Quadrilateral

“ If a quadrilateral, having two of its sides parallel, be described about a circle, a straight line drawn through the centre of the circle, parallel to either of the two parallel sides, and terminated by the other two sides, shall be equal to a fourth part of the perimeter of the figure.
199. A series of circles touch a fixed straight line at a fixed point: shew that the tangents at the points where they cut a parallel fixed straight line all touch a fixed circle. ”

“ It has been shown by Remy (in Crete s Journ., vol. iii.) that for any quadrilateral, if z be the spherical distance between the middle points of the diagonals, cos a + cos b + cos c + cos d = 4 cos Jar cos y cos z. ”

“ Rectilinear figures cannot be divided into triangles and quadrilaterals because there are rectilinear figures which have more than four sides. On the other hand, triangles can be divided into equilateral, isosceles and scalene, since no other kind of triangle can exist. ”

“ L. 1. I propound this easy process of calculation, delightful by its elegance, perspicuous with concise terms, soft and correct and pleasing to the learned.
*⁠*⁠*⁠*⁠* L. 139. A side is put. From that multiplied by twice some assumed number and divided by one less than the square of the assumed number a perpendicular is obtained. This being set aside is multiplied by the arbitrary number and the side as put is subtracted—the remainder will be the hypotenuse. Such a triangle is termed 'genuine.'
*⁠*⁠*⁠*⁠*
L. 189. Thus, with the same sides, may be many diagonals in the quadrilateral. ”

“ It can therefore be easily shown that a sphere is a quadric which touches the fundamental quadric along its intersection with the polar plane of the centre of the sphere.
In discussing the general case of the displacement of a rigid system it will simplify matters to suppose that the fundamental quadric has real rectilinear generators. It must, however, be understood that the results are not on that account less general. A displacement must not alter the quadric, and must not deform a straight line. ”

“ Perpendicular and horizontal straight lines are the more easy to examine, because their images fall directly between two opposing muscles. An oblique line is difficult to examine, and we instinctively turn the head, in order to bring it in the plane of rotation of one or the other set of muscles. In following a curved line with the eye, two muscles are used together, one contracting more rapidly than the other. A curve is therefore more difficult to observe, or run the eye over, than a straight line, and the difficulty increases with the subtileness of the curvature. ”

“ The tangent plane of a quadric surface meets it in a quadric curve having a node, that is, in a pair of lines; hence there are on the surface two singly infinite sets of lines. Two lines of the same set do not meet, but each line of the one set meets each line of the other set; the surface is thus a regulus in a twofold manner. The lines are real for the hyperboloid of one sheet and for the hyperbolic paraboloid; for the other forms of surface they are imaginary. ”

“ Consequently a quadric surface is covered by two sets of straight lines, a pair through every point on it; these are imaginary for the ellipsoid, hyperboloid of two sheets, and elliptic paraboloid.
A surface of any order is covered by two singly infinite systems of curves, a pair through every point, the tangents to which are all chief tangents at their respective points of contact. These are called chief-tangent curves; on a quadric surface they are the above straight lines. ”

“ THE SURGICAL DISSECTION OF THE STERNO-CLAVICULAR OR TRACHEAL REGION, AND THE RELATIVE POSITION OF ITS MAIN BLOODVESSELS, NERVES, &c.
The law of symmetry governs the development of all structures which compose the human body; and all organized beings throughout the animal kingdom are produced in obedience to this law. The general median line of the human body is characterized as the point of fusion of the two sides; and all structures or organs which range this common centre are either symmetrically azygos, or symmetrically duplex. ”

“ The curves of the vertebral column are, in quadrupeds, slightly different from those which characterize the human spine. First, instead of their being, as in the latter, curves in the antero-posterior aspect, because of the general attitude of the body, they are turned in the supero-inferior direction.
The cervical region is not a single curve, as in the human being. It presents two: one superior, with its convexity looking upwards; the other inferior, the convexity of which is turned downwards. ”

“ Each quarter will have two sides at right angles, and by inserting the apex into the opening of the angle drawn by the pupil, any incorrectness will be detected. A small brass or iron square will serve the same purpose, but does not satisfactorily show that a right angle is equal to a quarter of a circle, which is also called a quadrant.
25. Cross a right line with a perpendicular. (fig. 10.)
The right line should be drawn in various directions, to show the pupil that a perpendicular may be raised on any right line, whether horizontal or oblique. ”

“ When then one leg is advanced it becomes the hypotenuse of a right-angled triangle. Its square then is equal to the square on the other side together with the square on the base. As the legs then are equal, the one at rest must bend either at the knee or, if there were any kneeless animal which walked, at some other articulation. ”

