angles of a spherical triangle are greater than two, and less than six, right angles. XIII. If to the circumference of a great circle, from a point in the surface of a sphere, which is not the pole of that circle, arches of great circles be drawn; the greatest of these arches is that which passes through the pole of the first mentioned circle, and the supplement of it is the least; and of the other arches, that which is nearest to the greatest is greater than that which is more remote. XIV. In a right angled spherical triangle, the sides containing the right angle are of the same affection with the angles opposite to them, that is, if the sides be greater or less than quadrants, the opposite angles will be greater or less than right angles, and conversely. XV. If the two sides of a right angled spherical triangle about the right angle be of the same affection, the hypothenuse will be less than a quadrant; and if they be of different affection, the hypothenuse will be greater than a quadrant. COR. 1. Hence conversely, if the hypothenuse of a right angled triangle be greater or less than a quadrant, the sides will be of different or the same affection. 2. Since the angles of an oblique angled spherical triangle have the same affection with the opposite sides, therefore, according as the hypothenuse is greater or less than a quadrant, the oblique angles will be of different or of the same affection. 3. Because the sides are of the same affection with the opposite angles, therefore when an angle and the side adjacent are of the same affection, the hypothenuse is less than a quadrant; and conversely. XVI. In any spherical triangle, if the perpendicular upon the base from the opposite angle fall within the triangle, the angles at the base are of the same affection; and if the perpendicular fall without the triangle, the angles to the base are of different affection. COR. Hence, if the angles be of the same affection, the perpendicular will fall within the base; for if it did not they would be of dif ferent affection. And if the angles be of different affec. tion, the perpendicular will fall without the triangle; for, if it did not, the angles would be of the same affection, contrary to the supposition. XVII. If to the base of a spherical triangle a perpendicular be drawn from the opposite angle, which either falls within the triangle, or is the nearest of the two that fall without; the least of the segments of the base is adjacent to the least of the sides of the triangle, or to the

greatest, according as the sum of the sides is less or greater than a semicircle. XVIII. In right angled spherical triangles, the sine of either of the sides about the right angle, is to the radius of the sphere, as the tangent of the remaining side is to the tangent of the angle opposite to that side. XIX. In right angled spherical triangles the sine of the hypothenuse is to the radius as the sine of either side is to the sine of the angle opposite to that side. XX. In a right angled spherical triangles, the cosine of the hypothenuse is to the radius as the cotangent of either of the angles is to the tangent of the remaining angle. XXI. In right angled spherical triangles, the cosine of an angle is to the radius as to the tangent of the side adjacent to that angle is to the tangent of the hypothenuse. XXII. In right angled spherical triangles, the cosine of either of the sides is to the radius, as the cosine of the hypothenuse is to the other side. XXIII. In right angled spherical triangles, the cosine of either of the sides is to the radius, as the cosine of the angle opposite to that side is to the sine of the other angle. XXIV. In spherical triangles, whether right an gled or oblique angled, the sines of the sides are proportional to the sines of the angles opposite to them. XXV. In oblique angled spherical triangles, a perpendicular arch being drawn from any of the angles upon the opposite side, the cosines of the angles at the base are proportional to the sines of the segments of the vertical angle. XXVI. The same things remaining, the cosines of the sides are proportional to the ccsines of the segments of the base. XXVII. The same construction remaining, the sines of the segments of the base are reciprocally proportional to the tangents of the angles at the base.

MECHANICS.

MECHANICS is that branch of natural philosophy which treats of the equilibrium and motion of bodies. The two es sential properties of matter, both of which are inseparable from it, are extension and impenetrability. Extension in the three dimensions of length, breadth and thickness, belongs to matter under all circumstances; and impenetrability, or

the property of excluding all other matter from the space which it occupies, appertains alike to the largest body and to the smallest particles.

GRAVITY, is that property by which all terrestrial bodies tend towards the center of the earth. Gravity is a property of matter universally; and the force of gravity in any body is proportioned to its quantity of matter. Gravity, at different distances from the earth, varies inversely as the squares of the distance from its center. A body situated within a hollow sphere, would remain at rest in any part of the void. The force of gravity below the earth's surface is, at different distances from the center, directly as their dis

tances.

INERTIA. By the inertia of matter is meant, its resistance to a change of state, whether of rest or motion. The inertia of a body is proportioned to its quantity of matter, and of course to its weight.

