DRAWING.- No. 2. AVING in the previous number briefly shown the value, importance, and utility of Drawing as a grand preliminary step, indispensable for the successful completion of any design of importance, either in the manufactures or arts, and also its great value as a refining source of pleasure, I will proceed to describe the various styles of drawing. The modes of representing objects by Drawing are two parallel projection (that is, mechanical drawing) and perspective. The first, employed for engineering and architectural works, is that in which the eye is supposed to be at right angles with every part of the plane represented, and is on that account called parallel projection; and I regret that this style of drawing, which is so absolutely useful and necessary, should be so little studied, the preference being invariably given to that called perspective, which may, comparatively speaking, almost purely be said to be an accomplishment: it is useful to a certain extent, but small indeed in comparison with the former. If you desire a general view of an object, you can have it in no other style but perspectively or isometrically; but if you wish to enter into details, to know the true proportion, the exact dimensions of the various parts of an object, you must have recourse to parallel projection. If any one is at all sceptical as to the superiority of parallel projection over perspective for the representation of machinery, or not only machinery, but every object where it is sought to convey the actual proportion and dimensions, instead of a mere general notion, let them refer to the encyclopedia published in France and in this country above half a century back, where they will find everything illustrated in perspective, and compare such works with the valuable plates of the second edition of Tredgold on the Steam Engine, and Professor Barlow's Treatise on the Manufactures and Arts in the Encyclopedia Metropolitana. But perhaps you may imagine, from these remarks, that parallel projection is to be studied to the exclusion of perspective; that I wish to guard you against. You will recollect that I remarked in those objects, when the actual proportions and dimensions were intended to be conveyed, that parallel projection was to be used; but if a general view is required, perspective; consequently the architect, after representing the different plans, the front and side elevations, the longitudinal and transverse sections of a building, showing that the dimensions of the various rooms, the comfort, convenience, elegance, and economy of the design are such as desired, that the person for whom the design may be, may fully understand what the picturesque effect of the building will be when erected, then has recourse to perspective (under those circumstances distinguished by the name of geometrical perspective). Having thus slightly described the two principal styles of drawing, and the application of each mechanical or parallel projection for engineering, architectural, and useful purposes generally, and perspective for such objects and purposes which are not so generally and absolutely useful, but yet affording great delight, I will briefly describe the course to be pursued by those who aim at eventually being able to sketch from nature. After having acquired a certain facility of hand and correctness of eye, so that you can retain in your mind's eye the correct form of an object, and the hand to follow the dictates of the eye, and become conversant with the first principles of light, and shade, and perspective, -for the last two steps the assistance of a master is indispensable,—you may then proceed to the more agreeable and pleasing study of nature; and the subjects I more particularly recommend to be studied, as tending to the success most likely to ensue upon ordinary diligence and application, are weeds, such as the dock, foxglove, nettle, &c.; and to be successful in our first attempts at any study is an important point; for upon our success or otherwise in first attempting any study, we either prosecute it to a successful issue, stimulated by our previous success; but if unsuccessful, so few persons have sufficient energy and resolution to combat failure at the onset, that they discard and throw their studies aside in disgust. Presuming that you are successful in your study of weeds, you may then proceed to study all such objects in detail that are generally found in a landscape, before attempting composition; for what would be more absurd than to attempt to read before being able to spell, without knowing the alphabet. For this purpose you may then study the character of the foliage of the various trees, placing