GEOMETRICAL EXERCISES ON BOOK XII. 1 THEOREM I. If semicircles ADB, BEC be described on the sides AB, BC of a right-angled triangle, and on the hypothenuse another semicircle AFBGC be described, passing through the vertex B; the lunes AFBD and BGCE are together equal to the triangle ABC. It has been demonstrated (XII. 2.) that the areas of circles are to one another as the squares of their diameters; it follows also that semicircles will be to each other in the same proportion. Therefore the semicircle ADB is to the semicircle ABC, as the square BC is to the square of AC, squares of AB, BC are to the square of AC; squares of AB, BC are equal to the square of AC: (1. 47.) therefore the semicircles ADB, CEB are equal to the semicircle ABC. (v. 14.) From these equals take the segments AFB, BGC of the semicircle on AC, and the remainders are equal, that is, the lunes AFBD, BGCE are equal to the triangle BAC. THEOREM II. If on any two segments of the diameter of a semicircle semicircles be described, all towards the same parts, the area included between the three circumferences (called apßnaos) will be equal to the area of a circle, the diameter of which is a mean proportional between the segments. (Archimedis Lemm. Prop. 4.) Let ABC be a semicircle whose diameter is AB, and let AB be divided into any two parts in D, Then the area contained between the three semicircles, is equal to the area of the circle whose diameter is BD. Since AC is divided into two parts in C, the square of AC is equal to the squares of AD, DC, and twice the rectangle AD, DC; (11. 4.) and since BD is a mean proportional between AD, DC; the rectangle AD, DC is equal to the square of DB, (vi. 17.) therefore the square of AC is equal to the squares of ÀD, DC, and twice the square of DB. But circles are to one another as the squares of their diameters or radii, (XII. 2.) therefore the circle whose diameter is AC, is equal to the circles whose diameters are AD, DC, and double the circle whose diameter is BD; wherefore the semicircle whose diameter is AC is equal to the circle whose diameter is BD, together with the two semicircles whose diameters are AD and DC: if the two semicircles whose diameters are AD and DC be taken from these equals, therefore the figure comprised between the three semi-circumferences is equal to the circle whose diameter is DB. THEOREM III. Every section of a sphere by a plane is a circle. If the plane pass through the centre of the sphere, it is manifest that the section is a circle, having the same diameter as the generating semicircle. But if the cutting plane does not pass through the centre, let AEB be any other section of the sphere made by a plane not passing through the centre of the sphere. Take the centre C, and draw the diameter HCK perpendicular to the section AEB, and meeting it in D; draw AB passing through D, and join AC; take E, F, any other points in the line AĚB, and join CE, DE; CF, DF. Then since CD is perpendicular to the plane AEB, it is perpendicular to every straight line which meets it in that plane, therefore the angles CDA, CDE, CDF are right angles, and CA, CF, CE, being lines drawn from C, the centre of the sphere to points in the surface, are therefore equal to one another. Hence, in the right-angled triangles, CDA, CDF, CDE; the square of DA is equal to the difference of the squares of CA and CD; and the square of DF is equal to the difference of the squares of CF, and CD; also the square of DE is equal to the difference of the squares of CE and CD: therefore the squares of DA, DF, DE are equal to one another; and therefore the lines DA, DF, DE are equal to one another. But DA, DF, DE are three equal lines drawn from the same point D, in the same plane, hence the points A, F, E lie in the circumference of a circle of which D is the centre. THEOREM IV. There can be only five regular solids. If the faces be equilateral triangles. The angle of an equilateral triangle is one third of two right angles; and six angles, each equal to the angle of an equilateral triangle, are equal to four right angles; and therefore a number of such angles less than six, but not less than three are necessary to form a solid angle. Hence there cannot be more than three regular figures whose faces are equal and equilateral triangles. If the faces be squares. Since four angles, each equal to a right angle, can fill up space round a point in a plane. A solid angle may be formed with three right angles, but not with a number greater or less than three. Hence, there cannot be more than one regular solid figure whose faces are equal squares. If the faces be equal and regular pentagons. Since each angle of a regular pentagon is a right angle and a fifth of a right angle: the magnitude of three such angles being less than four right angles, may form a solid angle, but four, or more than four, cannot form a solid angle. Hence, there cannot be more than one regular figure whose faces are equal and regular pentagons. If the faces be equal and regular hexagons, heptagons, octagons, or any other regular figures; it may be shewn that no number of them can form a solid angle. Wherefore there cannot be more than five regular solid figures, of which, there are three, whose faces are equal and equilateral triangles; one, whose faces are equal squares; and one, whose faces are equal and regular pentagons. PROBLEMS. 1. In what sense is it said that the circle does not admit of quadrature? Describe generally the process by which Archimedes obtained a first approximation to the ratio of the circumference of a circle to its diameter. 2. Given a circle traced upon a plane, describe another whose area is exactly twice as great as that of the former. 