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angles A, B, and c of the triangle ABC be acute, then a perpendicular as cd drawn from any angle will fall within the triangle.
In the right-angled triangle Adc, the angles A and ACD are acute, and therefore (P. and R. 145.) Ac is less than a quadrant. By the same manner of reasoning from the triangle BDC, BC is less than a quadrant; and because the angles A and ABF are acute, AB is less than a quadrant. (P. and R. 145.)
Second. This part is demonstrated (O. 144.)
Third. If the three angles be obtuse, each side of a supplemental triangle will be less than 90° (Prop. vi.); if the three sides are less than 90°, the three angles will be acute, by this proposition, and the supplements of these three angles must necessarily be obtuse; but (U. 137.) they will be the sides of the original triangle, the angles of which were all obtuse. Q. E. D.
(G) COROLLARY I. In any spherical triangle having two obtuse angles and one acute, the sides are of the same species as their opposite angles.
Imagine the triangle AEC to have an acute angle at A, and two obtuse ones at E and c; then the triangle ceg has three atute angles, viz. the angle G is equal to the angle A, (Q. 135.) and the acute angles at E and c are supplements of the obtuse angles in the triangle AEC (M. 184.):
But when the three angles are acute, each side of the triangle CEG is less than a quadrant, therefore AE and ac, opposite to the obtuse angles c and E, are greater than quadrants, and CE opposite to the acute angle a, is less than a quadrant.
(H) COROLLARY II. If a spherical triangle have two sides cach less, and one greater than a quadrant ; the angles will be of the same species as their opposite sides.
In the triangle anc, let AH and hc be each of them less than a quadrant, and ac greater ; the supplemental triangle to it (U. 137.) will have two obtuse angles and one acute, and consequently the sides thereof, by this proposition, are of the same species as their opposite angles : but the supplements of these sides are the angles of the original triangle auc, therefore it has two acute angles HAC, ACH, and one obtuse, viz. AHC.
(I) COROLLARY III. The reverse of this proposition is true in all its parts. Vız. If the three sides be each less than a quadrant, each angle will be acute; if the three sides are each quadrants, the angles will be right-angles ; and if the three sides are each greater than quadrants, the three angles will be each obtuse.
These follow from considering the supplemental triangles. (U. 137.)
PROPOSITION XVIII. (K) If two sides of a spherical triangle be of the same species
and the angle included acute, the third side is less than a quadrant ; but if the two sides be of different species, and the included angle obtuse, the third side is greater than a quadrant. OR,
When two angles are of the same species, and the included side. greater than a quadrant, the third angle is obtuse ; but if the two angles be of different species and the included side less than a quadrant, the third angle is acute. DEMONSTRATION. Let the tri
B angles Bac and BDC be right-angled at A and D, then will bc be less than a quadrant (Q. 145.); but as the arc
D ABD approaches nearer to Act, it will diminish the angles A and D, and ci will consequently be less than cs, which has been proved to be less than a quadrant.
In the triangle cip right-angled, suppose at I, when ip is greater and ic less than a quadrant, the other side, or hypothenuse, cd is greater than a quadrant (Q. 145.); and as the angle dic increases, viz. when it becomes equal to dig, the side DC increases till it becomes equal to DG; hence the first part of the proposition is evident. And,
In the second part, the supplemental triangle (U. 137.) will have exactly the same properties as the first part of this proposition. G. E. D.
any right-angled spherical triangle ABC, if the sides be produced, viz. Ac to 1, AB to H, and bc to D, so that ci, CD, and Ah may be quadrants, or each 90°; and if from the point A, as a pole, the great circle HGFE be described, and from the point c, as a pole,
E IDE be drawn of such a length that GE may be a quadrant; then shall the triangles cGF and EDF have their respec
G tive sides and angles either equal to those of the triangle ABC, or they will be complements of each other, and
H eight right-angled spherical triangles
B will be formed, having (every two of them) equal angles at their bases.
DEMONSTRATION. Since A is the pole of the circle Hofe, AG and AĦ are quadrants and perpendiculars to FGH (H. 133.): and the arcs FGH, FCB, being each of them perpendicular to ABH, are each of them quadrants, and F is the pole of ABH: therefore the triangle cyf is right-angled at G; CG is the com
plement of ac, CF of BC; Bh, the complement of ab, is the measure of the angle F (D. 133.); and GH, the complement of FG, is the measure of the angle A.
Secondly. The arc GE being a quadrant, and perpendicular to cgi, E is therefore the pole of cgi, and Ei is a quadrant. The arcs CFD and cgi being quadrants, for c is the pole of IDE, are perpendicular to ide, therefore the triangle EdF is rightangled at D; Ed is the complement of in, and in measures the angle ACB equal to DCI (N. 135.); DF is equal to Bc; the hypothenuse EP is equal to the angle A, for it is equal to gh the measure of A, and for the same reason fed is equal to ac, and EFD is equal to the complement of AB. Lastly,
The right-angled triangles ABC, AHG, have the angle a common; FGC, FHB have the angle F common; cor, cid have the angle c common; and EDF, EiG have the angle E cominon. Q. E. D.