yourself at such a distance that the leaves do not appear too much in detail, but agreeably massed, and at such a distance that the light, shade, and demi-tint may be easily distinguished. Representations of landscape scenery have been made for many years by mechanical means, with the aid of the Camera Lucida, a species of box (provided with a mirror and a lens), into which the representations fall, and are therefrom copied by the attendant artist into his sketchbook. The idea of fixing representations of this kind on the surface on which they fall, by some chemical process, so as to save the trouble of copying, appears to have been entertained by ingenious men, both in France and in England, and at length the possibility of doing so was made known in Paris in January, 1839. The discoverer was Daguerré, aided by one or two other persons; and from that was called Dauguerrè's type, from the inventor's name. It is frequently called photography-a word expressive of the process, —that is, drawing by the action of the light. Some persons, I have no doubt, make use of those photographs in the composition of their pictures; but such a practice is more than questionable, being simply copying, in the best sense of the term, and not trusting to that higher form of tracing and imitating external nature, which must ever distinguish the legitimate artist from the empiric and pretender; besides, such a course of copying nature would be to cancel and put aside all those great and high advantages I have before alluded to in the previous number as resulting from observation, contemplation, &c., on the study of nature. The lens doing the work of the eye, and the camera generally being the copyist, except in the first place in fixing the machine, there is scarcely any occasion for lengthened contemplation; and many of those refining, purifying thoughts that most frequently arise upon any lengthened view of nature would be lost. For instance, should the object about to be copied be ruins, and it is well known how frequently they enter into the composition of pictures, and when artistically treated what sublimely beautiful pictures they make, such ideas as these, probably suggested by lengthened contemplation, would be lost. Everything is mutable, everything is perishable around us (I am supposing that the object being copied is a ruined abbey or castle); the forms of nature and works of art alike crumble away; and amid the gigantic forms that surround it, the soul of man alone is immortal; and from that, to what an extended, purifying, beneficial influence may the thought be carried. What geology is to us in relation to the early inhabitants of the earth, such are ruins as regards its human habitants. They are history in stone. Those shattered columns, those fallen capitals, and mutilated statues that once rose above the dwellings of the Hundred-gated Thebes, those mounds of rubbish (now shunned—at least until lately-even by the wild Bedouin Arab) that cover the wondrous relics of Nineveh. Those now silent mountains that look down on the lone and ruin-covered plain of Merdasht once echoed back the shouts of the royal Persepolis. Notwithstanding the great ends gained by steam and electricity, ruins are the voice of the past ages chiding the present for its degeneracy. These and similar thoughts have been suggested to me when copying and sketching a ruined ecclesiastical edifice, and I doubt not many others would experience similar thoughts and feelings, and no one, I can scarcely think, will be found to dispute their refining influence, tending, as they must necessarily do, to enlarge the mind, purify the thoughts, withdraw us from the present and less refining pleasures, inducing us to look into futurity, and preparing us for that brighter and better life which is the grand end and great object of our being. Oct. 7th, 1856. WILLIAM HOBDAY, Head Drawing Master, MORAL TEACHINGS. ILLUSTRATED BY ANECDOTES. - Perceptions of right and wrong belong to childhood, and, like other faculties, should be exercised and prompted to vigorous growth. This requires skilful training. Children tire of direct instruction, and unless charmed by the personal qualities of the teacher, become listless or mischievous. But tell them a story, a biographical anecdote, or historical fact, or let them tell one, and the case is altered. They see something tangible. The precept has its living type, and the moral lesson enters together with the story, and becomes for ever a part of the child's being. A story of this sort, and for this object, once a week, would do much for teachers and pupils, and would call attention to this much-neglected subject. It would compel investigation, and keep all on the alert for next week's lesson.