modified leaves. In certain plants-for example, the cucumberstipules undergo this metamorphosis, in others it is the petioles or the branches themselves which change; such, for example, are the tendrils of the vine (Fig. 56).

But the most curious modification of the leaf is seen in the pitcher-plant, one of which is represented in the diagram (Fig. 57).

Here the petiole gives off a tendril, at the extremity of which the pitcher is situated, the arrangement being such that the pitcher shall always retain its upright position. The pitcher is covered by a well-fitting lid, and "thereby hangs a tale" that shall now be told.

This pitcher-plant manifests a great longing for flies, with which it warms or nourishes itself. But how to catch the flies; that is the question. Had this problem been propounded to one of us, we suppose we should have smeared the plant with some glutinous body, a kind of bird-lime, or fly-lime, as we might call it. Nature manages things in a better way, as we shall see. Flies, as we all know, have a prying habit of crawling into little holes and corners for the purpose of seeing what they can steal. In this way we see them get into daffodils, buttercups, and many other flowers, into some of which, if they cannot go bodily, they thrust their noses. What wonder, then, if a hungry fly should crawl into the pitcher of the nepenthes (for such is the classical name of the plant), which lies so invitingly open, and tempts by its beautiful form? In the fly crawls, and down the cover pops, and the fly is caught. His sufferings are not long. The pitcher is not empty, but contains an acid liquid; so, partly suffocated, partly drowned, the fly comes to an untimely end. But this is not all; the pitcherplant is a good scientific farmer, and knows the way to make a good manure. The great chemist, Liebig, thought he showed the farmers a thing worth knowing when he taught them to soak bones in acid to make them into ma

tell you I happened to meet with what I consider as the greatest prodigy of the vegetable world. I had ventured some way from the party, when one of the Malay servants came running to me with wonder in his eyes, and said, 'Come with me, sir, come! a flower, very large, beautiful, wonderful!' I immediately went with the man about a hundred yards into the jungle, and he pointed to a flower growing close to the ground, under the bushes, which was truly astonishing. My first impulse was to cut it up and carry it to the hut. I therefore seized the Malay's parang (a sort of instrument like a woodman's chopping-hook), and finding that it sprang from a small root which ran horizontally (about as large as two fingers or a little more), I soon detached it and removed it to our hut. To tell you the truth, had I been alone and had there been no witnesses, I should, I think, have been fearful of mentioning the dimensions of this flower, so much does it exceed every flower I have ever seen or heard of; but I had Sir Stamford and Lady Raffles with me, and Mr. Palsgrave, a respectable man resident at Manna, who, though all of them are equally astonished with myself, yet are able to testify to the truth.

"The whole flower was of a very thick substance, the petals and nectary being in but few places less than a quarter of an inch thick, and in some places three-quarters of an inch; the substance of it was very succulent. When I first saw it a

nure. Farmers might have got the same knowledge from our friend the pitcher-plant centuries-ay, thousands of years ago. This liquid which the pitcher-plant contains is acid; thus it rapidly dissolves the fly-skin, bones, and all. His fate is like that of Mokanna, the veiled prophet, who, when conquered, jumped into a tub of aquafortis, and was dissolved.

We must not quit the subject of leaves without devoting a passing word to the gigantic leaf of the Victoria Regia, one of the tribe of Nymphaeacea, or water-lilies, and a native of Central America. A specimen of this truly wonderful plant is now flourishing in great vigour at Kew Gardens. Its leaves are from fifteen to eighteen feet in diameter, and its flowers and capsule, or seed-case, proportionately large. Fig. 58 is an engraving of this wonderful plant. A child is represented standing on one of its floating leaves, which, on account of its size, acts the part of a boat, and supports the child on the surface of the

water.