THE STEREOGRAPHIC PROJECTION OF THE SPHERE.
(M) 1. Definition. The Sterographic 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 great circle, to every point of the circle, to be projected, viz.
Conceive a point E (Plate IV. Fig. 8.) situated any where on the surface of a globe, and at the same time a plane cd, to stand at right angles to an imaginary line Eo, connecting the centre of the globe and the point E. Then, if an indefinite number of lines be supposed to be drawn from the point E, to every point of all the circles described upon the spherical surface, they will trace out upon the cutting plane cd a stereographical projection of the sphere.
(N) 2. The plane on which the points, lines, and circles of the sphere are represented, is called the plane of projection ; and the point from which all the lines are drawn through the several parts of the circles of the sphere, to this plane, is called the projecting point. (O) 3. The primitive circle is situated in the plane of
projection; and the projecting point, on the sphere, is one of the poles of this circle ; but on the plane of projection, the poles of the primitive are in its centre.
(Þ) 4. A circle, the plane of which is parallel to the plane of the primitive, is called a parallel circle, and is represented on
the plane of projection, by a circle parallel to, and comprehended within, the primitive.
(Q) 5. A circle, whose plane is perpendicular to the plane of the primitive, is called a right circle ; because, passing through the projecting point, its circumference becomes a straight line on the plane of projection.
(R) 6. A circle whose plane is oblique to the plane of the primitive, is called an oblique circle.
(S) 7. Lines drawn from the projecting point to every part of the circumference of a circle to be projected, will form the convex surface of a cone, whose vertex is the projecting point. Thus if AB be a diameter of a circle to be projected, and E the projecting point, then AEB will be a cone, of which E is the vertex,
(T) 8. All writers on conic-sections have demonstrated, that if a cone be cut by a plane ab parallel to the base AB, the section will be a circle; hence it follows, that all small circles parallel to the plane of projection will become circles in the projection. And the radii of all projected small circles, parallel to the primitive, will be equal to the semi-tangent of their distances from the remotest pole; thus ao=ob is the tangent of the angle ako, or the semi-tangent of the arc Ae where e is the remotest pole.
(U) 9. It is also shewn by writers on conic-sections that if a scalene, or oblique cone, be cut by a plane not parallel to the base, but in such a manner that the cone cut from the base towards the vertex be equi-angular with the original cone, the section shall be a circle.
Thus (Plate IV. Fig. 9.) if the cone ABE be so cut by a plane CD, that the angle cdE be equal to the angle BAE, and dce equal to ABE, then the section will be a circle, having cd for its diameter. And this is what is termed cutting a cone in a sub-contrary position.
Emerson's Conic Sections, Book I. Prop. 89th.
Proposition 1. (Plate IV. Fig. 10.) (W) Every circle of the sphere which does not pass through the poles of the primitive, is projected into a circle.
Letoc be the diameter of a circle to be projected on the plane of the primitive FB, from the point E. Lines from the point E to the circumference of that circle form a cone, whose triangular section is CED.
Now the extremity C, of the diameter cd, will be projected
in the plane of the primitive; therefore AB is the projected diameter.
Through p draw pd parallel to FB, that is, perpendicular to El, then will the arc Ed be equal to the arc ED; and since an angle at the circumference of a circle is measured by half the arc on which it stands, (Keith's Euclid 167 of VII). the angle ECD will be equal to the angle dde, or, which is the same thing, equal to the angle ABE, because AB is parallel to do.
Hence the cone Eco is equiangular with the cone EBA, and it is cut by the line AB in a subcontrary position, therefore (U.,153.) the section is a circle.
PROPOSITION 11. (Plate IV. Fig. 11.) (X) The distance of the centre of any oblique great circle from the centre of the primitive, is equal to the tangent of the angle formed on the sphere, by that circle and the primitive ; and the radius of the projected circle is equal to the secant of the same angle.
Let E be the projecting point, Fg the diameter of the primitive Efes, and co the diameter of a circle to be projected. The point c will appear at A, and D at B; therefore, by the 1st proposition, AB is the projected diameter. Bisect AB in p, and p will be the centre of the projected circle CAEB.
Now CED=CEB, being an angle in a semi-circle, is a right angle; and EPA and EPB are likewise right angles; hence the triangles AEB, APE and EPB are equiangular and similar. The angle epc is double of the angle eec, the one being at the centre and the other at the circumference of the circle, and for the same reason ape is double of ABE; but the angle eec has been shewn to be equal to ABE, therefore EPC=PPE, and CPF=PEP: and it is plain that pp is the tangent of Pep to the radius of the sphere, therefore it is equal to the tangent of the angle CPF, which is the spherical angle formed by the circle cd and the primitive FG.
It is very obvious that ap=ep, the radius of the projected circle, is the secant of the angle Pep or CPF.
PROPOSITION III. (Plate IV. Fig. 11.) (Y) The distance of the pole of any oblique great circle from the centre of the primitive, is equal to the semi-tangent of the ongle which that circle makes on the sphere with the primitive.
Let E be the projecting point, FG the diameter of the primitive, and co the diameter of the circle to be projected.
Make eM=CF, then cm will be a quadrant; draw MN, which