subtending half the arch. F. If two points be taken in the diameter of a circle, such that the rectangle contained by the segments intercepted between them and the center of the cir. cle be equal to the square of the radius: and if from these points two straight lines be drawn to any point whatsoever in the circumference of the circle, the ratio of these lines will be the same with the ratio of the segments intercepted between the two first mentioned points and the circumference of the circle. G. If from the extremity of the diameter of a circle, a straight line be drawn in the circle, and if either within the circle or produced without it, it meet a line perpendicular to the same diameter, the rectangle contained by the straight line drawn in the circle, and the segment of it, intercepted between the extremity of the diameter and the perpendicular, is equal to the rectangle contained by the diame. ter, and the segment of it cut off by the perpendicular. H. The perpendiculars drawn from the three angles of any triangle to the opposite sides intersect one another in the same point. K. If from any angle of a triangle a perpendicular be drawn to the opposite side or base; the rectangle contained by the sum and difference of the other two sides, is equal to the rectangle contained by the sum and difference of the seg ments, into which the base is divided by the perpendicular.

SUPPLEMENT TO THE ELEMENTS.

2.

BOOK I.-DEFINITIONS. 1. A chord of an arch of a circle is the straight line joining the extremities of the arch; or of the straight line which subtends the arch. The perimeter of any figure is the length of the line or lines, by which it is bounded. 3. The area of any figure is the space contained within it.

:

AXIOM. The least line that can be drawn between two points, is a straight line and if two figures have the same straight line for their base, that which is contained within the other, if its bounding line or lines be not any where convex towards the base, has the least perimeter. COR. 1. Hence the perimeter of any polygon inscribed in a circle, is less than the circumference of the circle. 2. If from a point two straight lines be drawn touching a circle, these two lines are together greater than the arch intercepted between them; and hence the perimeter of any polygon described

about a circle is greater than the circumference of the circle.

PROP. I. If from the greater of two unequal magnitudes there be taken away its half, and from the remainder its half: and so on; there will at length remain a magnitude less than the least of the proposed magnitudes. II. Equilateral polygons, of the same number of sides, inscribed in circles, are similar, and are to one another as the squares of the diameters of the circles. Cor. Every equilateral poly. gon inscribed in a circle is also equiangular: For the isosceles triangles, which have their common vertex in the cen. tre, are all equal and similar; therefore the angles at their bases are also equal, and the angles of the polygon are therefore also equal. III. The side of any equilateral polygon inscribed in a circle being given, to find the side of a polygon of the same number of sides described about the circle. IV. A circle being given, two similar polygons may be found, the one described about the circle, the other inscribed in it, which shall differ from one another by a space less than any given space. V. The area of a circle is equal to the rectangle contained by the semi-diameter, and a straight line equal to half the circumference. Cor. 2.

Hence a polygon may be described about a circle, the perimeter of which shall exceed the circumference of the circle by a line that is less than any given line. 3. Hence also, a polygon may be inscribed in a circle, such that the excess of the circumference above the perimeter of the polygon may be less than any given line. VI. The areas of circles are to one another in the duplicate ratio, or as the squares, of their diameters. Cor. 1. Hence the circum. ference of circles are to one another as their diameters. 2. The circle that is described upon the side of a right an. gled triangle opposite to the right angle, is equal to the two circles described on the other two sides. VII. Equiangu. lar parallelograms are to one another as the products of the numbers proportional to their sides. VIII. The perpendicu. lar drawn from the centre of a circle on the chord of any arch is a mean proportional between half the radius and the line made up of the radius and the perpendicular drawn from the centre on the chord of double that arch and the chord of the arch is a mean proportional between the diameter and a line which is the difference between the radius and the

foresaid perpendicular from the centre. IX. The circumference of a circle exceeds three times the diameter, by a line less than ten of the parts, of which the diameter contains seventy but greater than ten of the parts whereof the diameter contains seventy one.

BOOK II.-DEFINITIONS. 1. A straight line is perpendi cular or at right angles to a plane, when it makes right angles with every straight line which it meets in that plane. 2. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes, are pendendicular to the other plane. 3. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which, a perpendicular to the plane, drawn from any point of the first line, meets the same plane. 4. The angle made by two plains which cut another, is the angle contained by the two straight lines drawn from any, the same point in the line of their common section, at right angles to that line, the one, in one plane, the other, in the other. Of the two adjacent angles made by the two lines drawn in this manner, that which is acute is called the inclination of the planes to one another. 5. Two planes are said to have the same, or a like inclination to one another, which two other planes have, when the angles of inclination above defined are equal to one another. 6. A straight line is said to be parallel to a plane, when it does not meet the plane, though produced ever so far. 7. Planes are said to be parallel to one another, which do not meet, though produced ever so far. 8. A solid angle is an angle made by the meeting of more than two plane angles, which are not in the same plane in one point.