While we are calling attention to the enormous leaves and beautiful flowers of the Victoria Regia, we may direct the notice of the student to another giant member of the vegetable world, the Rafflesia Arnoldi, a plant which was discovered by a botanist of repute, Dr. Arnold, in 1818, when on an excursion into the interior of Sumatra with Sir Thomas Stamford Raffles and some other friends. The following is Dr. Arnold's account of the discovery of this monster plant and the general appearance of its blossoms. The plant was found on the banks of the Manna river, not far from Pulo Lebbar :

"Here," says Dr. Arnold in a letter to a friend, "I rejoice to

swarm of flies was hovering over the mouth of the nectary, and, apparently, laying their eggs in the substance of it. It had precisely the smoll of tainted beef. The calyx consisted of several roundish, darkbrown, concave leaves, which seemed to be indefinite in number, and were unequal in size. There were five petals attached to the nectary, which were thick, and covered with protuberances of a yellowishwhite, varying in size, the interstices being of a brick-red colour. The nectarium was cyathiform (cup-shaped), becoming narrower towards the top. The centre of the nectarium

gave rise to a large pistil, which I can hardly describe, at the top of which were about twenty processes, somewhat curved, and sharp at the end, resembling a cow's horn; there were as many smaller, very short processes. A little more than halfway down, a brown cord, about the size of common whipcord, but quite smooth, surrounded what perhaps is the germen, and a little below it was another cord, somewhat moniliform (shaped like a necklace).

"Now for the dimensions, which are the most astonishing part of the flower. It measures a full yard across; the petals, which were subrotund, being twelve inches from the base to the apex, and it being about a foot from the insertion of the one petal to the opposite one; Sir Stamford, Lady Raffles, and myself taking immediate measures to be accurate in this respect, by pinning four large sheets of paper together and cutting them to the precise size of the flower. The nectarium, in the opinion of all of us, would hold twelve pints; and the weight of this prodigy we calculated to be fifteen pounds."

This curious plant forms one of a distinct order called Rafflesiacea, which will be noticed in a future lesson. Like our mistletoe it is a parasite, and grows on the prostrate stems and roots of plants, especially on the trailing stems of a Sumatran vine, called Cissus Angustifolia; but unlike the mistletoe, the plant is peculiar in having no leaves, or any organ like the phyllodium, or enlarged petiole of the Australian acacia, that resembles a leaf. Another peculiarity of this remarkable par site is that it always blooms when the plant on which it gr is devoid of leaves.

LESSONS IN ARITHMETIC.-XII.

FRACTIONS (continued).

15. Multiplication of Fractions. To multiply by .

This means to take four-fifths of the fraction; that is, it is the same thing as finding the value of the complex fraction of .

Now, if be divided into five equal parts, i.e., if be divided by 5, we get; because, to divide a fraction by a whole number, we multiply the denominator by that number (Art. 5); and taking four of these fifth parts of -viz., four times we get as the required result.

This result is plainly got by multiplying the numerators together and the denominators together of and, to form a numerator and denominator respectively. The same method would evidently apply to any other two or more fractions. Hence the following

Rule for the Multiplication of Fractions.

Multiply together all the numerators for a numerator, and all the denominators for a denominator.

Obs. In multiplying fractions we can often simplify the operation by striking out or cancelling factors (as we are at liberty to do, Art. 6) which are common to the numerator and denominator of the fraction formed by multiplying the numera tors and denominators together. EXAMPLE.-Multiply together, I, 1, 18. 6 55 Their product is equal to

Dividing by a whole number is finding how many times the divisor is contained in the dividend. Now, a seventh is contained in unity 7 times, and therefore a seventh is contained in 3.x 7 times; 5 sevenths will be contained therefore in onefifth of this number of times, and therefore the quotient of by is, that is, , and the same method will be true for any other two fractions. Hence the following

Rule for the Division of Fractions.

Invert the divisor, and then proceed as in multiplication, i.e., multiply the numerators together for a numerator, and the denominators for a denominator.

