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b. Particularly noted for the blacklead found in its immediate neighbourhood.
e. Population in 1851, 2618.
V. Other places of note.
a. On the South Tyne River, in the most eastern part of the county.
b. Noted for its lead mines, which employ 1100 men.
c. Population in 1851, 2005.
2. Burgh, or Burgh-on-Sands.
a. Near the Eden; six miles north-west of Carlisle.
b. At this place King Edward I. died, when on his way to invade Scotland, in A.D. 1307.
a. Formerly a Roman station.
b. Population in 1851, 3074.
SALARIES AND CERTIFICATES OF MERIT.
JOHN J. GRAVES.
SIR,—As I do not think your correspondent's letter, published in March, on the above subject, has been satisfactorily answered, I enclose the accompanying extract from the Augmentation Broadsheet.-I am, &c. ARCANUS.
"The augmentation grants belong exclusively to the teachers, not to the general fund of the school. Their Lordships cannot sanction corresponding reductions in the previous salaries of the teachers, even though more than sufficient to fulfil the conditions of the particular grant."
PROVISION FOR OLD AGE.
SIR,-According to the request of your correspondent in the last Monthly Paper, I beg to forward some extracts from the Government Annuity Tables, No. III.
Money to be paid down in one sum at the time of purchase for an annuity of 1l., payable at the second quarter next following the ages of
These annuities may be purchased either at a savings-bank or direct from the Office for the reduction of the National Debt. I would venture to recommend these tables, which may be obtained from the "National Debt Office," or from the "Register of Friendly Societies, in England, London," to the attention of your readers. I think it ought to be generally known among schoolmasters that a man of twenty-two years of age, by investing his augmentation grant on account of a certificate in the lowest division of class III., will have secured at his twenty-sixth birthday an annuity of 301. a year, to commence at the age of sixty.—I am, &c. W. L.
April 21st, 1855.
SIR,-Will you kindly allow me space in your valuable Paper to offer a few remarks upon the communication of " W. L.," contained in your February Number, respecting Government Annuities.
I think your correspondent has placed the government annuities in a rather higher position than they ought to occupy. Certainly to a young man, unmarried, it would be no great sacrifice for him to do as "W. L." proposes; but I think it ought to be a consideration with every one, to choose the cheapest way (if it also be a safe one) to attain his object. As regards purchasing a deferred annuity, I think insurance offices offer greater advantages than government. Some time ago the purchase of a government annuity was recommended to me as the best course I could pursue; I therefore applied to the secretary of our savings-bank for information respecting them. He immediately wrote to London and procured the act and tables to which " W. L." refers. I had previously obtained the terms from a respectable insurance office upon which I could purchase an annuity, but had not accepted their terms, as I hoped to find the government scheme more advantageous.
I found, on consulting the government tables, that the terms to purchase a deferred annuity of 30%. per annum, by yearly premiums, to commence at 60 years of age (at the age I then was), to be 61. 2s. 6d., nothing being returned in case of my death previous to that time. The insurance office only required 67. 11s. 6d. annual premium for the same thing, and two-thirds of all paid to be returned to my executors, if I died before attaining 60 years. This did not show an advantage to the government; for by the payment of 9s. per annum more than government required, I secured an immense advantage, thus:-supposing I died at the age of 54, I would have paid to government 1471., all of which would be lost; to the insurance office in the same time 1571. 16s., and would leave to my relatives 105l. 4s., which the office would pay on receiving proof of my death.
I stated these facts to the secretary before mentioned, who informed me that they never had but three parties who had purchased annuities from government; and they had all sold out and purchased into insurance offices. He further added, that in Birmingham government annuities were universally rejected in favour of insurances, owing to the rates of the former being so much higher. If this be a truth, which I see no reason for doubting, it speaks for itself.
I did not inquire upon what terms insurance offices would grant a deferred annuity by payment of one sum, so that I might contrast it with government; but I have no doubt at all but they are more reasonable and advantageous than the latter. I would therefore strongly advise all who may think of following "W. L.'s" plan, to first convince themselves, by inquiring at some respectable, well recommended, or old established insurance office, which of the two is to be preferred. The secretary wherever they apply will at all times be glad to furnish every information, and will get the actuary to calculate for every contingency proposed. If the office would not accept a yearly payment for two or three years until the whole amount was paid, it is an easy matter to place the augmentation grant in the savings-bank until the sum required is obtained for purchasing an annuity by payment of one sum.
