may be done by making a wire-rather less than the tube-red- | Flasks with flat bottoms are required, and these may be had for hot, and then burning a hole through the cork. But by far the 4s. 6d. a dozen, having a capacity of 12 oz., 6d. less a dozen for best plan is to buy" cork-borers," which are pieces of thin brass / cach 2 oz. less in capacity, and Gd. more for every additional 4 oz. tubing, the edge of One of these latter which is sharpened at flasks may make a one end. They are very useful apparatus made in sets which fit -a wash bottle (Fig. into each other. A 11 b). By blowing set of two costs 10d.; down the open pipe at three, ls. 2d. There A, a jet of water issues are sets of six, but from the other at B, those of two or three which is a convenient answer every practical mode of filling a test purpose. tube, or adding a little Test tubes are in con water to a solution. stant requisition; their To make this bottle sizes range from two will be a good begininches in length and a ning for a student, as quarter of an inch in the tubes and cork diameter, to eight must fit tightly. inches in length and A Ring Stand (Fig. one and a-half inches 11 c), with three rings, diameter. Their prices is 5s., and is indisvary usually from 4d. pensable in the laboa dozen to 38. ratory. They may be When they have Fig. 12. had larger, but this is been used as with all quite sufficient. chemical apparatus—they should be cleaned. This is best effected Evaporating Basins (Fig. 11 d) are shallow basins of Berlin by a round brush, made of bristles held in twisted wire, which porcelain. A nest of seven, containing from 1} oz. to 18 oz., costs 3d., or a piece of tow at the end of a wire will answer the may be had for 6s, 5d. same purpose. Thuringian porcelain is thinner, and basins of this, more It is convenient to have a stand, such as is shown in Fig. 10, shallow than the last, are in nests of nine for 6s. in which to hold test tubes. The stand may be bought for 1s. 9d. Distillation is carried on by turning a liquid into vapour, and condensing this vapour again into liquid. A very useful condenser is the one shown in Fig. 12, Liebig's Condenser, price 14s. and upwards. a b is a glass tube, which passes through cold water held in the larger tin tube c; instead of the retort d, a Florence flask may be used, the tube which passes through a cork fitted into a being attached to its neck. The distilled liquid falls from the pipe e into the vessel f. The water in the condenser is kept cool by continually renewing it; the cold water entering from the barrel g by the funnel i; and, as warm water always rises, the warmest escapes by the pipe h. Liquids are often purified by filtering (Fig. 13); for this funnels of glass or porcelain are used, and white blotting paper is cut into a round disc, then folded thus, a, and again into half, b, opened, c, and placed in the funnel, d. The arrangement in this figure is simple and convenient. We advise students not to lay in a stock, but to get chemicals and apparatus as they are needed. The prices of preparations Fig. 11. will be found in catalogues, which will be forwarded by any working chemist. Of Glass Jars (Fig. 11 a), one ought to be graduated as a measure All acids must be kept in stoppered bottles, and no bottle -an 8 oz. “ measuring glass” is 1s. 8d. “Beakers" may be had should be without a label descriptive of its contents. 12 oz. with spouts, but if not, then a glass rod must be held at the place bottles, with ground-glass stoppers, are 6s. a dozen; ls. more over which the liquid is to run, as in (Fig. 9). per dozen for each additional 4 oz. Jars are required for collecting gases; those in which confec tioners keep LESSONS IN ARITHMETIC.-XXVII. the best; wide COMPOUND DIVISION. necked bottles 11. THE process of dividing a compound quantity may be also used by | regarded in two aspects. them are very (1.) We may divide a compound quantity by an abstract cheap, and an-number; that is, we may divide the compound number into swer every pur- a given number of equal parts, and thus find the magnitude of pose. one of these parts. Flasks, which (2.) We may divide a compound quantity by a compound also serve for quantity; that is, we may find how many times one given retorts, may be compound quantity is contained in another. had of any gro Thus £14 10s. 6. = 7 times £2 ls. 6d. either for no- gives as a result L2 ls. 6d. Here we have shown that if thing or for a trifling sum. They come from Italy, filled with £14 10s. 6. be divided into 7 equal parts, the magnitude of olive oil. The glass flasks are chiefly made at Florence—hence each part is £2 1s. 6d. their name, Florence flasks; they are covered with rushes, not Again, £14 10s. 6. divided by £2 18. 