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SECTION V.-1. Which is the cheapest, a hat which costs 158., and will last 9 months, or one which costs 12s., and will last 7 months; and how much will a man save in 20 years who wears the cheaper kind of hat, interest pot being reckoned ?
2. A. offers to give 21. 35. 4d. in charity for every pound B. gives to it; B. gives 51. 6s. 8d; what will A. have to give?
3. A farmer, who threshes 1000 quarters of wheat, annually saves 18. per quarter by using a threshing machine. In erecting this machine he sunk 2001., how much of that sum will his savings have paid back to him at the end of 5 years, allowing 5 per cent. interest on the sum which remains unpaid at the end of each year, and 5l. annually for repairs ?
SECTION VI.-1. Give and prove short rules for finding the price of a dozen, a score, and a hundred articles.
2. Give and prove short rules for finding the price of a lb. from the price of an ounce, and of a cwt. from the price of a lb.
3. Give and prove short rules for finding the interest of any number of pounds for a given number of months at 5 per cent, and at 6 per cent.
MENSURATION. SECTION I.-1. The side of a square field measures 120 yards, find the side of another square field six times the size of the first.
2. Find the area of a circular ring, whose inner diameter is 3 ft. 9 in., and its outer diameter 4 ft. 7 inches.
3. Find the solid content of a frustum of a right cone, the diameters of whose top and bottom are respectively 7 inches and 11 inches, and its height 13 inches.
SECTION II.-1. If a wall be 72 ft. 6 in. long, and 19 ft. 3 in. high, and 52 bricks thick, how many standard rods of brick-work are there in it ?
2. What must be the length of the cord used to strike a circle, which shall contain one acre ?
3. What is the solid content of a spherical segment whose height is 2, the radius of the sphere being 5?
HISTORY. SECTION I.-1. Under what circumstances, and in what century, did the Romans finally abandon Britain ?
2. In what way is the history of the great Constantine associated with that of Britain ?
SECTION II.-1. At what time and under what circumstances was the conquest of Ireland effected ?
2. What were the British possessions at the death of Henry V.? When, and under what circumstances, were those in France lost ?
3. What monuments are there of the original inhabitants of Ireland ? By what name was that country known from the fourth to the eleventh century? In what respects do its inhabitants, during that period, appear to have differed from the people whom they succeeded ?
Section III.--1. Who were the sovereigns between Edward III. and Henry VII., and what were their respective claims to the succession ?
2. What was the title and claim of George I. to the throne ?
3. Name some of the principal English writers of the eighteenth century, and give an account of the life, works, and style of one of the most distinguished among them.
SECTION IV.--1. When, and under what circumstances, were the colonies of New England and Virginia first settled? How long was this after the discovery of America ?
2. When, and under what circumstances, did the British first obtain permission to erect a factory in Bengal ?
3. What are the present British possessions in India; from what native powers and under what circumstances have they been severally conquered ?
SECTION V.-1. State shortly under what circumstances the four great empires of antiquity succeeded each other.
2. Give some account of the dissolution of the Macedonian empire. 3. Ġive some account of the life of one of the following eminent persons :
