Tangibility

“ The force with which a revolving body tends to fly from the centre of motion: a sling tends to fly from the hand in consequence of the centrifugal force. A tangent is a straight line touching a curved line in one point without cutting it, as m T, fig. 4. The direction of the centrifugal force is in the tangent to the curved line or path in which the body revolves, and its intensity increases with the angular swing of the body, and with its distance from the 432centre of motion. As the orbit of the moon does not differ much from a circle, let it be represented by m d g h, fig. ”

“ Find the angle which this tangent makes with the curve .
The slope of the tangent is the slope of the curve at the point where they touch one another (see p.77) ; that is, it is the of the curve for that point. Here and for , which is the slope of the tangent and of the curve at that point. The tangent, being a straight line, has for equation , and its slope is , hence . Also if , ; and as the tangent passes by this point, the coordinates of the point must satisfy the equation of the tangent, namely
,
so that and ; the equation of the tangent is therefore . ”

“ In a simple train of waves in a transparent medium, the potential energy, on the elastic theory, may be divided into two parts, of which one is due to that general deformation of the ether which is represented by the equations of wave-motion, and the other to those deformations which are caused by the interference of the ponderable particles with the wave-motion, and to such displacements of the ponderable matter as may be caused, in some cases at least, by the motion of the ether. ”

“ We would now refer to the general doctrine of motion and shape: the former arises from polarity, or constant duplicity of force; indeed, a polar tension lies at the bottom of all motion, which must, therefore, in its course, realize a sphere, which is the only shape in which the Creator and his various manifestations appear. Neither sun, moon, nor any of the fiery globes, or æriform vibrations, or sea, or cataract can ever assume any form but the spherical; indeed, there are no straight lines in life or nature, and, for that reason, there are no angles. ”

“ And therefore in mediums of equal density and elastic force, all the pulses are equally swift. Now if the density or the elastic force of the medium were augmented, then, because the motive force is increased in the ratio of the elastic force, and the matter to be moved is increased in the ratio of the density, the time which is necessary for producing the same motion as before will be increased in the subduplicate ratio of the density, and will be diminished in the subduplicate ratio of the elastic force. ”

“ Modern textbook makers find it convenient to define tangent circles as those that are tangent to the same straight line at the same point, and to define external and internal tangency by reference to their position with respect to the line, although this may be characterized as open to about the same objection as Euclid's. ”

“ One line through (x, y, z) is at right angles to the tangent plane. This is the normal
The tangent plane is cut by the surface in a curve, real or imaginary, with a node or double point at (x, y, z) . Two of the tangent lines touch this curve at the node. They are called the “chief tangents” (Haupt-tangenten) at (x, y, z) ; they have closer contact with the surface than any other tangents.
In the case of a quadric surface the curve of intersection of a tangent and the surface is of the second order and has a node, it must therefore consist of two straight lines. ”

“ And about all of them this has to be considered:—that the tendency of each towards its kindred element makes the body which is moved heavy, and the place towards which the motion tends below, but things which have an opposite tendency we call by an opposite name. Such are the causes which we assign to these phenomena. As to the smooth and the rough, any one who sees them can explain the reason of them to another. For roughness is hardness mingled with irregularity, and smoothness is produced by the joint effect of uniformity and density. ”

“ A tangent line curve, or surface; specifically, that portion of the straight line tangent to a curve that is between the point of tangency and a given line, the given line being, for example, the axis of abscissas, or a radius of a circle produced. See Trigonometrical function, under Function.
Artificial, or Logarithmic, tangent, the logarithm of the natural tangent of an arc. — Natural tangent, a decimal expressing the length of the tangent of an arc, the radius being reckoned unity. — Tangent galvanometer (Elec.) ”

“ It forms the surface of dissipated energy. If we consider all the points representing the volume, entropy, and energy of the body in every possible state, whether of equilibrium or not, these points will form a solid figure unbounded in some directions, but bounded in others by this surface. [10]
The lines traced upon the primitive surface by the rolling double tangent plane, which have been called the limit of absolute stability, do not end at the vertices of the triangle which represents a mixture of those states. ”

“ In the first place, there is no such thing as partial tangency. A tangent spoke is tangent, and that is all there is about it. A tangent is a definite thing, and means a line normal to a radius at the circumference; at least, we can accept this definition as well enough suited to the cycle art. And, in speaking of tangency, we ought rather to say tangent hub than tangent wheel, since the spokes are not tangent to the rim of the wheel, but to the hub. All cyclists know very well, nevertheless, what is meant by partial tangency in the cycle art, and I will therefore use the term. ”

“ When the establishment of equilibrium is rapid, the variation of the tension from its normal value will be manifested especially during the extension or contraction of the surface, the phenomenon resembling that of viscosity, except that the variations of tension arising from variations in the densities at and about the surface will be the same in all directions, while the variations of tension due to any property of the surface really analogous to viscosity would be greatest in the direction of the most rapid extension. ”

“ If the centre of the radiating lines is kept below the springing line, it adds greatly to the interest and beauty of the ornament. A succession of festoons or of drapery hanging from two points are examples of one species of curved radiation.{45}
Tangential Junction.—Euclid’s definition of a tangent is as follows:—“A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it,” and is obtained by drawing a line perpendicular to any radius from the point at which it touches the circumference. ”

