Axis (geometry)

“ The Axis or Axeltree is commonly thought to be proper to the sphere or globe, as here ae: But it is attributed to other kindes of solids, as well as to that.
2. A right solid is that whose axis is perpendicular to the center of the base.
Thus Serenus and Apllonius doe define a Cone and a Cylinder: And these onely Euclide considered: Yea and indeed stereometry entertaineth no other kinde of solid but that which is right or perpendicular.
3. If solids be comprehended of homogeneall surfaces, equall in multitude and magnitude, they are equall. 10 d xj. ”

“ This is called the central axis; if d be the shortest distance, θ the angle between it and a ray of the complex, then d tan θ = p, where p is a constant called the pitch or parameter. Any system of forces can be reduced to a force R along a certain line, and a couple G perpendicular to that line; the lines of nul-moment 722 for the system form a linear complex of which the given line is the central axis and the quotient G/R is the pitch. Any motion of a rigid body can be reduced to a screw motion about a certain line, i.e. ”

“ 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. ”

“ Hence a couple of moment G, whose axis has the direction (l, m, n) relative to a right-handed system of rectangular axes, is equivalent to three couples lG, mG, nG in the co-ordinate planes. The analytical reduction of a three-dimensional system can now be conducted as follows. Let (x1, y1, z1) be the co-ordinates of a point P1 on the line of action of one of the forces, whose components are (say) X1, Y1, Z1. Draw P1H normal to the plane zOx, and HK perpendicular to Oz. In KH introduce two equal and opposite forces ±X1. ”

“ The reader will at once perceive the analogy to a well-known phenomenon in ordinary geometry. The equation in rectangular coordinates a; 2 + 7/ 2 =0 denotes either a circle of which the radius is zero or the pair of lines in the latter case we are obliged to admit that the distance of any point on either of these lines from their intersection is equal to zero. We have now to consider the displacement of a rigid figure, and we shall for the present speak only of a plane movement. ”

“ We may in fact take, as a varied curve, the arc AQ of a neighbouring stationary curve, the straight line QP drawn from Q to a point of the arc AB, and the arc PB of the stationary curve AB (fig. 6) . The sign of the variation is then the same as that of the function E (x, y, tan φ, tan ψ) , where (x, y) is the point P, ψ is the angle which the straight line QP makes with the axis of x, and φ is the angle which the tangent at P to the curve APB makes with the same axis. We thus arrive at a new necessary (but not sufficient) condition for the existence of an extremum of the integral ∫Fds, viz. ”

“ In this system the three crystallographic axes are all at right angles, but they are of different lengths and not interchangeable. The parameters, or axial ratios, are a : b : c, these referring to the axes OX, OY and OZ respectively. The choice of a vertical axis, OZ = c, is arbitrary, and it is customary to place the longer of the two horizontal axes from left to right (OY = b) and take it as unity: this is called the “macro-axis” or “macro-diagonal” (from μακρός, long) , whilst the shorter horizontal axis (OX = a) is called the “brachy-axis” or “brachy-diagonal” (from βραχύς, short) . ”

“ This last equation, of which the left-hand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass.
There is strict duality in descriptive geometry between point-line-locus and line-point-envelope theorems. But in metrical geometry duality is encumbered by the fact that there is in a plane one special line only, associated with distance, while of special points, associated with direction, there are two: moreover the line is real, and the points both imaginary. ”

“ Now the values of the sine, cosine, and tangent (functions of the angles, as they are called) have been computed for the various angles, and some interest may be developed by obtaining them [Pg 235] by actual measurement, using the protractor and squared paper. Some of those needed for such angles as a pupil in geometry is likely to use are as follows:
It will of course be understood that the values are correct only to the nearest thousandth. ”

“ They are inclined to one another at equal oblique angles, and they are all equally inclined to the principal axis; further, they are all of equal length and are interchangeable. With such a set of axes there can be no statement of an axial ratio, but the angle between the axes (or some other angle which may be calculated from this) may be given as a constant of the substance. Thus in calcite the rhombohedral angle (the angle between two faces of the fundamental rhombohedron) is 74° 55′, or the angle between the normal to a face of this rhombohedron and the principal axis is 44° 361/2′. ”

“ We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties. In Lie’s transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle. A most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry. ”

“ In projective geometry any two straight lines in a plane intersect, and the straight lines are closed series which return into themselves, like the circumference of a circle. In descriptive geometry two straight lines in a plane do not necessarily intersect, and a straight line is an open series without beginning or end. Ordinary Euclidean geometry is a descriptive geometry; it becomes a projective geometry when the so-called “points at infinity” are added. ”

“ As in the triangle aei, the quadrate of the base ai, is greater in power, than the quadrates of the shankes ae, and ei, by double of the rectangle ar, which is made of ae, one of the shankes, and of eo, the continuation of the same ae, unto o, the perpendicular of the toppe i. ”

“ Similarly, if you draw the circle FGI from the center C, the amount of degrees in arc FG from circle FGI is the same as the degrees in vertex ABC according to arc DE from circle DEH. In the same manner, whatever the distance of the arc that lies between the two lines, it remains in proportion to the circumference, and the angle is the same, regardless whether the length of the lines are in cubits, handbreaths, or finger measurements.
Section 10 A right angle is one that contains 90 degrees and whose edges are perpendicular, not leaning towards either side. ”