Obs. In performing the process, the Obs. of Art. 15, with reference to cancelling factors which are common to both numerator and denominator, must be attended to.

17. By this and the foregoing rules we are able to simplify complex fractions.

718. 817 In a case like this it will be better to simplify each portion separately before performing the operation indicated.

Now

339-226 226 x 339

22. Find the quotients indicated by the following expressions:

DECIMALS.

1. FRACTIONS, the denominators of which are 10 or any power of 10, are called Decimal Fractions, or, more shortly, Decimals. Thus, are Decimal Fractions.

Such fractions are represented by a method of notation which is an extension of that employed for whole numbers.

In whole numbers the figures increase in a tenfold ratio from right to left; or, what is the same thing, decrease in a tenfold ratio from left to right. If we extend this method of representation towards the right beyond the units' place, any figure one place to the right of the units' place will be one-tenth of what it would be if it were in the units' place, and will thus really denote a decimal fraction; any figure two places to the right of the units' place will be one-hundredth of what its value would be if it were in the units' place; and so on for any number of figures and places. Hence, if we choose some means of indicating the point in any row of figures at which the units' place occurs, we can write down any decimal fraction without the trouble of expressing the decimal denominators. This is done by putting a dot, or decimal point, as it is generally called, between the figure in the units' place and the figure in the place to the right of it, which we may call the tenths' place. Thus, 14 would mean 1+; 3 would mean; 3.14159 would mean

3 + 1 + 1 + 1000 + 10000 + 100000

2. We generally speak of any figure in a decimal as being in such a place of decimals. Thus, in the last example we should say that the 5 is in the fourth place of decimals, the 9 in the Sfth place, and so on, reckoning from left to right.

Observe that the denominator of the fraction corresponding to the figure in any decimal place is unity followed by the same number of ciphers as the decimal place; or, what is the same thing, that the power of 10, which is the denominator, is the same as the number of the decimal place.

3. The figures 1, 2, 3, 4, 5, 6, 7, 8, 9 in a decimal are sometimes called significant figures, or digits. Thus in such a decimal as 0002356, we should say that 2 is the first significant digit, because it is the first figure which indicates a number, the ciphers only serving to fix the place in which the 2 occurs. 4. To express a Decimal as a Vulgar Fraction.

Rule for expressing a Decimal as a Vulgar Fraction. Write down the figures which compose the decimal (both integral and decimal part, if there is an integral part) for the numerator, omitting the decimal point; and for the denominator pat 1, followed by as many ciphers as there are decimal places in the given decimal.

5. Conversely, if we have a fraction with any power of 10 for its denominator, we can express it as a decimal by placing a decimal point before as many right-hand figures in the numerator as there are ciphers in the denominator.

Thus

BEFORE proceeding to the subject of the Centre of Gravity, I must direct your attention to two consequences which flow directly from the principles established in the last lesson, and furnish the basis on which the properties of that centre rest. You have seen there that the centre of a system of parallel forces is found by cutting in succession certain lines which join certain points in certain definite proportions, namely, inversely as the forces acting at their extremities. Now, such cutting can give for each line, and therefore for all, as final result, only one point. For example, the centre of two parallel forces of six and four pounds acting at two points, A B, of a body, as in the last lesson, is got by dividing A B into ten parts, and counting off four parts next to A, or six to B, and the result evidently can be only one point. If we now suppose a third parallel force of five pounds added, acting at some other point, c, of the body, and join the point last found with c, and divide the joining line into fifteen parts, taking ten next to c, here again only one point is the result. And so on for any number of forces it can be shown that there is but one centre.

But, lest it should be thought possible that, on cutting these lines in a different order of the points, A B C, etc., a second centre should turn up, let us think that possible, and apply forces at these points parallel to each other, but not parallel to the line joining these two centres. Their resultant then passes through both of these points, and therefore must act in the line joining them, which is impossible; since, as I have proved, it must be parallel to its components.

