Mathemati- 
cal Geogra- 
phy. 
PLATE 
CCLXVL# 
Fig. 1. 
Analemma. 
General 
properties, 
Construc- 
tion of a 
stereogra- 
phic polar 
162 
&e, according to.the number of meridians wanted ; then 
elliptic arches described through NaS, N6S, Ne S, 
&c. will represent the meridians, in this case, 10° distant 
from each other, or whose angles ofinclination to the pri- 
mitive, measured by the arches of the equator inkencepy: 
ed between them, are 10°, 20°, 30°, &c. respectively. 
Of these ellipses, NS is always the transverse axis, and 
fEa, Eb, Hic, &c. the semiconjugates. Hence, to 
find the foci of any given arch N/'S; from f the extre- 
mity of the semiconjugate, as.a centre, with the radius 
ZEN, half the transverse, describe an arch intersecting 
NS in F and F’; these points will be the foci required. 
The foci being found, the ellipse may be described 
according to the method explained under. Conic Sec- 
TIONS, vol. vii. p. 137, or by any of the elliptic instru- 
ments described under the article Drawine Instru- 
MENTS, vol. viii. p. 130, If the whole ellipse NS be 
described, the other. semicircumference will represent 
the corresponding meridian on the opposite side of NS. 
The points a,b,c, d, &c, may also be found by divi- 
ding EN or ES into nine equal parts, and letting fall 
erpendiculars from every division on E. Straight 
fare drawn through the divisions of EN, and parallel 
to EW, will represent parallels of latitude. 
When this projection is made upen the solstitial co- 
lure, the planisphere is distinguished by the name of 
Analemma, and is the foundation of a simple instru- 
ment of the same name used for the solution of various 
astronomical problems. See ANALEMMA, and Pro- 
JECTION OF THE SPHERE. 
The orthographical_ projection of the sphere on. the 
plane of the horizon, is seldom used in constructing 
maps, partly from the inaccuracy of representation 
common to it with the preceding methods, but chiefly 
from the difficulty of construction, both meridians and 
parallels of latitude being projected into ellipses. It is 
applied to the projection of solar eclipses. See AsTro- 
nomy, vol, ii. p. 744. 
Ill. By Stereographic Projection. 
In delineating maps according to the principles of this 
projection, the defects of the other methods are ina 
great measure avoided, both as to the accuracy,of re- 
presentation, and the facility of construction. These 
advantages are chiefly owing to the two following pro- 
perties, by which the stereographic projection is dis- 
tinguished from every other. 1s/, All circles are pro- 
jected into circles or straight lines ; and, 2d/y, The pro- 
jections of any two circles intersect one another in the 
plane of projection, at the same angle that the circles 
themselves do on the surface of the sphere. 
In maps of the world constructed on stereographic 
principles, the projection is generally made on the seed 
of a meridian, the eye being successively placed in the 
poles of that meridian, opposite the hemisphere to be 
projected. As the method, however, is of very exten- 
sive application, we shall give. an example of all the 
three cases, , 
1, The Polar. From P (Fig. 2.) with 60° from the 
line of chords, describe the primitive. WLEM, in this 
— equator, _ draw diameters for meridians as 
in the gnomonie, polar projection. To project the pa- 
rallels of latitude, take from the scale the ats 
_ of their complements of latitude, or distances from the 
le, and with these radii describe: concentric circles 
about P. Thus the semitangent of 10° will be the ra- 
dius of the parallel of 80°, the semitangent of 20° will 
be the radius of the parallel of 70°, &c. that is, PM by 
the intersections of the parallels, is converted into a line 
or plane of projection 3 but for the more convenient re« 
thus. 
: . MII ID Modzare ty > arialer ald a} 5 
Fron JE with theehord of 60° themeridian, A st 
WNES for the primitive, and throu hee ied draw, graphi 
WE and NS at Fiaht ang’ and thro ene eae 
ans wanted, and from these divisions as centres, describe 
arches of circles passit Hie tie tee P and S; 
these arches will be ons oF onidieds be- 
n Fig. 3. 
Jia, Hb, &c. ave the tangents of 10°, 20°,. there. 
fore the arch described from a, viz. NmS next to 
W, is 80° from N AES, the meridian passing thr 
the projecting point, that described from 8, viz. pew 
cond from, W is,70° from the same meridian, and so of 
the others, always measuring the distance between two 
meridians, or, the. angle which they make with each 
other, by the arch of the Sgpatar intercepted between 
them, Ifthe second from W be taken to represent the 
meridian. of London,. the primitive WNES will em« 
brace the whole of the eastern continent, or old world, 
except a small part of the north-east point of Asia, with« 
out including any part. of America, and the other me-« 
ridians ‘will be reckoned both ways, towards, the east 
and west. In thepresent case, however, as our obj 
is not to exhibit an actual map of the earth’s 
but only the imaginary lines with which it is su 
to be intersected, we shall assume N/ES as the first 
meridian, by which means our references to the Figure 
bs fe more obvious and distinct. ayaa. ty 
efore proceeding to the projection. of the parallels 
latitude, it may be proper ys a acanonae method, be- 
sides that already explained, of describing meridians, 
viz. by determining the points in which these circles 
must intersect the equator WE (Fig. 4.) This is done Fig.: 
by setting off from /E towards W and E, 4 10, Z 20, 
ZE 30, &c. equal to the semitangents of these arches res 
spectively. Then three points being given, viz. N, S, 
and 10, 20, or 30, in the equator WE, a fourth may be 
found, which shall be the centre of a circle. 
through the other three. To determine this fohith point 
or centre, draw lines. from N and S to the point in 
equator through which the meridian is to pass, bisect 
these lines, and erect perpendiculars at the points of 
bisection ; these perpendiculars . will meet in the. 
point required, See Grometry, Secr. II. Pros, XI. 
Though both these methods of describing meridians 
imply the use of lines of tangents, or semitangents, 
