HIGHER GEOMETRY. 23 
chords are obtained, and the triangle of these solved as a plane triangle ; or 
finally, the spherical triangle is treated as plane, in which case a correction 
is applied to the angles, each one being diminished by about the third part 
of the spherical excess. This latter rarely reaches five seconds. 
Knowing the angles and sides of the triangle, as a! so the relative positions 
of the signals, we have still to determine the angle which one of the lines 
makes with the meridian. To this fig. 120, pl. 3, has reference, where z is 
the zenith, p the pole, s the pole star, zs a great circle. ~ Hence the follow- 
ing problem is to be solved by means of the formule of spherical trigono- 
metry: From the sides ap, ab (fig. 121), and the angle pab of a spherical 
triangle abp, to determine the side pb, and the angles p and 0b, where pa 
and pb are the complements of the breadths of the positions A and B, and 
the angle p, the difference of their lengths. 
IV. HIGHER GEOMETRY, OR GEOMETRY OF CURVES. 
The higher Geometry treats, as above mentioned, of curved lines, curved 
surfaces, and the solids bounded by these. In applying Algebra and Analysis 
to Geometry, and establishing its principles by calculation, a marked 
difference is observed between it and the lower Geometry. This applica- 
tion of Analysis to Geometry is known as Analytical Geometry, which is by 
no means limited to the cases of the higher Geometry, since straight lines, 
the circle, and planes may be treated of analytically. The position of a 
poimt in a plane is indicated in Analytical Geometry by its co-ordinates 
(so called). By this is generally understood the distance of a point from 
two straight lines whose position is known, generally at right angles to each 
other, and called the axes (of ordinates and abscissas). The distances are 
parallel to the axes, and are known as the abscissa or ordinate of the point, 
accordingly as they are parallel to the axes of abscissas or of ordinates. 
The two together are called co-ordinates. The point of intersection of the 
two axes is called the origin of co-ordinates; since the two co-ordinates of 
a point form a parallelogram with the portions of the axes cut off by them, 
these latter may also be considered as co-ordinates; hence the ordinate 
only is generally drawn parallel to the corresponding axis, and the portion 
of the axis of abscissas cut off by it, called the abscissa. Thus if in pi. 3, 
g. 106, bc represent the axis of abscissas, and 0 the origin of co-ordinates, 
supposed to be rectangular; then the perpendicular fg let fall from f on 
bc, will be the ordinate, and bg the abscissa of the point f. 
Polar co-ordinates are different from the co-ordinates first explained. 
Here we assume only one fixed straight line, and a point in it (called the 
pole) as known, and determine the position of every other point by its 
distance from the pole, or the length of the connecting line (Radius vector) 
between point and pole, and the angle inclosed between it and the fixed 
straight line ; a point in space is known by its distance from these known 
planes, cutting each other in the origin of co-ordinates, and generally 
23 
