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Answered by Mr. Herbert, Mr. Hill, Mr. Sheppard, Mr. Henry

T. Sothern, and the Proposer. Here the solid units in the segment divided by the superficial units in the convex surface of the segment, are to be made a maximum. Let x = the height of the segment, and r = the radius of the sphere ; then we get

Convex surface segment = 2 x 2.

Solidity segment = 7 *2 (r = $x). .::q? (gr - 1x)

* (r - 3x)

- a max.; 2 ттх

2r

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dy

.. X = ir.

2

3 Whence the height of the segment must be lg time the radius of the sphere.

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= 0.

dx

LIST OF MATHEMATICAL ANSWERS. T. V. Henry, ans. 107, 108, 109, 110; H. Hill, Mortlake, ans. 107, 108, 110; Prismoid, ans. 107, 108, 110; T. Laurie, Oundle, ans. 108; J. Rowlatt, Evercreech, ans. 107, 108, 110; E. Carther, Roehampton, ans. 108; T. Sothern, Burtonwood, ans. 107, 108, 110; Mark, ans. 107, 108, 110; G. Morris, Gosport, ans. 107, 108, 110; J. Draper, Rochester, ans. 107, 108; J. Bolton, O. Malton, ans. 107, 108 ; Sam Dyer, Wanstead, ans. 107, 108, 110 ; W. Righton, ans. 107, 108, 110; A. M. ans. 107, 108, 110; H. P., Newcastle, ans. 107, 108, 110; J. Herbert, Wootton, ans. 107, 108, 109, 110; T. Abbott, Twickenham, ans. 107, 108.

NEW QUESTIONS,
TO BE ANSWERED IN our Number for November, 1851.

Ques. 111.--Proposed by P. T.
A silver shield in the form of an octagon weighs 150 oz.,

and is .09 inches thick; required the length of the side.

Ques. 112.-Proposed by Mr. Draper, Rochester. Given a and b the parallel sides of a trapezoid inscribed in a circle, and the obtuse angle double the acute angle; to find the area of the trapezoid.

Ques. 113.- Proposed by Mr. O'Clazey, Durham. Being a friend of the cold-water system, I intend to make a circular pond in the front of my house, and circular island in the centre, raised a feet above the surface of the water, which is to be d feet deep: I request the aid of my Journal friends, to tell me the breadth of the water, so that the excavated earth may just raise the island to the height required.

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To Correspondents.

K. P. in his communication about Sunday Schools, goes over the ground again in some respects, and his letter would need alteration before it could be printed. It is reserved for notice in a further paper upon the subject, so far as its suggestions can be used, and we shall be glad of any further remarks on the same subject from other quarters.

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THE GREAT EXHIBITION. The Great Exhibition of the Industry of all Nations, so far as it was only a vast and magnificent spectacle, has passed away from our sight like an airy pageant. It has no longer the charm which arose from the anticipated gratification of a rational and refined pleasure in beholding its wonders, but it now only appeals to our calm and deliberate reason as a great historical fact, of which all that remains to us is the name and the remembrance. We feel that a source of intellectual enjoyment of the highest order has been suddenly cut off from us, and we stand gazing in the same direction in which we had beheld the sight, as if it were still before our eyes. Kind Memory, however, still causes the gorgeous pageant to rise up before our minds, and patiently holds up the picture while we take another long and lingering look.

Our mind's eye eagerly seeks out those separate objects which most attracted our attention, and made the strongest impress upon our imagination. The Crystal Fountain, with its fairy lightness and rainbow hues, still seems to play in the centre of that wondrous fabric, surrounded by a living throng of all ages, admiring its beauty, and, as it were, doing homage to the genius of the palace. The glistening brilliancy of the Queen of Spain's and the Czarina's Jewels, and the steady blaze of lustre of the Golden Service from America, are still vividly before our minds. We still see the crowd of people round the Mountain of Light, gazing with a sort of desperation, as if to force themselves to believe that they were beholding something wonderful, and accusing their

eyes of attempting to deceive them; and again we see them drop away one by one, with disappointment pictured in their faces, ashamed of the injustice they have done to fame, and abashed at having vainly struggled against the wisdom of ages, which saith that seeing is believing.' a

