6. If two circles cut one another they cannot have the same centre.

7. Equal straight lines in a circle are equally distant from the centre; and those straight lines which are equally distant from the centre are equal to one another.

8. If from a 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 without the circle is equal to the square of the line which touches it.

9. To describe a circle about a given triangle. 10. To inscribe a circle in a given square. 11. To describe an isosceles triangle having each

angle at the base double the angle at the vertex. 12. If the first be to the second as the third to the fourth, and if the first be a multiple or a part of the second, the third is the same multiple or part of the fourth.

13. Give Euclid's definition of compound ratio: and

prove that equiangular parallelograms are to one another in the ratio compounded of the ratio of their sides.

14. Define a plane, and a perpendicular to a plane: and shew that from the same point in a given plane there cannot be two straight lines at right angles to the plane, upon the same side of it: and there can be but one perpendicular to a plane from a point above the plane.

15. Given the base and one side of an isosceles triangle to describe the triangle.

16. If two chords of a circle intersect each other at right angles, the sum of the squares describec. upon the four segments is equal to the square decribed upon the diameter.

17. Two parallel chords in a circle are respectively

six and eight inches long, and are one inch apart : how many inches in length is the diameter.

18. Find the Cosine of the angle of a triangle in terms of the sides.

19. Trace the variation of the tangent of an angle, as the angle increases from 0 to 360°.

20. At a given distance from the base of a tower,

the height of the tower subtends a known angle. Show how the height of the tower may be calculated.

X.

ARITHMETIC, ETC.

1. Find the value of 365 at £3. 17s. 6d.

2. Reduce 5s. 6d. to the fraction of a £: 3s. 9d. to the decimal of a guinea: and 31 days 10 hours to the decimal of a year of 365,25 days.

3. Required the value of ,06171 of a £, and of of of £126.

4. Find the amount of £841. 13s. 6d. in 3 years, first, allowing simple interest at the rate of 31 per cent. per annum: secondly, allowing compound interest at the rate of 5 per cent. per an

num.

5. What is the present worth of £843. 12s. 6d. due 11⁄2 years hence, discounting at the rate of 4 per cent. per annum.

6. If the carriage of 640 tons costs £3. 12s. 6d., how much must be paid for the carriage of 895 tons?

7. A contractor hires 248 men to complete some work in 60 days: but after they have worked 20 days he is desirous of finishing the remainder of the work in 30 days. How many more men must he employ?

3. Extract the square root of 8,961 to three places of decimals: and the cube root of 29791: and prove the rule for pointing in each operation.

9. Reduce the fractions, 4, 7 to a common denominator: add together 4 of 3 and 4 of 7: and reduce to its lowest terms.

10. Find the least common multiple of 188 and 972. 11. A ship and cargo are worth £17,600.; and the cargo is worth seven times as much as the ship: find the value of the ship and cargo.

12. Find the expense of insuring farming stock worth £2500., supposing the insurance to be th per cent.; and the commission upon the sum paid for insurance.

13. Prove that am. an = am+n.

and thence explain the construction of a table of logarithms of numbers.

14. Find the sum and difference of

16. Sum the arithmetical series

1 + } − − &c., to 10 terms, and write.

down the 20th term.

H

17. Sum the geometrical series

1 + 3 + 25 + &c., to 5 terms; and find the value of the series continued in infinitum. 18. Compute the present worth of an annuity of £10., the first payment to be made one year hence, and the payments continued annually for ten years; the rate of compound interest being 4 per cent. per annum.

19. There is a number composed of two digits, the second of which is double of the first: and if 27 is added to the number, the digits are inverted. What is the number?

8

20. Expand (a + b) ·

APPENDIX.

A CHARTER,

GRANTED TO THE UNIVERSITY OF DURHAM.

JUNE 1, 1837.

William the Fourth, by the Grace of God, of the United Kingdom of Great Britain and Ireland, King, Defender of the Faith, Co all to whom these presents shall come, greeting. Whereas by an Act passed in the thirty-first year of the Reign of our Royal Predecessor, His Majesty King Henry the Eighth, His Majesty was empowered to establish more Bishoprics and Collegial and Cathedral Churches, to the intent, among other purposes, that thereby God's Word might be better set forth, children brought up in learning, clerks nourished in the Universities, and readers of Greek, Hebrew, and Latin, might have good stipends; and His Majesty was also empowered to make Statutes for the Institutions by Letters Patent, under His Great Seal. And whereas His said Majesty, King Henry the Eighth, by Letters Patent, under the Great Seal, dated the twelfth day of May, in the year of our Lord Jesus Christ, one thousand five hundred and forty-one, after mentioning, among other inducements, "ut bonorum morum disciplina observetur, juventus in literis liberalibus instituatur, et cetera omnis generis pietatis officia illinc exuberantia in omnia vicina loca longe lateque demanent, ad Dei omnipotentis gloriam, et ad subditorum nostrorum communem utilitatem, felicitatemque," founded a Cathedral Church at Durham, to consist of one Dean and twelve Prebendaries, and other Ministers: and His said Majesty ordained that the said Dean and twelve Prebendaries should be one Body Corporate, and have perpetual succession under the name of "The Dean and Chapter of the Cathedral Church of Durham, of Christ, and of the Blessed Mary the Virgin," and that they should be governed by certain Statutes to be afterwards given to them. And whereas Statutes were afterwards given to the said Cathedral Church, one of which is entitled, "De pueris grammaticalibus et eorum informatoribus," by which, after the preface following, "Ut pietas et bonæ literæ perpetuo in dicta ecclesia suppullulescant, crescant, floreant, et suo tempore in gloriam Dei et Reipublicæ

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