square on FE is equal to the squares on DE and DF (1. 47), that is, to the squares described on A and B. And because the angle GFE is a right angle, the square on GE is equal to the squares on FE and FG, that is, to the squares described on A, B, and C. Q.E.F. By proceeding in this way a square can be constructed equal to any number of squares. ALGEBRA. 1. Add together 8ax-7by+3y2; ax2+2by -7y2; 9ax +4y2 − 2 ; 7+ 3by - 2y2 + 4ax; 6ax2 - y2. 2. From 4-7x+10a - 12b take away 7a + 2b − 15x + 20. 3. Multiply 8xy-3yx2 - 4xy by 7xy-8x+6y. 8x2y - 3yx2 - 4xy = 5x2y – 4xy 7xy - 8x+6y 5x2y- 4xy 35x3y2 – 40x3y +30x2y2 - 28x2y2+32x2y - 24xy2 35x8y2-40x3y + 2x2y2+32x2y - 24xy2.-Ans. 4. Divide x + 3x2y2+2y4 by x2 + 2y2, and x+(q − x)b -qb2 by b-1. x2+ 2y2)x4 + 3x2y2 + 2ya(x2 + y2.—Ans. x+2x2x2 6. A goods train travels at the rate of 8 miles an hour on a railway, at the end of 12 hours it meets a passenger train travelling in the opposite direction, which arrives at the original starting station 16 hours after the goods train left. At what rate is the passenger train following? Let x= rate passenger train is travelling. As it arrives 16 hours after the goods train left, it takes 4 hours to reach the station; 4 Then 4x number of miles travelled in hours. 4x=96 ..x = 24. passenger train is travelling at the rate of 24 miles an hour. Scholarship Examination, 1872. EUCLID. 3. Show how to bisect a given angle. Hence show how to divide an angle into four equal parts. Construction.-Bisect the given angle. Then bisect each half, and the given angle will be divided into four equal parts. ALGEBRA. 1. Add together 4x2 - 3xy + ay2; 6x2 + 3by - 4ay2; ax2 − 7xy +3112; 9xy - 8y2 - 10x2. 2. From 10a-7x+4y- 136 take away 7y+ 10X 3. Multiply 7x2 + 3xy3 – 4xy by 8y+6xy2 - x2y. = = x(3y3 − 41 + 7x) × y(6xy + 8 − x2) 18xy - 24xy2+ 46x2y + 2413 − 32y+ 56x - 3x2ys - 7x3 = xy(18xya + 24y3 – 3x2μ3 – 24xy2 – 32y + 46x2y +56x7x3).—Ans. 4. Divide x 16y4 by x-2y, and a2x2 - 4abxy + 462y2 by |