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modified leaves. In certain plants-for example, the cucumber— tell you I happened to meet with what I consider as the greatest stipules undergo this metamorphosis, in others it is the petioles prodigy of the vegetable world. I had ventured some way from or the branches themselves which change; such, for example, the party, when one of the Malay servants came running to me are the tendrils of the vine (Fig. 56).

with wonder in his eyes, and said, 'Come with me, sir, come! a But the most curious modification of the leaf is seen in the flower, very large, beautiful, wonderful !' I immediately went pitcher-plant, one of which is represented in the diagram with the man about a hundred yards into the jungle, and he (Fig. 57).

pointed to a flower growing close to the ground, under the Here the petiole gives off a tendril, at the extremity of which bushes, which was truly astonishing. My first impulse was to the pitcher is situated, the arrangement being such that the cut it up and carry it to the hut. I therefore seized the Malay's pitcher shall always retain its upright position. The pitcher parang (a sort of instrument like a woodman's chopping-hook), is covered by a well-fitting lid, and “thereby hangs a tale” that and finding that it sprang from a small root which ran horizon. shall now be told.

tally (about as large as two fingers or a little more), I soon This pitcher-plant manifests a great longing for flies, with detached it and removed it to our hut. To tell you the truth, which it warms or nourishes itself. But how to catch the flies; Lad I been alone and had there been no witnesses, I should, I that is the question. Had this problem been propounded to one think, have been fearful of mentioning the dimensions of this of us, we suppose we should have smeared the plant with some flower, so much does it exceed every Aower I have ever seen or glutinous body, a kind of bird-lime, or fly-lime, as we might call heard of; but I had Sir Stamford and Lady Raffles with me, and it. Nature manages things in a better way, as we shall see. Mr. Palsgrave, & respectable man resident at Manna, who,

Flies, as we all know, have a prying habit of crawling into though all of them are equally astonished with myself, yet are little holes and corners for the purpose of seeing what they can able to testify to the truth. staal. In this way we see them get into daffodils, buttercups, “The whole flower was of a very thick substance, the petals and many other flowers, into some of which, if they cannot go and nectary being in but few places less than a quarter of an bodily, they thrust their noses. What wonder, then, if a hungry inch thick, and in some places three-quarters of an inch; the fly should crawl into the pitcher of the nepenthes (for such is substance of it was very succulent. When I first saw it a the classical name of

swarm of flies was hoverthe plant), which lies

ing over the mouth of so invitingly open, and

the nectary, and, appatempts by its beautiful

rently, laying their eggs form? In the fly

in the substance of it. crawls, and down the

It had precisely the cover pops, and the ily

smoll of tainted beef. is canght. His suffer.

The calyx consisted of ings are not long. The

several roundish, darkpitcher is not empty,

brown, concave leaves, but contains an acid

which seemed to be inliquid; so, partly suffo.

definite in number, and cated, partly drowned,

were unequal in size. the fly comes to an un.

There were five petals timely end. But this

attached to the nectary, is not all; the pitcher

which were thick, and plant is a good scien

covered with protuber. tific farmer, and knows

ances of a yellowishthe way to make a good

white, varying in size, manure. The great che

the interstices being of mist, Liebig, thought

a brick-red colour. The he showed the farmers

nectarium was cyathi. a thing worth knowing

form (cup-shaped), bewhen he taught them 58. THE VICTORIA REGIA WATER-LILY, IN THE CONSERVATORY AT CHATSWORTH,

coming narrower toto soak bones in acid

wards the top. The to make them into ma

centre of the nectarium more. Farmers might have got the same knowledge from our gave rise to a large pistil, which I can hardly describe, at the friend the pitcher-plant centuries—ay, thousands of years ago. top of which were about twenty processes, somewhat curved, This liquid which the pitcher-plant contains is acid ; thus it and sharp at the end, resembling a cow's horn; there were as rapidly dissolves the fly-skin, bones, and all. His fate is like many smaller, very short processes. A little more than halfthat of Mokanna, the veiled prophet, who, when conquered, way down, a brown cord, about the size of common whipcord, jumped into a tub of aquafortis, and was dissolved.

