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ANSWERS TO THE MATHEMATICAL QUESTIONS.

QUES. 84.-Proposed by J. H.

FOUR labourers, A, B, C, and D, are engaged to dig a field containing 2 acres, 3 roods, 4 perches, 9 sq. yards. A digs 200 yards the first day, 205 the second, 210 the third, and so on; B digs 220 yards the first day, 216 the second, 212 the third, &c.; C digs 180 yards the first day, 186 the second, 192 the third, &c.; and D digs 225 yards the first day, 220 the second, 215 the third, &c. How many days will the work occupy them?

2.

Answered by Mr. Henry Hill, Mr. J. W. Spedding, Mr. Dungate, and Mr. G. Barnacle.

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Now these numbers are in arithmetical progression, the common difference being Let x = the number of days, then we have by the general formula for the sum of an arithmetical series

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.. x2+824 x = 13440,

... x = 16, the number of days required.

QUES. 85.-Proposed by the Editor.

Give a geometrical construction of the problem contained in question 81.

Answered by Mr. Righton, Prismoid, Mr. Salter, Mr. O' Clazey, Mr. Royds, Nero, and Mr. Stanwick.

Let A B C be the triangle. Upon the base A B describe a semicircle; divide A B into three equal parts in the points a and b ; and from a and b draw the perpendiculars a c and b d, meeting the semicircle in the points c and d. With the radius A c cut A B in D, and with the radius A d cut A B in E: then D and E are the points required. Draw D P and E H parallel to B C, meeting the side A C in the points P and H.

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QUES. 86.-Proposed by John C., Ecclesall School, Sheffield. Find the value of x in the following equation :

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8x+8x=4+.

Answered by Mr. Herbert, Mr. Salter, and the Proposer. Multiplying by 8, and transposing,

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Prismoid, Tewkesbury, ans. 84, 85, 86; J. Herbert, Little Woolton, ans. 84, 86; J.O'Clazey, Durham, ans. 84, 85, 86; Emma Royds, Hollingworth Free School, ans. 84, 85, 86; H. Hill, Chester, ans. 84, 86; W. Righton, Ripley, ans. 84, 85, 86; T. Dungate, Ightham, ans. 84; J. W. Spedding, Bradford, ans. 84, 86; W. Leonard Brown, Ightham, ans. 84; J. Webb, Ightham, ans. 84; J. Brown, Barton, ans. 84, 86; J. Royds, Belfield, ans. 84, 85, 86; W. H. Levy, Shelbourne, ans. 84, 85, 86; J. Stanwick, Gillingham, ans. 84, 85, 86; J. C. Ecclesall, Sheffield, ans. 84, 86; G. Barnacle, Empingham, ans. 84, 86; J. Fox, Burton, ans. 84, 86; T. Sothern, Burton-wood, ans. 84, 85, 86; W. Crow, Lockwood, ans. 84, 86; Thomas Brown, ans. 84, 86; J. Salter, Durham, ans. 84, 85, 86; A. Ford, Norham, ans. 84, 85; C. P., ans. 84, 85, 86; W. Wakely, Marlborough, ans. 86; Kerslake, ans. 84, 86; Nero, Berwick, ans. 84, 85, 86; G. Morris, Gosport, ans. 84, 86; A. M., ans. 85, 86; J. Sheppard, Gillingham, ans. 84, 85, 86; Quere, ans. 84, 85, 86; Sam. Dyer, Wanstead, ans. 84, 86.

NEW QUESTIONS,

TO BE ANSWERED IN OUR NUMBER FOR FEBRUARY, 1851.
QUES. 87.-Proposed by Mr. O'Clazey, Durham.

A man and a boy are engaged to dig a trench. The man could finish it in 12 days, and the boy in 20; supposing them to begin at the same time, one at each end of the trench, when should they exchange places so as to meet exactly in the middle?

QUES. 88.-Proposed by Mr. Dyer, Wanstead.

A cylindrical glass, six inches long, is forced into a tub of water until the top of the glass is on a level with the water in the tub. How high will the water ascend in the glass?

QUES. 89.-Proposed by Nemo.

How many cubic feet of water must descend a river every minute to drive a water-wheel of P effective horse-powers, by means of a fall of h feet, the wheel yielding the nth part of the work of the fall?

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Entelligence.

