g

may be done by making a wire-rather less than the tube-redhot, and then burning a hole through the cork. But by far the best plan is to buy "cork-borers," which are pieces of thin brass tabing, the edge of which is sharpened at one end. They are made in sets which fit into each other. A set of two costs 10d.; three, 1s. 2d. There are sets of six, but those of two or three answer every practical purpose.

α

d

Flasks with flat bottoms are required, and these may be had for 4s. 6d. a dozen, having a capacity of 12 oz., 6d. less a dozen for each 2 oz. less in capacity, and 6d. more for every additional 4 oz. One of these latter flasks may make a very useful apparatus -a wash bottle (Fig. 11 b). By blowing down the open pipe at A, a jet of water issues from the other at B, which is a convenient mode of filling a test tube, or adding a little water to a solution. To make this bottle will be a good beginning for a student, as the tubes and cork must fit tightly.

Fig. 13.

a

tioners keep their sweets are the best; widenecked bottles also used by them are very cheap, and answer every purpose.

Flasks, which also serve for retorts, may be had of any grocer or oilman, either for nothing or for a trifling sum. They come from Italy, filled with olive oil. The glass flasks are chiefly made at Florence-hence their name, Florence flasks; they are covered with rushes, not only for their preservation, but that a flat bottom may be provided on which they stand. For chemical use the rushes are cut off, and the flask cleaned with a little soda and warm water.

h

A Ring Stand (Fig. 11 c), with three rings, is 5s., and is indispensable in the laboratory. They may be had larger, but this is quite sufficient.

Evaporating Basins (Fig. 11 d) are shallow basins of Berlin porcelain. A nest of seven, containing from 1 oz. to 18 oz., may be had for 6s. 5d.

Thuringian porcelain is thinner, and basins of this, more shallow than the last, are in nests of nine for 6s.

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 will be found in catalogues, which will be forwarded by any working chemist.

12 oz.

All acids must be kept in stoppered bottles, and no bottle should be without a label descriptive of its contents. bottles, with ground-glass stoppers, are 6s. a dozen; 1s. more per dozen for each additional 4 oz.

LESSONS IN ARITHMETIC.-XXVII.
COMPOUND DIVISION.

11. THE process of dividing a compound quantity may be regarded in two aspects.

(1.) We may divide a compound quantity by an abstract number; that is, we may divide the compound number into a given number of equal parts, and thus find the magnitude of one of these parts.

(2.) We may divide a compound quantity by a compound quantity; that is, we may find how many times one given compound quantity is contained in another.

Thus £14 10s. 6d. = 7 times £2 1s. 6d.

Therefore £14 10s. 6d. divided by the abstract number 7 gives as a result £2 1s. 6d. Here we have shown that if £14 10s. 6d. be divided into 7 equal parts, the magnitude of each part is £2 1s. 6d.

Again, £14 10s. 6d. divided by £2 1s. 6d. gives 7 as a quotient. This is the same as saying that £14 10s. 6d. contains £2 18. 6d. 7 times.

Hence we see that a compound quantity divided by an abstract.

number gives a compound quantity, and that a compound quantity divided by a compound quantity of the same kind gives an abstract number as a quotient.

Obs. The last remark is the same thing as saying that the ratio (Art. 1, Lesson XXI., Vol. I., page 342) of two concrete quantities of the same kind must be an abstract number. It is of the nature of how many times.

Furthermore, notice that if two concrete numbers are to be compared--that is, if one is to be divided by the other-they must be of the same kind. The ratio of one sum of money to another sum can be found, or that of one weight to another weight; but money cannot be compared with weight or with length. To talk, for instance, of the ratio of 25 shillings to 13 lbs. would simply be an absurdity.

12. EXAMPLE.-Divide £87 10s. 74d. by 47. Beginning with the pounds, we find that £87 divided by 47 gives £1, with a remainder £40. Reducing these £40 to shillings, and adding in the 10 shillings of the dividend, we get 810 shillings, which, divided by 47, gives 17 shillings, with a remainder 11s. Reducing these 11 shillings to pence, and adding in the 7 pence of the dividend, we get 139 per.ce, which, divided by 47, gives 2 pence and a remainder 45 pence. Reducing the 45 pence to farthings, and adding in the 2 farthings of the dividend, we get 182 farthings, which, divided by 47, gives 3 farthings, and a remainder 41, which, divided by 47, gives a fraction of a farthing. The answer, therefore, in £1 17s. 2d. 3f.

