49. The centre of a circle being o, and Pa point in the radius, or in the radius produced; if the circumference be divided into as many equal parts a B, B C, C D, &c. as there are units in 2 n, and lines be drawn from P to all the points of division; then shall the continual product of all the alternate lines, viz. PA X PCX PE &C. be="" when P is within the circle, or = x" when P is without the circle; and the product of the rest of the lines, viz. PBX PD X PF, &c. where r = A o the radius, and x = o p the dis'tance of P from the centre.

=

A

B

50. A circle may thus be divided into any number of parts that shall be equal to one another both in area and perimeter. Divide the diameter QR into the same number of equal parts at the points s, T, v, &c. ; then on one side of the diameter describe semi- af circles on the diameters Q s, Q T, Q v, and on the other side of it describe semicircles on R V, RT, RS; so shall the parts, 1 7, 3 5, 5 3,7 1, be all equal, both in area and peri

meter.

SECTION V.-Of Planes and Solids.

Definitions.

1. The common section of two planes, is the line in which they meet, or cut each other.

2. A line is perpendicular to a plane, when it is perpendicular to every line in that plane which meets it.

3. One plane is perpendicular to another, when every line of the one, which is perpendicular to the line of their common section, is perpendicular to the other.

F

4. The inclination of one plane to another, or the angle they form between them, is the angle contained by two lines, drawn from any point in the common section, and at right angles to the same, one of these lines in each plane.

5. Parallel planes are such as being produced ever so far both ways, will never meet, or which are everywhere at an equal perpendicular distance.

6. A solid angle is that which is made by three or more plane angles, meeting each other in the same point.

7. Similar solids, contained by plane figures, are such as have all their solid angles equal, each to each, and are bounded by the same number of similar planes, alike placed.

8. A prism is a solid whose ends are parallel, equal, and like plane figures, and its sides, connecting those ends, are parallelograms.

9. A prism takes particular names according to the figure of its base or ends, whether triangular, square, rectangular, pentagonal, hexagonal, &c.

10. A right or upright prism, is that which has the planes of the sides perpendicular to the planes of the ends or base.

11. A parallelopiped, or a parallelopipedon, is a prism bounded by six parallelograms, every opposite two of which are equal, alike, and parallel.

12. A rectangular parallelopipedon is that whose bounding planes are all rectangles, which are perpendicular to each other

13. A cube is a square prism, being bounded by six equal square sides or faces, which are perpendicular to each other.

14. A cylinder is a round prism having circles for its ends; and is conceived to be formed by the rotation of a right line about the circumferences of two equal and parallel circles, always parallel to the axis.

15. The axis of a cylinder is the right line joining the centres of the two parallel circles about which the figure is de

scribed..

16. A Pyramid is a solid whose base is any rightlined plane figure, and its sides triangles, having all their vertices meeting together in a point above the base, called the vertex of the pyramid.

17. Pyramids, like prisms, take particular names from the figure of their base.

18. A cone is a round pyramid having a circular base, and is conceived to be generated by the rotation of a right line about the circumference of a circle, one end of which is fixed at a point above the plane of that circle.

19. The axis of a cone is the right line, joining the vertex or fixed point, and the centre of the circle about which the figure is described.

20. Similar cones and cylinders, are such as have their altitudes and the diameters of their bases proportional.

21. A sphere is a solid bounded by one curve surface, which is every where equally distant from a certain point within, called the centre. It is conceived to be generated by the rotation of a semicircle about its diameter, which remains fixed.

22. The axis of a sphere is the right line about which the semicircle revolves, and the centre is the same as that of the revolving semicircle.

23. The diameter of a sphere is any right line passing through the centre, and terminated both ways by the surface. 24. The altitude of a solid is the perpendicular drawn from the vertex to the opposite side or base.

Prop. 1. If any prism be cut by a plane parallel to its base, the section will be equal and like to the base.

2. If a cylinder be cut by a plane parallel to its base, the section will be a circle, equal to the base.

3. All prisms and cylinders, of equal bases and altitudes, are equal to each other.

4. Rectangular parallelopipedons, of equal altitudes, are to each other as their bases.

5. Rectangular parallelopipedons, of equal bases, are to each other as their altitudes.

6. Because, prisms and cylinders are as their altitudes, when their bases are equal: and, as their bases when their altitudes are equal. Therefore, universally, when neither are equal, they are to one another as the product of their bases and altitudes hence, also, these products are the proper numeral measures of their quantities or magnitudes.

M

L

7. Similar prisms and cylinders are to each other as the cubes of their altitudes, or of any like linear dimensions.

8. In any pyramid a section parallel to the base is similar to the base; and these two planes are to each other as the squares of their distances from the vertex.

9. In a cone, any section parallel to the base is a circle; and this section is to the base as the squares of their distances from the vertex.

10. All pyramids and cones, of equal bases and altitudes, are equal to one another.

11. Every pyramid is a third part of a prism of the same base and altitude.

12. If a sphere be cut by a plane, the section will be a circle.

13. Every sphere is two-thirds of its circumscribing cylinder.

14. A cone hemisphere, and cylinder of the same base and altitude, are to each other as the numbers 1, 2, 3.

15. All spheres are to each other as the cubes of their diameters; all these being like parts of their circumscribing cylinders. 16. None but three sorts of regular plane figures joined together can make a solid angle and these are, 3, 4, or 5 triangles, 3 squares, and three pentagons.

And therefore there can only be five regular bodies, the pyramid, cube, octaedron, dodecaedron, and icosaedron.

17. No other but only one sort of the five regular bodies, joined at their angles, can completely fill a solid space; viz. eight cubes.

18. A sphere is to any circumscribing solid B F, (all whose planes touch the sphere); as the surface of the sphere to the surface of the solid.

19. All bodies circumscribing the same sphere, are to one another as their surfaces.

20. The sphere is the greatest or most capacious of all bodies of equal surface.

SECTION VI.-Practical Geometry.

It is not intended in this place to present a complete collection of Geometrical Problems, but merely a selection of the most useful, epecially in reference to the employments of mechanics and engineers.

The instruments well known to be used in geometrical constructions, are the scale and compasses, the semicircular or the circular protractor, the sector, and a parallel ruler. To these a few other useful instruments may be added, which we shall describe as we proceed; speaking first of the Triangle and Ruler.

These are, as their names indicate, a triangle, that is to say, an isosceles right-angled triangle, and a ruler, both made of well seasoned wood, or of ivory, ebony, or metal. Each side A B, A C, of the triangle, about the right angle A, being 3, 4, 6, or 8 inches, according to the magnitude of the figures, in whose construction it is

likely to be employed. About the middle of the

triangle there should be a

A

circular orifice, as shown in the figures; and if a scale of equal parts be

B

E

placed along each of the three sides, all the better. The ruler may be from 12 to 18 inches in length; and it also may, usefully, have a scale along one of its sides. cation of these instruments is of great appear.

PROB. I. To bisect a given line.

The conjoined appliutility; as will soon

c

Let a b be the line proposed. Lay the longest side B C of the triangle so as to coincide with a b, and so that its angle в shall coincide with the point a; and along the side B A of the triangle draw a line a d. Then slide the base B C of the triangle along the line a b, until c coincides with b, and draw in coincidence with the side c A, the line bd intersecting the former in d. Next bring the ruler to coincide with a b, and

a

i

in contact with it lay one of the legs B A of the triangle; then slide the triangle along the ruler, until the other leg a C passes through the point d: draw along A c, so posited, the line di; it will be perpendicular to a b, and will bisect it in i, the point required.