Angle of repose

“ It is manifest from hence, that the cause of refraction consisteth only in the obliquity of the line of incidence, whether the incident body penetrate both the mediums, or without penetrating, propagate motion by pressure only.
Things thrown out of a thinner into a thicker medium are so refracted that the angle refracted is greater than the angle of inclination. ”

“ A single example will suffice. Suppose a section be drawn running north-east and south-west, and it crosses certain beds dipping north at 35° (or certain cleavage planes, or a vein, or a fault, or any other plane having that dip) , what angle ought we give to those beds in the section in order to give a true representation of the apparent dip they would have in a cliff running parallel to the section? In this case the angle between the direction of the dip and that of the section, or between north and north-east = 45° ”

“ It is only when the dome has to be moved through a considerable angle, as in turning to a fresh object, or in photographing some object which passes nearly through the zenith, that the need for a mechanical means of rotation is felt. ”

“ If the angle of inclination be semirect, and the line of inclination be in the thicker medium, and the proportion of their densities be the same with that of the diagonal to the side of a square, and the separating superficies be plane, the refracted line will be in the separating superficies.—8. If a body be carried in a strait line upon another body, and do not penetrate the same, but be reflected from it, the angle of reflection will be equal to the angle of incidence.—9. The same happens in the generation of motion in the line of incidence. ”

“ The rectangular cross, though more positive and effective, is no more potential in securing this unity than the crossing of lines at a long angle. A series of right angles will in time arrive at the same point as the tangent, but less quickly. ”

“ The too common mistake is to grind the lower or clearance side at too great an angle. Fig. 34p. 39 shows the correct angle, and the dotted line A illustrates the common tendency to grind the clearance.
The Proper Angle for Lathe Tools.—Now there is a reason why the angle of from 10 to 15 should be maintained in the clearance. The point of the tool must have suitable support for the work it is required to do, so it will not chatter or yield in the slightest degree. A bit ground along the dotted line has a cutting edge which will spring down, and consequently break or produce a rough surface. ”

“ The angle of b may be made such that the clearance will be the same to the actual surface it cuts at every point in the length of the cutting edge, making every point in that length equally keen and equally strong, the clearance being such as it is determined is the most desirable. ”

“ A right line drawn from the vertex and turning about it in the plane of the angle, from the position of coincidence with one leg to that of coincidence with the other, is said to turn through the angle, and the angle is the greater as the quantity of turning is the greater. Again, since the line may turn from one position to the other in either of two ways, two angles are formed by two lines drawn from a point. ”

“ Hence the angle made by the joint, of the plate-band arch or low segment arch, above a window or doorway, through which the fractures run, with the vertical, forms an approximate measure of the angle of emergence. ”

“ So always in practical operation that law is absolutely true which we observe to be approximated to more and more nearly as we consider smaller and smaller angles, that the versed sine of the angle is the measure of its deflection from the straight line of motion, or the measure of its fall toward the center, which takes place at every point in the motion of a revolving body. ”

“ A solid angle may be conceived two ways. First, for the aggregate of all the angles, which are made by the motion of a strait line, while one extreme point thereof remaining fixed, it is carried about any plain figure, in which the fixed point of the strait line is not contained. ”

“ Still later it learns to apply the same principle to uphill or downhill gradients, where the line of gravity makes, not a right angle, but an acute or obtuse angle with the visible surface. That is, it learns to take the constant line of its own centre of gravity as the base-line of its triangle and not the fickle flooring; and it still keeps in balance in movement by measuring the third side with its eye, in order to be assured that whatever angle its body in balance is making with the ground, be it uphill and acute, or downhill and obtuse, shall remain the same during its next movement. ”

“ A space that is arched looks wider than it actually is, for the eye unconsciously follows the lines of the arch, and a distance or width effect is the result. The same space treated with a straight line is quickly bridged. The same space treated with lines that come to an angle looks narrower, for the reason that the eye becomes focused by the apex of the angle, and a height effect, not a breadth effect, is the result. ”

“ The point of greatest curvature is situated nearer the summit than the point of minimum velocity. In the fire of mortar-shells, under great angles of projection, the trajectory may be considered as an arc, in which the angle of fall is slightly greater than the angle of projection. ”

“ But if four-seventeenths of a revolution had been taken for the angle of the creases, the ultimate polygon would have had thirty-four sides. In an angle taken at hazard the chances are that the number of ultimate sides will be large enough to present a circular appearance.
Any reader who chooses may amuse himself by trying results from three or more axes, whether all passing through one point or not. ”

“ Let a point move from the origin in a straight line at an angle of 45° with the axis of X to the point () , thence vertically in a straight line to the point () , thence obliquely in a straight line to the point () , etc The broken line thus described (continued indefinitely forwards and backwards) is the limiting form of the curve as the number of terms increases indefinitely. ”

“ The object which this sector proposes to attain, is the actual admeasurement of the dip angle; that is, to ascertain how much the visible horizon is depressed below the horizontal plane passing through the eye of the observer. The instrument is so contrived as to measure double the dip angle twice over, so that we obtain four times the required dip, and one quarter of this angle is what must be applied to vertical angles, measured from that part of the horizon which has been observed. ”

“ If the axis be so circumstanced that the pull is at any other angle to it, then the change of axis being always perpendicular to the pull, one component of the change rotates the axis as before, the other alters its inclination. ”

