Perpendicularity

“ In this case the geometries say right angles (or a perpendicular) are formed when one line meets another so as to make the adjacent angles formed equal. The dictionary should do the same. The Century Dictionary says a perpendicular is the shortest distance from a point to a line, and then defines right angle as one formed by a perpendicular. This definition for perpendicular is open to the same objection as the definition of a straight line, which says it is the shortest distance between two points. But the Century evidently does not fall into the silly course of Webster and the Standard. ”

“ When the weight is at a considerable distance from the point of support, its centre of gravity, in deviating either on one side or the other from a perpendicular direction, describes a larger circle, as at a, than when the weight is very near to the centre of rotation or the point of support, as at b. But, in a large circle, an arc of any determinate extent, such as an inch, for example, describes a curve which deviates much less from the perpendicular than if the circle were less ”

“ We may regard the foot of every perpendicular as the actual position of a moving point, and it can be proved that if the mass of each particle be multiplied into the area which the foot of its perpendicular describes in a second round any point in the plane, and then be added to the similar products from all the other particles, only observing the proper precautions as to sign, the sum will remain constant, i.e., in any other second the total quantity arrived at will be exactly the same. ”

“ If is the vector drawn from a fixed origin to any point on a curve, and the distance of that point measured on the curve from any fixed point, is a unit vector, tangent to the curve and having the direction in which increases; is a vector directed from a point on the curve to the center of curvature, and equal to the curvature; is the normal to the osculating plane, directed to the side on which the curve appears described counter-clockwise about the center of curvature, and equal to the curvature. ”

“ For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value , which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value i.e. a smaller value than , the difference being the more considerable, the greater is the radius of the circle in comparison with the radius R of the “world-sphere.” ”

“ A perpendicular is let fall from any point in a horizontal line; it is required to pass through any other point in this same horizontal line a plane which shall cut the perpendicular and along which a body will descend to the perpendicular in the shortest possible time. Such a plane will cut from the perpendicular a portion equal to the distance of the assumed point in the horizontal from the upper end of the perpendicular. ”

“ Just as in the measurement of distance we find a pair of fundamental points on each straight line, so in the measurement of angles we find a pair of fundamental rays in each plane pencil. These rays are the two tangents from the vertex of the pencil to the fundamental conic. In ordinary geometry the two fundamental points on each straight line coalesce into the single point at infinity; but it is exceedingly interesting to observe that even in ordinary geometry the two fundamental rays on each pencil do not coincide. ”

“ A plane perpendicular to a radius at its extremity is tangent to the sphere.
The student should always have his attention called to the analogue in plane geometry, where there is one. If here we pass a plane through the radius in question, the figure formed on the plane will be that of a line tangent to a circle. If we revolve this about the line of the radius in question, as an axis, the circle will generate the sphere again, and the tangent line will generate the tangent plane. ”

“ I allude to the word "perpendicular." Many people think that the word "perpendicular" means the position of a plumb-line. It means no such thing, The proper word for describing that position is " vertical." Perpendicular is a relative word, and it ought not to be used without reference to something else. Vertical is an absolute word. Thus, in speaking of the word perpendicular, with reference to that trough of water or prism in the experimental case before us, I mean a line perpendicular, not to the horizon, but to the surface of the trough. ”

“ And let io, be perpendicular to the ray unto the described periphery. This knit by a right line unto the center a, let eu, be drawne. I say, that eu, doth touch the periphery iu, assigned: Because it shall be perpendicular unto the end of the diameter. For the triangles eau, and oai, by the 2 e vij, seeing they are equicrurall; And equall in shankes of the common angle; they are equall in the angles at the base. But the angle aio, is a right angle: Therefore the angle eua, shall be a right angle. And therefore the right line eu, by the 13 e ij, is perpendicular to ao. ”

“ The sines of the angle of incidence and refraction (that is, the perpendicular drawn from the extremity of an arc to the diameter of a circle,) are always in the same ratio; viz: from air into water, the sine of the angle of refraction is nearly as four to three, whatever be the position of the ray with respect to the refracting surface. ”

“ Axes.—Conjugate diameters perpendicular to each other are called axes, and the points where they cut the curve vertices of the conic.
In a circle every diameter is an axis, every point on it is a vertex; and any two lines at right angles to each other may be taken as a pair of axes of any circle which has its centre at their intersection. ”

“ 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. ”

“ Run a line from A to C by standing at C in line with A and P. Then run two perpendiculars from A and C by any of the methods already given,—sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some point D, on CD, to P, putting a stake at B. Then run the perpendicular BE. Since DE : EB = BA : AP, and since we can measure DE, EB, and BA with the tape, we can compute the distance AP.
There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. ”

“ A line at a right angle to another is said to be perpendicular to it; thus, in Figures 39, 40, and 41, lines A are in each case perpendicular to line B, or line B is in each case perpendicular to line A.
A point is a position or location supposed to have no size, and in cases where necessary is indicated by a dot. ”

“ The solid angle subtended at a given point by a closed curve is measured by the area of a spherical surface whose centre is the given point and whose radius is unity, the outline of which is traced by the intersection of the radius vector with the sphere as it traces the closed curve. ”

“ To draw a plane through a given point A perpendicular to a given line p, we first draw through some point O in the axis lines γ′, γ′′ perpendicular respectively to the projections p1 and p2 of the given line. ”

“ Within the field of these curves we can construct the transversals of the family; that is to say, there is a finite area of the plane, through any point of which there passes one stationary curve of the field and one curve which cuts all the stationary curves of the field transversely. These curves provide a system of curvilinear co-ordinates, in terms of which the value of ∫Fdx, taken along any curve within the area, can be expressed. The value of the integral is the same for all arcs of stationary curves of the field which are intercepted between any two assigned transversals. ”

“ What are the differences between the geometry on a sphere and the geometry on a plane? If you make a triangle on the earth, whose sides are great circles, you will not find that the angles of the triangle add up to two right angles: they will add up to rather more. The amount by which they exceed two right angles is proportional to the size of the triangle. ”

“ 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. ”

“ We have only to select a number of such triangles and a system of trigonometry ensues, in which base, perpendicular, and hypotenuse of every triangle is a whole measure without fractions, and in which the nomenclature for every angle is clear and simple. ”

“ We do not now enter into the proof, but it is sufficiently obvious that a sphere of which the radius is the geometric mean between the greatest and least radii of curvature at each point will to a large extent osculate the surface, so that a portion of the surface in the neighbourhood of the point will, generally speaking, have the same curvature as the sphere. ”

“ Then, erecting any perpendicular RS, which may be considered as an ordinate of the curve required, in order to find the length of this ordinate, suppose the intervals HI, IK, KL, LM, &c., to be units, and let AH = a, - HS = p, 1⁄2p into - IS = q, 1⁄3q into + SK = r, 1⁄4r into + SL = s, 1⁄5s into + SM = t; proceeding, to wit, to ME, the last perpendicular but one, and prefixing negative signs before the terms HS, IS, &c., which lie from S towards A; and affirmative signs before the terms SK, SL, &c., which lie on the other side of the point S ”