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A C = 1 00 00
1 00 00
60,400 56,564 28,282 x 2
56,564 3,836 Here we shall never get at a finite answer, but the process may be carried on to as many places of decimals as may be deemed necessary.
58,57, &c. In looking back at Fig. 3, it is probable that the steps expressed algebraically, are more clear than any other to the eye of the scholar. The process of multiplying a + b by a + b, is as follows :
(a + b) x a a2 + a b
ao + 2 a b + 62 If the scholar does not see this, he will do well to try several examples in numbers, e.g. 8 x 8 64 (5 + 3) x 5 = 25 + 15
(5 + 3) x 3 = 15 + 9
25 + 30 + 9
EXPLANATION OF THE CUBE ROOT,
If we wish to discover the solid contents of any regular figure, as of a room, we must multiply the length by the breadth, and that by the height. But in a cube, these quantities are all equal to each other. The cube of 1 is
1 1 x 1 x 1 2
8 2 x 2 x 2 3
27 3 x 3 3 4
64 4 x 4 x 4 5
125 5 x 5 x 5 And so on
729 9x 9x 9 10
1,000 10 x 10 x 10 99
979,299 99 x 99 * 99 100
1,000,000 = 100 * 100 x 100 Here we see that each figure in the root never gives more than three places of figures in the cube ; therefore, where the object is to extract the root, we mark off every third figure, and not every second, as in the case of the square root.
If we consider the nature of a cube, and refer to Fig. 3, we shall observe that the base of the cube will be a square, composed of four parts, and that the thickness will be composed also of two parts, so that the whole will consist of eight parts. It will be (30 x 30) + 2 * (30 * 5) + (5 * 5) multiplied by (30 + 5) or (30 x 30 ~ 30) + 2 * (30 * 30 * 5) + (30 * 5 * 5.)
(30 * 30 * 5.) + 2 * (30 * 5 * 5) + (5 * 5 * 5.) 303
3 * 30× 5 3* 30 * 52 303
27,000 3 x 30" x 5
13,500 3 x 30 * 52
42,875 = 35 * 35 x 35
a (a + b) хь
+ 2 ab + b
a’b+ 2 ab + b3
(a + b)3
a ? + 3 a b + 3 ab + b3. If, therefore, I wish to proceed in the opposite direction, and to find out the cube root of 42,875, I must take every step inversely. I begin thus :
15,875 In this case a is 30; therefore 3 a' b 2,700 * b.
The question, then, which I ask myself is, how many times will 27 go in 1587
The answer is, 5 times. The 5 must be the greatest possible number, but it may be too great-but I try it. 3 * a' x 6 13,500 3 x 30* * 5
2,250 3 x 30 x 5*
Let us take a large number which is not an exact cube
59456 In this case a is 40; therefore 3 x a*b = 4,800 x b.
The question, then, which I ask myself is, how many times will 48 go in 594 ?
The answer is, 12 times. From the nature of the case, I know that b cannot be greater than 9, or a would have been 5 and not 4; therefore I try 9.
3 a*b 43,200
5,807,789 Now, a is 490; therefore 3 ab 720,300 * b.
The question, then, which I ask myself is, how many times will 72 go in 580 ? The answer is 8,* but I try 7, for it will only just go 8 times.
3 a? 6 5,042,100
693,316 The answer in whole numbers therefore is 497, and the cube of 497 is 122,763,473, which differs from the original number by the present remainder.
If the answer thus found be not sufficiently accurate for the practical question which we wish to answer, we must add three ciphers to the remainder, and proceed, as before, to find the first place in the decimals.
It is obvious that this whole proceeding is a very laborious one, and as all such questions can be much more easily answered by logarithms, the extraction of the cube and square roots can only be useful as an exercise for arithmeticians. A practical man, who needed to use them
With 8, the next subtraction would have been 5,856,992, which would have been too great.
frequently, and who had no knowledge of logarithms, would construct or obtain a table of roots sufficiently extensive for his own use.
A master who only required questions might easily make them for himself, by actually raising the powers ; but he would find it more beneficial to his scholars, if he carried them on to higher branches of knowledge, rather than employed them in working sums, which contain more of difficulty than instruction.
Ertracts from Charges.