1. Themistocles.--2. Alexander the Great.-3. Julius Cæsar.
GRAMMAR LEARNING AT OUR CLASSICAL SCHOOLS. Sir,—Allow me to address a few lines to your readers on the subject of grammar-learning at our classical schools. When I held the office of examiner at one of our universities, I had an excellent opportunity of forming a fair estimate of the state in which young men are generally sent from school to college. The inability of the greater number of those who came under my notice during three several examinations, to bear a searching grammatical trial, surprised me not a little. Many were profoundly ignorant, these of course were rejected as incompetent; many were barely passable, and only one here and there had a full knowledge, or could make a free use, of his grammar rules. The defect I speak of was often quite evident, even in the case of those whose general scholarship was showy; who could translate with considerable fuency, and who, I doubt not, could have produced a pretty copy of verses, if allowed to choose their own words, and thereby to escape the embarrassment which the necessary introduction of words exceptional in gender, declension, &c., might have occasioned them ; a great art in verse making, where gradus and dictionary are not allowed. And here let me suggest, before I proceed, as a corrective of this very delusive kind of fluency, that it would be well for masters, occasionally, preventing for the time all use of books of reference, to frame exercises which should necessitate the use of exceptional words; that is, giving the irregular word in its simple form, and so arranging the English sentence as to force the pupil to show whether he is aware of the irregularity or not. As for instance, to insist on the use of the word “maledicus,” giving as an English “more abusive.” Entire sentences might thus be framed of words anomalous in formation and government, both in Greek and Latin. Such "exercises in irregulars" would be invaluable for senior boys, and something of the same kind would form no bad ingredient even in university examination papers. I am quite certain, that very many, even of those who are candidates for high honours, would find themselves here and there at a loss, if thus brought to book. Does it not indeed stand to reason, that after the first two or three years of classical study, the attention ought to be far more directed to instances of deviation from the usual forms and laws of declension, structure, &c., than to instances in strict conformity with general rule? The latter are exemplified in every sentence which a boy learns to construe or parse; yet, if you stand and listen to the master, as I have frequently had the opportunity of doing, you will find that for the most part, even if he pretends to teach grammatically, he goes grinding on with the same questions over and over again, on some few of the more familiar rules, appearing quite as anxious to shirk those which are out of the common way as his pupils can be. Whether this arises, in the greater number of instances, from ignorance, obliviousness, or downright want of common sense in the instructor, I do not undertake to say.
With the various requirements of the present day, it is perhaps difficult for any master to persevere in teaching the elementary parts of language soundly, and what I should venture to call conscientiously, " line upon line, line upon line.” Some teachers fully acquit themselves when they have made their pupils learn the respective grammars twice, or, at the utmost, three times; and, as the pupils rise in the school, less and less reference is made to grammar rules. Even in many of the public schools, I have reason to think that, in this respect, there is often much slackness. It is true, that a weak point is now and then accidentally exposed : the master is surprised; his confidence in the general soundness of his boys is shaken; ihen come fitful grammatical questionings for a week or so, but with immense masses of construing to be accomplislied; there is no time for pursuing the practice consistently, and parsing is again dropped. Then, when the boy is placed before a new and searching interrogator, he is confounded at his own ignorance, and ashamed that he no longer knows what he once knew. If the main share of the disgrace of a pluck could be made to fall on the shoulders of culpable or inefficient masters, there would be many more cases of rejection at our university examinations; the average standard would be raised, which it ought to be, and schoolmasters would be obliged themselves to be adequately prepared, and to tighten the strings of discipline on their pupils ; respect for their own character would prevent them from allowing the resources of authority to be wrenched out of their hands by silly parents or tender theorists.
Allow me now to offer the following suggestion. With the partial management of a large school on my hands, I never allowed grammar learning to droop, or myself to think that even my head boys knew their grammars well enough : by keeping up the practice of grammar unfailingly, I kept up the knowledge perfectly, with a very slight expenditure of time. The first lesson in the morning was always a grammar lesson. Excepting only those who were learning their grammars in small fragments, for the first and second times, I divided all the pupils, irrespective of their other classifications, into three large classes, committing each to a several master. I found that the lowest of these, who had learnt the grammar twice before in small portions, could easily say a repetition of three or four pages-two of accidence, and two of syntax--a division unnecessary, except for young boys. Those in the class above learned a proportionately larger quantity, with the addition of all the more important matter which the notes of good grammars generally contain. I would especially mention Dr. Major's of King's College. The senior pupils had also their grammar lessons, with such additional notes as I judged might be useful to them. By this gradual process I got good grammar lessons, and a knowledge of all exceptional cases of importance, and that, without any sense of oppression on the minds of the boys. Many of them subsequently went to the highest public schools; and I never knew one who did not express himself as deeply indebted to me for persevering in the plan to which I have alluded, and who did not find himself in respect of grammatical accuracy superior to those into competition with whom he was thrown. Hoping that this experience may be useful to some of your readers, I remain, sir, yours very truly,