but quite smooth, surrounded what perhaps is the germen, and We must not quit the subject of leaves without devoting a a little below it was another cord, somewhat moniliform (shaped passing word to the gigantic leaf of the Victoria Regia, one of like a necklace). the tribe of Nymphæacere, or water-lilies, and a native of Central | “Now for the dimensions, which are the most astonishing America. A specimen of this truly wonderful plant is now part of the flower. It measures a full yard across; the petals, flourishing in great vigour at Kew Gardens. Its leaves are from which were subrotund, being twelve inches from the base to the fifteen to eighteen feet in diameter, and its flowers and capsule, apex, and it being about a foot from the insertion of the one or seed-case, proportionately large. Fig. 58 is an engraving of petal to the opposite one; Sir Stamford, Lady Raffles, and mythis wonderful plant. A child is represented standing on one self taking immediate measures to be accurate in this respect, of its floating leaves, which, on account of its size, acts the by pinning four large sheets of paper together and cutting them part of a boat, and supports the child on the surface of the to the precise size of the flower. The nectarium, in the opinion water.

of all of us, would hold twelve pints; and the weight of this While we are calling attention to the enormous leaves and prodigy we calculated to be fifteen pounds.” beantiful flowers of the Victoria Regia, we may direct the notice This curious plant forms one of a distinct order called Raffleof the student to another giant member of the vegetable world, siaceæ, which will be noticed in a future lesson. Like our the Rafflesia Arnoldi, a plant which was discovered by a botanist mistletoe it is a parasite, and grows on the prostrate stems and of repute, Dr. Arnold, in 1818, when on an excursion into the roots of plants, especially on the trailing stems of a Sumatran interior of Sumatra with Sir Thomas Stamford Rafiles and some vine, called Cissus Angustifolia; but unlike the mistletoe, the other friends. The following is Dr. Arnold's account of the dis- plant is peculiar in having no leaves, or any organ like the covery of this monster plant and the general appearance of its phyllodium, or enlarged petiole of the Australian acacia, that blossoms. The plant was found on the banks of the Manna resembles a leaf. Another peculiarity of this remarkable par river, not far from Pulo Lebbar :

site is that it always blooms when the plant on which it go “Here,” says Dr. Arnold in a letter to a friend, "I rejoice to is devoid of leaves.

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| And therefore the required result will be given by the following:FRACTIONS (continued). 15. Multiplication of Fractions.

To multiply by .
This means to take four-fifths of the fraction; that is, it is

EXERCISE 28. the same thing as finding the value of the complex fraction;

EXAMPLES IN MULTIPLICATION AND DIVISION OF of . Now, if be divided into five equal parts, i.e., if be divided

by 5, we get i ; because, to divide a fraction by a whole num 1. Find the following products :-
ber, we multiply the denominator by that number (Art. 5); and
taking four of these fifth parts of 3-yiz., four times we get

1. 3. * 15.
6. U x

11. * * * * *
2. 1. x &
7. 31 x 48.

12. 34} x of 68. as the required results

3. 81 x 27.
8. H x 1.

13. 3 of 65] x 46) This result is plainly got by multiplying the numerators

4. 123 x 8.

9.81 x 38 x . 14. of 16} of 9j. together and the denominators together of g and , to form a

5. 250,45 x 50. 10. ** ********.
numerator and denominator respectively. The same method
would evidently apply to any other two or more fractions. Hence 2. Perform the following divisions :-
the following

1. !: + 9.
4.1 .

7. 87 • 51. Rule for the Multiplication of Fractions.

2. 31 7.
5. = 1

8. 55 – 161 Multiply together all the numerators for a numerator, and all 3. 389 + 29.

