PROP. IV.

The angle formed by two arcs of great circles, is equal to the angle contained by the tangents drawn to these arcs at their point of intersection, and is measured by the arc described from their point of intersection or pole, intercepted by the arcs containing the angle.

Let ZPN, ZQN, arcs of great circles, intersect in Z.

Draw ZT, ZT, tangents to the arcs at the point Z.

With Z as pole, describe the arc PQ.

Take O the centre of the sphere, and join OP, OQ.

Then, the spherical angle PZQ is equal to the angle TZT', and is measured by the arc PQ.

For the tangent ZT drawn in the plane ZPN, is perpendicular to radius OZ.

N

T

And the tangent ZT' drawn in the plane ZQN, is perpendicular to radius OZ.

Hence, the angle TZT' is equal to the angle contained by these two planes, that is, to the spherical angle PZQ. (Geom. of Planes).

Again, since the arcs ZP, ZQ, are each of them equal to a quadrant;

.. Each of the angles ZOP, ZOQ, is a right angle,

.. The angle QOP is the angle contained by the planes ZPN, ZQN, and is =TAT.

.. The arc PQ, which measures the angle POQ, measures the angle between the planes, that is, the spherical angle PZQ.

Cor. 1. The angle under two great circles is measured by the distance between their poles. For the axis of the great circles drawn through their poles being perpendicular to the planes of the circles, the angles under these axes will be equal to the angle between the circles; but the angle under the axes is obviously measured by the arc which joins their extremities, that is, by the distance between their poles.

Cor. 2. The angle under two great circles is measured by the arc of a common secondary intercepted between them.

For, since the secondary passes through the poles of both, taking away from the equal quadrants of the secondary between each circle and its pole, the common arc intercepted between one circle and the pole of the other, the remainders are the intercept of the common secondary between the two circles, and the distance between their poles, and these are therefore equal. But the latter is, by the last Cor., the measure of the angle.

Cor. 3. Vertical spherical angles, such as QPW, QPS, are equal, for each of them is the angle formed by the planes QPS, WPR.

Also, when two arcs cut each other, the two adjacent angles QPW, QPR, when taken together, are always equal to two right angles.

W

R

PROP. V.

If from the angular points of a spherical triangle considered as poles, three arcs be described forming another triangle, then, reciprocally, the angular points of this last triangle will be the poles of the sides opposite to them in the first.

Let ABC be a spherical triangle.

From the points A, B, C, considered as poles, describe the arcs B'C', A'C', A'B', forming the spherical triangle A'B'C'.

Then, A' will be the pole of BC, B' of AC, and C' of AB.

For, since B is the pole of A'C', the distance from B to A' is a quadrant.

And, since C is the pole of A'B', the distance from C to A' is a quadrant.

Thus, it appears that the point A' is distant by a quadrant from the points B and C.

.. A' is the pole of the arc BC.

Similarly, it may be shown that B' is the pole of AC, and C' the pole of AB.

PROP. VI.

The same things being given as in the last proposition, each angle in either of the triangles will be measured by the supplement of the side opposite to it in the other triangle.

Produce the sides of the first triangle to D, E, F, G, H, K.

Then, since A is the pole of B'C', the angle A is measured by the arc EK.

For the same reason, the angles B and C are measured by the arcs DH and FG respectively. Because B' is the pole of FK, the arc B'K is a quadrant.

Because C' is the pole of DE, the arc C'E is a quadrant.

D

C'

H

K

C

B

F

B

G

But the arcs EK, DH, FG, are the measures of the angles A, B, C, respectively,... 180° B'C', 180° A'C', 180°

A'B', or the supplements of B'C',

A'C', and A'B', are the measures of these angles.

Again, since A' is the pole of HG, the angle A' is measured by GH.

For the same reason, the angles B', C', are measured by the arcs FK and

DE respectively.

Because B is the pole of A'C', the arc BH is a quadrant

Because C is the pole of A B', the arc CG is a quadrant.

And GH, FK, DE, are the measures of the angles A, B, C, respectively. These triangles ABC, A'B'C', are, from their properties, usually called Polar triangles, or Supplemental triangles.

PROP. VII.

In any spherical triangle any one side is less than the sum of the two others.

Let ABC be a spherical triangle, O the centre of the sphere. Draw the radii OA, OB, OC.

Then the three plane angles AOB, AOC, BOC, form a solid angle at the point O, and these three angles are measured by the arcs AB, AC, BC.

But each of the plane angles which form the solid angle, is less than the sum of the two others.

Hence each of the arcs AB, AC, BC, which measures these angles, is less than the sum of the two others.

C

PROP. VIII.

The sum of the three sides of a spherical triangle is less than the circumference

of a great circle.

Let ABC be any spherical triangle.

Produce the sides AB, AC, to meet in D.

Then, since two great circles always bisect each other (Prop. 1, cor.) the arcs ABD, ACD, are semicircles.

