ISBN: 9781130414943

Taschenbuch

RareBooksClub. Paperback. New. This item is printed on demand. Paperback. 70 pages. Dimensions: 9.7in. x 7.4in. x 0.1in.This historic book may have numerous typos and missing text. Purc… Mehr…

RareBooksClub. Paperback. New. This item is printed on demand. Paperback. 70 pages. Dimensions: 9.7in. x 7.4in. x 0.1in.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1858 Excerpt: . . . TREATISE SPHERICAL TRIGONOMETRY. GENERAL PROPERTIES OF CIRCLES OF THE SPHERE. DEFINITION OF A SPHERICAL TRIANGLE. 1. The boundary of every plane section of a sphere is a circle. If the cutting plane pass through the center, this is evident; and in this case the section is called a great circle, and is determined when any two points on the surface of the sphere through which it passes are given. All great circles are equal to one another, since they have the same radius, namely that of the sphere; and they all bisect one another, since their planes intersect in diameters of the sphere. Hence the distance of the points of intersection of two great circles measured on the sphere is a semi-circumference. If the cutting plane does not pass through the center of the sphere, from 0 (fig. 1) the center of the sphere drop upon it the perpendicular 00, and join C with any pointed in the boundary of the section; then AO lAO-OG which is invariable; therefore the boundary of the section is a circle whose center is C; and it is called a small circle. Arcs of small circles are very rarely used; and when, hereafter, an arc of a circle is mentioned, an arc of a great circle, unless the contrary be specified, is invariably intended; and in most cases it is employed to denote the angle which it subtends at the center of the sphere, no regard being had to the radius of the sphere. 2. If OC be produced both ways to meet the surface of the sphere in P and F, then the line PA JPC AC which is invariable. Also if PAM be an arc of a great circle passing through P and A, since in equal circles equal straight lines cut off equal arcs, the length of the arc PA is invariable. Therefore the distance of P is the same from every point in the perimeter of the circle AB, whether measu. . . This item ships from La Vergne,TN., RareBooksClub, RareBooksClub. Paperback. New. This item is printed on demand. Paperback. 74 pages. Dimensions: 9.7in. x 7.4in. x 0.1in.This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1889 Excerpt: . . . rectangle contained by the side, on which when produced the perpendicular falls, and the straight line intercepted between the perpendicular and the acute angle. If C be the obtuse angle of a triangle ABC, D and E the feet of the perpendiculars from A and B on the opposite sides: prove that the square on AB is equal to the sum of the rectangles contained by BC and BD, and A C and AE. 5. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. Find the condition that a circle may be inscribed in a quadrilateral. 6. If two straight lines cut one another in a circle, but not at right angles, one of which passes through the centre, and the other does not, the rectangles contained by their segments are equal to one another. PCQ is a chord of a circle passing through a fixed point C. CK is drawn at right angles to PQ to meet the circle described on PQ as diameter in R. On what curve does R lie 7. Inscribe a square in a given circle. Compare the magnitudes of the inscribed and circumscribed squares. 8. Make the construction for describing an isosceles triangle, having each of the angles at the base double of the third angle. Show that the base of the triangle is equal to the side of a regular pentagon inscribed in the smaller circle of the figure. 9. What is meant by a line being cut in extreme and mean ratio In right-angled triangles, the rectilineal figure described upon the side opposite to the right angle is equal to the similar and similarly described figures. upon the sides containing the right angle. 10. If the exterior angle CBD of a triangle be bisected by a straight line, which cuts the base produced in E, prove that the square on BE is equal to the difference of the rectangles contained by AE, EC, an. . . This item ships from La Vergne,TN., RareBooksClub<