Furthermore, you will observe that all these lines are cut only in reference to the magnitudes of the forces; no account is taken of their direction. Whether they pull upwards or downwards, or obliquely to left or to right, so long as the magnitudes remain the same, or even keep the same proportion-say that of six, four, and five-the centre cannot change. Of course, the points are supposed not to change. Whatever be the number of points and forces this is true; as for three, so for any other number. And mark, moreover, that it makes no difference how this change of direction is produced, whether, leaving the body in one fixed position, you make the forces change directions as at a and b (Fig. 17), or, preserving the direction, you turn the body round, as from a to c in the same Fig. In neither case does the centre change. These results may be summed up in the two following propositions:

1. A System of Parallel Forces acting at given points in a body, has ONE Centre of Parallel Forces, and only one.

2. The Centre of Parallel Forces does not change its position when the direction of the forces is changed in reference to the body.

THE CENTRE OF GRAVITY.

The centre of gravity is the particular case of the centre we have been last considering, in which the forces are those by which bodies on the earth's surface are drawn by attraction towards its centre. The smallest body, particle, or atom, is drawn in proportion to its mass, equally with the largest; and it is to the tendency of these bodies so to move downwards in obedience to this attraction, that we give the name of "weight." The term "gravity," carries a similar meaning, being derived from the Latin gravis, heavy.

Now, since every particle of matter is thus drawn to the earth's centre, it is evident that the weight of all large masses, such as of a block of marble, beam of timber, or girder of iron, is the joint effect, or the resultant, of the attractions of the separate atoms. But these attractions are all so many parallel forces; for, pulling, as they do, towards the earth's centre, which is nearly 4,000 miles away down in the ground, they incline, even in the largest objects, so little towards one another that practically they may be considered not to meet, that is, to be parallel. Hence you see that all the principles we have proved about parallel forces hold good of the earth's attraction of these atoms, and that we may affirm that

advantage in this simplification; for, instead of having to consider millions of diminutive forces acting at all its points, we direct our attention to only one force, acting at only one point.

You can now understand how it is that a piece of card or thin board may be supported on the point of a rod, wire, or needle. All that is necessary is to bring the point under the centre of gravity of the board; then, the resultant of all the forces by which its several parts are pulled downwards passing through that centre, will be resisted by the rod, and there will be equilibrium; the card will be balanced.

Another consequence follows. Let the body be of any shape, regular or irregular; and suppose that, having determined its centre of gravity, we fix or support that point in some way so that the body may freely turn round it, as on a pivot, in every direction. Then, since, as I have shown, the centre of gravity cannot change as the body turns round, whatever position I place it in, the centre remains supported, and the resultant weight, G P, passing through it, will be resisted by its supports, and the body will be in equilibrium, as in Fig. 18, where G is the supported centre of gravity. Now suppose that instead of this centre we make the body pivot round some other one of its points, o (as in Fig. 19). Then, if I place it so in the position o A B, that the centre of gravity, G, may lie exactly under o, as a plumb-line would hang, the weight acting along the line, o G, may be taken to have o for its point of application, by which, as it is fixed, it will be resisted. In such case there will be equilibrium, & being under

may be at rest, both on the vertical line through o; but one in the lowest position it can attain and the other in the highest. We thus learn that

1. If a body be suspended by or supported at its centre of gravity, it will be at rest, whatever be the position in which it is placed. 2. If the body be suspended by or supported at any other point, it will be at rest when the centre of gravity is in its highest or lowest possible position on the vertical line through the point of suspension or support.

C2

Fig. 20.

