AXIOM

, a self-evident truth, or a proposition assented to by every person at first sight. Such as, that the whole is greater than its part; that a thing cannot both be and not be at the same time; and that from nothing, nothing can arise.

Some Axioms are in effect, strictly speaking, no other than identical propositions. Thus, to say that all right angles are equal to each other, is as much as to say, all right angles are right angles; such equality being implied in the very definition, or the very name or term itself.

Axiom is also an established principle in some art or science. Thus, it is an axiom in physics, that nature does nothing in vain; that effects are proportional to their causes; &c. It is an axiom in geometry, that two things equal to the same third thing, are also equal to each other; that if to equal things equals be added, the sums will be equal. And it is an axiom in optics, that the angle of incidence is equal to the angle of reflection. In this sense also the general laws of motion are called axioms; as, that all motion is rectilinear, that action and reaction are equal, &c.

AXE or AXIS, in Geometry, the straight line in a plane figure, about which it revolves, to produce or generate a solid. Thus, if a semicircle be moved round its diameter at rest, it will generate a sphere, whose axis is that diameter. And if a right-angled triangle be turned about its perpendicular at rest, it will describe a cone, whose axis is that perpendicular.

Axis is yet more generally used for a right line conceived to be drawn from the vertex of a figure to the middle of the base. So the

Axis of a circle or sphere, is any line drawn through the centre, and terminated at the circumference, on both sides.

Axis of a cone, is the line from the vertex to the centre of the base.

Axis of a cylinder, is the line from the centre of the one end to that of the other.

Axis of a conic section, is the line from the principal vertex, or vertices, perpendicular to the tangent at that point. The ellipse and hyperbola have each two axes, which are sinite and perpendicular to each other; but the parabola has only one, and that infinite in length.

Transverse Axis, in the Ellipse and Hyperbola, is the diameter passing through the two foci, and the two principal vertices of the figure. In the hyperbola it is the shortest diameter, but in the ellipse it is the longest.

Conjugate Axis, or Second Axis, in the Ellipse and Hyperbola, is the diameter passing through the centre, and perpendicular to the transverse axis; and is the shortest of all the conjugate diameters.

, of a curve line, is still more generally used for that diameter which has its ordinates at right angles to it, when that is possible. For, like as in the conic sections, any diameter bisects all its parallel ordinates, making the two parts of them on both sides of it equal; and that diameter which has such ordinates perpendicular to it, is an Axis: So, in curves of the second order, if any two parallel lines each meeting the curve in three points; the right line which cuts these two parallels so, that the sum of the two parts on one side of the cutting line, between it and the curve, is equal to the third part terminated by the curve on the other side, then the said line will in like manner cut all other parallels to the former two lines, viz, so that, of every one of them, the sum of the two parts, or ordinates, on one side, will be equal to the third part or ordinate on the other side. Such cutting line then is a diameter; and that diameter whose parallel ordinates are at right angles to it, when possible, is an Axis. And the same for other curves of still higher orders. Newton, Enumeratio Linearum Tertii Ordinis, sect. 2, art. 1.

, in Astronomy. As, the Axis of the world, is an imaginary right line conceived to pass through the centre of the earth, and terminating at each end in the sursace of the mundane sphere. About this line, as an axis, the sphere, in the Ptolomaic system, is supposed daily to revolve.

Axis of the Earth, is the line connecting its two poles, and about which the earth performs its diurnal rotation, from west to east. This is a part of the axis of the world, and always remains parallel to itself during the motion of the earth in its orbit about the sun, and perpendicular to the plane of the equator.

Axis of a Planet, is the line passing through its centre, and about which the planet revolves.—The Sun, Earth, Moon, Jupiter, Mars, and Venus, it is known from observation, move about their several axes; and the like motion is easily inferred of the other three, Mercuty, Saturn, and Georgian planet.

Axis of the Horizon, Equator, Ecliptic, Zodiac, &c, are right lines passing through the centres of those circles, perpendicular to their planes.