“ In order to exhibit the correspondence between the short and methods of vector analysis and the more familiar formulae of Cartesian geometry, we take a right-handed system of three mutually perpendicular axes Ox, Oy, Oz, and adopt three fundamental unit vectors i, j, /1, having the positive directions of these axes respectively. ”

“ The unequal axis is spoken of as the principal axis or morphological axis of the crystal, and it is always placed in a vertical position; in five of the seven classes of this system it coincides with the single tetrad axis of symmetry.
The parameters are a : a : c, where a refers to the two equal horizontal axes, and c to the vertical axis; c may be either shorter (as in fig. 42) or longer (fig. 43) than a. The ratio a : c is spoken of as the axial ratio of a crystal, and it is dependent on the angles between the faces. ”

“ As here, when the diameter is continued it is perpendicular, and maketh twice two angles, within and without, the outter equall betweene themselves, and inner equall betweene themselves: But the outer unequall to the inner: And the angle of a greater semicircle is greater, then the angle of a lesser. Neither is this affection any way reciprocall, That all equall angles should bee right angles. For oblique angles may bee equall betweene themselves: And an oblique angle may bee made equall to a right angle, as a Lunular to a rectilineall right angle, as was manifest, at the 6 e. ”

“ Axes of Symmetry.—Consider the vertical axis joining the opposite corners a3 and ā3 of an octahedron (fig. 3) and passing through its centre O: by rotating the crystal about this axis through a right angle (90°) it reaches a position such that the orientation of its faces is the same as before the rotation; the face ā1ā2ā3, for example, coming into the position of a1ā2a3. During a complete rotation of 360° (= 90° × 4) , the crystal occupies four such interchangeable positions. Such an axis of symmetry is known as a tetrad axis of symmetry. ”

“ Any two interior opposite angles are equal to two right angles.
Let the straight line e f intersect the two parallel straight lines a b, c d, in the points m and n.
Then will any two interior opposite angles be equal to two right angles.
For the interior alternate angles Yellow, Red, are equal.
If to the angle Green we add the angle Yellow, we shall have the adjacent angles Green, Yellow.
If to the same angle Green we add the equal angle Red, we shall have the interior opposite angles Green, Red. ”

“ Then letting fall on the axis of the ellipsis the ordinate PR, from the proportion of the diameters of the ellipsis, the ordinate RQ of the circumscribed circle AQB will be given; which ordinate is the sine of the angle AOQ supposing AO to be the radius, and also cuts the ellipsis in P. It will be sufficient if that angle is found by a rude calculus in numbers near the truth. Suppose we also know the angle proportional to the time, that is, which is to four right angles, as the time in which the body described the arc Ap, to the time of one revolution in the ellipsis. ”

“ Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (ca. 180 B.C.) , the quadratrix of Hippias of Elis (ca. 420 B.C.) , or the spiral of Archimedes (ca. 250 B.C.) . But since this problem, stated algebraically, requires the solution of a cubic equation, and this involves, geometrically, finding three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection of any angle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. ”

“ Axis, aks′is, n. the axle, or the line, real or imaginary, on which a body revolves: the straight line about which the parts of a body or system are systematically arranged, or which passes through the centre of all the corresponding parallel sections of it, as of a cylinder, globe, or spheroid. The axis of a curved line is formed by a right line dividing the curve into two symmetrical parts, as in the parabola, ellipse, and hyperbola:—pl. ”

“ The demonstration of the converse is like unto the former. For if the tangent, or touch-line ae, be not perpendicular to the diameter iou, let oe, from the center o, be drawne perpendicular; Then shall the angle oei, be right angle: And oie an acutangle: And therefore by the 22 e vj, oi, that is oy, shall be greater then oye, that is the part, then the whole.
Therefore
20 If a right line doe passe by the center and touch-point, it is perpendicular to the tangent or touch-line. ”

“ For if beside io, and sr, there be supposed a third equall, the same also shall be unequall, because it shall be both nearer and farther off from the diameter. [192]
16 Of right lines drawne from a point in the diameter which is not the center unto the periphery, that which passeth by the center is the greatest: And that which is nearer to the greatest, is greater than that which is farther off: The other part of the greatest is the left. And that which is nearest to the least, is lesser than that which is farther off: And two on each side of the greater or least are only equall. ”

“ It should also be observed that in the degraded circumstances of ordinary geometry it would be impracticable to employ the same constant c for the purpose of both linear and angular measurement.
It is easy to see that the definition of a right angle in the generalized sense is embodied in the statement that “if two corresponding legs of an harmonic pencil touch the fundamental conic then the two other legs are at right angles.” We also see that all the perpendiculars to a given line pass through a point, i.e., the pole of the given line ”

“ All forms are referred to four axes, three equal axes inclined to each other at angles 60 degrees in a common horizontal plane, and a fourth vertical axis at right angles, and longer or shorter. The rhombohedral division comprises crystals having but three planes of symmetry, intersecting at angles of 120 degrees in the vertical axis. ”

“ If you multiply the height of any equilateral triangle by the moiety of the basis upon which the line marking the height stands perpendicularly, the product gives the area of that triangle.
In every equilateral quadrangle, the product of one diameter multiplied by the moiety of the other will be equal to the area of it.
In any circle, the product of its diameter, multiplied by three and one-seventh, will be equal to the periphery. This is the rule generally followed in practical life, though it is not quite exact. The geometricians have two other methods. ”