6. H. 1 9. 463 – 683. the denominators for a denominator. Obs.-In multiplying fractions we can often simplify the

3. Divide of į by 24. operation by striking out or cancelling factors (as we are at 4. Divide } + 1} + { by+ 3 liberty to do, Art. 6) which are common to the numerator and denominator of the fraction formed by multiplying the numera

5. Divide ? + 2} + $ + Á by } + tors and denominators together. EXAMPLE.-Multiply together j, k, i, . Their product is

6. Reduce to their simplest forms 2) + and equal to 2 x 5 6 X 55 2 x 5 x 3 x 2 x H x 5

7. Divide 2} + 11 + 3! by 1 of of 3 x 8 x 11 x 108 = 7x2x2x2 *# * 108.

8. Find the difference of 2$ x and

9. Simplify 69 x 14 of (11- )). And 2 occurs twice in both numerator and denominator,

10. What is that number of which is 27? 3 , once »

11. If a certain number be multiplied by 23 the result is 52. want _ 5 x 5 - 25

What is the number? Therefore the product = 2 x 108

12. By what must 29: be multiplied to obtain 671? 16. Division of Fractions.

13. Express the difference of the first two of the three folTo divide by .

lowing fractions as a fraction of the last two: I, 199,113 Dividing by a whole number is finding how many times the

times the ! 14. Perform the same operation on divisor is contained in the dividend. Now, a seventh is con 15. Of what quantity is

je š of seven-tenths ?

quantity 15 + 1 of 1 seven tained in unity 7 times, and therefore a seventh is contained in 3, 3 x 7 times; 5 sevenths will be contained therefore in s one

16. Find the products of the following fractions :fifth of this number of times, and therefore the quotient of a by is x 1, that is, #, and the same method will be true for

2. 38 x

5. * * * * * * * * any other two fractions. Hence the following

3. ' x 18 *

6. } of off of of Rule for the Division of Fractions. Invert the divisor, and then proceed as in multiplication, i.e.,

17. Find the products indicated by the following expressions:multiply the numerators together for a numerator, and the 1. 79

3. 1428 x tit. 1 5. 11 x 476. denominators for a denominator..

2. 86 x 16
4. t x 112.

6. x 2 Obs.-In performing the process, the Obs. of Art. 15, with reference to cancelling factors which are common to both nume 18. Find the products indicated in the following expressions:rator and denominator, must be attended to.

1. 4} 4 i 3. 49168,. I 5. 10003 * 111. 17. By this and the foregoing rules we are able to simplify 2.7 x 81 T 4. 2254 x 32. 1 6. 476859 3763. complex fractions. EXAMPLE. To sgo - 33

19. Divide the following fractions by each other, according to add and multiply the result the indicated expressions :-

1. + 1

3. # = $1. 2. 11 - 18. I 4 il

6. + gdje In a case like this it will be better to simplify each portion separately before performing the operation indicated. Now, 20. Find the quotients indicated by the following expressions:

339 - 226 113


+ 37. 1

3. M

5. 11

+ 11.
339 = 226 x 339 = 226 x 339 = 2 x 339

2. – 12.
4 11 < 52.

6. – 400.
21. Find the quotients indicated by the following expressions:-

| 5, 1; rồi: (120 being the L.C.M. of 24 and 40).

2. 172 =). 1_ 1 i

22. Find the quotients indicated by the following expressions: 1. 112 - 71. I 3. 1000 = 11.

5.1 + 13. 2. 160 = 9 4. 800 + 8001.

6. 2 – 2000!


1. 121.

I 2.500

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23. Find the quotients indicated by the following expressions:1. 173 - 7.

3. 14008) - 9. 1 5. 1000$ + 18. 2. 1001 - 12. 4. 478396577 + 112. 6.1; + 800.

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24. Find the quotients indicated by the following expressions:1. 7) + 51. I 3. 407? + 557.

5. 10001 - 105. 2. 981'= 17). I 4. 14283 4 8


ciphers between the decimal point and the first significant 1. FRACTIONS, the denominators of which are 10 or any power | figure does alter the value of the decimal, because this alters of 10. are called Decimal Fractions, or, more shortly, Decimals. | the places of the significant digits. Thus -23, 023, .0023 have Thus , 437, are Decimal Fractions.

| all different values, being respectively equal to , 2 TORO 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

MECHANICS.-V. right to left; or, what is the same thing, decrease in a tenfold

PARALLEL FORCES.–CENTRE OF GRAVITY. ratio from left to right. If we extend this method of represen

