PYRAMID
, a solid having any plane figure for its base, and its sides triangles whose vertices all meet in a point at the top, called the vertex of the pyramid; the base of each triangle being the sides of the plane base of the Pyramid.—The number of triangles is equal to the number of the sides of the base; and a cone is a round Pyramid, or one having an infinite number of sides. —The Pyramid is also denominated from its base, being triangular when the base is a triangle, quadrangular when a quadrangle, &c.
The axis of the Pyramid, is the line drawn from the vertex to the centre of the base. When this axis is perpendicular to the base, the Pyramid is said to be a right one; otherwise it is oblique.
1. A Pyramid may be conceived to be generated by a line moved about the vertex, and so carried round the perimeter of the base.
2. All Pyramids having equal bases and altitudes, are equal to one another: though the figures of their bases should even be different.
3. Every Pyramid is equal to one-third of the circumscribed prism, or a prism of the same base and altitude; and therefore the solid content of the Pyramid is found by multiplying the base by the perpendicular altitude, and taking 1/3 of the product.
4. The upright surface of a Pyramid, is found by adding together the areas of all the triangles which form that surface.
5. If a Pyramid be cut by a plane parallel to the base, the section will be a plane figure similar to the base; and these two figures will be in proportion to each other as the squares of their distances from the vertex of the Pyramid.
6. The centre of gravity of a Pyramid is distant from the vertex 3/4 of the axis.
Frustum of a Pyramid, is the part left at the bottom when the top is cut off by a plane parallel to the base.
The solid content of the Frustum of a Pyramid is found, by first adding into one sum the areas of the two ends and the mean proportional between them, the 3d part of which sum is a medium section, or is the base of an equal prism of the same altitude; and therefore this medium area or section multiplied by the altitude gives the solid content. So, if A denote the area of one end, a the area of the other end, and b the height; then 1/3 (A + a + √(Aa)) is the medium area or section, and 1/3 (A + a + √(Aa)) X b is the solid con- tent.
Pyramids of Egypt, are very numerous, counting both great and small; but the most remarkable are the three Pyramids of Memphis, or, as they are now called, of Gheisa or Gize. They are square Pyramids, and the dimensions of the greatest of them, are 700 feet on each side of the base, and the oblique height or slant side the same; its base covers, or stands upon, nearly 11 acres of ground. It is thought by some that these Pyramids were designed and used as gnomons, for astronomical purposes; and it is remarkable that their four sides are accurately in the direction of the four cardinal points of the compass, east, west, north, and south.
PYRAMIDAL Numbers, are the sums of polygonal numbers, collected after the same manner as the polygonal numbers themselves are found from arithmetical progressions.
These are particularly called First Pyramidals. The sums of First Pyramidals are called Second Pyramidals; and the sums of the 2d are 3d Pyramidals; and so on. Particularly, those arising from triangular numbers, are called Prime Triangular Pyramidals; those arising | from pentagonal numbers, are called Prime Pentagonal Pyramidals; and so on.
| The numbers | 1, | 4, | 10, | 20, | 35, | &c, |
| formed by adding the tri- | } 1, | 3, | 6, | 10, | 15, | &c, |
| angulars |
