Manual of Geometry And Conic Sections
Manual of Geometry And Conic Sections
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If the semicireumference is revolved about PQ, as an axis, through an angle AOC it will generate a lune whose angle is APC, or AOC, and if it is revolved through four right angles it will generate the surface of the sphere. Now, it is obvious that the surface generated whilst revolving through any angle is proportional to that angle ; hence, the surface of the lune, is to the surface of the sphere, as the angle of the lune, is to four right angles, which was to be proved.
184 MANUAL OF GEOMETBY....
Cor. 1. If we denote the area of the lune by L, ita angle, in terms of a right angle as a unit^ by A, the area of the tri-rectangnlar trian- gle by T, remembering that the surface of the sphere is equal to 8 tri-rectangolar triangles, we shall have, L : 8T :: A : 4, whence, L = 2A X T, that is, the area of a Inne is equal to twice the angle of the lune, multiplied by the area of the tri-rectangular triangle.
Cor. 2. Whilst the semi-circumference generates the lune . the semicircle generates the corresponding wedge, and whilst the semi-circumference generates the surface of the sphere, the semicircle generates the volume of the sphere ; hence, the volume of a spherical wedge, is to the volume of the sphere, as the angle of the wedge, is to four right angles.
What to read after Manual of Geometry And Conic Sections?
You can find similar books in the "Read Also" column, or choose other free books by William G William Guy Peck to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
If the semicireumference is revolved about PQ, as an axis, through an angle AOC it will generate a lune whose angle is APC, or AOC, and if it is revolved through four right angles it will generate the surface of the sphere. Now, it is obvious that the surface generated whilst revolving through any angle is proportional to that angle ; hence, the surface of the lune, is to the surface of the sphere, as the angle of the lune, is to four right angles, which was to be proved.
184 MANUAL OF GEOMETBY....
Cor. 1. If we denote the area of the lune by L, ita angle, in terms of a right angle as a unit^ by A, the area of the tri-rectangnlar trian- gle by T, remembering that the surface of the sphere is equal to 8 tri-rectangolar triangles, we shall have, L : 8T :: A : 4, whence, L = 2A X T, that is, the area of a Inne is equal to twice the angle of the lune, multiplied by the area of the tri-rectangular triangle.
Cor. 2. Whilst the semi-circumference generates the lune . the semicircle generates the corresponding wedge, and whilst the semi-circumference generates the surface of the sphere, the semicircle generates the volume of the sphere ; hence, the volume of a spherical wedge, is to the volume of the sphere, as the angle of the wedge, is to four right angles.
What to read after Manual of Geometry And Conic Sections?
You can find similar books in the "Read Also" column, or choose other free books by William G William Guy Peck to read onlineMoreLess
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