To those of my fellow-labourers who are married, I am afraid few could spare the whole of their government allowance for a few years. I would therefore advise them to purchase an annuity by annual payments in preference to paying down a large sum at Let them by all means make the contract so that some portion of what has been paid in may be returned to their families in case of their premature death. The whole of the premium paid in will be returned by some offices, but the rates are higher in proportion.
I am afraid I have trespassed largely on your valuable space; but if you think these hints may prove of service to the members of our profession, I hope you will pardon their length. I remain, &c. Αλφα.
1. Books on arithmetic, which were used in schools some years ago, contain a “rule” named Alligation, not recognised, at least in a separate form, in more recent treatises. It is the purpose of this paper to examine whether the problems which this rule embraced can be introduced with advantage or not into educational arithmetic.
2. Alligation, deriving its name from the linking of quantities in pairs (alligando) which the method prescribes, deals with the case where ingredients of different values are compounded in certain quantities into a mixture, whose value will in general be different from that of any of the components. Alligation is either medial or alternate.
3. Alligation medial proposes to compute the value of the compound,-the mean value, as the books term it, whence the title Medial,-if given quantities of materials of given value are combined. Such questions can be so easily and intelligibly solved, that they are an exercise constantly used. One example will suffice.
Three pounds of tea of 4s. 8d. the pound are mixed with two pounds worth 4s. 3d. the pound; what is the value of the mixture?
Here are 5 lbs. of mixture worth 17. 2s. 6d., and therefore worth 4s. 6d. the pound. 4. Alligation alternate proposes to compute the quantities of ingredients of different given values, which are to be combined to make a mixture of a proposed value.
The problem algebraically treated would be this: If a, b, c be the given values of units of the ingredients, x, y, z the quantities of these to be respectively taken to make a compound of value m, since the worth of the whole either in separate ingredients or in combination must be the same,
ax+by+ex+... =m(x+y+z+. .), or0=(a-m)x+(b−m)y+(c~m)x+...
This is the indeterminate equation, which is our only means for obtaining x, y, z,... The number of its solutions is unlimited. Certain of those solutions are found by the following method, which appears to have been the basis of the rule given in books of arithmetic.
To confine ourselves to four ingredients where
Since x, y, z, w are by the nature of the case all positive, of their coefficients some must be positive, some negative. Suppose a=m, b=m positive and the others negative, or m intermediate between α, b and C1 d. Then we can satisfy the equation by making the terms destroy one another in pairs, the 1st for instance, made antagonistic to the 3d, the 2d to the 4th; and the following group of values forms a solution
or by placing the terms in another way, a solution would be
or if m were intermediate to a, b, c and d, or the last term only were negative, this may be made to destroy the other three if
x=y=x=m-d, and wa-m+b¬m+c→m.
Such solutions as these are correct particular results, never claiming to be any general solution of the fundamental equation of the question.
5. On the method above given for solving the equation depends the rule of computa>. tion, which, usually without any explanation, books on arithmetic used to lay down as follows:
Place the given rates or values in a vertical line, and the mean value to the left of them. Link the values together in pairs, joining one greater to one less than the mean value. In each pair against each term write alternately (hence the name alligation alternate) the difference between the mean value and the others to which it is linked.
These differences are the quantities of the components against whose values they stand. Ex. Ingredients worth 30s., 35s., 42s, and 50s. the pound, are to be so mixed that the compound may be worth 36s. the pound. The proof would be
the conclusion being that 14 lbs of the first ingredient, 6 of the second, I of the third, and 6 of the fourth will make a compound of the required values, as may easily be proved. If it be remembered, that in this example, 30, 35, 42, 50, 36 are the a, b, c, d, m of the equation, it will be seen, that the process in fact leads to the values of a=m, b=m, m=c, m=d, with which, x, y, z, w were to be taken equal alternately in the pairs when they appear.
