and legs of the polar ▲, which, in this case, is right Zd, the similarity will be the same as before.

THEOREM XVII.

131. In any right-angled spherical ▲ the hypothenuse is less or greater than 90°, according as the two legs, or the two angles, or a leg and its adjacent angle, are like or unlike.

B

1st. If the ▲ A B C, right-angled at c, have its legs CA, C B each less than 90°, the hypothenuse A B will be less than 90°.

For make C P equal to a quadrant, and through the points P, B draw the arc of a great circle P B.

Then, because P is the pole of the great circle CBD, the arc P B is a quadrant, or 90° (def.)

And since, in the right A' PCB, A D B, the leg CB is less than 90°, and D B greater, the CPB or APB is also less than 90°, and the DA B, or PAB greater than 90° (theo. 16).

But the less side of every ▲ being opposite to the less, the hypothenuse A B is less than 90°, or than the quadrant P B.

2dly. If the ▲ a cb have its legs ca, cb each greater than 90°, the hypothenuse ab will, in this case also, be less than 90°.

For produce ca, cb till they meet at D, which will be a right angle, and through the points P, b draw the quadrant P b.

Then, since the legs D a, D b are each less than 90°, it may be shown, as before, that the hypothenuse a b, which is common to both the ▲a cb, a Db, is less than 90°, or than the quadrant p b.

3dly. If the ▲ a cв have one leg C B less than 90°, and the other ca greater, the hypothenuse a в will be greater than 90°.

For, since in the right

CB is less than 90°, and D B

A'ac B, PDB, the leg greater, the cа B, or

Pa B, is less than 90°, and the DP B, or a P B greater (theo. 12); whence, also, a в is greater than 90°, or than the quadrant P B.

Again, the 'in either of the AABC, abc, or a вC, being of the same kind as their opposite legs (theo. 16), it follows, that the hypothenuse a B, a b, or a B is less or greater than 90°, according as the two oblique', of the a to which it belongs, are like or unlike.

And because a leg and an in each of these are of the same kind as the two legs (the other leg being like its opp. 4), it is plain that the hypothenuse ab, ab, or a в is also less or greater than 90°, according as either leg and its adjacent are like or unlike. Q.E.D.

COR. It follows, reversedly, from this proposition, that in any right-angled spherical ▲, either leg is less or greater than 90°, according as its adjacent and the hypothenuse, or the other leg and the hypothenuse, are like or unlike.

Also, that either of the oblique angles is acute or obtuse, according as its adjacent leg and the hypothenuse, or the other and the hypothenuse, are like or unlike.

SCHOLIUM. This proposition and its corollaries will also hold for any quadrantal spherical ▲, observing to substitute the hypothenusal for the hypothenuse, and the terms greater or less for less or greater.

D

B

For the sides and angles of the quadrantal AABC are, evidently, like or unlike, according as the angles or legs of the right a polar ▲ DE F, which are their supplements, are like or unlike.

d

But the hypothenusal c, being the supplement of the hypothenuse DE, will consequently be greater than 90° when D E is less, and less than 90° when D E is greater; which is, therefore, the only change that takes place in the proposition.

OF THE

STEREOGRAPHIC PROJECTION OF THE SPHERE.

The stereographic projection of the sphere, is such a representation of the various parts of its surface, on the plane of one of its great circles, as would be formed by lines drawn from the pole of that circle to every point of the figure to be delineated.

Or, if taken in an optical sense, it is a view of the points and circles of the sphere, as they would appear on a transparent plane, passing through the centre, to

an eye placed at one of the extremities of a diameter, drawn perpendicular to that plane.

The place of the eye, is called the projecting point, and the plane, on which the points and circles of the sphere are to be represented, is called the plane of projection.

The primitive circle, is that which lies in the plane of projection; being the one to which all the other circles and points of the sphere are referred.

A right circle, is that which, passing through the eye, has its plane perpendicular to the plane of the primitive; and, being seen edgewise, is projected into a right line.

A parallel circle, is that which is parallel to the primitive; and an oblique circle is that which is seen obliquely by the eye.

It is also to be observed, that the projection of any point of the sphere, is that point in the plane of projection, which is cut by a right line drawn from the original point to the eye. And that lines flowing to the projecting point, or place of the eye, from every point of the circumference of a circle, form the convex surface of a cone.

LEMMA.

132. If a cone be cut by a plane parallel to its base,

the section will be a circle.

A

E

F

m

Let A B C D be a cone, either right or oblique, and EFG a section parallel to its base BCD; then will EFG be a circle.

For let the planes A cm, A Dm pass through the axis am of the cone, meeting the section in the points

F, G, n;

Then, because the section E F G is parallel to the base BCD, and the planes cn, Dn meet them, nF will be parallel to m c, and n G to m D.

And, because the ▲, formed by these lines, are similar, am: An::mc:nF or as m D: n G.

But mc is equal to m D, being radii of the same circle; whence, also, n F is equal to n G.

And the same may be shown for any other lines drawn from the point n to the circumference of the section E FG, which is therefore a circle. Q. E. D.

THEOREM I.

133. Every circle of the sphere, which does not pass through the poles of the primitive, is projected into a circle.

S

E

Let Am B be a circle to be projected on the plane LM, which passes through the centre of the sphere, at right to a radius drawn from the eye at E; then will its representation a n b, on that plane, be a circle. For through r, the centre of the circle Am в, draw the plane Fmo parallel to LM, and join Er, rm; the