6d. gives 7 as a only for their preservation, but that a flat bottom may be pro- quotient. This is the same as saying that £14 10g. 6d. contains vided on which they stand. For chemical use the rushes are cut £2 18. 60. 7 times. off, and the flask cleaned with a little soda and warm water. Hence we see that a compound quantity divided by an abstract. TIERE number gives a compound quantity, and that a compound quan- sively by them, instead of dividing at once by the whole divisor. tity divided by a compound quantity of the same kind gives an For instance, if it be required to divide 75 cwt. 2 grs. 8 lbs. by abstract number as a quotient. 35, which = 7 X 5, we can perform the operation thus :Obs.—The last remark is the same thing as saying that the cwt. qrs. lbs. ratio (Art. 1, Lesson XXI., Vol. I., page 342) of two concrete 7) 752 8 quantities of the same kind must be an abstract number. It is of the nature of how many times. 5) 10 3 57 Furthermore, notice that if two concrete numbers are to be compared that is, if one is to be divided by the other-they 2 0 17% + must be of the same kind. The ratio of one sum of money to Notice that the arises from the division of the by 5. another sum can be found, or that of one weight to another Adding and 36, we get 3 ; so that the required answer would weight; but money cannot be compared with weight or with be writtenlength. To talk, for instance, of the ratio of 25 shillings to cwt. qrs. lbs. 13 lbs. would simply be an absurdity. 0 1781 12. EXAMPLE.-Divide £87 10s. 7 d. by 47. And, if necessary, the ide of a pound could be further reduced Beginning with the pounds, we find that £87 divided by to ounces, etc. 47 gives £1, with a remainder £40. Reducing these £40 to 15. When it is required to divide one compound quantity by shillings, and adding in the 10 shillings of the dividend, we another of the same kind, we must reduce each to the same get 810 shillings, which, divided by 47, gives 17 shillings, with denomination, and then divide as in ordinary simple division; a remainder 11s. Reducing these 11 shillings to pence, and for, clearly, the number of times which one compound quantity adding in the 7 pence of the dividend, we get 139 perce, which contains another does not depend upon the particular denominadivided by 47, gives 2 pence and a remainder 45 pence. tion or denominations in which they happen to be expressed. Reducing the 45 pence to farthings, and adding in the 2 far- Supposing one man to have 5 sovereigns in his pocket, and things of the dividend, we get 182 farthings, which, divided by another 1 sovereign, the former would still have 5 times as 47, gives 3 farthings, and a remainder 41, which, divided by 47, much as the latter, if they had respectively 100 and 20 shillings gives a fraction of a farthing. The answer, therefore, in . instead of the sovereigns. £1 178. 2d. 34f. 16. EXAMPLE.—Divide £35 178. 6d. by £2 11s. 3d. The operation may be thus arranged : £35 175. 60. = 8610 pence. 47) £87 10 73 (£1 £2 lls. 3d. = 615 pence. 615 ) 8610 (14 615 20 2460 2460 47) 800 + 10 = 810s. (175. Hence 14 is the answer. We shall, however, return to this part of the subject when 340 we treat of fractions in connection with compound quantities. 329 EXERCISE 46.-EXAMPLES IN COMPOUND DIVISION. 11 Divide 12 1. £87 10s. 7 d. by 18, 27, and 39. 2. £33 by 96. 3, 312 lbs. 9 oz. 18 dwts. by 7, 43, 84, and 160. 8. 365 days 10 h. 40 min. by 15 and 48. 9, 111 yrs. 20 d. 13 h. 25 min, 10 sec, by 11, 19, 83, and 100, 141 10. 45° 17' 10" by 25, 35, and 45. 11. How much a day is £200 a year? Answer £1 178. 20. 334. 41 Remainder. How many times is 13. The principle upon the truth of which this process 12. 6s. 3!d. contained in £5? depends is the same as that mentioned in Art. 3, Lesson V. 13. 829 7s. 6d. contained in £523 15s, 3 d. ? (Vol. I., page 69), namely, that when a quantity is to be 14. 2 qrs. 13 lbs. 5 oz. contained in 4 tons 3 cwt. 2 qrs. 6 lbs.? divided, if we separate it into a number of parts, and divide Divide each part individually, the sum of all the quotients so obtained 15. 195 m. 7 fur. 30 ft. by 7 ft. 6 in. will be the required quotient. 16, 531 m. 2 fur. 10 p. by 17 m. 5 fur. 27 p. Here notice that we have, in reality, divided £87 10s. 7 d. 17. 950 days 1 h. 11 min. 6 sec. by 4 days 8 h. 6 min. 54 sec. into the following parts : *. The Key to Exercises 44, 45 (Vol. II., page 78), will be found af te £47 + 799s. + 94d. + 182 farthings; end of Lesson XXVIII. Or, £47 + (47 x 178.) + (47 28.) + (47 x 384 farthings). The quotients of each of these separate sums, divided by 47, LESSONS IN GEOGRAPHY.-XVII. are respectively £1, 178., 20., and 384 farthings. THE GREAT CIRCLES OF THE EARTH-THE MERIDIAN THE EQUATOR. Hence the required quotient is el 178. 20. 