Perpendicular

“ 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. ”

“ 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. ”

“ But the most easy and habitual manner of determining our want of perpendicularity, is by attending to the apparent motion of the objects within the sphere of distinct vision; for this apparent motion of objects, when we incline from our perpendicularity, or begin to fall, is as much greater than the real motion of the eye, as the diameter of the sphere of distinct vision is to our perpendicular height. ”

“ Mrs. B. The rule upon this subject is this; when a ray of light passes from a rare into a dense medium, it is refracted towards the perpendicular; when from a dense into a rare medium, it is refracted from the perpendicular. By the perpendicular is meant a line, at right angle with the refracting surface. This [Pg 181] may be seen in fig. 1, and fig. 2, where the lines A E, are the perpendiculars.
Caroline. But does not the attraction of the denser medium affect the ray before it touches it? ”

“ 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. ”

“ But when a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle, and the straight line which stands on the other is called a perpendicular to it. ”

“ What is a Scalene triangle ?
25.
When a perpendicular is raised on a right line, must the right line be a horizontal ? Ans. No. A perpendicular may be raised on any right line whether horizontal or oblique. ”

“ 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 ”

“ From the angles at the base of a triangle perpendiculars are drawn to the opposite sides, produced if necessary: shew that the straight line joining the points of intersection will be bisected by a perpendicular drawn to it from the middle point of the base. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ In perpendicular motion there is no refraction. If a body pass, or there be generation of motion from one medium to another of different density, in a line perpendicular to the separating superficies, there will be no refraction. ”

“ 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 ”

“ If a right line whose power is sesquialter unto the side of an equilater triangle, be cut after a double reason, the double segment perpendicular to the center of the triangle, knit together with the angles thereof shall comprehend a tetraedrum. 13 p xiij.
For a solid to be comprehended of right lines understand plaines comprehended of right lines, as in other places following. ”

“ To construct the made, on the surface of a right and vertical cylinder, by a plane perpendicular to one of the planes of projection.—To draw the tangent to the curve of inter.—To make the development of the cylindrical surface, and to refer to it the curve of inter, and also the tangent. ”

“ A perpendicular or vertical line is one perfectly upright, as a string, with a weight at the end of it, will hang from a nail, or from the hand.
In making horizontal lines, the pupil should make them parallel to the top or bottom of his slate or paper, and in making perpendiculars they should be parallel to the sides of the slate or paper. Parallel lines are lines running in the same direction equally distant from each other in every part ”

“ But if the centripetal power increases in so much less a degree, as to permit the line drawn from the center to the body, as it accompanies the body in its motion, at length to become more and more erect to the curve wherein the body moves, and in the end, suppose at K, to become perpendicular thereto; from that time the body shall rise again. ”

“ Vertical means an unvarying upright position. Perpendicular means that one line or plane meets another line or plane at right angles.
The fig. 295 on page 246 is a study in perspective, showing a water reflection. As rays from every visible part of the object are reflected, all following the same law, the reflection will appear to the eye inverted, and of the same size as the object. The arch itself forms the upper half of a hollow cylinder, and the reflection forms the lower half. The reflection shows much more of the interior of the arch than can be seen directly. ”