a sphere, is a part of its surface, cut off by a plane, or intercepted between two parallel planes; and the intercepted solid is called a spherical segment. A spherical triangle, is a portion of the surface of a sphere, bounded by three arcs of great circles, that is, circles whose planes pass through the centre.

The convex surface of a cylinder, is equal to the circumference of its base, multiplied by its altitude: that of a cone, is equal to the circumference of its base, multiplied by its slant height, or distance from the vertex to any point of the circumference just named: and the surface of a sphere, is equal to the product of its diameter by the circumference of a great circle, that is, of the sphere itself. The measure of the solidity of a cylinder, is the product of its base by its altitude; that of a cone, is the product of its base by one-third of its altitude; and that of the sphere is the product of its surface by onethird of its radius.

§ 6. Descriptive Geometry, relates to the representation of geometrical figures on planes, and the construction of graphical problems thereby. It includes, therefore, the principles of perspective, and Spherical Projections. If we suppose the eye to be placed at a very great height, and looking vertically down upon an object situated above a horizontal plane, then the object will hide a part of the plane, of the same shape or outline, as that which the object itself presents to the eye. This representation of the object on the plane, is called its horizontal projection; and the plane itself is called the horizontal plane of projection. In like manner, if we suppose the eye to be placed in front of an object, and a vertical plane behind it, we may have a vertical projection, of the object, on the vertical plane of projection. When these two planes of projection are both used, they intersect each other in a line called the ground line and if we suppose one of them, with any projections made upon it, to be revolved about the ground line as an axis, till it coincides with the other plane, we shall then have both the projections of any object, on one and the same plane; as that of the paper or drawing board. Plate VII., Fig. 9, represents the horizontal plane GHIB, as revolved about the ground line AB, until it coincides with the vertical plane ACDB, prolonged downwards to EF.

The projection of any point, is another point, directly above or below it, or else directly before or behind it; and is found by drawing a perpendicular from the given point to one or the other plane of projection. The projection of a line, is another line, lying in one or the other plane of projection; and is found by joining the projections of two of its points on that plane. The position of any plane, in space, is known, if we have its intersections with the two planes of projection; which intersections are called its traces. If a given plane be revolved about one of its traces as an axis, until it coincides with the plane of projection in which that trace lies, each point of the given plane will revolve in a circular arc, and take the same relative position in the given plane, after the revolution, as it had before. It is by an ingenious application of these and similar principles, that lines, surfaces and solids may be delineated on a single plane, and their dimensions or relations determined to a surprizing extent. In

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 projections 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 different 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 opposite 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 cylindrical 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 developed, 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 Mathematics in which Algebra is employed in determining the relations and properties of Geometrical figures; or, in other words, it is the application of Algebra to Geometry. We venture to propose, for this branch of Mathematics, the name of Ancylometry, suggested by Judge Woodward, and derived from the Greek ayxv2os, a curve, and METOV, 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

parts of triangles; by which, certain parts being given, the others. may be determined. Conic Sections, is the name applied to the study of the curves formed by the intersections of a plane and a cone; that is, the circle, ellipse, parabola, and hyperbola. These curves are often referred to, particularly in Astronomy and Navigation; while Trigonometry is also of frequent service, in these studies, and in the practice of Surveying and Mensuration.

Trigonometry, derives its name from the Greek piywvos, a triangle, and usτpov, a measure. It is said to have been first investigated by Hipparchus but the oldest work extant upon it, is that of Menelaus of Alexandria; and the earliest trigonometrical tables which have been preserved, are those of Ptolemy, in his Almagest. The Arabians simplified Trigonometry, by the introduction of sines, or the half chords of double arcs, as the means of expressing angles: a method employed in the writings of Albategnius, about A. D. 880; though its invention is also claimed for the Hindoos. The Arabian astronomer, Geber ben Aphla, in the 11th century, compiled three or four theorems, which became the basis of modern trigonometry. Müller, of Germany, called also Regiomontanus, farther improved Trigonometry by the use of tangents: and he was the first to resolve spherical triangles, by finding the relations of their sides and angles. To Napier, we are indebted, for his rules or Analogies, which assist us in remembering the more difficult formulas; and especially for the invention of Logarithms, by which trigonometrical calculations are so greatly simplified. Other improvements have been made by Euler and others; and the formulas of Trigonometry are now become so general, and complete, as to leave but little more to be expected, or even desired.

The first examination of Conic Sections, has been attributed by some writers to Menechmus, a friend of Plato; and by others to Aristæus, whose writings are lost. The earliest work extant, on this subject, is that of Apollonius of Perga, who flourished about 150 B. C.; and who ranked next to Archimedes, as a geometer. Dr. Wallis, in 1655, introduced the method of studying these curves without any necessary reference to their being sections of a cone. Galileo, Kepler, and Newton, discovered that the orbits of the planets are curves of this class; since which discovery they have been very extensively and carefully studied.

The invention of the modern Analytic Geometry, is attributed to Vieta, and Descartes. Vieta applied it only to the construction of the roots of equations; but Descartes, in 1637, by the invention of coördinates, found the means of designating geometrical curves by algebraic equations; in which the essence of this branch consists. Descartes applied this system to curves of double curvature, by means of their two projections; but Maclaurin discovered a more direct method, by means of triple coördinates, parallel to three different axes, and related to each other by the nature of the curve, or surface. It is only since the time of Descartes, that Trigonometry and Conic Sections have been treated analytically, and thus become a part of this branch of Mathematics; which has thus aided the study of pure Geometry.