We find ourselves once more standing before the Milanese Window, in presence of the great Dante and his Beatrice. The material medium and the painter's art have vanished, and these figures have been transferred to our memories, and will continue there as real as if we had seen the bodily forms and moving life of the poet and his “donna pietosa ;" whereas, before, our ideas of them were too spiritual to have a permanent place in our imaginations. The scenes surrounding them have fallen back into their places in the truly divine song, for no prominence conferred on separate parts can make us regard it otherwise than as a great and marvellous whole. Yet a light has been shed over all the other scenes, which now present themselves more vividly to our minds, as they pass in quick succession, and have altogether a deeper meaning and more lively interest for us than they had before. Again we picture to ourselves the magnificence of the princely house of Esterhazy, when we recall those doors of costly malachite; and we meditate upon the mysterious changes which have, under the providence of God, been wrought in the fortunes of mankind by at very Industry to which this glorious temple had been raised, when we think that this splendour has passed from the palace of a prince whose nobility is derived from the remote ages of Chivalry to the man

VOL. IX.-NO, XI.

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sion of a merchant, the representative of the true aristocracy of the Industrial centuries.

Nor do we allow ourselves to be carried away entirely by the more gaudy parts of the picture, though these at present have the greatest hold upon our imaginations; our judgment after this first éblouissement will not allow us to forget that significant collection of the implements of British Industry. We do not there stand gazing at particular objects, for wonderful and ingenious though we acknowledge them to be, they have no charm for the outward eye; but the whole of them rise before the mind as a great general aggregate, with a collective importance, which throws every mere sight into the remotest regions of the memory. The genius of Britannia presides over this department, and triumphantly declares it the source of all her greatness. With the large soul and impartial spirit of a loving mother, she commands her handmaids, Art and Commerce, to provide for the wants of all her children, and not to confer splendour and luxury upon the few, and leave the many to their destitution. She is not ashamed of the rude implements with which this lofty purpose is to be accomplished ; she points to her people as her only care, and declares that whatever administers to their welfare shall be ennobled by her peculiar love and protection. She is ready to teach the nations gathered around her the secrets by which she has conferred a lasting prosperity on hier people, and in her turn to reap the benefit of their experience ; but whatever knowledge she gains from them she purposes to apply to the improvement of what she has already begun, for she knows that she would mistake her destiny, if she were lightly to abandon her present course, and seek to rival others in matters for which she has no vocation.

Having thus selected for our contemplation objects which represent in the highest degree the two main features of the Exhibition, the grandeur of its display, and its practical bearing upon the economy of labour, we once more seek to realise the whole spectacle in all its vastness and magnificence. Even the picture which Memory thus conjures up, though but a faint representation of the reality itself, we cherish with a fond regret; and the consciousness that it is a picture strikes us with regret still deeper, for we may not conceal from ourselves that it cannot withstand the hand of Time, before which all traces of the splendour we beheld will " vanish clean out of our sight.”

It was indeed most appropriate for the members of the Executive Committee, on the closing day of the Exhibition, to take their leave of the public in the words of Prospero :

• Our revels now are ended; these our actors,

As I foretold you, were all spirits, and
Are melted into air, into thin air :
And, like the baseless fabrick of this vision,
The cloud-capp'd towers, the gorgeous palaces,
The solemn temples, the great globe itself,
Yea, all which it inherit, shall dissolve ;
And, like this insubstantial pageant faded,

Leave not a rack behind.”
It now behoves us to leave regrets, and set ourselves seriously to in-
quire what will be the beneficial influence of this great World's Fair,

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so far, at least, as our own nation is concerned. Already is this question engaging the minds of our countrymen, for Englishmen are not much given to regretting. The broad general results of the Exhi. bition, which concern the progress of Manufactures and Commerce, have been discussed at great length by our contemporaries. These do not come within our province: we have only to deal with such as have a direct reference to Education ; but these are by no means few in number or small in importance.

