MASTERS.

In my last Report I stated my intention of opening a class to teach the masters what I should expect them to teach their pupils. This has accordingly been done, and after the usual hours of teaching, the masters attended me twice a week, and I believe much of the improvement perceptible in the school classes is due to the care with which the masters themselves studied the principles of arithmetic, and actually practised the rules. The text book in this class was De Morgan's Arithmetic, a work of which it would be impossible to speak too highly.

Before concluding I ought to mention that a regular series of astronomical and meteorological observations will commence with the new year, from which I hope much advantage will arise indirectly, in training a body of skilled observers, and, perhaps directly, in adding to our existing stock of information. The instruments at present in our possession are-transit instrument, sextant, barɔmeter, wet and dry bulb thermometers; and I feel confident that the Board of Education, with their accustomed liberality, will add a wind and rain-gauge.

Subjoined are a list of the questions proposed at the late examination, with specimens of the answering, and a table showing the value of the answers of each student. To Mr. Green I have to return my thanks for his valuable assistance in preparing the papers for the West and Normal classes.

JOSEPHI PATTON,

Professor of Mathematics and Natural Philosphy.

MATHEMATICS.

CLARE SCHOLARS.

Viva Voce.

1. State Newton's binomial theorem.

2. Apply it to (a-x)-2.

3.

4.

Prove the rule for summing an arithmetic series.

Prove the rule for summing a geometric series.

5. Define triangular numbers, and give the rule for summation. 6. Define pyramidal numbers, and give the rule for summation. 7. Given two equations and two unknown quantities they can be found?

8. Whatever number of unknown quantities there must be an equal number of independent equations?

9. General rule for writing down the value of the denominator? 10. How is the numerator of the value of each unknown quantity deduced from this denominator?

11. If there be three equations and two unknown quantities, what must be inferred?

12. When are the roots of a quadratic equation ax2 + bx + c = 0 equal?

13. If a be one of the imaginary cube root of unity, a 2 is the other?

14. Define a logarithm, and thence prove the ordinary rules for multiplication, &c. by means of logarithms.

15. How do the Napierian and common logarithms differ? 16. What is the value of the modulus of the common system? 17. Generally how can one system of logarithms be changed into another?

18. Prove the rule for finding the greatest common measure. 19. Prove that the diagonal and the side of a square are incommensurable.

20. Prove the segments of a line cut in extreme and mean ratio. 21. The algebraic method of proving two quantities to have no common measure is the same as the geometric method of proving two lines to be incommensurable?

22. In the figure of 47th the lines joining base angles with the angles of squares meet in the perpendicular from the right angle?

23. The rectangle under the segments of the base of an isosceles

triangle equal to the difference of the squares of the side and a line drawn from the vertex to the point of section?

24. The line bisecting the sides of a triangle is parallel to the base, and half of it?

25. Describe a maximum paralellogram in a triangle.

26.

When can a circle be inscribed in a quadrilateral?

27. Find a point in a line so that sum of its distances from the two given points shall be a minimum.

28.

vertex.

29.

Given the base and sum of squares of sides, find the locus of

Given the base and difference of squares of sides, find the locus of vertex.

30. Given the base and ratio of sides, find the locus of vertex. 31. Find the locus of the point from which equal tangents can be drawn to two circles.

32. On what problem does the trisection of an angle depends? Define similar figures so as to include curves.

33.

34. What is a harmonic pencil?

35. How is the harmonic ratio of a pencil represented?

36. What is property of transversals?-prove it in the case of quadrilaterals.

37. State distinctly the converse of this.

38. Draw three lines from the vertex of a triangle to meet the sides, the ratio of segments of one side is compounded of the segments of the other?

39. Hence prove that the lines from the angles of a triangle to the points of contract of inscribed circle pass through the same point.

40. Define the centre of similitude of two circles.

41. The centres of similitude of three circles are three and three

in right lines?

42. The radical axis of three circles meet in a point?

43. Define the terms pole and polar.

44. Show that any line passing through the pole is cut harmonically by the polar and the circle. Prove this when the point is inside the circle.

45. Prove this when the point is outside.

46. Show that the intersection of two lines is the pole of the line joining their poles.