“ This their trained sense continues to adjust at such an inclination to the shifting but relatively fixed point of their feet as may always give their hand a second relatively fixed point of judgment by which they can estimate the angle that then body is making, at any instant of rest or motion, with the line of gravity ”

“ When the elevation is determined by sights the angle of elevation is the angle between the line of sight and the axis of the piece, when these lines are in the same vertical plane,—or the angle between the line of sight and a plane containing the axis of the piece and a horizontal line intersecting it at right angles, when they are not. The graduations of tangent scales and fixed breech-sights give this angle in degrees. The graduation of the pendulum hausse gives the angle correctly only when the line of sight is horizontal. ”

“ I think I could get the vertices and arms to coincide in the way you describe.
Nie. But the one angle would not be on the other; one would extend round one-fourth of the circle, and the other round the remaining three-fourths.
Min. Then, after all, the angle is a mysterious entity, which extends from one of the Lines to the other? That is much the same as Euclid's Definition. Let us now take your definition of a Right Angle.
Nie. We first define 'one revolution,' which is the angle described by a Line revolving, about one extremity, round into its original position. ”

“ CHAPTER XXIII. THE EDGE AND FILLET.
§ I. The decoration of the angle by various forms of chamfer and bead, as above described, is the quietest method we can employ; too quiet, when great energy is to be given to the moulding, and impossible, when, instead of a bold angle, we have to deal with a small projecting edge, like c in Fig. LI. In such cases we may employ a decoration, far ruder and easier in its simplest conditions than the bead, far more effective when not used in too great profusion ”

“ Reflection Angle
"When you throw a ball against a wall at an angle, it bounds away at the same angle. That is reflection, and is just exactly what light does when a ray strikes a mirror. If, on the other hand, the glass had no mercury on it to reflect the light, the ray would not go straight through, but would bend, just as you have seen a stick in a glass of water appearing as though it was bent below the water line. That is refraction."
Two weeks of very vigorous work had now been put in since the yaks had disappeared, and the wagon was still at the edge of the forest. ”

“ The weight which suspends itself in the angle will give from it some weight at the sides of this angle which will be between them in the same proportion as is that of the obliquity of their sides. Or: one such weight will distribute itself between its supports in the same proportion as is that of the two angles born of the division of the angle where this weight sustains itself, division of angle which is made by the vertical which descends in the center of the suspended heavy body ”

“ So angulus solidus, a solid angle, in the common section of three surfaces at the least.
[But the learned B. Salignacus hath observed, that all angles doe not consist in the common section of the bounds, Because the touching of circles, either one another, or a rectilineal surface doth make an angle without any cutting of the bounds: And therefore he defineth it thus: Angulus est terminorum inter se invicem inclinantium concursus: An angle is the meeting of bounds, one leaning towards another.] So is aei. ”

“ Another method which can at times be employed for accurately locating a jig-plate in different positions on an angle-plate, is shown in Fig. 29. The angle-plate is, of course, set at right angles to the spindle and depth gages and size blocks are used for measuring directly the amount of adjustment. Both the angle-plate and work should have finished surfaces on two sides at right angles to each other, from which measurements can be taken. ”

“ If the pull is exerted at an angle both to the base and the inclined face, we have a problem that is slightly more complicated and we need not go into it here because it involves a bit of trigonometry. In all cases, however, it may be noted that the amount of rope that is taken in, in hauling the weight up the incline, bears a definite relation to the amount of effort required to raise the weight. ”

“ Considering now the case of a prism or cylinder growing longer in a horizontal direction, we must find out in what ratio the moment of its own weight increases in comparison with its resistance to fracture. ”

“ The Surface of every Round Solid is equal to a Rectangle, whose Base is the Circumference of the Figure, by the Rotation whereof the Solid is begot, and the Height equal to the Circumference, which the Center of Gravity of the Perimeter of the Figure describeth. ”

“ It will be seen that if a horizontal line is drawn from the foot of one of these figures, say G, to point O on the edge of the incline, then dropped vertically to o·, then again carried on to o·· where the other figure g is, we find it is the same height and also that the other vanishing scale is the same width at that distance, so that we can work from either one or the other. In the event of the rising ground being uneven we can make use of the scale on the level plane. ”

“ At a distance of nearly 7 feet comes a right angle, and the passage goes on for 6 feet, then turns to the left by an obtuse angle and pursues its course for 12 feet, then again turns to the right by another obtuse angle, and for 15 feet more one is still half under water, till N is reached, after which the level of the floor rises, as does also the ceiling; one is able to stand erect alongside of another person. ”

“ The cause of refraction may be illustrated by considering a line of men moving across a field and occupying at equal time intervals the successive positions 1, 2, 3, etc., indicated in Fig. 369. Suppose that the upper and lower parts of the field have a smooth hard surface, while at the center is a strip of newly ploughed ground. The line will move more slowly over the ploughed field than over the hard field. ”

“ From the theory of resistance of profiles we know that the resistance of a line to exterior normal forces is directly proportional to its degree of curvature, and consequently inversely proportional to the radius of the curve. ”

“ This phenomenon is the impossibility of stretching a rope in such a way that it shall be at once straight and parallel to the horizon; the fact is that the cord always sags and bends and that no force is sufficient to stretch it perfectly straight. ”