3. ABC is a circle of given radius, describe another concentric circle abc whose area shall be equal to 3 times the area of ABC. 4. Divide a circle into any number of equal parts by means of concentric circles. 5. To divide a circle into any number of equal parts, the perimeters of which shall be equal to the circumference of the circle. 6. Two circles touch each other internally, and the area of the lune cut out of the larger is equal to twice the area of the smaller circle. Required the ratio of the diameters of these circles. 7. If on one of the radii of a quadrant a semicircle be described ; and on the other, another semicircle so described as to touch the former and the quadrantal arc; compare the area of the quadrant with the area of the circle described in the figure bounded by the three curves. 8. The centres of three circles A, B, and C are in the same right line, B and C touch A internally, and each other externally; P, Q being the points where A is touched by B, C respectively: to find a point R on A such that the portion of the lune PR intercepted between B and A may be equal to the portion of QR between C and A. 9. The diameter of a circle is divided into two parts, upon each of which as diameters circles are described; when the remaining area of the great circle is equal to that of one of these two circles, find the ratio which the parts of the diameter bear to one another. 10. Describe a sphere about a given regular tetrahedron ; and the edge of the tetrahedron being 1, find the radius of the sphere. 11. Having given an irregular fragment, which contains a portion of spherical surface: shew how the radius of the sphere, to which the fragment belongs, may be practically determined. 12. Let three given spheres be placed on a horizontal plane in mutual contact with each other; find the sides of the triangle formed by joining the points in which the spheres touch the plane. 13. Construct the five regular solids. 14. Having given six straight lines, of which each is less than the sum of any two, determine how many tetrahedrons can be formed, of which these straight lines are the edges. 15. Inscribe a sphere within a tetrahedron. 16. Find the dihedral angle contained by two adjacent faces of a regular octahedron: and find its solidity. 17. Shew how to find the content of a pyramid, whatever be the figure of its base, the altitude and area of the base being given. 18. A pyramid of triangular base is composed of ten spherical balls of given radius; the base is composed of six, the next layer of three, and the remaining one is placed upon them. Find the distance of the upper ball from the ground. THEOREMS. 5. The area of a circle is equal to half the rectangle contained by two straight lines which are equal to its circumference and radius. 6. Compare the angle in a segment, which is one fourth part of a circle, with a right angle. 7. In different circles the radii which bound equal sectors contain angles reciprocally proportional to their circles. 8. Prove that the sectors of two different circles are equal, when their angles are inversely as the squares of the radii. 9. If the diagonals of a quadrilateral inscribed in a circle cut each other at right angles, and circles be described on the sides ; prove that the sum of two opposite circles will be equal to the sum of the other two. 10. If circles be inscribed in the triangles formed by drawing the altitude of a triangle right-angled at the vertex, the circles and the triangles are proportional. 11. Two straight lines are inclined to each other at a given angle, find the area of all the circles which can be described touching each other and the two given lines, the position of the centre of the last circle being given. 12. If two chords of a circle intersect each other either within or without the circle at right angles; and if on these segments as diame, ters, circles be described, the areas of these four circles are together equal to that of the original circle. 13. Shew that the semicircles described on the diagonals of a right-angled parallelogram together equal the sum of the semicircles described on the sides. 14. The diameter of a semicircle is divided into any number parts, and on these parts semicircles are described. Shew that their circumferences are together equal to that of the given semicircle. 15. If two circles be so placed that their planes are parallel, and the straight line which joins their centres perpendicular to the plane of each; the straight lines which join the opposite extremities of any pair of parallel diameters will all intersect the straight line which joins the centres in the same point: if the circles be equal, that point will be the bisection of the aforesaid straight line. 16. If through a point without the plane of a circle straight lines be drawn to its circumference and a plane be drawn parallel to the circle on either side of the point, the points of intersection of the lines with the plane will be in a circle, and the area of this and of the first circle will be as the squares of their distances from the point given. 17. If the arc of a semicircle be trisected, and from the points of section lines be drawn to either extremity of the diameter, the difference of the two segments thus made will be equal to the sector which stands on either of the arcs. 18. The centres of three circles A, B, and C are in the same right line, B and C touch A internally, and each other externally. Shew that the portion of the area of A, which is outside B and C, is equal to the area of the semicircle described on the chord of A which touches B and C at their point of contact. 19. If three equal circles intersect, so that each of the circumferences pass through the centres of the other two, the spaces bounded by the circumferences intercepted will be all equal. |