332 MATHEMATICS. 



Plate VII., Fig. 9, aa! and bb', are supposed to be perpendicular to 

 the plane ABDC ; and a' b' represents the projection of the line 

 ab, on that plane. In like manner, b" and a" are the projections of 

 b and a on the plane GHIB, and d' and c' are the same projec- 

 tions when this plane is revolved down into the position GKFB. 



When all the projecting lines and planes are perpendicular to the 

 planes of projection ; that is, when the eye is supposed to be at an 

 infinite distance above, or in front of the object, the projection is then 

 said to be Orthographic. When the eye is supposed to be placed 

 comparatively near to the object, or objects, so that the projecting 

 lines diverge from the eye as a focus or centre, then the projection is 

 said to be Scenographic, which is the same as Perspective ; and the 

 plane of projection, is, in this connection, termed the perspective 

 plane. The name of Spherical Projections, is applied both to the 

 orthographic and scenographic projections of the sphere, with its dif- 

 ferent circles: and, in this case, the plane of projection is called the 

 primitive plane ; and its intersection with the sphere is called the 

 primitive circle. In the Stereographic projection, the primitive 

 plane is supposed to pass through the centre of the sphere, and the 

 eye to be placed at one pole of the primitive circle, viewing the oppo- 

 site hemisphere. In this case, all circles of the sphere are projected 

 either as circles or right lines ; as in Plate VII., Fig. 11. If the eye 

 were revolved down to A, the point d would evidently be seen as 

 if it were at d"; but if the eye were revolved about AB as an axis, 

 to the point D, then the point d would appear in the direction d'. 

 Thus, the parallels and meridians are determined. In the cylin- 

 drical projection, or developement of the sphere, which is that used 

 in the Mercator Charts, the eye is supposed to be placed at the centre, 

 and the surface is projected on a circumscribed cylinder, tangent to 

 the sphere, around the equator ; which cylinder is afterwards deve- 

 loped, or spread out, as a plane. Of the other projections, and of 

 warped surfaces, and surfaces of revolution, we have no farther 

 room to speak. 



CHAPTER IV. 



ANCYLOMETRY. 



ANCYLOMETRY, or Analytic Geometry, is that branch of Mathema- 

 tics in which Algebra is employed in determining the relations and 

 properties of Geometrical figures ; or, in other words, it is the appli- 

 cation of Algebra to Geometry. We venture to propose, for this 

 branch of Mathematics, the name of Jlncylometry , suggested by 

 Judge Woodward, and derived from the Greek oyxvAoj, a curve, and 

 pcrpov, a measure ; it being extensively employed in the measure of 

 curves. Under this head, we comprehend not only Conic Sections, 

 which it is generally made to include; but also Trigonometry; which, 

 though sometimes considered as a distinct branch of Mathematics, 

 may rather be regarded as a sub-branch, of limited extent, but of 

 high importance. The object of Trigonometry, is the relation of the 



