Solid angle

“ Any face angle of a trihedral angle is less than the sum, but greater than the difference, of the supplements of the other two face angles.
2. A solid angle cannot be formed of equal plane angles which are equal to the angles of a regular polygon of n sides, except in the case of n = 3, 4, or 5. ”

“ The second species of solid is formed out of the same triangles, which unite as eight equilateral triangles and form one solid angle out of four plane angles, and out of six such angles the second body is completed. ”

“ 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. ”

“ Each of the eight solid angles of this figure, therefore, is composed of two portions of the common prism, fig. 196; giving, by the common goniometer, along the edges a b c, an angle somewhat more than 112°, which, generally speaking, is the same as that l m f; fig. 200; the plane 1 on 1, over the edge between them, which is that of the common prism, also gives an incidence of 90° corresponding with that of the planes 1 on 1, fig. ”

“ As an experiment on the angle of contact only gives us the difference of the surface-tensions at the solid surface, we cannot determine their actual value. It is theoretically probable that they are often negative, and may be called surface-pressures.
The constancy of the angle of contact between the surface of a fluid and a solid was first pointed out by Dr Young, who states that the angle of contact between mercury and glass is about 140°. ”

“ But the angle B D F, being at the circumference, is measured by half the arc B F:
Then the equal of B D F, or B A C, must be measured by half the arc B F, or half the difference between the intercepted arcs B C and D E.
159 APPENDIX. Note A.—To those teachers who think that the line should be derived from a surface, and the surface from a solid, the author would say, that, according to his experience, children apprehend the ideas conveyed by the terms line and surface as readily as they do any ideas whatever ”

“ 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. ”

“ Angle of Incidence.—Sixty degrees is safe for most solids, but for bright or polished surfaces, such as varnishes, polished metals, etc., the angle must be lessened as the degree of smoothness increases. ”

“ Suppose OP of fixed length, equal to OA, and let PM, PN be drawn perpendicular to A A, B B, " respectively; then OM and ON, taken with their proper signs, are the projections of OP on A A and B B. The ratio of the projection of OP on B B to the absolute length of OP is dependent only on the magnitude of the angle POA, and is called the sine of that angle; the ratio of the projection of OP on A A to the length OP is called the cosine of the angle POA. The ratio of the sine of an angle to its cosine is called the tangent of the angle, and that of the cosine to the sine the cotangent of the angle ”

“ In many cases the result of a number of particular facts, or the collective instances rising out of them, can only be discovered by geometry, which so far becomes necessary to complete the work of induction. For instance, in the case of optics, when light passes from one transparent medium to another, it is refracted, and the angle which the ray of incidence makes with the superficies which bounds the two media determines that which the refracted ray makes with the same superficies. ”

“ These are right angles if they are equal, and they will be equal if one line is perpendicular to the line it crosses. If the angle be less than a right angle, it is called an acute angle; if more, it is called an obtuse angle. Acute means sharp, and obtuse means blunt. ”

“ Let us then take instead of the tangents of the angles to the radius and which are to each other as the sines of the angles that is, as and
We have then M x RQ [errata 2] , that is, if squaring both sides,
It appears then that is in general greater than its ultimate value by the quantity supposing the angle to be small. ”

“ Refraction by a Prism.—In optics a prism is a piece of transparent material bounded by two plane faces which meet at a definite angle, called the refracting angle of the prism, in a straight line called the edge of the prism ”

“ We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= δs) in the directions of the tangent and normal respectively. If T, T + δT be the tensions at P, Q, and δψ be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + δT) cos ψ − T, or δT, ultimately; whilst for the component along the normal at P we have (T + δT) sin δψ, or Tδψ, or Tδs/ρ, where ρ is the radius of curvature. ”

“ A few illustrations, as of an open book, the wall and floor of a room, and a swinging door, serve to make the concept clear, while a plane at right angles to the edge shows the measuring plane angle. So manifest is this [Pg 299] relationship between the dihedral angle and its measuring plane angle that some teachers omit the proposition that two dihedral angles have the same ratio as their plane angles.
Theorem. If two planes are perpendicular to each other, a straight line drawn in one of them perpendicular to their intersection is perpendicular to the other. ”

“ All right angles are equal to one another.
This can be proved as follows:—Let there be two right lines AB, CD, and two perpendiculars to them, namely, EF, GH, then if AB, CD be made to coincide by superposition, so that the point E will coincide with G; then since a right angle is equal to its supplement, the line EF must coincide with GH. Hence the angle AEF is equal to CGH.
xii. ”

“ Notwithstanding this, a self-luminous surface, plane or curved, viewed through a hole in a plate of metal, is of the same brilliancy at all possible distances as long as it subtends a sensible angle, because, as the distance increases, a greater portion comes into view; and, as the augmentation of surface is as the square of the diameter of the part seen through the whole, it increases as the square of the distance. ”

“ Min. Then why omit that case? However, it matters little. You say 'a plane angle,' I observe. You limit an angle, then, to a magnitude less than the sum of two right angles.
Nos. No, I can't say we do. A little further down we assert that 'two angles are formed by two straight lines drawn from a point.'
Min. Why, these are like Falstaff's 'rogues in buckram suits'! Are there more coming?
Nos. No, we do not go beyond the sum of four right angles. These two we call conjugate angles. 'The greater of the two is called the major conjugate, and the smaller the minor conjugate, angle.'
Min. ”

“ When a ray of light leaves a denser medium, such as glass, to enter a rarer medium, such as air, it becomes refracted from the perpendicular. In such case, the angle of refraction being greater than the angle of incidence, its sine will also be greater than that of the latter ”

“ Let us examine the case in which the particle is placed at a distance from a curved stratum of the substance, whose principal radii of curvature are and . Let (fig. 2) be the particle and a normal to the surface. Let the plane of the paper be a normal section of the surface of the stratum at the point , making an angle with the section whose radius of curvature is . ”

“ The decimal system measurement of angles has never come into ordinary use. In analytical trigonometry the circular measure of an angle is employed. In this system the unit angle is the angle subtended at the centre of a circle by an arc equal in length to the radius. ”

“ An angle is commonly represented as a pure number, but it has not really this character; an angle can only be measured in terms of a unit of angle, just as a length is measured in terms of a unit of length. ”

“ This requires a new proof, as we must substitute for the words 'the bisector of the angle FAC,' the words 'the perpendicular to FC drawn through A.'
(5) Vertical angles together greater than two right angles.
This also requires a new proof, as we must insert, after the words 'the bisector of the angle FAC,' the words 'produced through A,' and must then prove (by your Th. 1) that the angles OAC, OAF, are equal.
On the whole, I take this to be the most cumbrous proof yet suggested for this Theorem. ”