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In the sight of our law the African slave-trader is a pirate and a felon; and in the sight of heaven, an offender far beyond the ordinary depth of human guilt.

What hope of liberty is there remaining, if whatever is their pleasure, it is lawful for them to do; if what is lawful for them to do, they are able to do; if what they are able to do, they dare do; if what they dare do, they really execute; and what they execute, is in no way offensive to you?

It is not the use of the innocent amusements of life which is dangerous, but the abuse of them; it is not when they are occasionally, but when they are constantly pursued; when the love of amusement degenerates into a passion; and when, from being an occasional indulgence, it becomes an habitual desire.

The prevailing colour of the body of a tiger is a deep tawny, or orange yellow; the face, throat, and lower part of the belly are nearly white; and the whole is traversed by numerous long black stripes.

The horse, next to the Hottentot, is the favourite prey of the lion; and the elephant and camel are both highly relished; while the sheep, owing probably to its woolly fleece, is seldom molested.

The horse is quick-sighted; he can see things in the night which his rider cannot perceive; but when it is too dark for his sight, his sense of smelling is his guide.

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The smile of gaiety is often assumed while the heart aches within: though folly may laugh, guilt will sting.

There is no mortal truly wise and restless at the same time: wisdom is the repose of the mind.

Nature felt her inability to extricate herself from the consequences of guilt: the gospel reveals the plan of Divine interposition and aid. Nature confessed some atonement to be necessary: the gospel dis covers that the atonement is made.

Law and order are forgotten: violence and rapine are abroad: the golden cords of society are loosed.

The temples are profaned: the soldier's curse resounds in the house of God: the marble pavement is trampled by iron hoofs: horses neigh beside the altar.

Blue wreaths of smoke ascend through the trees, and betray the half-hidden cottage: the eye contemplates well-thatched ricks, and barns bursting with plenty: the peasant laughs at the approach of

37. The semicolon is sometimes used as a note of interroga- winter. tion, and sometimes as an exclamation.

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fellow-subjects?

Oh, it was impious; it was unmanly; it was poor and pitiful! Have not you too gone about the earth like an evil genius; blasting the fair fruits of peace and industry; plundering, ravaging, killing without law, without justice, merely to gratify an insatiable lust for dominion?

Art thou not, fatal vision, sensible to feeling as to sight? Or art thon but a dagger of the mind; a false creation, proceeding from the heat-oppressed brain?

By such apologies shall man insult his Creator; and shall he hope to flatter the ear of Omnipotence? Think you that such excuses will gain new importance in their ascent to the Majesty on high; and will you trust the interests of eternity in the hands of these superficial advocates?

And shall not the Christian blush to repine; the Christian, from before whom the veil is removed; to whose eyes are revealed the glories of heaven?

Why, for so many a year, has the poet and the philosopher wandered amidst the fragments of Athens or of Rome; and paused with strange and kindling feelings, amidst their broken columns, their mouldering temples, their deserted plains? It is because their day of glory is past; it is because their name is obscured; their power is departed; their influence is lost!

Where are they who taught these stones to grieve; where are the hands that hewed them; and the hearts that reared them?

Hope ye by these to avert oblivion's doom; in grief ambitious, and in ashes vain ?

Can no support be offered; can no source of confidence be named? Is this the man that made the earth to tremble; that shook the kingdoms; that made the world like a desert; that destroyed the cities ?

Falsely luxurious, will not man awaken; and, springing from the bed of sloth, enjoy the cool, the fragrant, and the silent hour, to meditation due, and sacred song?

But who shall speak before the king when he is troubled; and who shall boast of knowledge when he is distressed by doubt ?

Who would in such a gloomy state remain longer than nature craves; when every muse and every blooming pleasure wait without, to bless the wildly devious morning walk?

What a glorious monument of human invention, that has thus triumphed over wind and wave; has brought the ends of the earth in communion; has established an interchange of blessings, pouring into the sterile regions of the north all the luxuries of the south; diffused the light of knowledge and the charities of cultivated life; and has thus bound together those scattered portions of the human race, between which nature seems to have thrown an insurmountable barrier!

Who that bears a human bosom, hath not often felt how dear are all those ties which bind our race in gentleness together; and how sweet their force, let fortune's wayward hand the while be kind or cruel ?