PROP. I. One part of a straight line cannot be in a plane and another part above it. II. Any three straight lines which meet one another, not in the same point, are in one plane. III. If two planes cut one another, their common section is a straight line. IV. If a straight line stand at right angles to each of two straight lines in their point of intersection, it will also be at right angles to the plane in which these lines are. V. If three straight lines meet all in one point, and a straight line stand at right angles to each of them in that point: these three straight lines are in one

and the same plane. VI. Two straight lines which are at right angles to the same plane, are parallel to one another. VII. If two straight lines be parallel, and one of them at right angles to a plane; the other is also at right angles to the same plane. VIII. Two straight lines which are each of them parallel to the same straight line, though not both in the same plane with it, are parallel to one another. IX. If two straight lines meeting one another be parallel to two others that meet one another, though not in the same plane with the first two; the first two and the other two shall contain equal angles. XI. From the same point in a plane, there cannot be two straight lines at right angles to the plane, upon the same side of it: and there can be but one perpendicular to a plane from a point above it. XII. Planes to which the same straight line is perpendicular, are parallel to one another. XIII. If two straight lines meeting one another, be parallel to two straight lines which also meet one another, but are not in the same plane with the first two: the plane which passes through the first two is parallel to the plane passing through the others. XIV. If two para!lel planes be cut by another plane, their common sections with it are parallels. XV. If two parallel planes be cut by a third plane, they have the same inclination to that plane. XVI. If two straight lines be cut by parallel planes, they must be cut in the same ratio. XVII. If a straight line be at right angles to a plane, every plane which passes through that line is at right angles to the first mentioned plane. XVIII. If two planes cutting one another be each of them perpendicular to a third plane, their common section is perpendicular to the same plane. XX. If a solid angle be contained by three plane angles, any two of these angles are greater than the third. XXI. The plane angles which contain any solid angle are together less than four right angles. BOOK III.-Definitions.

1. A solid is that which has length, breadth, and thick. ness. 2. Similar solid figures are such as are contained by the same number of similar planes similarly situated, and having like inclinations to one another. 3. A pyramid is a solid figure contained by planes that are constituted betwixt one plane and a point above it in which they meet. 4. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one

another; and the others are parallelograms. 5. A paral lelogram is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel. 6. A cube is a solid figure contained by six equal squares. 7. A sphere is a solid figure described by the revolution of a semicircle about a diameter, which remains unmoved. 8. The axis of a sphere is the fixed straight line about which the semicircle revolves. 9. The center of a sphere is the same with that of the semicircle. 10. The diameter of a sphere is any straight line which passes through the center, and is terminated both ways by the superficies of the sphere. 11. A cone is a solid figure described by the revolution of a right angled triangle about one of the sides containing the right angle which side remains fixed. 12. The axis of a cone is the fixed straight line about which the triangle revolves. 13. The base of a cone is the circle described by that side, containing the right angle, which revolves. 14. A cylinder is a solid figure described by the revolution of a right angled parallelogram about one of its sides, which remains fixed. 15. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. 16. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. 17. Similar cones and cylinders are those which have their axis, and the diameters of their bases proportionals.

PROP. I. If two solids be contained by the same number of equal and similar planes similarly situated, and if the inclination of any two contiguous planes in the one solid be the same with the inclination of the two equal, and similarly situated planes in the other, the solids themselves are equal, and similar. II. If a solid be contained by six planes, two and two of which are parallel, the opposite planes are similar and equal parallelograms. III. If a solid parallelopiped be cut by a plane parallel to two of its opposite planes, it will be divided into two solids, which will be to one another as their bases. IV. If a solid parallelopiped be cut by a plane passing through the diagonals of two of the opposite planes, it will be cut into two equal prisms. V. Solid parallelopipeds upon the same base, and of the same altitude, the insisting straight lines of which are terminated in the same straight lines in the plane opposite to the base, are equal to one another. VI. Solid parallelopipeds upon the same base,