The operation may be thus arranged:

47) £87 10 7 (£1

47

And, if necessary, the of a pound could be further reduced to ounces, etc.

15. When it is required to divide one compound quantity by another of the same kind, we must reduce each to the same denomination, and then divide as in ordinary simple division; for, clearly, the number of times which one compound quantity contains another does not depend upon the particular denomination or denominations in which they happen to be expressed.

Supposing one man to have 5 sovereigns in his pocket, and another 1 sovereign, the former would still have 5 times as much as the latter, if they had respectively 100 and 20 shillings instead of the sovereigns.

16. EXAMPLE.-Divide £35 17s. 6d. by £2 11s. 3d.
£35 17s. 6d. = 8610 pence.
£2 11s. 3d. =
615 pence.

615) 8610 (14

14. From the above remarks we see the truth of the following

Rule for Compound Division when the Divisor is an Abstract Number.

Beginning with the highest denomination, divide each separately and in succession. When there is a remainder, reduce it to the next lower denomination, adding the number of that denomination contained in the dividend, and divide the sum as before. Proceed in this manner through all the denominations. Obs. It is sometimes convenient, when the divisor is a composite number, to separate it into factors, and divide succes

Hence 14 is the answer.

We shall, however, return to this part of the subject when we treat of fractions in connection with compound quantities. EXERCISE 46.-EXAMPLES IN COMPOUND DIVISION. Divide

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.

4. 410 lbs. 4 oz. 5 dwts. 6 grs. by 8, 25, 39, 73, and 210.

5. 786 bshs. 18 qts. by 17, 19, 21, 25, 48, and 97.

6. 216 yds. 3 qrs. by 20.

7. 500 yds. 3 qrs. 2 nls. by 54, 63, and 108.

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.

10. 45° 17′ 10′′ by 25, 35, and 45.

11. How much a day is £200 a year?

How many times is

12. 6s. 3 d. contained in £5?

13. £29 7s. 6d. contained in £523 15s. 3fd. ?

14. 2 qrs. 13 lbs. 5 oz. contained in 4 tons 3 cwt. 2 qrs. 6 lbs.? Divide

15. 195 m. 7 fur. 30 ft. by 7 ft. 6 in.

16. 531 m. 2 fur. 10 p. by 17 m. 5 fur. 27 p.

17. 950 days 1 h. 11 min. 6 sec. by 4 days 8 h. 6 min. 54 sec. The Key to Exercises 41, 45 (Vol. II., page 78), will be found at the end of Lesson XXVIII.

LESSONS IN GEOGRAPHY.-XVII. THE GREAT CIRCLES OF THE EARTH-THE MERIDIANTHE EQUATOR.

On the globe of the earth, or terrestrial globe, as it is called, the first great circle of importance is the meridian; this is a great circle which passes through the two poles, P P (Fig. 6), of the axis of the earth, and through any point, as M, on the earth's surface. It is called meridian, because when the sun in our climate shines on a gnomon or style (the pin of a sun-dial), and casts its shadow in the direction of this line on the surface of the earth, it is then (meridies) mid-day or noon; and whenever any heavenly body appears in the plane of this circle, as determined by the position of the style and its shadow at noon, it is said to be on

definition, the circle P M T P N S.

But as every spot on the surface of the globe has its own meridian, if we wish to have a proper notion of the distance of the meridian of any place from that of the place where we dwell, we must fix upon the meridian of some one place as a standard to which we shall refer the distance of every other meridian. Accordingly, the meridian of Greenwich has been fixed upon by common consent in this country as the standard or FIRST MERIDIAN, to which we are to refer all others in point of P distance.

E

M

S

R

Р

Fig. 6.

