to be represented een the middle and ex- 
treme meridian of the map, and a’ the amplitude of the 
same arch in the projection, as described from the cen- 
tre of the parallel, or the angle which a straight line, 
drawn from the extremity of the parallel to the centre, 
makes with the middle meridian, we have 
cost 5 : sin, / " 
Bae Sid - radius: 1 = raps BO. radius r ; hee. 
fore a’ =“ "and by logarithms, 
€ 
log. a’= log. a +4-.log. sin. ? — log. r. 
~ Let now half the longitude of the map be as above 
40°, and let it be required to find the extremity of the 
middle parallel Mm, (Fig. 10.) rs 
In case @ = 40 and / = 50°, therefore 
_»» log. a’ = log. 40 + log. sin. 50°— log. r. 
Logisinibo*! ey) wl! ie. 465 9.884254 
11.486314 
Tags Oe Sipe el) gig L, 10.000000 
Log. aon Bea i 1.486814 
A ae ee) 809.641 = 30°38! 27” 
- therefore a straight line. drawn from C, and making 
an angle with MC = 30° 38’ 27”, will intersect Mm in 
the point through which the meridian must pass, whose 
distance from NS= 40°. By a similar, though a more 
GEOGRAPHY. 
tedious calculation, the amplitudes of the other paral- Mathemati- 
cal Geogra-- 
167 
lels may be determined: but without entering upon 
these calculations, we shall proceed to find what is the 
real difference between the two methods in point of 
accuracy. In the case of the middle parallel, it has: 
been shewn, that the angle MC m = 30° 38’ 27", and 
joining mM, we have in the isosceles triangle CMm an 
angle'C, and a side mC. If the triangle therefore be 
resolved, the base m M, or the chord of the arch mM, 
will be found to be 1.27 in. which gives for the excess 
of the former method .015 in.: an error which in al- 
most all cases may be safely overlooked. 
The characteristic property of this projection is, that 
all the quadrilaterals formed by meridians and paral« 
lels of latitude have nearly the same ratio to one an« 
other on the map, that the corresponding quadrilate« 
rals have to each other on the sphere. It is also a con« 
sequence of this property, that distances on the map 
may be readily and correctly measured by a scale of 
= parts. This scale may be constructed as fol- 
ows. 
phy. 
From any point A (Fig. 11.), draw a straight line-AB, scale of the: 
equal to any number of the assumed degrees of latitude, map. 
as for example 60, and from the same point draw an Ps.ATE 
indefinite straight line AC, making any angle with AB. Seger 
Then, suppose the seale is to be divided so as to repres 8 +» 
sent English miles, the whole will contain 69.045 x60 
«= 4142:7, or nearly 4140. From any-scale of equal 
parts, set off from A towards C 4 divisions, and:.14 of 
another division, and let them terminate at D» © Join 
DB, and through the divisions of AD draw straight 
lines parallel to DB, and intersecting AB:in the points 
1, 2, 8,45 each of these divisions will represent 1000 
English miles, except the last, which will be 140, and 
the distance between two places on the map: applied 
to this scale will give their distance in miles, 
II? Of the Cylinder. 
The principle of this developement may be explained 
ina eit aber: prt sty to that oF the cone Let WN C} seibcnia 
(Fig. 1, Plate CCLXVII.) be the eighth part of a sphere, Cie 
a portion of which it-is proposed te develope, and let o¢Lxvqr, 
Mw be the radius of the middle parallel of that ion» Fig. 1. 
Then if a cylinder ABCN, equal in diameter to.the radi~ 
us of the middle parallel, be partly inscribed in the 
sphere, and partly circumscribed about it, a zone of the 
cylinder to a short distance, on each side of Mm, may 
be considered as .very nearly coinciding with the cor- 
responding zone of the sphere. If the former, therefore, 
be developed, or spread out, the parallels of latitude will 
be straight lines parallel and equal to M m, and the me- 
ridians will also. be straight lines, cutting the parallels of 
latitude at right angles ; that is, they will be parallel to 
one another, and equal in length to the breadth of the 
zone. Upon this-principle is constructed the Plane 
Chart, as follows. 
Suppose the chartis required to extend from 40° to 60° Plane chart. 
north latitude, and from,10° west to 10° east longitude ; 
that is, to neo pl of aenaie and 20° of longitude. 
Describe a parallelogram ABCD, (Fig. 2,) making BC y; 
of an: jonauied nak and AB “BC . oinhat the 
middle latitude (50°); radius, Divyide-AB and BC 
each into four, equal, parts, and straight, lines drawn 
through these points parallel to BC and AB, will be 
meridians and. parallels of latitude five degrees distant 
from one another. If necessary, intermediate parallels 
and meridians may be drawn.in the same.way. . 
: 3 