GOVERNMENT EXAMINATION FOR CERTIFICATES OF MERIT AND FOR QUEEN'S SCHOLARSHIPS. Christmas, 1850.-We have ascertained that about four hundred and fifty candidates for certificates, and upwards of fifty competitors for Queen's Scholarships, attended the recent examination before H. M. Inspectors at the various Training Colleges. The last-named class of candidates are pupil teachers who have completed the term of their apprenticeship, and are thus eligible to compete for the advantages held out to them under the minutes of 1846, consisting of exhibitions to the amount of 201. or 251. towards defraying the cost of their residence in any training school, to be selected by themselves, under certain limitations.

We subjoin the names of the Institutions at which an examination has been held at Christmas, with the name of the Inspector by whom it was conducted in each case:- -Battersea, Rev. J. J. Blandford; St. Mark's, Chelsea, Rev. W. H. Brookfield; Home and Colonial, Rev. M. Mitchell; Whitelands, Rev. F. C. Cook; Cheltenham, Rev. H. W. Bellairs; Chester, Rev. J. P. Norris ; York, Rev. F. Watkins; Durham, Rev. D. J. Stewart; Salisbury, Rev. E. D. Tinling; Carmarthen, Rev. H. L. Jones; Warrington, Rev. W. J. Kennedy; Kneller Hall, the Rev. F. Temple, Principal; Borough Road, J. Fletcher, Esq.

OXFORD UNIVERSITY COMMISSION. An additional circular has been issued by Mr. Stanley, the Secretary to the Oxford University Commission, requesting information on the following heads of inquiry :—

1. Is your society governed by statutes? If not, are there any orders or rules by which it is governed?

2. If the society is governed by statutes, were those statutes given by the founder? Are the original statutes in force, wholly or in part?

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If they are not in force, by what authority and when have they been altered?

3. Is there any provision in your present statutes for their alteration or amendment? or was there, in your original statutes, any such provision?

4. Will you state in what respects, if any, your statutes have ceased to be observed-whether owing to lapse of time or other causes?

5. In what case is the non-residence of your head or your fellows permitted by the statutes? and how many of your fellows are non-residents? Would the University or the college be benefited, in your opinion, by the general enforcement of residence?

6. Is the marriage of the head of your college permitted by the statutes? If not, by what authority is such permission granted? Is that permission applicable to any other members of the foundation besides the head?

7. Does the college consist of several foundations? If so, do the fellows on the several foundations enjoy the same rights and advantages ?

8. Are there in your college any unincorporated or by-fellows? If so, by what statutes are they governed? Do you consider such fellowships beneficial to the society? Or do you think their present position might be altered with advantage?

9. How many of your fellowships, studentships, scholarships, exhibitions, or the like, are at present open to competition without restriction; and how many confined to particular places or schools, or to persons of the kin or name of founders?

10. Will you quote the clauses of your statutes on which any such restriction rests, together with any special reasons which the founder or framer of your statutes may have had for this restriction?

11. Is the restriction absolute? If not, has the college availed itself

of any facilities which the statutes allow for opening the foundation?

12. If the statutes give a "preference" to certain candidates, how do you interpret such preference?

13. Do you consider the present restrictions, if any, on the election of your fellows, students, scholars, demies, or the like, to be beneficial to the promotion of education or learning in the University, in your own society, and in the particular place, school, or family, if there be any, which is supposed to be favoured by those restrictions?

14. Are your fellowships, studentships, scholarships, demyships, or the like, disposed of strictly according to merit? Is such merit tested

by examinations?

15. What is the statutable connection between your fellowships and your scholarships, demyships, or the like? What is the practice in this

respect?

16. If your fellowships are limited to those who are or have been scholars, or the like, of your society, has this system been found beneficial to the college?

17. Are the fellows of your college obliged to proceed to the higher degrees? If so, in what faculties ?

18. Do your statutes enjoin that your fellowships, studentships, scholarships, or the like, be increased or diminished in number as the revenues of your college vary? Has such provision of the statutes been acted upon? Do you conceive that the enforcement of such provision at the present time would be beneficial to the society?

19. Do your statutes contemplate the residence in college of any undergraduates not on the foundation? Do they forbid it? Will you quote the clauses in your statutes which such permission or prohibition rests?