Pirst of these we may, perhaps, place that to which His Royal Highness Prince Albert made allusion, in his reply to the Executive Committee on the occasion of the closing ceremony, when, speaking of the grant of the council medal for beauty of design and for excellence in the fine arts, His Royal Highness said : “ Valuable as this Exhibition has proved in many respects, it appears to the Commissioners that there is no direction in which its effects will be more sensibly and immediately perceived than in the improvement which it may be expected to produce in taste, and the impulse it has given to the arts of design." The public are aware that a Committee has been appointed by the Lords of the Board of Trade for the purchase of such articles in the Exhibition as may be calculated to diffuse a correct and refined taste, and that the collection when formed will be made available to the Government Schools of Design which have been established in the manufacturing districts. We may thus expect that the reproach that has hitherto lain upon us, of being behind our neighbours the French in art-manufacture, will ere long be removed, especially as the Government have taken measures, as intimated in this Journal, for developing the Schools of Design, by establishing a connection between them and common elementary schools in the study of drawing. The æsthetic training of the mind is not, as many suppose, a thing of small moment. Not only do we confer, by this means, a gratification to the eye and the intellect of the purest and most refined description, but we influence the very mode of thought, and consequently the whole character ; for we all know and feel that the “hue of thought depends greatly upon the environment in which we are placed, and even upon particular objects which accidentally meet onr view. Who doubts for a moment that there is a deadening and almost demoralizing effect produced upon the minds of the poor in our large towns by the squalid and ungainly hovels and rookeries in which they are condemned to pass their days; or that, on the contrary, the staiely mansions and princely squares inhabited by the wealthy exert a correspondingly great amount of influence for good upon their minds, and contribute largely to their refinement ? Besides this, if we enter the dwellings of the poor, we find them surrounded by objects of the rudest and coarsest design; and here again they are debarred from a source of enjoyment and education always open to the wealthier classes. But it is possible to “ clothe the articles required for the use of daily life with beauty that can please the eye and instruct and elevate the mind;" and there is no doubt that the encouragement which the Exhibition has given to the application of taste in this direction, and the means which the Board of Trade have adopted in order to promote art-education among the people, will lead to results

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of greater value than would be anticipated by careless thinkers. The country will cordially agree with the Prince, that "a special acknowledgment was justly due to those who had afforded the best examples of art, whether pure or applied, and led the way in this interesting career of improvement.'

Another educational result which some people expect to follow from the Exhibition is thus referred to by one of our contemporaries : Some think that we must effect a radical change in our educational systemthat we must substitute living science for dead literature, and distribute the honours and rewards of life in channels where they may fructify to the use of the commonwealth, instead of being limited to the learned professions, the military and naval services, and the residents of our universities." It is indeed much to be regretted that our universities should have so long neglected to provide efficiently for the teaching of physical science; but the improvement in this respect lias already commenced-new professorships, with this object, have been founded, though with endowments, perhaps, scarcely commensurate with their importance. Much good may be expected to result in particular from the University Commissions, which will make the provision for teaching science a matter of special inquiry. Nothing could more forcibly impress upon cur minds the necessity for this change than the experience of the Exhibition. When we see that our manufactures and commerce, our agriculture and inland communication, in fact, our whole outward prosperity, depend so materially upon the application of science, it becomes us as a nation to diffuse the knowledge of scientific truths widely among the members of the community, so that the country in return may reap the advantage of new discoveries and new inventions to the fullest possible extent. In many of our elementary schools, we are glad to be able to say that attention has been given to teaching the science of common things, after the example of Battersea and King's Somborne; but much more may and should be done, for such teaching is especially needed by the industrial classes, both because their occupations are often in the midst of scientific operations, and because a genius for mechanical invention exists, as a natural result of these occupations, to a greater extent among them than among the higher ranks of society. The expense of purchasing apparatus need be no obstacle, for everything that is required for performing ordinary experiments may be constructed by the common blacksmith and carpenter of the village for the most trifling outlay. It would be very desirable that means should be afforded to the students in our training schools for acquiring sufficient skill in carpentering to enable them to make little things of this kind for themselves. There are many of our schoolmasters who already possess this skill, and they find it a great advantage to them in preparing lessons on experimental science.

Our middle schools are in this respect very far behind-hand : there are, indeed, very few of them in which any attempt has been made to teach the most elementary matters of science. This would soon be effected, however, if parents, understanding themselves the utility of such knowledge, would require it to be taught to their children, as in the case of other matters of practical importance. This will prove of more real use to their children than a superficial acquaintance with Greek

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