BEFORE proceeding to the subject of the Centre of Gravity, I tation towards the right beyond the units' place, any figure one

must direct your attention to two consequences which flow place to the right of the units place will be one-tenth of what

directly from the principles established in the last lesson, and it would be if it were in the units' place, and will thus really | furnish the basis on which the properties of that centre rest. denote a decimal fraction; any figure two places to the right of You have seen there that the centre of a system of parallel the anita' place will be one-hundredth of what its value would

forces is found by cutting in succession certain lines which join

forces is found be ontting in succession be if it were in the units' place; and so on for any number of

certain points in certain definite proportions, namely, inversely figures and places.

as the forces acting at their extremities. Now, such cutting can Hence, if we choose some means of indicating the point in

give for each line, and therefore for all, as final result, only one any row of figures at which the units' place occurs, we can

| point. For example, the centre of two parallel forces of six and write down any decimal fraction without the trouble of express- four pounds acting at two points, A B, of a body, as in the last ing the decimal denominators. This is done by putting a dot, lesson, is got by dividing A B into ten parts, and counting off or decimal point, as it is generally called, between the figure in four parts next to A, or six to B, and the result evidently can be the units place and the figure in the place to the right of it, only one point. If we now suppose a third parallel force of which we may call the tenths' place. Thus, 1:4 would mean

five pounds added, acting at some other point, c, of the body, 1 + $; 3 would mean %; 3:14159 would mean

and join the point last found with c, and divide the joining line 3+ to + Tô + Tols + Todos + Todo.

into fifteen parts, taking ten next to c, here again only one 2. We generally speak of any figure in a decimal as being in point is the result. And so on for any number of forces it can such a place of decimals. Thus, in the last example we should be shown that there is but one centre. say that the 5 is in the fourth place of decimals, the 9 in the But, lest it should be thought possible that, on cutting these fifth place, and so on, reckoning from left to right.

| lines in a different order of the points, A B C, etc., a second Observe that the denominator of the fraction corresponding. centre should turn up, let us think that possible, and apply to the figure in any decimal place is unity followed by the same forces at these points parallel to each other, but not parallel to sumber of ciphers as the decimal place; or, what is the same the line joining these two centres. Their resultant then passes thing, that the power of 10, which is the denominator, is the through both of these points, and therefore must act in the line same as the number of the decimal place.

joining them, which is impossible; since, as I have proved, it 3. The figures 1, 2, 3, 4, 5, 6, 7, 8, 9 in a decimal are some must be parallel to its components. times called significant figures, or digits. Thus in such a decimal | Furthermore, you will observe that all these lines are cut as '0002356, we should say that 2 is the first significant digit, only in reference to the magnitudes of the forces; no account is because it is the first figure which indicates a number, the taken of their direction. Whether they pull upwards or downciphers only serving to fix the place in which the 2 occurs. wards, or obliquely to left or to right, so long as the magnitudes 4. To erpress a Decimal as a Vulgar Fraction.

remain the same, or even keep the same proportion-say that of *347 = 1 + Too + Toon

six, four, and five—the centre cannot change. Of course, the Or (reducing the fractions to a common denominator, 1000)

points are supposed not to change. Whatever be the number

of points and forces this is true; as for three, so for any other - 300 + 40 +7 1000 =

number. And mark, moreover, that it makes no difference how • 0237 = + Too + Tivo + Tobos

this change of direction is produced, whether, leaving the body

in one fixed position, you make the forces change directions as at Or (reducing the fractions to a common denominator, 10000)

a and b (Fig. 17), or, preserving the direction, you turn the body 0 + 200 + 30 + 7

round, as from a to c in the same Fig. In neither case does the 10000 100m

centre change. These results may be summed up in the two Again 43.25037 = 43 + Lo + b + Theo + Todos + Tomo following propositions :Or (reducing the fractions to a common denominator, 100000) 1. A System of Parallel Forces acting at given points in a 4300000 + 20000 + 5000 + 0 x 100 + 30 + 7

body, has ONE Centre of Parallel Forces, and only one. 100000

2. The Centre of Parallel Forces does not change its position Hence we see the truth of the following

when the direction of the forces is changed in reference to the

body. Rule for expressing a Decimal as a Vulgar Fraction.

