extremity, so as not to cut the circle; or, which is the same thing, no straight line can make so great an acute angle with the diameter at its extremity, or so small an angle with the straight line which is at right angles to it, as not to cut the circle. 11. The opposite angles of any quadrilateral figure described in a circle, are together equal to two right angles. 12. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it. EUCLID.-(B.) DEFINE a plane superficies, a right angle, a semicircle, a parallelogram. 1. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angles contained by the two sides equal to them, of the other. 2. The greater side of every triangle subtends the greater angle. 3. If from the ends of a side of a triangle, there be drawn two right lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle. 4. If a right line falling upon two other right lines, make the exterior angle equal to the interior and opposite upon the same side of the line; or make the interior angles upon the same side together equal to two right angles; the two right lines shall be parallel to each other. 5. The right lines which join the extremities of two equal and parallel right lines towards the same parts, are also themselves equal and parallel. 6. Parallelograms on the same base and between the same parallels, are equal to each other. 7. If a right line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. 8. Describe a square that shall be equal to a given rectilineal figure. 9. Find the centre of a given circle. 10. Draw a right line from a given point, either without or in the circumference, which shall touch a given circle. 11. The angle at the centre of a circle is double of the angle at the circumference, upon the same base, that is, upon the same part of the circumference. 12. If two right lines within a circle cut one another, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. EUCLID. Books I. II. III.—(A.) For all Candidates for a B.A. Degree. 1. WHAT is meant by an axiom? Give an example of an axiom. 2. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another. Prove that equilateral triangles are equiangular. Can you extend the proposition to polygons? 3. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles. 4. Describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle. Is this a determinate problem? 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, is equal to the square of half the line. Prove this proposition algebraically. 6. Find the centre of a given circle. Shew that if in a circle a straight line bisects another at right angles the centre of the circle is in the line which bisects the other. 7. The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote: and the greater is nearer to the centre than the less. BOOKS IV. VI. For Candidates for Honors. 1. INSCRIBE in a given circle a triangle equiangular to a given triangle. When it is possible, shew how to inscribe in a given circle a quadrilateral figure equiangular to a given quadrilateral figure. 2. Describe a circle about a given equilateral and equiangular pentagon. If two diagonals of such a pentagon cut one another, shew that the greater segments are equal to a side of the pentagon. 3. If the angle of a triangle be divided into two equal angles by a straight line which also cuts the base; the segments of the base shall have the same ratio which the other two sides of the triangle have to one another. Having given the vertical angle, and the ratio of the sides containing it and also the diameter of the circumscribing circle, construct the triangle. 4. If three straight lines be proportionals, the rectangle contained by the extremes is equal to the square on the mean. 5. Describe a rectilineal figure which shall be similar to one and equal to another given rectilineal figure. 6. If from any angle of a triangle a straight line be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle. The rectangle contained by the two sides can never be less than twice the triangle. EUCLID. Books I. II. III.—(B). For all Candidates for a B.A. Degree. 1. Is there any distinction between a postulate and an axiom? Give an example of the former. 2. If two angles of a triangle be equal to each other, the sides also which subtend, or are opposite to, the equal angles shall be equal to each other. Prove that equilateral triangles are equiangular. Can you extend the proposition to polygons? 3. The opposite sides and angles of parallelograms are equal to each other, and the diameter bisects them, that is, divides them into two equal parts. What distinction is there between a parallelogram and a rhomboid? 4. If a square described on one of the sides of a triangle be equal to the squares described on the other two sides of it; the angle contained by these two sides is a right angle. 5. If a right line be bisected, and produced to any point; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the right line which is made up of the half and the part produced. Prove this proposition algebraically. 6. If any two points be taken in the circumference of a circle, the right line which joins them shall fall within the circle. Shew that if in a circle a straight line bisects another at right angles the centre of the circle is in the line which bisects the other. 7. Draw a right line from a given point, either without or within the circumference, which shall touch a given circle. Books IV. VI. For Candidates for Honors. 1. ABOUT a given circle describe a triangle equiangular to a given triangle. Shew how to describe about a given circle a quadrilateral figure equiangular to a given quadrilateral figure. 2. Inscribe an equilateral and equiangular quindecagon in a given circle. Shew that the number of the sides of any regular figure that can be inscribed in a circle by Euclid's propositions is expressed by one of the numbers 2", 2"+2±2′′, or 2′′+4—2′′. 3. If the outward angle of a triangle, made by producing one of its sides, be divided into two equal angles by a straight line which also cuts the base produced; the segments between the dividing line and the extremities of the base, have the same ratio which the other sides of the triangle have to one another. 4. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means. Shew that the greatest and least of four such lines are together greater than the other two. 5. 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. 6. The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides. If the diagonals cut one another at an angle equal to one third of a right angle, the rectangles contained by the opposite sides are together equal to four times the quadrilateral figure. MECHANICS.—(A.) (Trigonometrical forms of solution are not excluded.) 1. DEFINE Weight, Density, and Mass. If 5 and 6 cubic inches respectively of two substances, whose densities are as 3 2, together weigh as much as 12 cubic inches of copper, whose density is 9; find the densities of the substances. 2. Define Force and Resultant. When may a force which acts on a body be said to be known? Two forces act on a particle, and their greatest and least resultants are 7.3 lbs. and 5.6 lbs.; find the forces. 3. Enunciate the parallelogram of forces and prove it for the magnitude of the resultant. If one of two forces which act on a particle be 5lbs. and the resultant be also 5lbs. and at right angles to the known force, find the magnitude and direction of the other force. 4. If two forces acting perpendicularly on a straight lever in opposite directions and on the same side of the fulcrum balance each other, they are inversely as their distances from the fulcrum and the pressure on the fulcrum is equal to their difference. If the lever were a heavy beam, what alteration would be introduced in the condition? 5. Enumerate the different kinds of levers. Explain the term advantageous, as applied to the lever. The arms of a lever are as 4: 3, and the forces act on opposite sides of the fulcrum: find the forces when the pressure on the fulcrum is 33 lbs. 6. In a system of pullies in which the same string passes round any number of pullies, and the parts of it between the pullies are parallel, there is equilibrium when P: W:: 1: the number of strings at the lower block. Draw two figures to represent this system, according as the number of strings at the lower block is odd or even. 7. The weight W being on an inclined plane, and the force P acting parallel to the plane, there is equilibrium when P : W :: the height of the plane: its length. What is the condition of equilibrium when the power acts parallel to the base of the plane? 8. Define velocity. If P and W balance each other on the wheel and axle and the whole be set in motion, P: W:: W's velocity: P's velocity. Prove this, assuming that the arcs which subtend equal angles at the centres of two circles are as the radii. 9. Define "centre of gravity." Find the centre of gravity of two heavy particles. If O be a point in the line joining m and m' two heavy particles, shew that M. Om2 + M'. Om'2 is least when O coincides with the centre of gravity of the particles. 10. If a body balance itself upon a line in all positions, the centre of gravity of the body is in that line. If a body were found to balance in two positions upon a line, could it be inferred that the centre of gravity was in the line? 11. Find the centre of gravity of a triangle. Find the locus of the centres of gravity of all triangles upon the same base and between the same parallels. MECHANICS.-(B.) (Trigonometrical forms of solution are not excluded.) 1. DEFINE Weight, Density, and Mass. If 3 and 4 cubic inches respectively of two substances, whose densities are as 21, together weigh as much as 2.3 cubic inches of silver whose density is 10; find the densities of the former substances. |