Ex. 2. When 4d, 6d., 10d., are the values of the ingredients, of which a compound is to be made of value 8d.
so that 2 units of the first, 2 of the second, and 6 of the third, make a compound of the kind required.
In such examples as these, every system of linkings or alligations which can be made in conformity with the rule produces a solution of the problem.
6. Now (1st) solutions such as these show nothing of their reasons, but stand as mere applications of rules, which can hardly be made intelligible to a puvil until he can
compare them with the algebraic question; and (2d) when solutions are thus obtained, we have no assurance, and can make no profession of all the solutions being obtained.
On these grounds, alligation alternate seems to have been wisely dismissed from arithmetic books, if the questions which it professes to embrace are such that arithmetic cannot give an intelligible or complete solution.
7. Where, however, the ingredients to be compounded are not more than two, then the problem is determinate, and can fairly be bought under the power of arithmetic. The fundamental equation in this case is→→
or the quantities of the ingredients are inversely proportional to the differences between their values and the mean value.
The following method treats such cases in a manner likely to be understood by children. Ex. How should ingredients worth 8s. and 12s. the pound be mixed to produce a compound worth 9s. 4d. the pound?
Suppose a pound of the cheaper ingredient taken; and let any aliquot part of it, say an ounce, be withdrawn, and replaced by as much of the other. Since an ounce is withdrawn worth 6d., and replaced by the same quantity worth 9d., the pound of mixture is worth 3d. more than the original. Interchange 2 ounces, and the pound is worth 6d. more than it was originally. While thus every ounce interchanged raises the value 3d., the question becomes, how many ounces must be so changed to raise the value Is. 4d.: 3d. 16d. : 1 oz.: answer,
the result being, or 53 ounces, which quantity of the dearer ingredient, with 103 oz. of the cheaper, makes 1 lb. of the value required.
ON TEACHING Reading.
St. Andrew's National School, Halstead. SIR,-1 entirely agree with the declaration of Mr. Flint, that "nothing is more rare in our elementary schools than good reading." Amidst the rage for teaching highsounding subjects, we seem comfortably to have reclined upon honest Dogberry's opinion, that "reading comes by nature;" and in consequence, National Schools are becoming notoriously inferior in this branch of education to the humble dame-schools. Is it not absurd that we, who are ever boasting of the excellence of our language, and affect such horror at hearing the vitiated or imperfect pronunciation of the lower classes, should neglect the chief, if not the only means of improvement, by a careful training of the speech of the young operative or peasant? I cannot believe that the tedium of giving instruction in this art can have led to its neglect, but am rather of opinion, that an idea of its being so very easy has caused us all to think too superficially on the matter, and to lose sight of some really formidable difficulties which beset it.
It occurs to me that there is a striking similarity between vocal music and reading; and that in order to secure success in teaching the latter, we must follow the same methods as are practised in imparting a knowledge of the first. In both we have sound, time, accent, and expression. We should laugh at a music master who allowed his pupils to produce their intervals right or wrong, as they best could; to overleap a rest or two at pleasure; to whine, sniffle, and drawl, ad libitum; transform pp into ff; and to flounder on from piece to piece, without obtaining accuracy in any. Yet I ask: Is not this exactly what we are doing in reading?
I do not pretend that the following method is perfect; but having found considerable advantage from its use, I beg leave to submit it to the consideration of your readers, many of whom can doubtless offer suggestions for its extension and improvement.
To begin with a third class. Having selected the lesson for the day, say, for example, from the 40th page of the 2d Irish book, my first care is to cull from the first five or six sentences such words as are supposed to be unfamiliar, as: country, perhaps, another, notice, reddish, &c. These I write syllabically upon the black-board, and they are learned in silence by the class, a quarter of an hour being allowed for the exercise. The books are then passed, and reading commences, when we reap our reward by passing smoothly over what must otherwise have proved serious stumbling-blocks. The sentence in question having been read two or three times individually, is next gone over simultaneously, in a low, clear tone and strict time. And now, a thorough acquaintance with the sounds of the passage in question being obtained, another exercise ensues. Hitherto we have had a dead level of pronunciation. I therefore read the first sentence myself, marking the points of expression pretty strongly. "All little boys and girls who live in the country must often have seen a robin." A boy is selected to imitate
me. Generally speaking he fails, when I reproduce his faults, remark on his defects, and correct them by another reading. A second attempt on the part of the child usually succeeds, and the entire class then simultaneously follows. Sentence succeeds sentence in this manner, till the portion designed for the lesson is completely mastered.