If two points A, B (Fig. 20), are fixed, all the points of the line A B are fixed, but the body is free to turn round that line; and if in that case the centre of gravity is somewhere on A B, as G, it also is fixed, and the weight there concentrated will be borne by the two points of support, A B, divided between them in two portions inversely proportional to their distances, A G, B G, from the centre of gravity. The body will, therefore, round the line A B. be in equilibrium in every position into which it can be turned centre is not on the line A B, it is free to move round it. There But if, when two points are fixed, this are, therefore, two positions, G,, G, in a plane vertically passing rest, and the result is similar to that stated in the above prothrough this line-one below, the other above, in which it may positions. Familiar examples of this are furnished by all pieces of machinery in which bodies move round fixed axles, such as the fly-wheel of a steam-engine, or the smaller wheels round which the bands pass, which set the printing presses at work in the machine-room-all the points along the line which runs down the centre of the axle are at rest. A trap-door, which opens both downwards and upwards, is another instance; in that case the centre of gravity is under or above the axle-line of the hinges when the door hangs in equilibrium.

of fixing points within their substance. A horse poised in the But bodies may be kept in equilibrium in other ways than that air, as it is about to be lifted into a transport ship, by a rore which descends from the top of a crane and is attached to a belt which goes round his body, is an instance. Here the centre of gravity of the lifted animal is under the point of support and on the line of direction of the rope which transmits its only one position of equilibrium-namely, the lowest. The rope weight to the crane above. But observe, in this case, there is not being rigid, you cannot wheel the horse half round, heels up in the air (Fig. 21) until he reaches the highest position the chain would allow him to reach, and make his weight thence take the place of the rope. press downwards on the crane. To do this a rigid bar should

But bodies are most commonly kept at rest by being supported from below by the earth, either on the ground itself, or on some floor, table, etc. What conditions will secure a steady equilibrium? First, there must be some base or bottom to the body on which it may rest, such as the bottom of a teapot or candlestick. Secondly, it must be broad enough to keep the body steady, to prevent its upsetting or rocking. A candlestick resting on the socket into which the candle is put, would soon overturn, and the slightest touch would set an egg rocking.

Fig. 21.

Now, in order to ascertain the equilibrium and stability of bodies so placed, let us suppose two of the forms in Fig. 22 to rest on a level table, touching it on the two perfectly flat bases X Y Z, X, Y, Z, there represented. Let G be the centre of gravity of that to the right, and GP the perpendicular to the table through that point. Let, moreover, G, and G, P, be the

corresponding centre and perpendicular of the body to the left. Now, since the table, by its resistance distributed equally over the base X Y Z of the first body, prevents its moving downwards, and this resistance at every point is perpendicular to the floor, these resistances, taken together, are a system of parallel forces, and have a parallel centre somewhere in that base. Let this centre be o. Join now OP; and, as the same reasoning holds good of the body to the left, let O, P, be the corresponding line in it. Moreover, let X, X,, be the points in which the lines o P, O, P, cut the circumference, or boundary of the bases X Y Z, X, Y, Z,. The body to the right is thus acted on by two forces; the resistance at o upwards supporting it, and the weight at G pulling downwards. But, as the point P falls, in this case, outside the base x Y Z, there is nothing to prevent the body obeying it by turning over on its edge at x.

Fig. 22.

But, in the other case, where P, is within the base, the weight at G, tends to make the body fall inwards, turning on its edgo at x,. But then, there is the resistance of the table at o,, acting upwards to prevent that motion; and consequently the body remains at rest, or is in equilibrium.

And this statement holds equally good when the plane on which the body rests is sloped or inclined to the horizontal plane; as is evident from Fig. 23, where the cylindrical body on the slope A B must upset if G P falls outside the base X Y Z. We may, therefore, conclude generally both as to horizontal and inclined planes that a body will rest in equilibrium on a plane, if the vertical line, passing through its centre of gravity, meets the plane within the base. If it meets it outside the base, the body will overturn.

Fig. 23.