SPACE, TIME and VELOCITY. The space described by a body moving with a uniform velocity, increases in a compound ratio of the time and velocity of its motion. The space equals the product of the time into the velocity. The time equals the space divided by the velocity. The velocity equals the space divided by the time. The momentum of a body is its quantity of motion, and is the product of its quantity of matter and velocity. The momentum equals the product of the quantity of matter into the velocity. The quantity of matter, equals the momentum divided by the ve locity. The velocity equals the momentum divided by the quantity of matter. If two bodies move with velocities which are inversely as their quantities of matter, then their momenta will be equal. Force is any cause which moves or tends to move a body, or which changes, or tends to change its motion. If a force acts instantaneously, and then ceases, it is called an impulsive force. When a force acts incessantly, it is called a constant force.

LAWS OF MOTION I. A body continues always in a state of rest, or of uniform rectilinear motion, till by some external force, it is made to change its state. II. Motion, or change of motion, is proportional to the force impressed, and is produced in the right line in which that force acts. III. When bodies act upon each other, action and reaction are equal and in opposite directions.

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FALLING BODIES. The spaces described by bodies falling from rest under the influence of gravity, are to each other as the squares of the times, in which they are described, or as the squares of the last acquired velocities, or as the times and last acquired velocities conjointly. If a body after it has fallen from rest through any space, should then proceed on uni. formly with the last acquired velocity, it would describe twice the space in the same time, as that in which it has fallen to acquire that velocity. The space described in any time by a body projected downward with a given velocity, is equal to the space which would be described with that velocity continued uniformly for that length of time together with the space through which a body would fall from rest by the action of gravity in the same time. The space described by a body ascending for a given time is equal to the difference between the space which would be described by the body moving uniformly for that time with the velocity of projection, and the space through which a body would fall from rest, by the action of gravity in the same time. To find the space fallen through in feet. Multiply the square of the time by 16, or divide the square of the velocity by 641. To find the time of falling. Divide the velocity by 321, or divide the space by 16, and take the square root of the quotient. To find the velocity. Multiply the time by 321; or multiply the space by 16, and double the square root of the product.

Composition and resolution of motion. Two impulses which when communicated separately to a body, would make it describe the adjacent sides of a parallelogram in a given time, will, when they are communicated at the same instant, cause it to describe the diagonal in that time; and the motion of the diagonal will be uniform. If a body acted upon by two forces, one of which would cause it to move uniformly over one side of a triangle, and the other over another side of the triangle, then by the joint action of those forces it will be made to describe the third side, in the same time that it would have described either of the sides by the forces acting separately. If a body be impelled by any number of forces which acting separately, would, in a given time make it describe all the sides of a polygon except the last side; when all the forces act at the same instant, it will be made to describe the remaining side in the same given

time. If all the sides of a polygon except the last, represent the quantity and direction of several forces, acting at the same instant upon a body, the remaining side will repre sent the quantity and direction of a force equivalent to them all. A given force may be resolved into an unlimited number of others acting in all possible directions. When the resolved forces are required to make a given angle with one another, the number of pairs of forces will be limited to the number of triangles which can be described in a segment of a circle containing the given angle, and described upon the line representing the given force. A body acted upon at the same time by three forces represented in quantity and direction by the three sides of a triangle taken in order, (or by lines parallel to these three,) will remain at rest. If a body be kept at rest by three forces, those three forces will be rep.. resented by the three sides of a triangle formed by lines drawn in their respective directions. The proportion of three forces which keep a body at rest will be represented -by the three sides of any triangle drawn parallel or perpendicular to the sides of the triangle which are in the direction of the forces. Any one of three forces which keep a body at rest, is as the sine of the angle included between the other two, and conversely. I. The resultant of two parallel forces is in a direction constituting another parallel; and is equal to their sum. II. If a straight line be drawn perpen. dicular to the directions of their three forces, (viz. the two components and their resultant,) each of the components will be represented by the part of the perpendicular contained between the direction of the two others. All the forces which can possibly act upon a body, may be resolved into equivalent forces acting in the direction of three straight lines or axes at right angles to each other.

CENTER OF GRAVITY. The center of gravity of a body is that point about which, if supported, all the parts of a body (acted upon only by the force of gravity,) would balance each other in any position. In regular plane figures, such as squares, parallelograms, circles, polygons inscribed in circles, &c. the center of gravity is the same with the center of the figure. Two weights or pressures, acting at the extremities of an inflexible rod, void of gravity, will be in equilibrio about a given point, when their distances from that point are to each other inversely as their weights or