N

The second great circle of importance on the terrestrial globe is the equator. This is a great circle which passes through all the points Q on the earth's surface situated at an equal distance from the two poles, P P, of the earth's axis; it is called the equator, because when the sun's rays are vertical to this line there is no shadow to the gnomon or style at noon, and there is an equalisation of light all over the globe (on the days when this takes place), this position of the sun being the equaliser (equator). The equator is also made the starting place for the measurement of the distances of places on the surface of the earth as to their position in the northern or southern hemisphere; for the equator divides the globe into two equal parts, called the northern and southern hemispheres or half-globes: that hemisphere in which we live is called the northern hemisphere, because our Saxon ancestors called the point opposite to the sun at noon the north; and that hemisphere in which the point opposite to the sun in the contrary direction is seen, is called the southern hemisphere, because they called the point where the sun is seen at noon the south. The distance of any place on the earth's surface in the northern or southern hemisphere from the equator is usually measured in degrees of the quadrant of the meridian of that place. Thus, the meridian of the place or point м on the surface of the globe (Fig. 6) being the circle P M PS; the distance of M from the equator, E Q, is measured by the number of degrees of the quadrant, E P, contained in the arc E M, the extent of opening of the angle E C M. Now the quadrant E P is divided into 90° from E to P, that is, from the equator to the north pole at P, and the degrees are reckoned from E, which is marked 0° (no degrees), to P, which is marked 90° (ninety degrees). Hence, if I be any point on the earth's surface to which the rays of the sun are vertical on the 21st of June as shown in the diagram of the seasons (Fig. 4, page 80) at R at mid-summer-then the distance of the point M from the equator is 23° 28′ N.; that is, 23 degrees 28 minutes north. The reason of this is plain; for, if from the right angle or 90° formed between the plane of the earth's orbit and the perpendicular to that plane (see Fig. 4, page 80), and also from the right angle or 90° formed between the plane of the earth's equator and the perpendicular to that plane in the axis of the earth, we take away the common angle NO E, the inclination of the earth's axis to the plane of the earth's orbit, which is 66° 32', we shall have 23° 28′ in either case; and this is the distance between the polar circle and the pole, or the inclination of the plane of the earth's equator to the plane of the earth's orbit, and consequently the distance M E (Fig. 6). The distance of any point on the earth's surface, measured in degrees, from the equator is called its latitude (from the Latin latitudo, breadth), because the extreme distance of the earth from north to south was by early geographers reckoned less than its extreme distance from east to west; the term longitude (from the Latin longitudo, length) being applied to distances reckoned east or west from the first meridian. The latitude of the point м on the earth's surface is thus reckoned 23° 28′ N. In like manner the latitude of the point T on the earth's surface, or any point to which the rays of the sun are vertical on the 21st of December, is reckoned 23° 28′ S.; that is, 23 degrees 28 minutes

south.

The two circles of the greatest importance in Geography and

Navigation are the meridian and the equator. We proceed now to show their use.

A

P

Fig. 7.

X

To do this clearly let us suppose that a golden treasure was hid in a field, and that two of its boundaries consisted of one fence lying north and south, or such that at noon its shadow coincided with itself—that is, lay in the same direction; and another fence lying east and west, or such that it intersected or crossed the former fence at right Y angles, as in Fig. 7. In this figure, the straight line AY represents the north and south fence, and the straight line A x the east and west fence; that is, if you go from A to Y you go north, and if you go from Y to A you go south; but if you go from A to X you go east, and if you go from x to A you go west. The directions of the fences being thus understood, suppose that you were told the exact distance of the place where the golden treasure lay from the fence A X, say 20 yards; this would not be enough to enable you to find it, because there are ever so many points in the field, all at 20 yards distance from the fence A X. Now suppose you were also told the exact distance of the place where the golden treasure lay from the fence A Y, say 25 yards; this alone would not be enough to enable you to find it, because there are ever so many points in the field, all at 25 yards distance from the fence A Y. Among these latter points, however, there can be only one which is at the exact distance of 20 yards from the fence A x; so that if you were told both distances at once, you could evidently, by some means or other, determine the place where the golden treasure lay hid. It is necessary, therefore, and sufficient, to inform you of the exact distances of the place in the field from both fences, in order to enable you to find it.

Y

With the information now supposed to be given, the next question is, how should you proceed to determine the exact place of the golden treasure. A little reflection would suggest the following method. In Fig. 8, measure off from the point A, along the fence A X, the given distance of 25 yards, at which the place is said to be situated from the fence A Y; let this distance be a M. Then, from the point м, draw a straight line, M P, parallel to the fence A Y; or, which is easier in this case, draw M P a perpendicular to the fence A X from the point м; for M P and A Y being both at right angles to A X, are parallel to each other. Lastly, measure off from the point м, along the straight line MP, the given distance of 20 yards, at which the place is said to be situated from the fence A X; and the point P will be the place in the field where the golden treasure is to be found.

