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N.B.-Algebraic symbols must not be used.

(a) The straight line drawn at right angles to the diameter of a circle from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle. (III 16.)

(b) Draw a common tangent to two given circles. How many can be drawn? (Apollonius.)

2. (a) The opposite angles of any quadrilateral figure inscribed in & circle are together equal to two right angles. (III 22.)

(b) If straight lines be drawn from any point on the circumference of a circle perpendicular to the sides of an inscribed triangle, their feet are in the same straight line. (M. F. Jacobi.)

3. (a) If the chord of a circle be divided into two segments by a point in the chord or in the chord produced, the rectangle contained by these segments will be equal to the difference of the squares on the radius and on the line joining the given point with the centre of the circle. What propositions in Euclid follow immediately from this?

(b) Describe a circle which shall pass through a given point and touch two straight lines given in position. (Apollonius.)

4. (a) To describe an isosceles triangle, having each of the angles at the base double of the third angle. (IV 10.)

(b) Construct a triangle having each of the angles at the base equal to seven times the third angle.

5. (a) If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base have the same ratio which the other sides of the triangle have to one another; and, if the segments of the base have the same ratio which the other sides of the triangle have to one another, the straight line drawn from the vertex to the point of section shall bisect the vertical angle. (VI 3.)

(b) The points in which the bisectors of the external angles of a triangle meet the opposite sides, lie in a straight line.



N.B.-Algebraic symbols must not be used. Candidates who take Book II will omit Questions 1, 2, and 3, marked *.




16 *1. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles on the other side of the base shall be equal to one another. Where does Euclid require the second part of this theorum? 2 If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by two sides of one of them greater than the angle contained by the two sides equal to them of the other, the base of that which has the greater angle shall be greater than the base of the other.


Why the restriction" Of the two sides DE, DF, let DE be the side which is not greater than the other "?

16 *3. If two triangles have two angles of the one equal to two angles of the other, each to each, and have also the sides adjacent to the equal angles in each, equal to one another, then shall the other sides be equal, each to each; and also the third angle of the one to the third angle of the other. (Prove by superposition.)


What propositions in Book I are thus proved?

16 4. 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 angle on the same side; and also the two interior angles on the same side together equal to two right angles.


What objection may be taken to the twelfth axiom?
What is its converse ?

16 5. In any right-angled triangle, the square which is described on
the side subtending the right angle is equal to the squares
described on the sides which contain the right angle.
Prove also by dissection and superposition.


18 6.

18 7.

20 8.

Draw through a given point between two straight lines not parallel a straight line which shall be bisected in that point. The perpendiculars from the angles of a triangle on the opposite sides meet in a point.

Given the lengths of the lines drawn from the angles of a triangle to the points of bisection of the opposite sides, construct the triangle.

20 9. If a straight line be divided into two parts, the square on the whole line is equal to the squares on the parts, together with twice the rectangle contained by the parts.

20 10. In every triangle, the square on the side subtending an acute angle is less than the squares on the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle, and the acute angle.

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1 Emm.,22.,35.,46.
Sid.,30. Trin.,37.

2 Trin.,40. Cai. ,57.

8 Trin. ,32.,37.,50.
T. H.,52. Joh.,54.
S. H.,54.

4 Sid. ,30.,43.


Qu. ,34.
Trin. ,40. Cla.,47.
Emm. ,56.
5 Emm.,21. Qu.,23.
,40.,42. Trin.,26.
,27.,29. C. C.,30.
,55. Pem.,32.,38.
6 S. H.,17. Trin.
,24.,37. Qu.,25.
Emm.,27. ,48.
Cath.,29.,48. Pem.
,39.,47. Sid.,40.
Chr.,45. Cla.,56.

7 S. H.,19.

,29. Qu.,35. Pem.
,44. Jes.,49. B. S.

8 Qu.,26.,28. S. H.

,49.,50. Pet.,56.
Emm. ,50. C. C.
,57. Cai.,55.
9 Mag.,38. Joh.,58.
10 Emm.,34.

11 Cai.,40. Joh.,50.
12 S. H.,40.,54.

13 Cath.,31. S.H.,50.

15 Pet.,57.

16 Cath.,22.,33. Trin.
17 Cai.,57.
18 Chr.,56.
19 Emm.,56.
20 T. H.,51.
21 Jes.,58.
22 S. H.,50.
23 Qu.,19. T. H.,51.
Emm.,51. Pem.

