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QUESTIONS IN EUCLI D.
1. In a given circle to inscribe an equilateral and equi- 1828 angular pentagon.
2. If a straight line be at right angles to a plane, every plane passing through that straight line is at right angles to the same plane.
3. In a circle the angle in a semicircle is a right angle, but 1829 the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle.
4. If two triangles have the sides about equal angles reciprocally proportional, they are equal.
5. If a straight line be at right angles to two straight lines at their point of intersection, it is at right angles to the plane passing through them.
6. In any right-angled triangle, the square which is described 1830 upon the side subtending the right angle, is equal to the sum of the squares described upon the sides containing the right angle.
7. The sides about the equal angles of equiangular triangles are proportional, and those which are opposite to the equal angles are homologous sides,
8. If two straight lines meeting one another be parallel to two others which meet one another, but are not in the same plane with the first two, the plane which passes through them, is parallel to the plane passing through the others.
9. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.
10. Describe an equilateral and equiangular pentagon about a given circle.
11. Triangles and parallelograms of the same altitude are to one another as their bases.
12. Planes to which the same straight line is perpendicular are parallel to one another.
13. Every prism having a triangular base, may be divided into their pyramids that have triangular bases, and are equal to one another.
14. The angles which one straight line makes with another upon one side of it are either two right angles, or are together equal to two right angles.
15. Draw a straight line from a given point either without or in the circumference which shall touch a given circle.
16. If the sides of a rectangle be a and b, what is meant by saying that the rectangle ab? Illustrate this by considering the case where a = 5 and b
What two propositions of Euclid may then be adduced to prove the area of a triangle to be equal to
base . altitude
? 17. 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.
18. If a straight line be at right angles to a plane, every plane passing through it shall be at right angles to that plane.
19. In the demonstration of the fourth proposition of the First Book of Euclid, is it assumed that two straight lines cannot have a common segment? Give a definition of a straight line which shall supersede the necessity of any axiom respecting it.
20. One circle cannot touch another in more points than one, whether it touches it on the inside or outside.
21. If an angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle.
22. The perimeters of similar rectilinear figures are in the simple ratio, and the areas in the duplicate ratio of their homologous sides.
23. Segments of right lines intercepted between parallel planes are proportional.
24. Through a given point draw a straight line parallel to 1834 a given straight line.
25. If any two points be taken in the circumference of a circle, the straight line which joins them falls within the circle.
26. Triangles and parallelograms of the same altitude are to one another as their bases. State Euclid's definition of proportion, and the algebraic definition, and shew that they coincide.
27. If two straight lines be at right angles to the same plane, shew that they are parallel to one another. What is Euclid's definition of parallel straight lines, and what other definition has been proposed ?
28. If a circle be described touching the base of a triangle and the sides produced, and a second circle be inscribed in the triangle; prove that the points where the circles touch the base are equidistant from its extremities, and that the distance between the points where they touch either one of the sides is equal to the base.
29. The opposite side and angles of parallelograms are 1835 equal to one another, and the diameter bisects them. If both the diameters be drawn, the parallelogram will be divided into four equal parts.
30. Upon the same base and upon the same side of it there cannot be two similar segments of circles not coinciding with one another.
31. In equal circles sectors have the same ratio which the circumferences on which they stand have to one another.
32. If a solid angle be contained by three plane angles, any two of them are greater than the third. How magnitudes of two solid angles be compared ?
33. It is impossible to divide a quadrilateral figure (except it be a parallelogram) into equal triangles by lines drawn from a point within it to its four corners.
34. Similar triangles are to each other in the duplicate ratio of their homologous sides.
35. In a given rectangle inscribe another, whose sides shall bear to each other a given ratio.
36. If ab, cd be chords of a circle at right angles to each other, prove that the sum of the arcs ac, bd is equal to half the circumference.
37. A common tangent is drawn to two circles which touch each other externally; if a circle be described on that part of it which lies between the points of contact, as diameter, this circle will pass through the point of contact of the two circles, and will touch the line which joins their centres.
38. The diameter of a semicircle is divided into two parts, on each of which as diameter a semicircle is described, and on the same side as the given one. If a circle be described touching each of these three semicircles, the distance of its centre from their common diameter is equal to twice its radius.
39. Define a parallelogram. Describe one which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
40. Describe a square about a given circle.
QUESTIONS IN ARITHMETIC AND ALGEBRA, NOT INCLUDING
THE GENERAL NATURE OF EQUATIONS.
1. The number 803 expressed in a different scale of nota- 1821 tion becomes 30203; required the radix of the scale. 2. Prove that
1 n n.(n − 1)" + n (n − 2)" — &c. = 1.2.3....N.
2 3. Find x from the equation
2a Viva + x =
4. The common difference of 4 numbers in arithmetical progression is 1, and their product 120; find the numbers.
5. Insert three harmonic means between a and b.
6. Investigate the rule for transposing a number from one scale of notation to another. 7. If P A
+ (x + a)"Q (x + a)" (x+a)n-1 (x +a) + Q where P and Q are rational functions of t, to determine the values of A, A7, A, ... An-, and P'.
8. A tetrahedron, whose base is an equilateral triangle, the side of which is equal to one half of each of the remaining edges, is thrown upon a horizontal table : what is the probability of its resting upon its base, excluding all consideration of the mechanical action, which arises from the rotation of the solid?