M. A. OXON.
PROFESSOR HALL'S ELEMENTS OF ALGEBRA. Sir,-In your November number, the reviewer, while bestowing general praise upon Professor Hall's Elements of Algebra, is very severe in his remarks upon particular parts of the work, and, as it seems to me, with some injustice; the portion which he selects for his especial castigation being the proof of the method for finding the greatest common measure of three numbers; which, while I certainly think it is not the clearest that can be given, still has by no means the illogical character imputed to it. As your reviewer quotes the proof, it stands thus:To find the greatest common measure of three numbers, a, b, C.
Let D be the greatest common measure of a and 6
of D and c
of a, b, c. For every com. meas. of a and b measures D, therefore E measures a and b; and it is the greatest that measures D and c, therefore it is the greatest which measures a, b, c.
And then, taking this as his text, he proceeds to liken it to the syllogism,
Every horse is a quadruped,
therefore a cow is a horse. Now it seems plain at once that in the first sentence there must be a misprint in the stopping, either of the original text, or of the quotation, the semicolon being in its wrong place. It should be,
For every com. meas. of a and b measures D;
Therefore, &c. Of course the fact of E measuring a and 6 following, not from what goes before, but because E itself is a measure of D, and ... necessarily of a and b, of which D is a common factor.
The fault in the proof is, that it is too short, as will at once be seen by expanding it a little, thus :
(Now) every com. meas. of a and b measures D; (1)
And (since) it is the greatest which measures D and c, (and by (1) all the com, meas. of a and b measure D, and .. all the com. measures of a, b, and c measure D and c, or are identical with
the com, meas. of D and c., since all the com. measures of D and c, also, measure a, b,
.:. It is also the greatest which measures a, b and c. I must say, however, that it is not even then so clear a proof as the following :
To find the greatest common measure of three numbers, a, b and c, (having first shown, of course, that all the measures of two numbers measure their great com. meas.) Let d be the greatest common measure of a & 6
of d & c then x is the
æ being a com. meas. of d and c is also a com. meas. of a, 6 and c. Also all the other com. meas. of a and b measure d,
and .. all the other com. meas. of a, b and c measure d and c; and
.. also measure x, which is the greatest com. mea. of d and c. Since then x is a com, mea. of a, b and c, and all the other com. measures of a, b, c measure it, it must be the greatest com. measure.
[This defence of Professor Hall's logic is more plausible than sound. If we do not misunderstand “C. C.'s” • little expansion of the professor's reasoning, he will have it that the word since' is to be understood twice, the passage then standing as follows:
“For every common measure of a and b measures D, therefore (since) E mea. sures a and b, and (since) it is the greatest that measures D and c, therefore it is the greatest which measures a, b, c."*
But, under this point of view, one of the words therefore' is superfluous, and will have to be rejected; but we do not see what right any one has to omit it. In fact, the word “therefore' occurring twice, there must be two inferences. The inference corresponding to the first therefore' cannot be 'E is the greatest number which measures a, b, c, for this is clearly the inference corresponding to the second. What then is the inference corresponding to the first 'therefore?' It evidently can be no other than E measures a and b;' and the word 'since' cannot therefore be understood before “E.” Since then, “E measures a and b," is an inference; and since moreover the only previous portion of the passage is, “Every common measure of a and b measures D,” are we not compelled to conclude that the professor intended the former of these propositions to be inferred from the latter ?
We must emphatically deny therefore that we have done Professor Hall any injus. tice. We maintain that we have criticised his work fairly, and that the passage commented on can only be understood in the sense in which we took it. "C.C." entirely begs the question, and his remarks are fairly open to further comment; but we are content, after the preceding observations, to leave the whole matter to be decided by each reader for himself.]
* By some inadvertence, a semicolon was inserted in our quotation in the last number instead of a comma. There is no semicolon in the original text, either in the last edition or the previous one.