24 Jes.,58.
25 S. H.,14. Cla.,55.
26 Cai. .41.

27 Chr. ,26.,41.,52
Jes.,52. Joh.,31.
Pet.,38. Trin. ,39,
,50. Mag.,51.
28 S. H.,58.
29 C. C.,53. S. H.

30 C. C.,53. Qu.,54.

31 Trin.,31.
32 S. H.,36.,48. Mag.
47. Chr.,54,
33 Emm.,25.

34 Joh.,19. Qu.,25.
35 Chr.,28. Pem.,42.

36 Trin.,26. Sid.,43.
C. C.,57.

37 Pem.,29.

,48. Qu.,52.

38 Qu.,50.

B. S

39 Qu.,31. Cath. 5.



,38. B. S.,40.

40 Trin.,34.
41 S. H.,55.
42 S. H. ,04. C. C.
,23. Chr.,29.,50.
Cath.,35. Jes.,52.
Pet.,36. Qu.,39.
Trin. ,37. ,49. Cai.
40. Pem.,48.

43 Trin.,54. Emm.,54.

44 Trin. ,58.

45 Cai.,55.

46 Pet.,58.

47 Chr.,55.
48 Cai.,49.

49 Jes. 54.

50 S. H.,53.

51 Trin.,39.,51. Pem.

52 Trin.,43.

53 Joh. 26. Pem.,47.


54 Cai.,46. Qu.,48.
55 Cai.,31. Joh.,30.

57 Jes. 52. Cai.,56.
58 Jes. 55.

59 Pet ,51.
60 Chr.,39.

61 Pet.,36.

62 Trin.,52.,54. T.H.
63 Pet.,51.
64 Trim.,51.
65 Jes. 54.
66 Pet.,51.
67 S. H.,48.


69 T. H.,54.

70 Trin.,40.


72 Cai.,33. Qu.,33.
73 Trin.,49.

74 Qu.,31. Chr.,56.
Sid.,36. Pet.,53.

75 Qu.,19.
76 Qu.,24.

77 Cla.,51.
78 Qu.,32. Jes.,36.
S. H.,49.,50.
79 S. H.,49. Mag.,52.
80 Qu.,37.

81 Trin.,48.
82 Chr.,58.
83 Chr.,52.
84 Trin.,52.
85 Cath.,49.

86 Cla.,57.
87 Pet.,46.

88 C. C. ,50. Cai. ,63.
89 Mag. ,51.,58.
90 Jes.,54,

91 Cath.,49. S. H.,54.

92 Jes.,55.
93 Cai.,46.

94 Jes.,41.
95 Chr.,43.
96 Joh.,31.

97 Cai. ,36. Cath.,55.

98 Emm.,30. Cath.

99 Trin.,59.

100 Pet.,51.

101 Qu.,29.,35.,37.
B. S.,39.

103 Mag.,52.
104 Chr.,47. Cla.,48.
105 Pet.,51.

106 Sid.,45. Chr.,47.'

107 S. H.,52.

108 Emm.,57.

109 S. H. ,04.


,34. Emm.,39.

110 Qu.,25. Trin.
,25.,38. Pet.,39.
Jes.,52. Pem.,42.
111 S. H. ,03.,18.
Trin. ,25.,44. Cla.

112 Qu.,29.,37.,26.
Trin. ,27.,33.,36.
,40.,49.,50. Chr.
,44. Pem.,45.
Cath.,58. Emm.
,54. Jes.,52. S. H.
,50. C. C.,58.
113 Emm.,32.
114 Qu.,19.,37.Emm.
,25.,53. Mag.,29.
,32. Cai.,34. Trin.
,37.,38. Pet.,44.

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115 S. H.,48.
116 Qu.,39. Mag.,54.
S. H.,59.
117 Trin.,29.

118 Emm.,22. C. C.

119 Pet.,45.

120 S.H.,35.,48. Joh.

[blocks in formation]

,52.,53.,56. Chr.
,50. T. H.,52.

139 Cla.,36.

140 Mag.,49.

141 Cla.,36.

142 Joh.,58. Chr.,58.
143 Trin.,53.,54.
144 Joh.,16. Qu.,30.
Pem.,33.,49. Jes.
,46. Trin.,47.,58.
C. C.,58.

145 Pet.,27.
146 S. H.,36.

147 Chr.,54.

148 Cla.,56.

149 Jes.,20. Qu.,32.
,48. Cath.,35.
S. H.,59.

150 Trin. ,40.

151 Pet.,32.,35.
152 Pet.,49.
153 S. H.,55.

154 Jes.,53.

155 Chr.,56.
156 T. H.,52.
157 Joh.,20.



158 Sid.,46. Mag.,58.
159 Cai.,37.

160 Emm. 32. Qu.
C. C.

,36.,59. Mag.,39.
B. S.,47.
161 Trin.,21.,50.
162 Jes.,35.

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