Axis

“ If the major axis and foci be given, there is a convenient mechanical construction based on the property that the sum of the focal distances of any point is constant and equal to the major axis. Let AA′ be the axis and S, S′ the foci. ”

“ The diameter or axis of the figure X being drawn, the infinitesimal elements taken are parallel sections of X in general, but not always, at right angles to the axis or diameter, so that the centres of gravity of all the sections lie at one point or other of the axis or diameter and their weights can therefore be taken as acting at the several points of the diameter or axis. ”

“ Weiss, in 1809, published a Dissertation On the mode of investigating the principal geometrical character of crystalline forms, in which he says,31 “No part, line, or quantity, is so important as the axis; no consideration is more essential or of a higher order than the relation of a crystalline plane to the axis;” and again, “An axis is any line governing the figure, about which all parts are similarly disposed, and with reference to which they correspond mutually.” ”

“ When the elevation is determined by sights the angle of elevation is the angle between the line of sight and the axis of the piece, when these lines are in the same vertical plane,—or the angle between the line of sight and a plane containing the axis of the piece and a horizontal line intersecting it at right angles, when they are not. The graduations of tangent scales and fixed breech-sights give this angle in degrees. The graduation of the pendulum hausse gives the angle correctly only when the line of sight is horizontal. ”

“ This turning, as the plane creature conceives it, we should call turning about an axis perpendicular to the plane. What he calls turning about the null point we call turning about the white line as it stands out from his plane. There is no such thing as turning about a point, there is always an axis, and really much more turns than the plane being is aware of. ”

“ If the axis be so circumstanced that the pull is at any other angle to it, then the change of axis being always perpendicular to the pull, one component of the change rotates the axis as before, the other alters its inclination. ”

“ The figure below represents a model illustrating the rotation of the earth on its axis, and the concurrent revolution of the sun round the earth once a year. To represent the downward force it is perfectly fair if I add a weight. The moment this is done the axis of the gyroscope representing the earth's axis, instead of retaining its direction to the same point as it did before, now describes a circle round the pole of the heavens. ”

“ Let Δs denote the arc QP, and ψ the angle which the tangent at P to the varied curve, in the sense from A to B, makes with the axis of x (fig. 5) . Also let φ be the angle which the tangent at P to the stationary curve AP, in the sense from A′ to P, makes with the axis of x. We evaluate (A′Q) − (A′P) approximately by means of a result which we obtained in connexion with the problem of variable limits. ”

“ Let us imagine the cube in our space, which is the base of a tesseract, to turn about one of its edges. The rotation will carry the whole body with it, and each of the cubic sections will rotate. The axis we see in our space will remain unchanged, and likewise the series of axes parallel to it about which each of the parallel cubic sections rotates. The assemblage of all of these is a plane.
Hence in four dimensions a body rotates about a plane. There is no such thing as rotation round an axis. ”

“ If any line in a plane revolves about an external axis in the plane, the area of the curved surface generated by the revolution is equal to the product of the length of the line and the distance travelled by the centroid of the line. ”

“ If two of the principal axes through the centre of gravity have equal moments, every axis in their plane has the same moment, and is to be considered equally as a principal axis. In this case the body would revolve on any of these axes without pressure. ”

“ Lines balance when in opposite attitudes. We get Balance in all inversions, whether single or double.
Fig. 129
Here similar lines are drawn in opposite attitudes and we get Measure and Shape-Balance. In the above case the axis of balance is vertical. The balance is, therefore, symmetrical. Symmetrical Balance is obtained by the single inversion of any line or lines on a vertical axis. Double inversion gives a Balance of Measures and Shapes on a center. We have no Symmetry in double inversions. All this has been explained.
Fig. 130
We have Measure and Shape-Balance on a center in this case. ”

“ When one of the axes of elasticity occurs in the plane of the plate, one of the nodal figures always consists of two straight lines, which intersect each other at right angles, and one of which invariably places itself in the exact direction of this axis ”

“ A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis—considered as a circle of infinite radius. ”

“ If the medium has any axis of symmetry, the ellipsoid or hyperboloids which represent the values of will have an axis in the same direction. If the medium after a revolution of less than 180° about any axis is equivalent to the medium in its first position, the ellipsoid or hyperboloids will have an axis of revolution in that direction. ”