VI. THE COLON.

:

38. The Colon is composed of two periods, placed one above the other. 29. Sometimes the passage ending with a colon is to be read

The necessaries of life are few, and industry secures them to every man: it is the elegancies of life that empty the purse: the superfluities of fashion, the gratification of pride, and the indulgence of luxury, make a man poor.

My dear children, I give you these trees: you see that they are in

good condition. They will thrive as much by your care as they will decline by your negligence: their fruits will reward you in proportion to your labour.

A bee among the flowers in spring is one of the most cheerful objects that can be looked upon. Its life appears to be all enjoyment: so busy and so pleased: yet it is only a specimen of insect life, with which, by reason of the animal being half-domesticated, we happen to be better acquainted.

'Tis a picture in memory distinctly defined, with the strong and unperishing colours of mind: a part of my being beyond my control, beheld on that cloud, and transcribed on my soul.

Yet such is the destiny of all on earth: so flourishes and fades majestic man.

Let those deplore their doom whose hopes still grovel in this dark sojourn: but lofty souls, who look beyond the tomb, can smile at fate, and wonder why they mourn.

If for my faded brow thy hand prepare some future wreath, let me the gift resign: transfer the rosy garland: let it bloom around the temples of that friend beloved, on whose maternal bosom, even now, I lay my aching head.

Do not flatter yourselves with the hope of perfect happiness; there is no such thing in the world.

But when old age has on your temples shed her silver frost, there's no returning sun: swift flies our summer, swift our autumn's fled, when youth, and spring, and golden joys are gone.

A divine legislator, uttering his voice from heaven; an almighty governor, stretching forth his arm to punish or reward: informing us of perpetual rest prepared hereafter for the righteous, and of indignation and wrath awaiting the wicked: these are the considerations which overawe the world, which support integrity, and check guilt.

It is not only in the sacred fane that homage should be paid to the Most High: there is a temple, one not made with hands, the vaulted firmament: far in the woods, almost beyond the sound of city-chime, at intervals heard through the breezeless air.

As we perceive the shadow to have moved along the dial, but did not perceive its moving; and it appears that the grass has grown, though nobody ever saw it grow: so the advances we make in knowledge, as they consist of such minute steps, are perceivable only by the distance gone over.

MECHANICS.-IV.

TWISTED POLYGON-FORCES APPLIED TO TWO POINTSPARALLEL FORCES.

THE method given in the last lesson of finding the resultant of several forces holds good, whether they act all in the same plane, or some of them upwards or downwards from it in different directions. For example, five forces, represented by the lines O A, O B, O C, O D, O E, in Fig. 9, are thus applied to a point o of a body on the floor of a room; two of them, O A, O D, along the floor in two different directions; another, O B, pointing to a picture on the left wall; a fourth, o c, to the cross on the top of a steeple, seen through the open window, and the fifth and last, o E, obliquely downwards, pressing the body against

the floor. On constructing, in such a case, the polygon of forces, we should have the figure as represented in perspective below, one of whose sides, o A, is on the floor, while the others, A R, RR, R, R2, and R, R., are in the air. A figure of this kind is termed a twisted polygon, as though its sides had been all originally in the same plane, but, by a twist, some of them bad been pulled from it. You can see that, since such a polygon cannot be drawn on paper, so as to have the magnitudes of its sides and angles there accurately represented, it can be of no practical use in finding resultants. You might make one by fastening five rods of the proper lengths together at the proper angles, but the structure would probably break down before you arrived at your resultant, and at the best the operation would be very troublesome. Calculation alone can help in such cases; but the "twisted polygon" has the educational value of giving the student mechanical ideas.

KRI

ΤΑ

Fig. 9.

EXAMPLES FOR PRACTICE.

1. Three forces act on a point o A, equal to 3 pounds, oв to 5, and The second lies between the other two, making with o A an

o c to 7.

angle of 30 degrees, and with o c 45 degrees. Find the pounds in the

resultant, and the angle it makes with the least force o A.

2. A roller of a hundred-weight is supported on an incline, the gradient of which is one foot in two, by a force which acts along its slope. Find the magnitude of this force and the pressure of the roller on the plane.