THE CENTRE OF GRAVITY. Write down the figures which compose the decimal (both integral and decimal part, if there is an integral part) for the,

The centre of gravity is the particular case of the centre we numerator, omitting the decimal point; and for the denominator

have been last considering, in which the forces are those by pat 1, followed by as many ciphers as there are decimal places

which bodies on the earth's surface are drawn by attraction in the given decimal.

towards its centre. The smallest body, particle, or atom, is 5. Conversely, if we have a fraction with any power of 10 for

drawn in proportion to its mass, equally with the largest; and ita denominator, we can express it as a decimal by placing a

it is to the tendency of these bodies so to move downwards in decimal point before as many right-hand figures in the nu.

obedience to this attraction, that we give the name of "weight.” merator as there are ciphers in the denominator. Thus

The term “gravity,” carries a similar meaning, being derived 13888 = 5:3459.

from the Latin gravis, heavy.

Now, since every particle of matter is thus drawn to the If the figures in the numerator be fewer than the ciphers in the

: be fewer than the ciphers in the earth's centre, it is evident that the weight of all large masses, denominator, we must place before the left-hand figure of the such as of a block of marble, beam of timber, or girder of iron, numerator ciphers equal in number to the excess of the number

the excess of the number is the joint effect, or the resultant, of the attractions of the sepaof ciphers in the denominator over the number of figures in the

rate atoms. But these attractions are all so many parallel Dumerator, and then prefix the decimal point. For example

forces ; for, pulling, as they do, towards the earth's centre, Tovoro = *00235.

which is nearly 4,000 miles away down in the ground, they in068.-It will be perceived from the foregoing remarks that cline, even in the largest objects, so little towards one another placing ciphers on the right of a decimal does not alter its value, that practically they may be considered not to meet, that is, to for this does not alter the place of any of the significant figures. be parallel. Hence you see that all the principles we have Thus, -23, 230, 2300 are all equal in value, for, expressed as proved about parallel forces hold good of the earth's attraction fractions, they are respectively

2006 But prefixing of these atoms, and that we may affirm that

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n one Centre of Gravity, and only one.

may be at rest, both on the vertical line through o; but one in Wirkutru of Gravity is not changed by the body being the lowest position it can attain and the other in the highest. nad after any manner in any direction.

We thus learn that It haus appears that the weights of all the separate atoms of 1. If a body be suspended by or supported at its centre of 1.1) 910 of matter are equal to a single weight supposed to act gravity, it will be at rest, whatever be the position in which it is hot mome point within that mass, or, as sometimes happens (and placed. ww shall sec), even without, equal to their sum. There is great. 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.

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

Fig. 20.
weight there concentrated will be
Fig. 17.

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,

be in equilibrium in every position into which it can be turned advantage in this simplification ; for, instead of having to con.

round the line AB. But if, when two points are fixed, this sider millions of diminutive forces acting at all its points,

centre is not on the line A B, it is free to move round it. There we direct our attention to only one force, acting at only one

are, therefore, two positions, G,, G,, in a plane vertically passing point.

through this line one below, the other above, in which it may You can now understand how it is that a piece of card or thin

rest, and the result is similar to that stated in the above prcboard may be supported on the point of a rod, wire, or needle.

positions. Familiar examples of this are furnished by all pieces All that is necessary is to bring the point under the centre of

of machinery in which bodies move round fixed axles, such as gravity of the board; then, the resultant of all the forces by

the fly-wheel of a steam-engine, or the smaller wheels round which its several parts are pulled downwards passing through

which the bands pass, which set the printing presses at work in that centre, will be resisted by the rod, and there will be equili.

the machine-room-all the points along the line which runs down brium; the card will be balanced.

the centre of the axle are at rest. A trap-door, which opens both Another consequence follows. Let the body be of any shape,

pe: downwards and upwards, is another instance ; in that case the regular or irregular; and suppose that, having determined its

centre of gravity is under or above the axle-line of the hinges centre of gravity, we fix or support that point in some way so

when the door hangs in equilibrium. that the body may freely turn round it, as on a pivot, in every