Another subject is then introduced, and at the end of an hour we recur to our reading lessons, which now becomes an exercise in dictation, and thus enables us to test the amount of attention bestowed upon the spelling task. A trial or two of reading from the slate, and this passage in question is considered in all its points as mastered.
A different mode of proceeding, but based upon the same principles of imitation and repetition, is practised with my first class, which, with your permission, I will endeavour to describe in my next.—I am, &c. JOHN BRION..
ON TEACHING READING.
Derbyshire, 12th March, 1855. SIR,-I will this month, with your kind permission, draw the attention of your readers to the methods I have followed in order to prevent and correct the faults which Mr. Flint mentions in your January number; namely, incorrect pronunciation, punctuation, and emphasis, and song or tone reading.
I. Incorrect Pronunciation. I make the class read round word by word, boy by boy, through the lesson. The teacher and scholars thus hear each word distinctly pronounced, and any shuffling over the hard words is prevented; moreover, by bringing every word under notice in its turn, it gives an opportunity of examining each closely, and thus acts as a sort of spelling lesson. Any difficult word is paused at, and pronounced and spelled simultaneously several times. The lowest class in reading (the one using the detached alphabet, as mentioned in my last letter) spell every word in the lesson, and pronounce each simultaneously after me, as a preliminary step to this.
And here I will take occasion to mention a difficulty, which I am sure many besides myself must have met with. When at home or at play, the children of the poorer classes (and these form the bulk of our national schools) will scarcely ever be heard speaking save in the particular dialect of the locality. The speech is thus loaded with provincial phrases and pronunciations, the use of which, however harmless they may be in themselves, is plainly detrimental to a good pronunciation. It would be well, therefore, if a correct way of speaking were insisted upon, not only while in school, but when at play ; and if the same practice were encouraged when away from school. This latter, however, requires no small degree of moral courage on the part of the children. I have been told by some of my lads, that their playmates laugh and jeer at them when they attempt to "talk fine." If this or any other plan has been tried by any of your readers, I, with, I may venture to say, many others in the same difficulty, will feel obliged if your paper is made the medium of communication of such plan and its results.
II. Incorrect Punctuation. The first boy reads to the first point, whatever it may be; the next to the second; and so on from point to point round the class. After the lesson has been gone through in this way, each boy reads a period; and should he make a mistake in punctuation, he begins again. Reading simultaneously is another good plan; the children then correct each other. All mistakes are checked by the perpetrator's finding himself reading alone. The relative value of each point is of course taught. Practice and imitation teach the difference between grammatical and reading stops.
III. Incorrect Emphasis and Song or Tone. The children are impressed with the fact that the pieces found in books are neither more nor less than talking put upon paper; and therefore, in reading them, they must talk and not sing (read in a monotone). Now, before a boy can talk his reading, that is, read with proper emphasis and tone, it is necessary that he should understand what he reads. This must be the teacher's chief care, as it is the main object of learning to read; the practice of reading as a part of elocution is secondary to this. When the lesson is explained I proceed thus: Reading the first clause, I require the class to read after me, imitating me in every respect; and so on clause after clause through the lesson; laying particular and marked stress on such words as ought to be emphasised, and taking care that the tone of voice is natural (in a talking style). After the lesson has been gone through in this manner, begin again, reading a whole period without interruption, the class reading after me simultaneously, as before, till the whole lesson has been again read. The boys then read to full stops individually, the class and myself taking care that no blunders are made.
To unite all these plans in one lesson, I take them for a lower class in the following somewhat synthetical order: (1) Read through by words; (2) stopping at every point, explaining and questioning; (3) by periods, questioning on former explanation; (4) by