3. From two points on a ceiling, five feet apart, a sixty-pound weight is suspended by two strong cords, which meet at the point of suspension. The lengths of the cords are three and four feet respectively. Find the magnitudes of the forces by which they are strained. 4. Three weights of three, four, and five pounds are attached to three cords, which are knotted together at their other ends. The two cords bearing the lesser weights are thrown over two pulleys fastened at a distance of 10 feet from each other, and at the same height, into a wall, the greatest weight hanging between them. Find the position in which the cords and weights will settle into equilibrium.

You will observe that these problems are to be done by rule and compass, etc. We have not yet come to the more effective method of solving them by calculation. The geometric way, however, of drawing and measuring is the best for giving you accurate ideas of the subject, and therefore indispensable in the first stages. The lines you must carefully lay down by a ruler, and the angles by a circular protractor, keeping in mind, as to the latter, that in every right angle there are ninety degrees. The distances representing the forces you must take from an ordinary scale; and observe, as to this, that you need not make in every case your drawings so large that a whole inch be given to every pound of force. You may allow a quarter of an inch to each pound, or hundredweight, or ton, or even a tenth, if the numbers be large. All that is necessary is to keep the proportion of your figures right, whether they be on a large or a small scale, as is done in mapping or drawing plans of buildings. For the above examples a scale of a quarter of an inch for each pound will be quite sufficient. Perhaps for the third example tenths of an inch will best answer. In the next lesson the answers to these problems will be given. I now proceed to

FORCES APPLIED TO TWO POINTS.

Three cases present themselves for consideration.

A R

one; but, as their directions meet outside the body, it is neces-
sary to show that their effect is the same as though the point
of meeting was a real point of application. This, in a future
lesson, can be demonstrated by a perfect proof; but, in the mean-
time, the following considerations will satisfy you that it is true.
Let A P and BQ be the two forces applied to the points A and
B (as in Fig. 10), and o the outside point in which their direc-
tions meet. Also, let o R be the direction which their resultant
would take were the body extended to o and the forces there
applied. Suppose now that, in order to
extend it, a round bar of iron of uniform
thickness is firmly soldered to it, so as
to include the line o R within its sub-
stance. The body being thus extended,
o may be considered a point of applica-
tion of both forces, which we may con-
ceive to be transferred to it by two
thin but strong wires, o A, O B, the mass
of which is so small that it may be neglected in comparison
with that of the body. The forces AP and B Q then evidently
become one force, acting along O R on rod and body together,
and producing the same effect on both as though they acted
at A and B. But the effect taken separately of the resultant on
O R, and therefore of AP and B Q, is evidently the same-
namely, a pressure along its length. Their effects, therefore,
on the body itself taken separately must be the same; and o,
although outside, may be considered a point of application.
The two forces are reducible to one applied to the body at any
point on the line OR within the body.

TWO PARALLEL FORCES.

P

A

Fig. 10.

D

D

M

B

Third Case. The resultant single force can be determined in this case also by the parallelogram of forces, but the proof given by the greatest mechanician of antiquity-Archimedes of Syra. cuse-is, with a slight alteration, much preferable, on account of its simplicity. I shall first take two equal parallel forces, which act in the same direction. Let A and B (Fig. 11) be the points of application, and their directions those of the arrow-heads P and Q. Suppose, moreover, that in magnitude they are each one pound, or ounce, or ton-say one pound. Now, in the first place, the resultant, whatever it be, must pass through the middle point of a B. The best reason I can give you for this is, that the resultant cannot, since the forces be equal, be nearer to one than to the other. If it were a tenth of an inch nearer to A, it should be also a tenth nearer to B.

P

Fig. 11.

Now, in order to find its magnitude and direction, let us suppose that two other forces, A C, B D, each equal to a pound, are applied to the body along the line A B in opposite directions. These being equal, and therefore of themselves balancing each other, can neither add to nor take from the effect of A P and BQ, which may consequently be considered equivalent to the four forces A P, B Q, A C, B D. Let the two at a be now compounded into one, acting in some direction between them (I care not which), and let the same be done with the two at B. Now produce these resultant directions backwards, until they meet at o, and transfer the resultants themselves to that point. Now resolve them back into their original components, and you have two pounds, o c, and o D,, acting against each other parallel to A B, and two separate pounds pulling from o down. wards parallel to A P and B Q. The two former cancel each other, and there remain two pounds acting parallel to A P. Hence we can say, that—

1. If two equal parallel forces act on a body in the same direction, their resultant is parallel to either, and bisects, or

1. When the lines of direction of the two forces meet within divides equally, the line joining their points of application. the body.

