The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

[CO. V.

CHAPTER V.
MEASUREMENTS OF LENGTH AND AREA, WITH THE
AID OF THE INFINITESIMAL CALCULUS.
~66. In this Chapter we shall apply the Trigonometrical
Formulae found in Chapter IV. to the measurements of Length
and Areas of Curves.
The first thing to be done is to obtain the expression for
the element of arc of a plane curve.
The Element of Arc in Cartesian Coordinates.
In the Euclidean Plane
ds2 = dx2 + dy2.
We shall now prove that in the Hyperbolic Plane
ds2 -cosh2Y dX2 + dy2.
kLet P, Q be the points (x, y), (x + Sx, y + dy).
Draw PM and QN perpendicular to the axis of x.
Then OM =x, MP=y, ON=x + x, and NQ =y + y.
From P draw PH perpendicular to QN.
Let PQ =8s, PH=q, HQ=p, and NH=z.
Then, in the right-angled triangle PHQ,
6s2=p2 +q2,* to the lowest order.
*This follows from ~65, where we have proved that the Euclidean
Formulae hold in Infinitesimal Geometry. If we start with
cosh = cosh cosh q
k k k
we obtain the same result when we neglect terms above the lowest
order.