37f. On the globe of the earth, or terrestrial globe, as it is called, the 14. From the above remarks we see the truth of the fol. first great circle of importance is the meridian ; this is a great lowing circle which passes through the two poles, PP (Fig. 6), of the ans Rule for Compound Division when the Divisor is an Abstract of the earth, and through any point, as m, on the earth's so Number. It is called meridian, because when the sun in our climate shines Beginning with the highest denomination, divide each sepa- on a gnomon or style (the pin of a sun-dial), and casts its s rately and in succession. When there is a remainder, reduce it in the direction of this line on the surface of the earth, to the next lower denomination, adding the number of that i then (meridies) mid-day or noon; and whenever any heaven. denomination contained in the dividend, and divide the sum as body appears in the plane of this circle, as determined by before. Proceed in this manner through all the denominations. position of the style and its shadow at noon, it is said to be Obs.-It is sometimes convenient, when the divisor is a com- the meridian. posite number, to separate it into factors, and divide succes. The meridian of the point m in Fig. 6 is, according to this bado definition, the circle PMTPN s. But as every spot on the Navigation are the meridian and the equator. We proceed now all others in point of angles, as in Fig. 7. In this figure, the straight line A Y represents the north and A X great circle which passes east, and if you go from x to A you go west. Fig. 7. through all the points The directions of the fences being thus Q on the earth's surface understood, suppose that you were told the exact distance of situated at an equal dis- the place where the golden treasure lay from the fence A x, say tance from the two poles, 20 yards; this would not be enough to enable you to find it, P P, of the earth's axis ; because there are ever so many points in the field, all at 20 it is called the equator, yards distance from the fence A X. Now suppose you were also because when the sun's told the exact distance of the place where the golden treasure rays are vertical to this lay from the fence A Y, say 25 yards; this alone would not be line there is no shadow enough to enable you to find it, because there are ever so P to the gnomon or style many points in the field, all at 25 yards distance from the Fig. 6. at noon, and there is an fence A y. Among these latter points, however, there can be equalisation of light all only one which is at the exact distance of 20 yards from the over the globe (on the days when this takes place), this position fence AX; so that if you were told both distances at once, you of the sun being the equaliser (equator). The equator is also could evidently, by some means or other, determine the place made the starting place for the measurement of the distances where the golden treasure lay hid. It is necessary, therefore, of places on the surface of the earth as to their position in and sufficient, to inform you of the exact distances of the place the northern or southern hemisphere; for the equator divides in the field from both fences, in order to enable you to find it. the globe into two equal parts, called the northern and southern With the information now supposed to be given, the next hemispheres or half-globes: that hemisphere in which we live question is, how should you proceed to determine the exact is called the northern hemisphere, because our Saxon ancestors place of the golden treasure. A little reflection would suggest called the point opposite to the sun at noon the north; and that the following method. In Fig. 8, measure off hemisphere in which the point opposite to the sun in the con- from the point A, along the fence A X, the trary direction is seen, is called the southern hemisphere, because given distance of 25 yards, at which the they called the point where the sun is seen at noon the south place is said to be situated from the fence The distance of any place on the earth's surface in the A Y; let this distance be A M. Then, from Fig. 8. of M from the equator, E Q, is measured by the number of degrees both at right angles to ax, are parallel to of the quadrant, E P, contained in the arc EM, the extent of each other. Lastly, measure off from the point , along the opening of the angle ECM. Now the quadrant E P is divided straight line MP, the given distance of 20 yards, at which the into 90° from E to P, that is, from the equator to the north pole place is said to be situated from the fence AX; and the point at P, and the degrees are reckoned from E, which is marked 00 P will be the place in the field where the golden treasure is to (no degrees), to p, which is marked 90° (ninsty degrees). Hence, if be found. u be any point on the earth's surface to which the rays of the sun That this mode of determining the place of the golden are vertical on the 21st of June as shown in the diagram of the treasure is correct may