2. When they meet without.

3. When the two forces are parallel to each other. First Case. This is easily disposed of. When two forces meet within a body, the point of meeting may be taken as the point of application of both forces, which can there be compounded into one; and the case thus becomes that of a single force applied to a single point.

Second Case.-Here also the two forces may be reduced to

2. The resultant is in magnitude equal to their sum, or to twice either force.

As an example to illustrate, take two equally strong horses pulling a carriage; two equal forces are applied to the splinter bar, which give one force equal to double the strength of either horse acting at its middle point. When the carriage is backed, these forces are applied in the opposite direction directly to the centre through the pole.

We are now in a position to find the resultant of any two

even.

↑↑

parallel forces, the first step towards which is to determine the resultant of any number of equal ones applied to a body at equal distances along a line. The number may be either odd or We shall consider each separately. First, take odd; and let it be seven, as in Fig. 12. Now, supposing each to be one pound, if we take the middle one, which is evidently at the middle of the line A B, we find that there are three pounds on either side of it acting in pairs at equal distances from M. The resultant of the nearest pair gives, as proved above, two pounds at м; the next pair also give two, and so does the third. These make six pounds of resultant at м, which, with the single one already there, are seven pounds-the sum of all the forces for resultant. Were the number thirteen the conclusion would be the same. There would be six on either side of the middle one, and you would have a resultant of thirteen pounds; and the same holds good of any other odd number you select, be it large or small.

M

Fig. 12.

M

Now, suppose we have an even number of such forces, say six, as in Fig. 13, counting them from either end towards the middle, there will be no middle pound; and the middle of the line A B will be in the middle of the space between the middle pair of forces. What have we then? The inside pair gives two pounds at M, so does the next outside, and so the next; and there are evidently thus six pounds of resultant at the centre of A B. Take any other even number, and the result is the same; and thus, for both odd and even numbers, we arrive at this conclusion:-The resultant of any number of equal parallel forces acting on a body at equal distances along a line, is equal to their sum, and bisects the line joining the points of application of the extreme forces.

Fig. 13.

An instance of this is the working of a hand fire-engine. Suppose seven men at the lever on either side, that is, fourteen hands on each lever; supposing the men to be equally arranged and of equal strength, this makes fourteen equal forces applied at equal distances, the resultant of which is the muscular power of seven acting at the centre on either side.

Now we shall, without difficulty, find the resultant of two unequal parallel forces. As before, let A and B be their points of application, and let us first suppose that they act in the same direction. Measuring the forces by pounds, or ounces, or even grains, there are three cases which may occur. The number, Bay of pounds, in the forces may be both even, or both odd, or sne odd and the other even. 1. We shall take "both even" first, and, for simplification sake, let them be six at A and four at B. Divide now the line A B into ten equal parts, that is, into as many parts as four and six together make. Exten 1 also A B on either side, as represented (Fig 14) by the dotted lines, and measure off on the extensions any number of portions you please, each equal to one of the subdivisions of A B. Beginning at A, suppose you apply a pound force at the end of the

A

M

Fig. 14.

B

first subdivision to the right, another pound at the end of the third, another at that of the fifth, and so on until you come to B. You will find then that there will be a pound at the end of the first division from B. Put pounds now at the end of the first division from A on the dotted line, on the third, and on the fifth, and do the same on the dotted line from B, on the first and third. Count all the pounds you have; they are ten, five inside and five outside. Calling the points occupied by the extreme pounds and Q, the resultant of these ten, so distributed at equal distances, must pass through the middle, M, of P Q, and be ten pounds, by the principle last established. But if we take separately the three outside and the three inside A, they make six pounds acting at A. Also the two pair on either side of B make four pounds at B. The ten pounds at м must therefore produce the same effect on the body as the six at A and the four at B, and therefore must be the resultant of these forces; that is to say, the resultant is the sum of the components.