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20. What amount of property vacates a fellowship, scholarship, or the like, according to your statutes? Is the rule enforced equally with regard to real and personal property? Do you conceive that the enforcement of such a rule at the present time would be beneficial to the society?

21. Is the head of your society statutably required to enter into holy orders? How many of your fellows, students, or the like, are subject to the same rule? If the statute be not observed, on what authority does the non-observance or dispensation rest? Is the obligation to enter into holy orders expressly laid down by statute, or is it deduced from an injunction to study theology, from an injunction to discharge clerical duties now disused, or from any other like provision.

22. Are your fellowships confined to persons of a certain University standing? Is the admission of undergraduates to fellowships, or the restriction of fellowships to persons of particular degrees, productive of inconvenience?

23. Are clergymen excluded from presenting themselves as candidates for your fellowships? Are laymen?

24. Are fellows or other members on your foundation allowed by statute or other authority to hold ecclesiastical preferment? and, if so, to what amount?

25. What statutable restrictions limit the selection of your head?

26. How many benefices in the gift of your society have been added since the original foundation? Will you state at what time and in what manner each was acquired? Have you at present a fund for the purchase of advowsons?

27. Are there any prælectorships founded in your college for the benefit of the whole University? Are fellowships connected with such prælectorships? If so, do the statutes allow any special liberty of choice in the fellowships so connected?

28. Has the college the nomination to the masterships of any schools? What control does the college exercise over such schools?

29. Will you quote the clauses of your statutes which set forth the powers and duties of the visitor of your college? Has the visitor ever interposed his authority to relieve the college from the observance of any of the statutes, or to make new statutes or ordinances?

30. Are gentlemen commoners in your society called upon to pass the

same examination at entrance as other persons? Do they follow the same course of studies, and are they subjected to the same discipline as other persons in statu pupillari? To what charges are they liable beyond those borne by other independent members?

31. Can you state how many members of your society receive assistance from exhibitions or the like, not in the gift or under the administration of your society? What are the sources and what is the amount of the assistance so received?

32. How many persons are supported, wholly or in part, in your society as batellers, servitors, Bibleclerks, or the like? What are their duties, and what are their stipends or other emoluments or immunities?

How are they chosen? Are they marked by any particular dress? Was the number ever greater? If so, can you state why it has been reduced? What do you consider to be the advantage or disadvantage of such a body of scholars?

33. How many tutors are there in your society? How many lecturers, catechists, or other instructors, who are not tutors? Does the head of your society take any direct part in the instruction?

34. Are there any tutors in your society who are not, or have not been, on the foundation? Do they all reside within the walls?

35. Is each tutor expected to lecture on all subjects; or is there a division of subjects?

36. During how many weeks in the year are lectures given in your society? Will you state the average number of lectures given weekly, and the subjects? How many undergraduates attend mathematical lectures beyond arithmetic and the elements of Euclid and algebra ?

37. Are any members of the college required to attend any professor's lectures? And are any means adopted by the college to secure profitable attendance by examination or otherwise?

38. Can you state how many members of your foundation, and how many ndependent members of the

society, are engaged as private tu

tors ?

39. Can you state how many under.. graduate members of your society are now reading with private tutors ? 40. What attendance at chapel is required by your statutes? What attendance is actually enforced? And by what means? Is attendance at chapel ever enforced as a punish

ment?

41. What is the nature and extent of religious instruction given in your society, distinguishing lectures and sermons delivered in chapel, and instruction given in other ways?

42. What is the average amount of the "battels" of each independent member of your society? What was the highest and what was the lowest amount in the year 1849?

43. Will you have the goodness to supply the commissioners with a weekly battel bill of the average amount, and with a quarterly battel bill for each of the four quarters of 1849, also of the average amount?

44. What is the lowest yearly sum for which you have known an undergraduate to live in your society? What is the lowest amount which you have known an undergraduate to expend, from his matriculation to his graduation?

45. Do you conceive that the college expenses could be materially diminished? If so, will you state in what respects?

46. Is the college library open to all members of the college, and what fees are paid to the library by each member?

47. What number of under-graduates is your college capable of accommodating?

The two following circulars have been issued by the Commission to the Master and Fellows of each College, and to the authorities of the ViceChancellor's Court::

"Oxford University Commission, Downing-street, Nov. 1850. "Gentlemen-Her Majesty's Commissioners for the University of Oxford will feel much obliged for any information which you may be

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