But bodies may be kept in equilibrium in other ways than that direction. Then, since, as I have shown, the centre of gravity

vity of fixing points within their substance. A horse poised in the cannot change as the body turns round, whatever position I

air, as it is about to be lifted into a transport ship, by a rope place it in, the centre remains supported, and the resultant

which descends from the top of a crane and is attached to a belt weight, G P, passing through it, will be resisted by its supports,

which goes round his body, is an instance. Here the centre and the body will be in equilibrium, as in Fig. 18, where G is the

of gravity of the lifted animal is under the point of support supported centre of gravity. Now suppose that instead of this centre we make the body

and on the line of direction of the rope which transmits its

weight to the crane above. pivot round some other one of its points, o (as in Fig. 19). Then,

But observe, in this case, there is

only one position of equilibrium-namely, the lowest. The rope if I place it so in the position 0 A B, that the centre of gra

not being rigid, you cannot wheel the horse half round, heels op vity, G, may lie exactly under o, as a plumb-line would hang,

in the air (Fig. 21) until he reaches the highest position the the weight acting along the line, o G, may be taken to have o

chain would allow him to reach, and make his weight thence for its point of application, by which, as it is fixed, it will be

press downwards on the crane. To do this a rigid bar should resisted. In such case there will be equilibrium, G being under

take the place of the rope.

But bodies are most commonly kept at rest by being sup-
ported 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 candle-
stick. Secondly, it must
be broad enough to keep
the body steady, to pre-
vent its upsetting or
rocking. A candlestick
resting on the socket
into which the candle is

Fig. 21.
Fig. 18.

Fig. 19.

put, would soon overo. And so, also, if G were exactly above o, as in o cd, in the turn, and the slightest touch would set an egg rocking. vertical line produced upwards, the weight would press down. Now, in order to ascertain the equilibrium and stability of wards on o, and be there resisted. But if I put it in any other | bodies so placed, let us suppose two of the forms in Fig. 22

EF, where G will not be either above or below o, to rest on a level table, touching it on the two perfectly flat 'ng downwards, in the direction G P, will not be bases X Y Z, X, Y, 2, there represented. Let G be the centre of "ne o G of resistance of o, and there cannot be gravity of that to the right, and GP the perpendicular to the iere are thus two positions in which the body table through that point. Let, moreover, G, and G, P, be the

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corresponding centre and perpendicular of the body to the left. all the space within the oblong or triangle got by joining these
Now, since the table, by its resistance distributed equally over points.
the base x y z of the first body, prevents its moving down. There is another class of cases to be noticed, those which are
Fards, and this resistance at every point is perpendicular to round all over their surface like a ball, or egg, or sea-shore
the floor, these resistances, taken together, are a system of pebble, and have no flat bases to rest on—that is, which can be
Parallel forces, and have a parallel centre somewhere in that supported at only one point of their surface; or, whero there are
base. Let this centre be o. Join now op; and, as the same hollows on them, along a line of points surrounding the hollow.

reasoning holds good of This latter case we
the body to the left, let need not consider, for
0, P, be the corresponding such bodies rest, like
line in it. Moreover, let | those we have already
X, X, be the points in examined, so far as the
which the lines o P, O, P, hollows are concerned
cut the circumference, or (as in d, Fig. 25), on
boundary of the bases wide bases.
X Y Z, x, y, 2. The body Confining attention,

to the right is thus acted therefore, to cases in
Fig. 22.
on by two forces; the re- which there are no

Fig. 25. sistance at upwards hollows, or the surface supporting it, and the weight at g pulling downwards. But, is convex all round, if you place such a body, say an oval, as the point p falls, in this case, outside the base x y z, there in the position represented at a (Fig. 25), the perpendicular, is nothing to prevent the body obeying it by turning over on GP, from its centre of gravity, G, on the plane will fall outits edge at x

side its base, or point of support, 0, and it will roll over Bat, in the other case, where p, is within the base, the weight until, after rocking for a few turns, it settles into the position at a, tends to make the body fall inwards, turning on its edgob, in which G is above o. Now move it further from this until at x,. But then, there is the resistance of the table at 0,, acting it reaches the position c, in which again a will be over the opwards to prevent that motion; and consequently the body point of support, o; and again you will have a possible equi. remains at rest, or is in equilibrium.

