Tangent

“ 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. ”

“ 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. ”

“ Conceive the point of contact of any tangent removed in infinitum, and the tangent will degenerate into an asymptote, and the constructions of the preceding problem will be changed into the constructions of those problems wherein the asymptote is given. ”

“ Mental time, as I remarked a while ago, is an infinite circle with materiality a line running tangent to it. The point of tangency interprets it to the physical senses, and so creates what we call physical time. Since a line can be tangent to a circle at only one point, our physical existence is single. ”

“ The pu- pils in the Monitorial School have various methods of marking circles, without the aid of dividers, the most  (8)
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14, Describe a circle and draw a tangent to it.
Tangent comes from a latin word, which means to touch. A tangent is a right line which touches a cir- cle, but does not cut any of it off. If a right line be drawn from the centre of the circle or arc to the point of contact (that is, the point where the tangent touches the circle) the two right lines, (that is, the radius and the tangent) will be perpendicular to each other, (fig.9.) ”

“ Exactly the same result may be obtained from the well-known proposition which states that ‘the angle between the tangent from any external point and the focal distance is equal to the angle between the other tangent and the focal distance.’
3. The same opinions are often held by individuals in quite different walks and classes of life. Let these individuals be represented by points on an ellipse. Join these, and we have a system of parallel chords. Draw a straight line through the middle points of these chords, and lo! it will always pass through the centre. ”

“ You begin by dragging the unfortunate beginner straight into the most difficult part of the subject. Your first chapter positively bristles with difficulties about 'direction.' Then comes a long chapter on circles, including some very complicated figures, and a theory of tangents which depends upon moving lines and vanishing chords—all most disheartening to a beginner. What do you suppose he is likely to make of such a sentence as 'the direction of the motion of the generating point of any curve is that of the tangent to the curve at that point'? ”

“ Find the equation of the tangent at to the ellipse .
Ans. .
6. Find equations of tangent and normal to the witch as at the point where .
Ans. .
7. Prove that at any point on the catenary the lengths of subnormal and normal are respectively.
8. Find equations of tangent and normal, lengths of subtangent and subnormal, to each of the following curves at the points indicated:
9. Prove that the length of subtangent to is constant and equal to .
10. Get the equation of tangent to the parabola which makes an angle of 45° with the axis of X.
Ans. . ”

“ Among his discoveries were a solution by infinite series of the Keplerian problem, a method of drawing tangents to curves geometrically, and a rule, founded on the principle of exhaustions, for the direct and inverse method of tangents. ”

“ When, therefore, it is asked, Does the sphere of sense ever fall within itself? this is equivalent to asking, Do the senses themselves ever become sensations? Is that which apprehends sensations ever itself apprehended as a sensation? Can the senses be seized on within the limits of the very circle which they prescribe? If they cannot, then it must be admitted that the sphere of sense never falls within itself, and consequently that an objective reality—i.e., a reality extrinsic to that sphere—can never be predicated or secured for any part of its contents. ”

“ Everything, line, point, shape, angle and figure, corresponds to something in both the Spirit and the Soul-world (the outward and inward Soul-life) , and are, independent of magnitude, absolute and arbitrary symbols, embodying an absolute and fixed principle: and every line, dot, point, shape or angle has a fixed definition in the lexicon of the Starry Heavens. [1] All pure and good thoughts, being themselves full of symmetry and beauty, can only be outwardly conveyed or expressed ”

“ This fact of reflection and the fact that the so-called angle of reflection is the same as the angle of incidence, or, in other words, the angle at which the light falls upon the object, explains a familiar sight. Have you never seen, late in the afternoon, when the sun is above the horizon, a blaze upon a hill side, so bright that your first thought is it must be a house on fire? ”

“ The parts of the rational explanation or notion, into which that notion is divided, are prior to the notion, at least in some instances. But the notion of a right angle is prior to that of an acute angle or is one of the elements into which the notion of an acute angle is divided; for you cannot define an acute angle without introducing the right angle into your definition, nor can you define the semicircle without introducing the circle, nor the finger without introducing the man — the finger being such and such a part of the man. ”

“ We get it when we pass from a short vertical line to a long horizontal one, from a small square to a large circle, as well as when we pass between those figures whose shapes are congruous. But when the shapes are congruous our consciousness of the relation is a good deal more distinct, and it is most distinct of all when, in the exercise of our analytic attention, we notice, first, a part, and then the whole, of a single line or shape. ”

“ You think yourself the perfection of existence, while you are in reality the most imperfect and imbecile. You profess to see, whereas you can see nothing but a Point! You plume yourself on inferring the existence of a Straight Line; but I can see Straight Lines and infer the existence of Angles, Triangles, Squares, Pentagons, Hexagons, and even Circles. ”

“ Some thoughts are light and fantastic, like bubbles on the sea; they are beautiful while the sun shines, but the very ray that reveals their beauty also seals their doom—for the heat kills them; they burst, and forever disappear, being hollow and of but little substance: other thoughts are round, heavy, and solid as a cannon-ball, and like them, too, their mission is to batter down the mounds erected by unwise men. Words are but the garments of thought. Geometry is the Soul of all Sciences—order, symmetry, and form! ”

“ Why a curve is more pleasing to the eye than a straight line may be too deep a question; but none the less may one safely appeal to the universal experience that there are certain lines, certain forms, certain colors, certain sounds which give us pleasure. ”