B

Between these two, it should be observed that there is an intermediate case, in which the perpendicular meets the plane neither within nor without the base, but on its circumference. When this happens, the body is equally disposed to stand or upset; but, in fact, it will overturn; for in such an unsteady position the slightest touch or shake would send it over. It is a case of unstable equilibrium. In interpreting and applying this principle to practice, you must be on your guard as to the meaning of the word "base;" else you may imagine some day you have discovered that a body does not upset when the vertical from the centre of gravity falls outside the base. Suppose the base to be bent inwards into a horse-shoe form, as in the cone, a (Fig. 24), or into the form of the semi-circular wall, b, in which latter case the centre of gravity is with out the substance of the body; then the point P is on the floor, outside the spaces along which the bodies are in contact with it. Still, neither body will upset; for the advanced spurs of the bases at Y and z will act as props, and in order to upset they must turn over the line YZ joining them. This shows that the real base includes all the open space within Y z; and you learn that, whenever the base of contact bends inwards, you must measure the base of support from one projecting point to another all round, making no account whatever of the inward bends. A common table touches the floor only at four points, and a round table at three; but in both the base of support is

Fig. 24.

all the space within the oblong or triangle got by joining these points. There is another class of cases to be noticed, those which are round all over their surface like a ball, or egg, or sea-shore pebble, and have no flat bases to rest on-that is, which can be supported at only one point of their surface; or, where there are hollows on them, along a line of points surrounding the hollow. This latter case we b need not consider, for such bodies rest, like those we have already examined, so far as the hollows are concerned (as in d, Fig. 25), on wide bases.

Confining attention, therefore, to cases in which there are no hollows, or the surface

a

Fig. 25.

d

is convex all round, if you place such a body, say an oval, in the position represented at a (Fig. 25), the perpendicular, GP, from its centre of gravity, G, on the plane will fall outside its base, or point of support, o, and it will roll over until, after rocking for a few turns, it settles into the position b, in which a is above 0. Now move it further from this until

it reaches the position c, in which again G will be over the point of support, o; and again you will have a possible equilibrium, that is, possible in imagination, for the body is supported from below. But actually to produce equilibrium in this case is the celebrated problem of Columbus, which that great navigator solved after so summary a fashion. So unsteady is it, that the body drops immediately into the position b.

Of this unsteady, or unstable equilibrium, we shall have more in the next lesson; my object here is to point out the fact that in both positions, b and c, the line Go is perpendicular to the surface of the body. It is evidently perpendicular to the plane on which the oval rests; but, since the latter's surface touches, or coincides at o, with that plane, G o must be perpendicular also to that surface. Hence we learn that, whatever be the number of points at which a convex body can rest, steady or unsteady, on a horizontal plane, for every one of these points the lines connecting them with the centre of gravity must pierce its surface at right angles; or

The number of positions of equilibrium of a conver body, supported on a horizontal plane, is equal to that of the perpendiculars to its surface which can be drawn from its centre of gravity.

A few instances in illustration of the principles explained in this lesson will now be useful. When a man stands upright, the base by which he is supported is so much ground under him as is covered by his feet, together with the space between them. If he widens that space to left and right, he makes himself more steady as to being thrown sideways, but is more easily cast on his face. If he puts one foot before the other, he becomes steadier at front and back, but less so to his sides. A twowheeled gig, or Hansom, to be properly balanced, should have its centre of gravity over the line joining the points at which the wheels touch the ground. If it be in advance of that line, it will throw a weight on the horse's back; if behind it, the gig will upset backwards should the belly-band break.

a

A body may be made to roll up an incline by loading it at one side. Take a round ball of cork, for instance, and put some lead into a hollow scooped out near its surface, closing the hole so as to leave the ball perfectly round. The centre of gravity will then no longer be at the centre of the ball, but to one side, let it be at G (in a, Fig. 26). Put the ball now on the incline, with the leaded side looking up the slope; the perpendicular G P will meet the incline above o, and the ball will roll upwards until G comes over the point of support.

Fig. 26.

This experiment may be tried in another form without the use of the lead, by simply scooping a hollow on one side, or as in the following example:-Get a round cylinder of cork-a