librium, that is, possible in imagination, for the body is supported And this statement holds equally good when the plane on from below. But actually to produce equilibrium in this case is which the body rests is sloped or inclined to the horizontal the celebrated problem of Columbus, which that great navigator plane; as is evident from Fig. 23, where the cylindrical body solved after so summary a fashion. So unsteady is it, that the on the slope A B must upset if G P falls outside the base X Y Z. | body drops immediately into the position b. We may, therefore, conclude generally both as to horizontal Of this unsteady, or unstable equilibrium, we shall have more and inclined planes that a body will rest in equilibrium on in the next lesson; my object here is to point out the fact that plane, if the vertical line, passing

in both positions, 6 and c, the line G o is perpendicular to the through its centre of gravity, meets

surface of the body. It is evidently perpendicular to the plane the plane within the base. If it

on which the oval rests; but, since the latter's surface touches, meets it outside the base, the body

or coincides at o, with that plane, G O must be perpendicular will overturn.

also to that surface. Hence we learn that, whatever be the Between these two, it should be

number of points at which a convex body can rost, steady or observed that there is an interme

unsteady, on a horizontal plane, for every one of theso points diate case, in which the perpendicu.

the lines connecting them with the centre of gravity must pierce lar meets the plane neither within

its surface at right angles; ornor without the base, but on its

The number of positions of equilibrium of a convex body, circumference. When this happens,

supported on a horizontal plane, is equal to that of the perpenthe body is equally disposed to A

diculars to its surface which can be drawn from its centre of stand or upset; but, in fact, it will

gravity. overturn; for in such an unsteady

Fig. 23.

A few instances in illustration of the principles explained in position the slightest touch or shake

this lesson will now be useful. When a man stands upright, the would send it over. It is a case of unstable equilibrium.

base by which he is supported is so much ground under him as In interpreting and applying this principle to practice, you is covered by his feet, together with tho space between them. must be on your guard as to the meaning of the word “base;" If he widens that space to left and right, he makes himself more else you may imagine some day you have discovered that a body steady as to being thrown sideways, but is more easily cast on does not upset when the vertical from the centre of gravity his face. If he puts one foot before the other, he becomes falls outside the base. Suppose the base to be bent inwards steadier at front and back, but less so to his sides. A twointo a horse-shoe form, as in the cone, a (Fig. 24), or into the wheeled gig, or Hansom, to be properly balanced, should have form of the semi-circular wall, b, in which latter case the centre its centre of gravity over the lino joining the points at which

of gravity is with the wheels touch the ground. If it be in advance of that line,
out the substance it will throw a weight on the horse's back; if behind it, the gig
of the body; then will upset backwards should the
the point P is on belly-band break.
the floor, outside A body may be made to roll up
the spaces along an incline by loading it at one
which the bodies side. Take a round ball of cork,
are in contact with for instance, and put some lead
it. Still, neither into a hollow scooped out near its
body will upset; for surface, closing the hole so as to

the advanced spurs leave the ball perfectly round.
Fig. 24.

of the bases at y The centre of gravity will then
and z will act as no longer be at the centre of the

Fig. 26. props, and in order to upset they must turn over the line ball, but to one side, let it be at I z joining them. This shows that the real base includes G (in a, Fig. 26). Put the ball now on the incline, with the all the open space within y z; and you learn that, when- | leaded side looking up the slope; the perpendicular G P will over the base of contact bends inwards, you must measure meet the incline above o, and the ball will roll upwards until G the base of support from one projecting point to another comes over the point of support. all round, making no account whatever of the inward bends. This experiment may be tried in another form without the A common table touches the floor only at four points, and a use of the lead, by simply scooping a hollow on one side, or as round table at three; but in both the base of support is in the following example :-Get a round cylinder of cork—

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