be proved thus : in Fig. 9 let P be the seasons (Fig. 4, page 80) at R at mid-summer-then the distance place in question ; from p draw in perpenof the point m from the equator is 23° 28' N.; that is, 23 dicular to A Y, and P M perpendicular to Ax; y degrees 28 minutes north. The reason of this is plain ; for, if then, according to the data (things given), from the right angle or 90° formed between the plane of the PM is a distance of 20 yards, and PN is a earth's orbit and the perpendicular to that plane (see Fig. 4, distance of 25 yards. But by the nature of page 80), and also from the right angle or 90° formed between the construction, the figure AMP N is a rectthe plane of the earth's equator and the perpendicular to that angular (right-angled) parallelogram, and its plane in the axis of the earth, we take away the common angle opposite sides are therefore equal; whence A M X NOR, the inclination of the earth's axis to the plane of the A M is equal to NP, and An equal to M P. It Fig. 9, earth's orbit, which is 66° 32', we shall have 23° 28' in either follows, therefore, that the point p is found case; and this is the distance between the polar circle and the by the method shown in the preceding paragraph. In mathe pole, or the inclination of the plane of the earth's equator tomatical language, the distances P N and PM of the point p from the plane of the earth's orbit, and consequently the distance M E the fences A Y and Ax, are called the rectangular co-ordinates (Fig. 6). The distance of any point on the earth's surface, mea- of that point; but the distances AM and MP, which are equal sured in degrees, from the equator is oalled its latitude (from to the former, are more usually denominated the rectangular the Latin latitudo, breadth), because the extreme distance of the co-ordinates of the P; and by these co-ordinates we can always earth from north to south was by early geographers reckoned determine the position of any point, when their exact lengths less than its extreme distance from east to west; the term longi- are given. The straight lines AX and A Y, from which the tude (from the Latin longitudo, length) being applied to distances given distances are measured, are called rectangular axes, and reckoned east or west from the first meridian. The latitude of the point a, where these axes intersect each other, is called the point x on the earth's surface is thus reckoned 23° 28' N. the origin of the rectangular axes. With the origin and the In like manner the latitude of the point t on the earth's surface, direction of the rectangular axes (in our figures, the fences at or any point to which the rays of the sun are vertical on the 21st right angles), and the lengths of the rectangular co-ordinates, of December, is reckoned 23° 28' S.; that is, 23 degrees 28 minutes all given, in reference to any point on a given surface, we com south. always find the true position or place of that point The two circles of the greatest importance in Geography and required. LESSONS IN DRAWING.-XVII. mode of treatment of the form, as we have already remarked; but now it is the strength and quantity of the work we more TREATMENT OF TREES AND FOLIAGE (concluded). especially allude to. If the same tree were drawn on a dull BEFORE concluding our observations upon trees and foregrounds, heavy day, there might be much more leaf character introduced we will offer a few additional remarks upon that which we have both in the lights and in the shades. There is a very common so often maintained to be of the utmost importance, and which and well-known custom when in difficulties as to the true extent of our pupils will by this time begin to realise. It is because, in light and shade ; when the pupil is in doubt as to where the light this particular instance of trees, there is some difference of treat- ends and shade begins, let him half close his eyes when looking ment in making the outline, to that required by a solid object, at the object; the minor tones, or those which seem to belong to the form of which is unmistakable, that we think it unnecessary neither light nor shade, will apparently disappear, and the true to offer any excuse for this repetition. The power of drawing extent and force of both extremes become distinct, and so far is the rock upon which the whole superstructure of art is based; evident as to enable him to determine their shape and character. in other words, it is practically the foundation of all that after- Fig. 110 is the general character of a fir-tree, in which we have wards commands admi. endeavoured to show ration or praise. To how the foregoing inwhatever point of excel structions are to be oblence we may hereafter served. Fig. 109 in the attain, we shall invari last lesson will also ably Fig. 110. look back with illustrate our meaning. satisfaction upon the