(Continued from page 18.) UNANIMITY OF (HURCHMEN WITH REGARD TO EDUCATION. That our confidence in the principle here asserted upon this important question, is neither misplaced nor presumptuous, we have much stronger grounds for believing, than we could possess if it were the result of any individual judg. ment. But upon what other point, within the recollection of any now living, has the church of England exhibited such perfect unanimity? from the metropolitans of our provinces, descending through all the various orders of the clergy, I have not heard that there has been a second opinion expressed: and they have carried their flocks with them in an agreement quite extraordinary upon this point; that the only system of education either safe or becoming for the members of the church of England to be placed under, is a system which admits them at all times unrestrictedly to receive, under the direction of their clergy, instruction in the holy Scriptures, and in the creeds, catechisms, and other authorised formularies of the Christian faith, as it was preached to the world by the apostles, and through God's great mercy, restored to us at the reformation. Neither is the unanimity here spoken of confined to England. It has been expressed with equal earnestness of feeling by the clergy and laity in every colonial diocese; so that we have the consent of our church throughout the world, recorded by an orerwhelming majority against any such general scheme of education, as it seemed but recently in contemplation to impose upon it. Indeed, even more than this may be urged; for we have of late perceived that some dissenting bodies, convinced of the injustice and impracticability of such a scheme, have expressed themselves adverse to the attempt to introduce it.From the Lord Bishop of Australia's Charge, 1844.
THE STATE À GREAT GAINER BY CHURCH EDUCATION. On the other hand, there are not wanting causes of regret and apprehension. I think (and wherefore should my opinion be suppressed ?) that there is very great reason to regret the extreme difficulties which have been thrown by our government in the way of obtaining aid for the support of schools, in cases where they were plainly wanted, and might by a very small outlay have been instituted and supported. It is little known to the publie in general, what labours and privations have been undergone by many of the clergy in their endeavours to maintain schools through their own exertions, when public support was either withheld, or dispensed with much unwillingness in concession only to a naked claim of right, such as the very extreme letter of the regulations could not be so interpreted as to resist. There is, indeed, a mode of dealing out the public bounty amounting to a discouragement, which the meek in spirit, and they who set a becoming estimate on their own position and expectations, are equally incapable of encountering; and this, I know from experience, is the feeling which the government regulations for granting aid to schools, have,
through their practical working, been instrumental in producing. It is the more to be regretted and wondered at, that an uncordial feeling should have been admitted towards a school establishment, not only pertaining to the numerical majority of the inhabitants of the colony, but which had, during many years, been yielding a return of benefits to the community, exceeding, in a fourfold proportion, the cost bestowed on their production. It does, indeed, the more excite my astonishment that a public advantage, so cheaply to be purchased, should have been declined, when I find it recorded that the experience of England itself has furnished an undeniable test of the value of a system of general education, conducted according to the principles of the established church, and under the direction of its clergy. In proof of this, I will read to you a short extract from the last report of the National Society. This statement may be relied on, having been set forth in the face not only of the church, but of parliament, of the entire body of dissent-in a word, of the nation, and of the world; yet not contradicted. Indeed, being founded on ascertained facts, its correctness could not be called in question.—“Our national schools,” it is said in this report, “ may not be in every instance what they ought to be, nor what we hope to make them; and yet, even in their present state, they have been the means of instilling Christian principles—the great sources of peace and order, of social happiness and of hope for eternity—into the minds and hearts of our manufacturing population. Of this important fact, proofs the most gratifying and incontestible have recently been afforded. During the late disturbances, the question how far the influence of the church and church schools was beneficially exerted in support of law and order, and in what degree the check which the spirit of anarchy received, and its ultimate suppression, were owing to the early dissemination of religious and moral principles among the working classes, may be considered as set at rest by the evidence which the Society has laid before the public. From the statements of about one hundred and fifty correspondents, lay as well as clerical, within the disturbed districts, it appeared that in every case, the effect of education, whether in Sunday or daily schools, was salutary in proportion to its completeness. Wherever means of church instruction were best provided, there the efforts of the disaffected were least successful. In whatever districts church principles predominated, no outbreak took place, however grievous the privations of the people, except in cases where the rightly disposed inhabitants were overpowered by agitators from a distance." It may be objected, I am well aware, that this representation does not bear upon our circumstances here, where we have no
disaffected," and no “disturbed districts.” It is true, we have not—and may God accept our praises accordingly. But we bave the same corrupt human nature to contend against ; and the presumption is, that the remedy (namely church education) which has proved so effectual against its evil propensities, manifested in one particular shape in England, will tend elsewhere to repress the extension of evil, taking its rise from the same source, whatever may be the special form which it may have a tendency to assume. The principle of church education is every where one and the same. It conduces to hold in check that diseased action to which the mind of society is inevitably subject, when abandoned too freely to its own impulses, from having been left to pick up a scanty and erroneous acquaintance with the articles of the Christian faith, instead of being trained in the knowledge of them, and in the fear of God, by the agency of the Church, as the appointed witness and keeper of holy writ. I can but express iny own confirmed and painful conviction, that the adoption of any of the now favorite theories of general education, founded upon an exclusion of the church from its appointed province, would but aggravate the evil which it is designed to remove. In place of the opposition which truth has now to encounter from rooted ignorance, there would be substituted a more embittered spirit of opposition from unsanctified knowledge ; vice, in the meanwhile, being not diminished in amount, but rendered only more specious and refined.-Ibid.