-Canadian Journal of Education. A RATIONAL METHOD OF TEACHING FRACTIONS. BY CHARLES DACUS HERMANN. (Continued from page 423.) XIII. Multiplication of Fractions. M. How much are 5 yards taken 3 times?—A. 15 yards. M. £8 multiplied by £7 ?—A. £56. M. 4×9 pieces or parts of a loaf ?—A. 36 pieces. M. 3x4 parts called sevenths?—A. 3 × 4 sevenths=12 sevenths. M. 5x-A. 20, or 23. 9 M. Here we have a multiplication of fractions: repeat the last example? What was the fraction to be multiplied?-A.. M. What was the product ?-A. 2o. M. How did you obtain it ?-A. By multiplying the numerator. M. We consider, as we often did before, the denominator as a mere name for certain parts of a whole, and this consideration fully justifies our way of multiplying by increasing the real number of the parts. Now multiply the following fractions; if the product is more than 1, reduce it into wholes, and do not forget to reduce the remaining fractions to lower terms, if possible. In mixed numbers, first multiply the fraction as above, then the wholes. 100 1 × 5, 7 × 3, 18 × 10, 81 × 4, 97 × 8, 2517 × 83, 23588 × 24, × 2, 3, 4, 5, &c.—87 × 2, 3, 4, 5, &c. — 5218 x 2, 3, 4, 5, &c. 8543 × 36. The master may occasionally observe that there is another mode of multiplying a fraction which may sometimes be made use of, i. e., by dividing the denominator. Reminding the pupils of the nature of that term, and that by lessening or dividing it the parts become greater, he may induce them to give some examples where this mode of multiplication is available, and reason about them like this: × 2. A part twice as large as is; therefore, are twice as much as . Or like this: }×2 = 3 = 1; x 2-3 times more, XIV. Division of Fractions. The division of fractions offers no particular difficulties, after all that has preceded. There are, as for multiplication, two different ways of doing it. The most simple and natural is that of dividing the numerator; but the pupil, after having done several such sums given by the master, will soon perceive that in most cases the numerator is not divisible by the given number, and we must then have recourse to the second method, by multiplying the denominator. In reasoning about these cases, we always first reduce the numerator to 1; as, for instance, ÷3? ÷÷÷3=21,÷÷÷3=21 M. Which are the two different ways of dividing a fraction? Give examples of each. Which way is always possible? which not? Do the following divisions. (In mixed numbers, you begin by dividing the wholes, and reduce the remaining ones to a fraction of the given denominator.) 23, ÷2, 2÷17, 18, 20, 25, 64, 100. 4, 21÷2, 3, 7, 18; 1715÷4, 12, 17, 260; 3768231÷17, 294. TO MULTIPLY BY A FRACTION. a.-A Whole Number. M. How much are 7 x 11 lb. ?-A. 7 x 11 lb. =77 lb. M. Multiply 8 by 7.-A. 8x7=56. M. 8 by 3-A. 8×3=24. M. 8 by 1-A. 8×1=8. M. Now 8 by. Do you know what this means, to multiply 8 by? -A. I suppose it means that I have to take of 8. M. And how do you find of 8? or one-half of anything else?—A. By dividing it into 2 equal parts, or by 2. M. Therefore 8x is the same as 8 divided by 2? Now, how would you find 8×, or what is this like to?-A. 8x=8÷3=23. M. What does it mean, to multiply 20 by?—A. 20 × means to take of 20, or divide 20 by 5. M. Remember, then, that in all these expressions, when we say, "To multiply by a fraction,,,, &c.," we really mean a division. It is of course not an exact way of speaking to say "multiply" when we mean "to divide ;" but it is excusable, and in many cases useful, to retain that expression; and I will show you how it happens that we say so. A man has, for instance, to buy several quantities of cloth at 18s. a yard; he wants 7 yards, 4 yards, 1 yard, yard, and yard. How much has he to pay You will say, 7 x 18, 4x 18/, 1x 18/; and then-instead of changing your way of speaking, and saying of 18/, of 18/, you continue 18/, × 18/; or we must multiply 18/ by 7, 4, 1, 1, and. In mathematics, we often make use of such expressions, which really mean the contrary of the common signification of the word; thus we say He owes me -8£, which means, I owe him £8. Now you will at once be able to understand and answer the following and all similar questions :-8 × 1, 21 × 1, 30 × 1, 13 × 1, 54 × 1, 593 × 1? Say and write like this: 8×4 (or 8÷1) = 2, &c. Write the following series : times times 1£. x2/ 103d. &c. 33/1. = 2/ 63d., &c. and many similar series, as well as promiscuous examples. M. How much is 9x-4. 9x=3. M. Instead of multiplying by, I now want you to multiply by 3. Will the product be more, or less?-A. If I multiply by 3, I must get the double of what I get in multiplying by |