“ I have a sensation, or perception, of varieties of colour, and of all the dimensions of matter. I cannot separate the colour from the length and breadth of the trunk of a large oak before me; but equally impossible is it for me, to separate the colour from the convexity and the magnitude; and, from this equal impossibility, I might conclude, with equal force, that the perception of the convexity and the magnitude is immediate and original, as the perception of mere length and breadth. Where all things are equal, we cannot justly deny to one what we allow to another. ”

“ As vision was said to be, in some sense, perceptive of the visible and the invisible, and the other senses equally of their opposites, so Touch may be said to be perceptive of the tangible and the intangible; and by intangible is meant as well what differs but slightly from what is tangible, as air for instance, as what is in such excess as to be destructive of all sensation. ”

“ It cannot very well be done by reasoning. It must be done by visual feeling. The principle of Balance being clearly understood, finding the center of any tone-contrasts is a matter of experimental practice in which those persons succeed best who are most sensitive to differences of tone, and who make the greatest effort to feel the centers and to indicate them accurately. Experience and practice are necessary in all cases.
Fig. 228
Here, within this circle, are the attractions to be balanced. ”

“ What an infinitely complex play of subtle waves having their origin in our [58] minds may spread itself over the gross lower realm of action, perceptible to the outward eye and ear, like the fine ripples on the large waves of a pond, or the flat designs on the surface of a closely woven carpet, which takes from them its whole beauty and higher meaning. The physicist, however, recognizes and follows only the action of the lower exterior order, and does not concern himself with the finer, which he does not perceive. ”

“ Neither can a like ‘imaginary distance between tangible and solid bodies.’ ‘Suppose two cases, viz. that of a man supported in the air, and moving his limbs to and fro without meeting anything tangible; and that of a man who, feeling something tangible, leaves it, and after a motion of which he is sensible perceives another tangible object. ”

“ So much of law and order, of symmetry and beauty in every part, implies, it may be thought, that every part has been framed with a view to some use;—that its symmetry and its beauty are the marks of some noble purpose. ”

“ The raison d'être of Matter is the notion of tangibility. But scientific men now tell us that no atom ever touches another. If this be so, scientific tangibility disappears and the raison d'être of Matter disappears, with it. But it is so natural a figment that we shall all probably talk about it, and most of us probably will believe in it, until human nature is very much changed. ”

“ The Harmony is, therefore, the Harmony of equivalent contrasts which are not the same contrasts.
183. We have more or less movement in every composition of tones which [Pg 185] is unbalanced, in which the eye is not held between equivalent attractions, either upon a vertical axis or upon a center. In all such cases, of tones unbalanced, the movement is in the direction of the greatest contrast. Unless the movement is regular and marked in its measures, as I think it is in all regular and perfect gradations, the movement is not rhythmical. ”

“ Every sensation has a degree or intensity, by which it may fill more or less the same time, that is, the inward sense relatively to the representation of an object, until it be reduced to nothing = 0 = negation. ”

“ All our ideas of sensation of course resemble the corresponding sensations, but they do so with very different degrees of exactness and of reliability. No one, I presume, can recall in imagination a colour or an odour with the same distinctness and accuracy with which almost every one can mentally reproduce an image of a straight line or a triangle. ”

“ But there are other modes in which infinity may be represented, which are confused by no associations of the kind, and which would, as being in mere matter, appear trivial and mean, but for their incalculable influence on the forms of all that we feel to be beautiful. The first of these is the curvature of lines and surfaces, wherein it at first appears futile to insist upon any resemblance or § 14. The beauty of curvature.suggestion of infinity, since there is certainly in our ordinary contemplation of it, no sensation of the kind. ”

“ There came a day when, for simple exhaustion, if symmetry should really demand just one more, he was ready to take symmetry into account. Symmetry was harmony, and the idea of harmony began to haunt him; he said to himself that harmony was of course every thing. ”

“ The emotions are dual as well as the sensations, as we have seen in the first chapter. All the operations of the intellect are dichotomic, and in mathematical logic must be expressed by an equation of the second degree. Subject and object must be understood as polar pairs, and in physical science polarization, contrast of properties corresponding to contrast of position, is a universal phenomenon. Analogy, therefore, vindicates the assumption that the unknown, like the known, is the field of the operation of contradictory powers. ”

“ Depth is the path which leads from what-is, to our own body. In music it is not otherwise. Tone is existence, without which nothing at all can be. Symmetry and Numbers are the manner in which tones exist, one with another. ”

“ As far as these triangles are individual, they are made so by the will of God, and not by the substance you call triangle. The universal—the abstract right angle, or any other abstract form—is only an idea, a concept, to which reality, individuality, or what we might call energy is wanting. The only true energy, except man's free will, is God. ”

“ But here already “the rhythm, phrasing, and articulation are entirely faultless, exquisitely clear, melodious, and rare.” [74] It were superfluous labour to point at special beauties, at the exquisite rendering of nature; and copious commentaries exist to explain the course of the argument, if a series of moods is to be called an argument. One may note such a point as that (xiv.) where the poet says that, were he to meet his friend in life,
“I should not feel it to be strange.” ”

“ Yet these arts likewise aim, by bringing into prominence certain relations of symmetry in form as perceived by the eye, or in aerial vibrations as perceived by the ear, to excite in us the states of feeling with which these species of symmetry are by subtle laws of association connected. They, too, imitate, not literally, but under the guidance of a predominating sentiment or emotion, relations which really exist among the phenomena of nature. And here, too, we estimate excellence, not in proportion to the direct, but to the indirect imitation. ”