“ Roof Frame: any Polygon Fig. 68. Tangents 30. Tangents; Miter Cuts of the Plate.—Before the principles involved in the laying out of rafters on any type of roof can be understood, a clearer idea of the term tangent as used in roof framing must be had. A tangent of an angle of a right triangle is the ratio or fractional value obtained by dividing the value of the side opposite that angle by the value of the adjacent side. The tangent at the plate, to which reference was made is the tangent of the angle having for its adjacent sides the run of the common rafter and the run of the hip or valley. ”

“ The tendency of the dark mass is to hold the eye at its center. The problem is to find the balance-center between the motion and the mass. I have done this, and the balance-center is indicated by the symmetrical outline of the diagram.
180. Some shapes hold the eye with peculiar force, and in such cases the [Pg 179] attractions of tone or measure or shape on the other side have to be increased if we are to have a balance. ”

“ Can a triangular harmony form the substance of the elements? Is fire, as Plato affirms, an acute triangle, and light and heat properties of this triangle? Air and water, are they rectangular and equilateral triangles? Is the form of the terrestrial element a square, because, being the least perfect of all the four elements, it recedes as much as possible from a triangle without losing its essence? Do the male and female embrace only to complete the triangle of generation? These platonic ideas have two very different [334] aspects. ”

“ The physical equilibrium which exists in the balanced “seesaw” of our childhood and the optical balance which is the result of the proper adjustment of masses within the confining edges of a design are similar, in that each is an equalizing of forces of attraction. In the former the force is gravity; in the latter, the attraction to the eye, which varies with the size and tone of the mass. ”

“ As the tangent increases at a greater rate than the angle, the increase of the refraction soon exceeds 1′′ for each degree. At 45° from the zenith .the tangent is 1 and the mean refraction is about 58′′. As the horizon is approached the tangent increases more and more rapidly, becoming infinite at the horizon; but the refraction now increases at a less rate, and, when the observed ray is horizontal, or when the object appears on the horizon, the refraction is about 34′, or a little greater than the diameter of the sun or moon. ”

“ But the strained normal piling which surrounds the surface, and propagates with the surface, being of less density than the mean density of the medium, represents a displacement of the negative mass of the inequality, i.e., of the grains absent, and in whatever new direction the surface is propagating, the motion of the medium, outside the surface, is such as represents equal and opposite momentum, as when a bubble rises in water. ”

“ For when points and tangents and the centre are given, as many other points and as many other tangents are given at an equal distance on the other side of the centre. And an asymptote is to be considered as a tangent, and its infinitely remote extremity (if we may say so) is a point of contact. ”

“ On the other hand, if the main body of the root points vertically down while the sensitive tip is horizontal, a curvature results, because as long as the tip is horizontal it is stimulated, and the stimulus is transmitted to the motor region.
This experiment proves not only that the tip of the root is the sense-organ for gravity, but also that the motile part is not directly sensitive; in other words, that gravitation is perceived exclusively in the tip of the root. ”

“ The theoretical substitution of ideal rigid bodies with geometrically regular surfaces, sliding in contact under pressure at the common regular surface, for the aërated surfaces in the actual circumstances, and the theoretical substitution of the absolute independence of the resistance of the rate of sliding for the limited independence in the actual circumstances, prove the general acceptance of the conceptions— (1) that matter can slide over matter under pressure at a geometrically regular surface ”

“ The triangulum characteristicum is similar to the triangle formed by the tangent, the ordinate of the point of contact, and the sub-tangent, as well as to that between the ordinate, normal, and sub-normal. ”

“ Every tremulous body in an elastic medium propagates the motion of the pulses on every side right forward; but in a non-elastic medium excites a circular motion. Case. 1. The parts of the tremulous body, alternately going and returning, do in going urge and drive before them those parts of the medium that lie nearest, and by that impulse compress and condense them; and in returning suffer those compressed parts to recede again, and expand themselves. ”

“ A globe in these extremely fluid mediums meets with a resistance that is to the force by which its whole motion may be destroyed or generated in the time wherein it can describe, with that motion uniformly continued, eight third parts of its diameter, as the density of the medium to the density of the globe. ”

“ The proposition of the greatest thickness if Solids, beyond which encreased they sink. When ever the excess of the Gravity of the Solid above the Gravity of the Water, shall have the same proportion to the Gravity of the Water, that the Altitude of the Rampart, hath to the thickness of the Solid, that Solid shall not sink, but being never so little thicker it shall. ”

“ What part of the diameter o d is the radius o c?
Name a chord that is produced without the circle.
It is called a “secant.”
A secant is a chord produced.
Name two secants.
If the chord d i were made a secant, would it become longer or shorter?
In how many points does the straight line l m touch the circumference?
It is called a “tangent.”
A tangent is a straight line that touches a circumference in only one point. ”

“ The circle is, of course, a perfect illustration of Measure and Shape-Balance on a center. The balance is also symmetrical. We have a Harmony of Directions in the repetition of the same change of direction at every point of the outline, and we have a Harmony of Distances in the fact that all points of the outline are equally distant from the balance-center, which is unmistakably felt.
Fig. 167
The Ellipse is another example of Measure and Shape-Balance on a center. In this attitude it is also an illustration of Symmetry.
Fig. 168
In this case we still have balance but no symmetry. ”

“ If we draw a line upon the primitive surface, dividing it into parts which represent respectively stable and unstable equilibrium, in respect to continuous changes, i.e., dividing the surface which is concave upward in both its principal curvatures from that which is concave downward in one or both, this line, which may be called the limit of essential instability, must have a form somewhat like that represented by in figure 2. ”