Y

A

A

That this mode of determining the place of the golden treasure is correct may be proved thus: in Fig. 9 let P be the place in question; from P draw P N perpendicular to A Y, and P M perpendicular to a x; then, according to the data (things given), PM is a distance of 20 yards, and PN is a distance of 25 yards. But by the nature of the construction, the figure A M P N is a rectangular (right-angled) parallelogram, and its opposite sides are therefore equal; whence A M is equal to N P, and A N equal to M P. It follows, therefore, that the point P is found by the method shown in the preceding paragraph. In mathe matical language, the distances P N and P M of the point P from the fences A Y and A X, are called the rectangular co-ordinates of that point; but the distances A M and M P, which are equal to the former, are more usually denominated the rectangular co-ordinates of the P; and by these co-ordinates we can always determine the position of any point, when their exact lengths are given. The straight lines A X and A Y, from which the given distances are measured, are called rectangular axes, and the point ▲, where these axes intersect each other, is called the origin of the rectangular axes. With the origin and the direction of the rectangular axes (in our figures, the fences at right angles), and the lengths of the rectangular co-ordinates, all given, in reference to any point on a given surface, we can always find the true position or place of that point required.

LESSONS IN DRAWING.-XVII. TREATMENT OF TREES AND FOLIAGE (concluded). BEFORE Concluding our observations upon trees and foregrounds, we will offer a few additional remarks upon that which we have so often maintained to be of the utmost importance, and which our pupils will by this time begin to realise. It is because, in this particular instance of trees, there is some difference of treatment in making the outline, to that required by a solid object, the form of which is unmistakable, that we think it unnecessary to offer any excuse for this repetition. The power of drawing is the rock upon which the whole superstructure of art is based; in other words, it is practically the foundation of all that afterwards commands admiration or praise. To whatever point of excellence we may hereafter attain, we shall invariably look back with satisfaction upon the exertions we have used, and the time we have devoted to ensure our success in making a really learned and carefully-constructed outline; it must be the one only starting-point of all who are ambitious to excel, though the subjects they may eventually choose will vary according to their individual tastes, wishes, and circumstances.

In the case of foliage it is necessary to explain what we mean by outline, and how it is to be treated when subject to the various changes caused by sun and shade

under which the tree is

found. Let us suppose ourselves to be standing opposite a tree on a dull, cloudy day. The force of light and depth of shadow will each be less than if the sun were shining upon it, and the half-tints will be more apparent and varied. All round the tree against the grey sky behind there will be the same distinctive and uniform character throughout; but let the sun break out, and then observe what a remarkable change takes place. The general or

larger masses of light and shade will be more decided, the neutralising tones among the half-tints will in a great measure have disappeared; the shadow side of the tree will be distinctly made out against the sky, whilst the details in light will be less definite than they were before the sun shone, owing to the radiation of light from the leaves; the half-tints and small shadows in the light will have less strength than they had before they will be of a warmer tone, and partake of the light and colour around them; the corresponding half-tints on the shadowed side will follow the same course on the same principle —that is, become more general and less distinct in form. We therefore advise the pupil, when "massing in the foliage" of a tree in sunshine, to use his pencil less vigorously on the lights, and not to be betrayed into leaf-drawing and making dark heavy lines. The kind of tree he may be drawing will suggest its own

mode of treatment of the form, as we have already remarked; but now it is the strength and quantity of the work we more especially allude to. If the same tree were drawn on a dull heavy day, there might be much more leaf character introduced both in the lights and in the shades. There is a very common and well-known custom when in difficulties as to the true extent of light and shade; when the pupil is in doubt as to where the light ends and shade begins, let him half close his eyes when looking at the object; the minor tones, or those which seem to belong to neither light nor shade, will apparently disappear, and the true extent and force of both extremes become distinct, and so far evident as to enable him to determine their shape and character. Fig. 110 is the general character of a fir-tree, in which we have

leaves of the acanthus plant. This, whether true or not, is highly suggestive, and tells us there are beautiful combina tions to be found in nature, which the designer would do well to cultivate. To point out a few of them will be sufficient to direct the way in which the lover of nature and art may select examples for himself without fearing to exhaust the supply. The most graceful of all the wild plants are those which cling to others for support. Who has not noticed the wild convolvulus, with its elegant elongated leaves, and its simple symmetrical flowers twined about the stem of a brier or hazel? The hop plant, also, the black bryony, and others may be named whose spiral twistings round stems of various kinds produce natural combinations which no mind could suggest, or power of invention could supply. The leaves alone are models for imitation. The ancient Greeks saw this, and proved it by their frequent appli

cation of the vine-leaf, the oak, and the ivy. In fact, as Mr. Redgrave has said, "He that would be a great designer must be in the hedgerows and fields at all times, sketching with patient diligence the form and curvatures of leaves, fruits, and fowers, their groupings and foreshortenings; studying them as a whole, and in their minutest details, together with their growth and structure. Not to repeat as a mere imitator, but to display them as ornament; to dispose them geometrically, to arrange them to suit the various fabrics or manufactures for which they may be called on to design, and to give them life and words, as it were, by using them as emblems of some living thought or poetical allusion."