Axis of a Magnet, or magnetical Axis, is a line passing through the middle of a magnet, lengthwise; in such manner, that however the magnet be divided, provided the division be made according to a plane passing through that line, the magnet will then be cut into two loadstones. And the extremities of such lines are called the poles of the stone.

Axis in Mechanics.—The axis of a balance, is the line upon which it moves or turns.

Axis of Oscillation, is a line parallel to the horizon, passing through the centre about which a pendulum vibrates, and perpendicular to the plane in which it oscillates.

Axis in Peritrochio, or wheel and axle, is one of the five mechanical powers, or simple machines; contrived chiefly for the raising of weights to a consider- | able height, as water from a well, &c. This machine consists of a circle AB, concentric with the base of a cylinder, and moveable together with it about its axis CD. This cylinder is called the axis; and the circle, the peritrochium; and the radii, or spokes, which are sometimes fitted immediately into the cylinder, without any circle, the scytalæ. About the axis winds a rope, or chain, by means of which great weights are raised by turning the wheel.—The axis in peritrochio takes place in the motion of every machine, in which a circle may be conceived as described about a fixed axis, concentric with the plane of a cylinder about which it is placed; as in Crane wheels, Mill wheels, Capstans, &c.

The chief properties of the Axe-in-peritrochio, are as follow:

1. If the power F applied in the direction AF a tangent to the circumference, or perpendicular to the spoke, be to a weight E, as the radius of the axis Ce is to the radius of the wheel AD, or the length of the spoke; the power will just sustain the weight; that is, the power and the weight will be in equilibrio, when they are in the reciprocal proportion of their distances from the centre.

2. When the wheel moves, with the power and weight; the velocities of their motion, and the spaces passed over by them, will be both in the same proportion as above, namely, directly proportional to their distances from the centre, and reciprocally proportional to their own weights when they are in equilibrio.

3. A power, and a weight, being given to construct an axis-in-peritrochio, by which it shall be sustained and raised. Let the axis be taken large enough to support the weight and power without breaking: then, as the weight is to the power, so make the radius of the wheel to the radius of the axis. Hence, if the power be very small in respect of the weight, the radius of the wheel will be vastly great. For example, suppose the weight 4050, and the power only 50; then the radius of the wheel will be 81 times that of the axis; which would be a very inconvenient fize. But this inconvenience is provided against by increasing the number of the wheels and axes; making one to turn another, by means of teeth or pinions. And to find the effect of a number of wheels and axes, thus turning one another, multiply together, all the radii of the axes, and all the radii of the wheels, and then it will be, as the product of the former is to the product of the latter, so is the power to the weight. So, if there be 4 wheels and axes, the radius of each axis being 1 foot, and the radius of each wheel 3 feet; then the continual product of all the wheels is 3X3X3X3 or 81 feet, and that of the axis only 1; therefore the effect is as 81 to 1, or the weight is 81 times the power. And, on the contrary, if it be required to find the diameter of each of four equal wheels, by which a weight of 4050lb shall be balanced by a power of 50lb, the diameter of each axis being 1 foot: dividing 4050 by 50, the quotient is 81; extract the 4th root of 81, or twice the square root, and it will give 3, for the diameter of the four wheels sought.

Axis of a vessel, is that quiescent right line passing through the middle of it, perpendicular to its base, and equally distant from its sides.

Axis in Optics.—Optic Axis, or visual axis, is a ray passing through the centre of the eye, or falling perpendicularly on the eye.

Axis of a lens, or glass, is the axis of the solid of which the lens is a segment. Or the axis of a glass, is the line joining the two vertices or middle points of the two opposite surfaces of the glass.

Axis of Incidence, in Dioptrics, is the line passing through the point of incidence, perpendicularly to the refracting surface.

Axis of Refraction, is the line continued from the point of incidence or refraction, perpendicularly to the refracting surface, along the further medium.