“ The revolving axis communicates its motion to the indices, which are so adjusted that the index on the right-hand dial passes through a division for every fathom of vertical descent whether quick or slow, and makes an entire revolution for 15 fathoms; while the left-hand index passes through a division on the circle for 15 fathoms, and makes an entire revolution during a descent of 225 fathoms. ”

“ Imagine curve lines to be described on any plane, and to revolve about any given axes passing through the centre of force, and by that revolution to describe curve superficies; and that the bodies move in such sort that their centres may be always found in those superficies. If those bodies reciprocate to and fro with an oblique ascent and descent, their motions will be performed in planes passing through the axis, and therefore in the curve lines, by whose revolution those curve superficies were generated. ”

“ It is a matter of common knowledge that the mathematical geometrician defines the position of any point in space with reference to three planes, which in the simplest case are all mutually at right angles to each other like the faces of a cube, and which intersect in three rectangular axes a, b, c, the third c being the vertical axis, b the lateral one, and a the front-and-back axis. ”

“ The axis, or axle, of the earth is only an unseen line which scientists have thought of as if it ran straight through the center of the earth. The place in the middle of the top of the globe where this line, if there were one, would come out, is named the North Pole, and the same place at the opposite end is called the South Pole. ”

“ If a point lies in the horizontal plane, its elevation lies in the axis and the plan coincides with the point itself. If a point lies in the vertical plane, its plan lies in the axis and the elevation coincides with the point itself. ”

“ Let x and y be the cartesian coordinates of a point as determined by A, the axes of reference being the major and minor axes of the ellipse in which the balls move. Let and be the coordinates of the same point as determined by B. Then the circular path of motion, as determined by B, has the equations
the angle θ being measured from the major axis of the ellipse. ”

“ Thus, if a cone be cut by a plane at right angles to its axis we get a circle. If, instead of being perfectly at right angles, the plane subtends with the axis an angle of 89° 59', we have an ellipse, which no human eye, even when aided by an accurate pair of compasses, can distinguish from a circle. ”

“ The top, of course, turns round its axis very fast, and this is like the Earth turning round its axis every 24 hours.
Now, if you place the blunt end of a pencil on the upper axis of the spinning top, as shown in Fig. 77, and try to tilt it in some other direction than that it took when it began to spin, you will find it rather a hard thing to do. In other words, once a body is rapidly turning on its own axis it very strongly tends to keep its axis pointing in the same direction. ”

“ A cone is cut by every plane through the vertex in lines, and if it is a cone of revolution by planes perpendicular to the axis in circles.
A cylinder is cut by every plane parallel to the axis in lines, and if it is a cylinder of revolution by planes perpendicular to the axis in circles.
A sphere is cut by every plane in a circle. ”

“ We may take a horizontal line as an axis of abscissas, and lay off as an abscissa measured to the right of any origin, and erect as a corresponding ordinate, thus determining a certain curve. Again, taking a different origin, we may lay off as abscissas measured to the left, and determine a second curve by erecting the ordinates . ”

“ If a wheel stand upon the inside of a concave globe at right angles thereto, and revolving about its own axis go forward in one of the great circles of the globe, the length of the curvilinear path which any point, given in the perimeter of the wheel, hath described since it touched the globe, will be to the double of the versed sine of half the arc which in all that time has touched the globe in passing over it, as the difference of the diameters of the globe and the wheel to the semi-diameter of the globe. ”

“ That, if any Plain Surface have allocation about its Axis in any Situation whatsoever, and at any distance whatsoever, or none, it produceth a Round Solid equal to an Upright Solid, whole Base is the begetting Figure, and Height is equal to the Circumference described by its Center of Gravity. ”

“ Thus every diameter of a conic is an axis of skew-symmetry, the conjugate direction being the direction of the chords which it bisects. § 20. We state a few properties of these figures useful in mechanics, but we omit the easy proofs:-
I f 'a plane area has an axis of skew-symmetry, then the mass-centre (centre of mean distances or centre of inertia) lies on it. If a figure undergoes a shear, the mass-centre of its (area remains the mass-centre; and generally-In
parallel projection the mass-centres of corresponding areas (or of groups of points, but not of curves) are corresponding points. ”