IV.

✓ Buclid

[N.B. Two Propositions at least from the Second Book are expected.]

1. Define plane rectilineal angle, point, rhombus, gnomon, parallelogram.

2. Distinguish between a Problem and a Theorem.

3. What is Euclid's test for ascertaining when two straight lines will meet, and when two straight lines will be in one and the same straight line.

4. To a given right line apply a parallelogram which shall be equal to a given triangle and have one of its angles equal to a given rectilineal angle.

5. If a straight line be divided into two equal parts, and also into two unequal parts, the rectangle contained by the unequal parts together with the square of the line between the points of section, shall be equal to the square of half the line.

Give the Corollary.

6. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by these two sides is a right angle.

7. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

8. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the parts together with twice the rectangle contained by those parts.

Give the Corollary.

9. Describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

10. In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side on which, when produced, the perpendicular falls, and the straight line intercepted without the triangle, between the perpendicular and the obtuse angle.

11. Find the value of all the interior angles of any rectilineal figure; and also of all the exterior angles, made by producing the sides successively in the same direction.

2. Multiply a⭑+a3x+x by a3 +ax3+x1.

3. Divide x13 — 3x3y2 + 3x2y3 —x3y1 by —x2y2; and at-16 by a

-2.

4. Find the square root of

16m2 + 13 m2n+8m2 + — n2 + n + 1.

what would be the result if a were m2 ?

9. A grocer would mix spice at 10s. a pound with another sort at 6s. a pound, to make 100 lb. to be sold at 78. a pound. What quantity of each must he take?

10. The difference between a number and its square is 18 more than the same difference would be if the number were one less. Find the number.

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DEC. 1873.