Now the pupil must exertions we have used, clearly understand that and the time we have whilst we advocate a devoted to ensure our breadth of treatment on success in making a the whole, characteristio really learned and care details must not be fully-constructed out omitted; these details line; it must be the one may be expressed in only starting-point of all such a way (without who are ambitious to descending to littleness excel, though the sub of manner) as not to jects they may eventu destroy that breadth, ally choose will vary and yet be sufficiently according to their indi carried out to enable us vidual tastes, wishes, to say whether the tree and circumstances. be an oak, a poplar, & In the case of foliage fir, or one of any other it is necessary to explain description. what we mean by out. We will now introline, and how it is to be duce & few practical treated when subject to hints respecting some the various changes of the rises to which the caused by sun and shade knowledge of drawing under which the tree is trees, shrubs, or wild found. Let us suppose plants may be applied, ourselves to be standing especially by designers opposite a tree on a dull, of patterns and ornacloudy day. The force ment. Our country lanes of light and depth of and hedgerows afford shadow will each be less abundance of material than if the sun were to supply us.with an shining upon it, and the endless variety of form half-tints will be more and culture especially apparent and varied. applicable for the decoAll round the tree ration of our walls, and against the grey sky for the enrichment of behind there will be articles of ornament and the same distinctive use. The Corinthian and uniform character capital is said to have throughout; but let the had its origin from the sun break out, and then circumstance of a tile observe what a remark having been placed on able change takes the top of a basket, place. The general or around which grow the larger masses of light and shade will be more decided, leaves of the acanthus plant. This, whether true or not, is the neutralising tones among the half-tints will in a great highly suggestive, and tells us there are beautiful combinameasure have disappeared; the shadow side of the tree will be tions to be found in nature, which the designer would do well distinctly made out against the sky, whilst the details in light to cultivate. To point out a few of them will be sufficient to will be less definite than they were before the sun shone, owing direct the way in which the lover of nature and art may select to the radiation of light from the leaves; the half-tints and examples for himself without fearing to exhaust the supply: small shadows in the light will have less strength than they had The most graceful of all the wild plants are those which cling to before-they will be of a warmer tone, and partake of the light others for support. Who has not noticed the wild convolvulus, and colour around them; the corresponding half-tints on the with its elegant elongated leaves, and its simple symmetrical shadowed side will follow the same course on the same principle flowers twined about the stem of a brier or hazel ? The hop -that is, become more general and less distinct in form. We plant, also, the black bryony, and others may be named whose therefore advise the pupil, when "massing in the foliage" of a spiral twistings round stems of various kinds produce natural tree in sunshine, to use his pencil less vigorously on the lights, combinations which no mind could suggest, or power of invention and not to be betrayed into leaf-drawing and making dark heavy could supply. The leaves alone are models for imitation. The lines. The kind of tree he may be drawing will suggest its own ancient Greeks saw this, and proved it by their frequent appu. cation of the vine-leaf, the oak, and the ivy. In fact, as Mr. We now take up another portion of our subject relating to Redgrave has said, “ He that would be a great designer must landscape-the principles of the reflection of objects in water, as be in the hedgerows and fields at all times, sketching with | by reflection only can water be represented. patient diligence the form and curvatures of leaves, fruits, and It has been frequently said that a landscape is incomplete fowers, their groupings and foreshortenings; studying them as without water; it is certainly an element which contributes a whole, and in their minutest details, together with their much additional beauty and effect to any scene, be it ever so growth and structure. Not to repeat as a mere imitator, but simple; yet we cannot go so far as to say that it must neces. to display them as ornament; to dispose them geometrically, sarily be introduced in all cases. Independently of itself, there to arrange them to suit the various fabrics or manufactures