Now, in the triangle BCD,

BCBD + DC, by Prop. vII.;

.: AB+ AC+ BC ≤ AB + BD + AC + DC

ABD + ACD

circumference of great circle.

D

OF MOTION, FORCES, &c.

DEFINITIONS.

Art. 1. BODY is the mass, or quantity of matter, in any material substance; and it is always proportional to its weight or gravity, whatever its figure may be. 2. Body is either Hard, Soft, or Elastic. A Hard Body is that whose parts do not yield to any stroke or percussion, but retains its figure unaltered. A Soft Body is that whose parts yield to any stroke or impression, without restoring themselves again,—the figure of the body remaining altered. And an Elastic Body is that whose parts yield to any stroke, but which presently restore themselves again, and the body regains the same figure as before the stroke.

We know of no bodies that are absolutely or perfectly either hard, soft, or elastic; but all partaking these properties, more or less, in some intermediate degree.

3. Bodies are also either Solid or Fluid. A Solid Body, is that whose parts are not easily moved amongst one another, and which retains any figure you give it. But a Fluid Body is that whose parts yield to the slightest impression, being easily moved amongst one another; and its surface, when left to itself, is always observed to settle in a smooth plane at the top.

4. Density is the proportional weight or quantity of matter in any body. So, of two spheres, or cubes, &c., of equal size or magnitude; if the one weigh only one pound, but the other two pounds, then the density of the latter is double the density of the former; if it weigh three pounds, its density is triple, and so on.

5. Motion is a continual and successive change of place. If the body move equally, or pass over equal spaces in equal times, it is called Equable or Uniform Motion; but if it increase or decrease, it is Variable Motion,—and it is called Accelerated Motion in the former case, and retarded motion in the latter. Also, when the body moved is considered with respect to some other body at rest, it is said to be Absolute Motion; but when compared with others in motion, it is called Relative Motion.

6. Velocity, or Celerity, is an affection of motion, by which a body passes over a certain space in a certain time. Thus, if a body in motion pass uniformly over 40 feet in 4 seconds of time, it is said to move with the velocity of ten feet per second; and so on.

7. Momentum, or Quantity of motion, is the power or force incident to moving bodies, by which they continually tend from their present places, or with which they strike any obstacle that opposes their motion.

8. Force is a power exerted on a body to move it. If the force act constantly, or incessantly, it is a Permanent force,-like pressure or the force of gravity; but if it act instantaneously, or but for an imperceptibly small time, it is called Impulse, or Percussion,-like the smart blow of a hammer.

9. Forces are also distinguished into Motive, and Accelerative or Retarding. A Motive or moving force, is the power of an agent to produce motion; and it

is equal or proportional to the momentum it will generate in any body, when acting, either by percussion, or for a certain time as a permanent force.

10. Accelerative, or Retardive force, is commonly understood to be that which affects the velocity only; or it is that by which the velocity is accelerated or retarded; and it is equal or proportional to the motive force directly, and to the mass or body moved inversely.—So, if a body of 2 pounds weight, be acted on by a motive force of 40; then the accelerating force is 20. And if the same force of 40 act on another body of 4 pounds weight; then the accelerating force in this latter case is only 10: and so is but half the former.

11. Gravity, or Weight, is that force by which a body endeavours to fall downwards. It is called Absolute Gravity, when the body is in empty space; and Relative Gravity, when immersed in a fluid.

12. Specific Gravity is the proportion of the weights of different bodies of equal magnitude; and so is proportional to the density of the body.

AXIOM S.

13. EVERY body naturally endeavours to continue in its present state, whether it be at rest, or moving uniformly in a right line.

14. The Change or Alteration of Motion, by any external force, is always proportional to that force, and in the direction of the right line in which it acts. 15. Action and Re-action, between any two bodies, are equal and contrary that is, by action and re-action, equal changes of motion are produced in bodies acting on each other, and these changes are directed towards opposite or contrary parts.

GENERAL LAWS OF MOTION, &c.

PROP. I.

16. The quantity of matter, in all bodies, is in the compound ratio of their magnitudes and densities.

That is, b is as md; where b denotes the body or quantity of matter, m its magnitude, and d its density.

For, by article 4, in bodies of equal magnitude, the mass or quantity of matter is as the density. But, the densities remaining, the mass is as the magnitude; that is, a double magnitude contains a double quantity of matter, a triple magnitude a triple quantity, and so on. Therefore, the mass is in the compound ratio of the magnitude and density.

17. Corol. 1. In similar bodies, the masses are as the densities and cubes of the diameters, or of any like linear dimensions. For the magnitudes of bodies are as the cubes of the diameters, &c.

18. Corol. 2. The masses are as the magnitudes and specific gravities. For, by articles 4 and 12, the densities of bodies are as the specific gravities.

19. SCHOLIUM.-Hence, if b denote any body, or the quantity of matter in it, m its magnitude, d its density, g its specific gravity, and a its diameter or other dimension; then, co being the mark for general proportion, from this proposition and its corollaries we have these general proportions: