. 
168 
It is obvious; from inspecting Fig. 1, that a chart 
ap constructed. on this principle may, for a few degrees on 
phy: each side of the equator, be tolerably correct ; butthat 
Mercator’s the-inaccuracy increases with the distance from the 
chart. equator, and in high latitudes becomes excessive. To 
obviate this inconvenience, another method of develepe- 
ment has been invented, known by the name of Merca- 
tor’s projection. In this method, as in the former, me- 
ridians and parallels of latitude are straight lines cut- 
ting one another at right angles, the degrees of longi- 
tude being of course the same in all latitudes; but in 
order that the degrees of latitude and longitude may 
preserve their true proportions to one another, the for- 
mer are|made:to increase on the map, in the same ratio 
that the latter:diminish on the sphere. The distances 
from the: equator; or from, one another, at which the 
-parallels of latitude ought to be drawn according to this 
principle, can only be determined accurately by the ap- 
‘plication of the fluxional calculus: (see Fruxtons, 
vol. ix, p, 463); butan approximation to these distan~ 
ees may be found as follows: 
General: | ‘ Let: PE (Fig: 3.) represent the quadrant of a meri- 
principle of dian, CE the equator, and DE any arch of PE; them 
waar DG. will be the sine, CG or'DH the cosine, EF the: 
7 ent, and CF the secant of the arch DE. Now by 
PLATE similar triangles (see Grometry ) CG :CD::CE:CF, 
CCLXVIII. op taking / for the latitude of D, cos. J: rad. :: rad.: 
Fig. 3. see, /, But since circles, or arches of circles, are to one 
another as their radii, rad. : cos. : : an arch of the equa« 
tor :.a corresponding arch of the parallel J; or su 
sing the earth to be a sphere, rads: cos, J::: an arch of 
‘the meridian: a ding arch of the» parallel J. 
Hence if d:represent the length of a d of the me- 
ridian, and d’ the length ofa degree-of the parallel /on 
the globe; 
sec. 1: rad, :2 di: d’. 
But on the map, the natural degree of the meridian d, 
must be increased in the same ratio as d’ is diminished 
on the sphere ; thats, taking 9 to denote the lengthen- 
ed degree of the meridian. 
ppg tiesee.t 
rad. : sec. 132d: => — a — . 
When d and rad. are both =1, the formula becomes 
d=sec. 1; that/is, when the natural degree of the meri- 
dian and radius are both assumed =1, the length of 
any de; of latitude will be expressed by the secant 
-of that latitude. But no ‘degree of the meridian, nor 
indeed any arch of a definite length, can ‘have all the 
same latitude /, and therefore in the equation 3=see. J 
3 is to be understood as the projection of an indefinitel 
small arch d, assumed eqiiat 0 unity. Now any tack 
of the meridian DE, is made up ofan indefinite num- 
ber’ of such arches, and therefore the projection of DE, 
or the distance of the. parallel / from the equator, is 
ponte the-sum ofthe secants of an indefinite number: 
arches, each of which is assumed’ equal to unity. 
This distance, as was formerly observed, can only be 
found accurately by fluxions, but an approximation is 
obtained by dividing DE into anumber of smallarches, 
each being reckoned unity, and finding the: sum of 
their secants. The greater the number of parts, the 
ater also will be the-accuracy of the a imation. 
his principle was: first explained, and applied to the 
construction of charts, by Mr Wright in 1599, who-de- 
termined the distance ofeach parallel to 1 minute of 
the quadrant, by finding the sum of the secants, of all 
the arehes of 1 minute, from the equator to that: 
lel. These distances he-arranged in a table which is 
denominated a-table- of meridional parts, and which is 
GEOGRAPHY. 
still employed in constructing charts, as in the following Math 
examples. te lee Ce, Cae 
1. Let it be required to construct a chart of the _ F 
world, according to Mercator’s projection. __ hav 
Through the point C (Fig. 4.) intended tobe the cen- world 
tre of the map, draw two indefinite straight lines WE, Mere 
NS at right angles to one another, the former repre- projecti 
senting the equator, and the latter the first meridian: 
From C by means of any convenient scale of 
parts, set off towards W and E, 18 equal parts, each 
representing 10 degrees of longitude. Find then, in 
the: Table, the meridional parts, aia waaay bey 10°, 
20°, 30°, &e. divide each by 60,;and taking 10+ 
tients from the same scale of equal parts, set them > eae 
longitude, and from 30° to 60° N. latitude. ae 
raw AB (Fig. 5.) to represent. the parallel of 30°, 
and from the extremity, erect the perpendicular BC for 
the first meridian. From B, by means. of any conves 
nient scale of equal parts, set off five divisions, towards 
A, and from. these points erect perpendiculars, for ‘the 
other: meridians, 10° distant from each other, Take 
then from the Table the meridional parts,corr i 
to 40°; 50°, and 60°, Te a om 
parts corres ing to 300, west’ 
chart, and Rae ee remainders by 60° the quotients 
taken from the:same scale-of equal parts, and:set from 
B-to C, will give the distances of the respective parallels, 
Thus, to find, the distance: of the: parallel of 409: 
Méridional parts of! 400... ss « « 2622.7 
Meridional parts of 30°. 2... . 4. 1888.40 
784.3 
and a = 12, 24 parts of the scale from “which the 
divisions of AB were:taken. , 
To facilitate the construction of charts acc to 
this projection, the flat rulers commonly called Gunter’ 
scales, are provided with two lines adjacent and-parallel 
to one another marked Mer. and E: P, the first being 
meridional parts, previously divided by-60, so as to re= 
duce: hep to leareete anette second a, scale, sre 
s, or degrees tude, corresponding to the. 
Redes on the other. Tees if altace of a.chart 
be taken from the line E: P, the distance. of any paral- 
lel from the equator is found by extending the com- 
passes from the-extremity of the line Mer. to the num~ 
ber denoting the latitude, and applying that) distance 
frony thescommencement of the line E: P. In. like 
manner. to find the distance between any two parallels, 
take the distance between the latitudes on Mer. and 
apply it'to E: P. Thus the distance between.the. pa« 
ria. of 30° and 40° on Mer. will be equal to 12.24 
on E:P; the same as in the preceding example, 1 
Such is the principle of the method, originally in- Jmpr 
vented by Wright, and still frequently employ in meni 
constructing a chart, according to Mercator’s projec. Greg\ 
tion: It was. soon discovered, however, and s Ha 
quently demonstrated by- ory and Halley, that the 
meridian line, divided according’ to this pon oh be- 
comes:a line of logarithmic cotangents, to half the co- 
1 