for are associations connected with water that cannot be passed over which they may be called on to design, and to give them life without notice, and which bear an important part in the whole and words, as it were, by using them as emblems of some living composition whenever it forms a portion of the picture, such as thought or poetical allusion.” shipping, barges, boats, fishermen, and picturesque bridges. Why It is the application of the graceful forms of the vegetable is it that, in our choice of a walk, we generally prefer a stroll kingdom that constitutes the most important part of the study near some stream? We attribute it to the variety of scenery of the designer and decorator: the power of drawing, important afforded by the winding river, and the numberless points of inteas it is, is only the means; the adaptation is the end sought for. | rest that catch the eye as we ramble along its banks. The life Here it is, we can say with truth, that it requires the mind of and motion connected with water have no limit; and besides, we an artist to accomplish it, to be imbued with an originality of cannot forget, when it is clear and calm, its capability of reflecting thought, that can make the simplest object do duty for worthy every object near it in full perfection, and increasing our admirapurposes. tion by the It Fery fre fidelity quently with which occurs, in it reverses art univer form, and sally, that reflects by contrast colour, or applica light and tion we dis Fig. 111. shade, thus corer excel making a lences not double before ob picture. Eerved: re There are specting several the use of phenomena this idea, resulting how many from the ap. times, may pearances we ask, of reflec. hare we tions upon trod on the the surface decaying of water leaf in our which unpathway, doubtedly without require having had more atthe atten. tention tion in the than is least di. generally rected to it devoted to as capable such subof suggest jects by ing either many who an original aim at reform or a presenting fresh ar them. A rangement of colour? However insignificant and valueless an course of study is necessary which some would suppose to be object the fallen leaf may seem to be, it is capable of teaching beyond the limits pursued by artists generally, but which we us a lesson of great practical utility. It has been supposed contend is indispensable; for every one who undertakes or by some that the shape of the vase owes its origin to a leaf ; hopes to paint Nature as she is, must go deeply into her mys. it may be so or not, but it is sufficient for us to know its teries, and endeavour as far as possible to understand them, and capability of suggesting it, and it leads us to where the designer not abide by a mere superficial following of outward appearances. may apply if any new form is required. Such resources, when Why is it that the sculptor and the historical painter seek the regulated by a disciplined and scientific taste, must produce advantages to be gained in the dissecting-room? Because they something as beautiful as it is original. In search for hints for feel that a knowledge of anatomy is of the utmost importance to decorative purposes it is not absolutely necessary to confine our them when engaged upon the human form. Similarly the landchoice to the floral varieties of a conservatory or greenhouse, scape painter wisely looks about for aid when difficulties arise, however valuable they may be for the purpose; the green lanes which have their remedy often beyond the limits of his own legiti. and hedgerows can boast of gems of form amongst nettles and mate art; and he will meet with an abundant source of difficulty wild flowers, from which articles of ornament and utility may with regard to reflections. There are incidents so puzzling conborrow their simple elegance either to decorate a palace or nected with these, that unless he possesses a little geometrical perform gome humble service in a cottager's dwelling. Nature knowledge, he cannot avoid falling into endless mistakes. We everywhere offers hints that are useful as well as beautiful, and must again have recourse to geometrical perspective, which the designer need never sigh for a model. As an illustration of will not only assist us in our explanations, but will set at rest the way in which plants may be adapted to ornament and many doubts which might arise in the minds of our pupils with design, we have introduced one for a candlestick in Fig. 111, regard to facts that seem to be impossibilities, unless we employed the socket of which is a lily; the extinguisher inserted in this conclusive help in rendering them intelligible. Sir Joshua the side is a dead blossom of the same plant, emblematic of Reynolds said, “ The rules of art are not fetters to genius ; they are fetters only to men of no genius.” its use. |