Introduction to Quaternions, by P. Kelland and P. G. Tait

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1873 edition. Excerpt: ...points of section diameters be drawn to both circles, their other extremities and the other point of section will be in a straight line. 2. If a chord be drawn parallel to the diameter of a circle, the radii to the points where it meets the circle make equal angles with the diameter. 3. The locus of a point from which two unequal circles subtend equal angles is a circle. 4. A line moves so that the sum of the perpendiculars on it from two given points in its plane is constant. Shew that the locus of the middle point between the feet of the perpendiculars is a circle. 5. If 0, (y be the centres of two circles, the circumference of the latter of which passes through 0; then the point of intersection A of the circles being joined with 0' and produced to meet the circles in C, D, we shall have AC.AD = 2A0'. 6. If two circles touch one another in 0, and two common chords be drawn through 0 at right angles to one another, the sum of their squares is equal to the square of the sum of the diameters of the circles. 7. A, B, G are three points in the circumference of a circle; prove that if tangents at B and C meet in D, those at C and A in E, and those at A and B in F; then AD, BE, CF will meet in a point. 8. If A, B, C are three points in the circumference of a circle, prove that V(AB. BC. CA) is a vector parallel to the tangent at A. 9. A straight line is drawn from a given point 0 to a point P on a given sphere: a point Q is taken in OP so that OP.OQ = V. Prove that the locus of Q is a sphere. 10. A point moves so that the ratio of its distances from two given points is constant. Prove that its locus is either a plane or a sphere. 11. A point moves so that the sum of the squares of its distances from a number of given points is constant. Prove...