“ But an important aid to locomotion may be obtained from the artificial muscles, whose elastic tension is of such value in overcoming contractions. The quadriceps extensor, the most frequently paralyzed, may be supplemented by an India-rubber band and chain passing down the front of the thigh from a point on the pelvic girdle corresponding to the anterior iliac spine to a point on a leg-band, imitating the tibial insertion of the quadriceps tendon. Analogous bands stretched on the posterior aspect of the thigh simulate the hamstring muscles. ”

“ When, proceeding from this point, the rings without the cross and with the black spot in the centre were formed, this spot, on the heat being increased, divided itself into two, which removed themselves from the centre into the second and fourth quadrants, and after having passed through the figure in the circular light, closed into a circle with the arcs proceeding from the first and third quadrants, so as to produce the system of rings with a bright centre, which would have been obtained at the very beginning by turning the polarizing prism 90°. ”

“ The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix. (2) A right cylinder having for its base an Archimedean spiral is intersected by a right circular cone which has the generating line of the cylinder passing through the initial point of the spiral for its axis. From every point of the curve of intersection, perpendiculars are drawn to the axis. ”

“ If the girl sits at school with one elbow on the desk, the head will be turned to the opposite side and the spine will be inclined from the perpendicular, and a lateral curvature be likely to result. If she carries her books always on the same side, it will tend to increase the curvature. If she sits with both elbows supported, her shoulders will be pushed up. If her body is twisted as she sits, a strain comes upon the muscles, and some ligaments will be lengthened and others shortened, thus producing a lateral curvature. ”

“ The tissue of the neural arches does not any longer form a continuous sheet, but is divided into (1) a series of arches encircling the spinal cord, and (2) a basal portion resting on the cartilaginous sheath of the notochord. There are two arches to each muscle-plate, one continuous with the basal portion of the arch-tissue and forming the true arch, which springs opposite the centre of a vertebral body, and the second not so continuous, which forms what is usually known as the intercalated piece. Between every pair of true arches the two roots of a single spinal nerve pass out. ”

“ Why was the vertebral column at the back of the body as in quadrupeds, when it would have been more logical, in creating a man who stands on his feet, to place it in the centre of the body as a strong support, thus avoiding the curvatures and weakness of the spine that are now suffered by this disequilibrium in the support of its weight? ”

“ At the other end of the scale is man, whose spine is in a vertical line, and his face occupies a like plane. Between these two extremes are types which tend in various degrees, from the lower to the higher form, and just in proportion as the spinal axis approaches a vertical line from one side, the plane of the face approaches it from the other. ”

“ The amount of the flexural couple for either principal plane at any section is the product of the flexural rigidity for that plane, and the resolved part in that plane of the curvature of the central line at the centroid of the section; the resolved part of the curvature along the normal to any plane is obtained by treating the curvature as a vector directed along the normal to the osculating plane and projecting this vector. The flexural couples reduce to a single couple in the osculating plane proportional to the curvature when the two flexural rigidities are equal, and in this case only. ”

“ In Man, in consequence partly of the disposition of the articular surfaces of the vertebra, and largely of the elastic tension of some of the fibrous bands, or ligaments, which connect these vertebræ together, the spinal column, as a whole, has an elegant S-like curvature, being convex forwards in the neck, concave in the back, convex in the loins, or lumbar region, and concave again in the sacral region; an arrangement which gives much elasticity to the whole backbone, and diminishes the jar communicated to the spine, and through it to the head, by locomotion in the erect position. ”

“ That is, if we suppose P with its tangent to move along the curve to P', at the point of inflection Q the curvature is zero, the rotation of the tangent is momentarily arrested, and as the direction of rotation changes, the center of curvature moves out indefinitely and the radius of curvature becomes infinite. ”

“ Long practice and training, begun in the schools and continued in the experience of daily life, enable us to discriminate at once by the sense of touch, between the angles of an equal-sided Triangle, Square, and Pentagon; and I need not say that the brainless vertex of an acute-angled Isosceles is obvious to the dullest touch. It is therefore not necessary, as a rule, to do more than feel a single angle of any individual ”

“ Each vertebra except the Atlas is composed of a body and an arch; the arch is made up of two pedicles, short, thick plates of bone extending outward and backward from the postero-lateral surface of the body nearer its upper than its lower border, two laminae, thin plates of bone extending backward and inward from their union with the pedicles and joining behind to form the spinous process, and has projecting from it seven processes, two transverse, one spinous, and four articular, two of which are superior and two inferior. ”

“ Each trapezium (or triangle) is weighed where it is against another area hung at a fixed point of an assumed lever; thus the whole of the inscribed and circumscribed figures respectively are weighed against the sum of an indefinite number of areas all suspended from one point on the lever. The result is obtained by a real integration, confirmed as usual by a proof by the method of exhaustion. ”