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.

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.; A

.: AB+ AC+ BC ≤ AB + BD + AC + DC

ABD+ACD

circumference of great circle.

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, ∞ being the mark for general proportion, from this proposition and its corollaries we have these general proportions:

20. The momentum, or quantity of motion, generated by a single impulse, or any momentary force, is as the generating force.

That is, m is asƒ; where m denotes the momentum, and ƒ the force. For, every effect is proportional to its adequate cause. So that a double force will impress a double quantity of motion; a triple force, a triple motion; That is, the motion impressed, is as the motive force which pro

and so on.

duces it.

PROP. III.

21 The momenta, or quantities of motion, in moving bodies, are in the compound ratio of the masses and velocities.

'That is, m is as bv. For, the motion of any body being made up of the motions of all its parts, if the velocities be equal, the momenta will be as the masses; for a double mass will strike with a double force; a triple mass, with a triple force; and so on. Again, when the mass is the same, it will require a double force to move it with a double velocity, a triple force with a triple velocity, and so on; that is, the motive force is as the velocity; but the momentum impressed, is as the force which produces it, by Prop. II.; and therefore the momentum is as the velocity when the mass is the same. But the momentum was found to be as the mass when the velocity is the same. Consequently, when neither are the same, the momentum is in the compound ratio of both the mass and velocity.

PROP. IV.

22. In uniform motions, the spaces described are in the compound ratio of the velocities and the times of their description.

That is, s is as tv.

For, by the nature of uniform motion, the greater the velocity, the greater is the space described in any one and the same time; that is, the space is as the velocity, when the times are equal. And when the velocity is the same, the space will be as the time; that is, in a double time a double space will be described; in a triple time, a triple space; and so on. Therefore universally, the space is in the compound ratio of the velocity, and the time of description.

23. Corol. 1. In uniform motions, the time is as the space directly, and velocity reciprocally; or as the space divided by the velocity. And when the velocity is the same, the time is as the space; but when the space is the same, the time is reciprocally as the velocity.

24. Corol. 2. The velocity is as the space directly and the time reciprocally; or, as the space divided by the time. And when the time is the same, the velocity is as the space; but when the space is the same, the velocity is reciprocally as the time.

25. SCHOLIUM.-In uniform motions generated by momentary impulse, Let bany body or quantity of matter to be moved,

ƒ = force of impulse acting on the body b,

v

the uniform velocity generated in b,

m = the momentum generated in b,

the space described by the body b,

t = the time of describing the space s with the velocity v.

Then from the last three propositions and corollaries, we have these three general proportions, namely, ƒ m, m œ bv, and sa tv; from which is derived the following table of the general relations of those six quantities, in uniform motions, and impulsive or percussive forces:

By means of which, may be resolved all questions relating to uniform motions, and the effects of momentary or impulsive forces.

PROP. V.

26. The momentum generated by a constant and uniform force, acting for any time, is in the compound ratio of the force and time of acting.

That is, m is as ft,

For, supposing the time divided into very small parts, by Prop. 11., the momentum in each particle of time is the same, and therefore the whole momentum will be as the whole time, or sum of all the small parts. But, by the same proposition, the momentum for each small time, is also as the motive force; consequently, the whole momentum generated, is in the compound ratio of the force and time of acting.

27. Corol. 1. The motion, or momentum, lost or destroyed in any time, is also in the compound ratio of the force and time. For whatever momentum any force generates in a given time, the same momentum will an equal force destroy in the same or equal time, acting in a contrary direction.

And the same is true of the increase or decrease of motion, by forces that conspire with, or oppose the motion of bodies.

28. Corol. 2. The velocity generated or destroyed, in any time, is directly as the force and time, and reciprocally as the body or mass of matter. For, by this and the third proposition, the compound ratio of the body and velocity, is as that of the force and time; and therefore the velocity is as the force and time divided by the body. And if the body and force be given, or constant, the velocity will be as the time.