It is the application of the graceful forms of the vegetable kingdom that constitutes the most important part of the study of the designer and decorator: the power of drawing, important as it is, is only the means; the adaptation is the end sought for. Here it is, we can say with truth, that it requires the mind of an artist to accomplish it, to be imbued with an originality of thought, that can make the simplest object do duty for worthy

purposes.

It very frequently occurs, in art universally, that by contrast or application we discover excel

lences not

before observed: respecting the use of

this idea,

how many times, may we ask, have we trod on the decaying leaf in our pathway, without having had the atten

tion in the least di

rected to it as capable of suggesting either an original form or a fresh ar

rangement of colour? However insignificant and valueless an object the fallen leaf may seem to be, it is capable of teaching us a lesson of great practical utility. It has been supposed by some that the shape of the vase owes its origin to a leaf; it may be so or not, but it is sufficient for us to know its capability of suggesting it, and it leads us to where the designer may apply if any new form is required. Such resources, when regulated by a disciplined and scientific taste, must produce something as beautiful as it is original. In search for hints for decorative purposes it is not absolutely necessary to confine our choice to the floral varieties of a conservatory or greenhouse, however valuable they may be for the purpose; the green lanes and hedgerows can boast of gems of form amongst nettles and wild flowers, from which articles of ornament and utility may borrow their simple elegance either to decorate a palace or perform some humble service in a cottager's dwelling. Nature everywhere offers hints that are useful as well as beautiful, and the designer need never sigh for a model. As an illustration of the way in which plants may be adapted to ornament and design, we have introduced one for a candlestick in Fig. 111, the socket of which is a lily; the extinguisher inserted in the side is a dead blossom of the same plant, emblematic of its use.

We now take up another portion of our subject relating to landscape-the principles of the reflection of objects in water, as by reflection only can water be represented. It has been frequently said that a landscape is incomplete without water; it is certainly an element which contributes much additional beauty and effect to any scene, be it ever so simple; yet we cannot go so far as to say that it must necessarily be introduced in all cases. Independently of itself, there are associations connected with water that cannot be passed over without notice, and which bear an important part in the whole composition whenever it forms a portion of the picture, such as shipping, barges, boats, fishermen, and picturesque bridges. Why is it that, in our choice of a walk, we generally prefer a stroll near some stream? We attribute it to the variety of scenery afforded by the winding river, and the numberless points of interest that catch the eye as we ramble along its banks. The life and motion connected with water have no limit; and besides, we cannot forget, when it is clear and calm, its capability of reflecting every object near it in full perfection, and increasing our admira

tion by the fidelity with which it reverses form, and reflects colour, light and shade, thus making a double picture. There are several

phenomena resulting from the ap pearances of refl ections upon the surface of water which undoubtedly require more attention than is generally devoted to such subjects by many who aim at representing them. A

course of study is necessary which some would suppose to be beyond the limits pursued by artists generally, but which we contend is indispensable; for every one who undertakes or hopes to paint Nature as she is, must go deeply into her mysteries, and endeavour as far as possible to understand them, and not abide by a mere superficial following of outward appearances. Why is it that the sculptor and the historical painter seek the advantages to be gained in the dissecting-room? Because they feel that a knowledge of anatomy is of the utmost importance to them when engaged upon the human form. Similarly the landscape painter wisely looks about for aid when difficulties arise, which have their remedy often beyond the limits of his own legitimate art; and he will meet with an abundant source of difficulty with regard to reflections. There are incidents so puzzling connected with these, that unless he possesses a little geometrical knowledge, he cannot avoid falling into endless mistakes. We must again have recourse to geometrical perspective, which will not only assist us in our explanations, but will set at rest many doubts which might arise in the minds of our pupils with regard to facts that seem to be impossibilities, unless we employed this conclusive help in rendering them intelligible. Sir Joshua. Reynolds said, "The rules of art are not fetters to genius; they are fetters only to men of no genius."