But count now the number of subdivisions on either side, from м to A and B. There are four on the side of A and six on B's side that is to say, the resultant cuts the line A B in the proportion of the numbers 4 and 6, with this peculiarity, however, that the smaller number is on the side of the greater force. This is what we might expect, for the resultant ought naturally to tend towards the greater, on account of its preponderance. When a line is cut in this way, the smaller portion being on the side of the greater number of pounds, it is said to be cut inversely as the two numbers-that is, in the contrary order.

2. Now let us take the case of two odd numbers; let them be 9 and 7. It is evident that if we put another 9 pounds at A, and 7 at B, the resultant of this second 9 and 7 should in every respect agree and coincide with that of the first, and that the resultant of the four should be the sum of two nines and two sevens. But the double 9 at A is 18 pounds, and the double 7 at B 14 pounds. The case, therefore, becomes one of even numbers, and the line A B, as proved above, must be cut by the resultant in the inverse proportion of 18 to 14. But to divide a line so that there may be 18 parts one side and 14 on the other becomes, by throwing every two of the subdivisions into one, the same thing as dividing it so that 9 may be on one side and 7 on the other. In this case then, also, A B is divided inversely as the forces.

3. When the numbers are one odd and the other even, say 4 and 7, the result is the same. By doubling each force you get 8 and 14 pounds, both even numbers; the line A B is divided by the resultant inversely as 14 to 8, which is the same as 7 to 4 inversely as the forces.

B

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We have supposed in all these cases that the forces contained an exact round number of pounds; but what should we do if there were fractions of a pound in either or in both ? I say, reduce the forces to ounces, and work by round numbers in ounces. If there were fractions of ounces, work in grains. You can thus still secure round numbers, and the above proofs will hold good. But what are you to do if there are fractions of grains? Work them by tenths, or hundredth, or thousandth parts of grains, or by even far smaller fractions, and you will still have round numbers, and you can say that the resultant cuts A B inversely as these numbers, however great they be, and therefore inversely as the forces. To trouble you about smaller fractions would only get you into a cloud of metaphysics for no practical purpose.

Fig. 15.

I have proved this important principle only for particular even numbers, 6 and 4, but you will find that the reasoning will be the same whatever be the even numbers you choose. The rule simply is to divide the line A B into as many equal parts as there are pounds in both forces, and then to distribute all the pounds at A in two batches on either side of that point, and to do the same at B with the pounds there acting, observing to place the pounds as you go from A or B in any direction, at the first, third, fifth, and so forth, points of division.

X

2

B

You are now in a position to find the resultant of three or more parallel forces acting, say, at the points A, B, C, D, as in Fig. 15. First A join a with B, and cut it inversely as the forces which are there applied; next join the point x so found with c, and cut the joining line at y inversely as the sum of the two first forces to that at c; join this again with D, and cut it inversely as the three first forces to that of D; and so proceed until you have exhausted all the forces. The point z last found is that through which the resultant of all passes, and is called the centre of parallel forces.

D

Fig. 16.

Suppose, for example, that the centre was required in the case of parallel forces of 1, 2, 3, and 4 pounds applied to the four corners of a square board, A, B, C, D (Fig. 16). First divide A E into three parts, and take two next to A and one to B. The point x so found is the parallel centre for these two forces. Join x now with c, and cut x c into six parts (the sum of 1, 2, and 3), and take three next to c and three to x. The centre Y

so found, which evidently will be the middle of c x, is the centre of the three. Now join Y with D, and divide Y D into ten parts (the sum of 1, 2, 3, and 4), and take four next Y and six next D. This last point, z, is the centre of all the given forces. Try your own hands row on the following Examples, and in the next lesson we shall have for subject the centre of gravity, which is a centre of parallel forces.

Examples.

1. Three equal parallel forces act at the corners of a triangle; find the centre through which their resultant passes.

2. A force of a pound is applied to one end of a beam, of three at the other, and of two at the middle; find the centre of these forces, they being parallel to each other.

3. A weight of one pound and three-quarters hangs from one end of a rod which is two feet in length, and of three and a-half from the other; find the magnitude of the resultant, and the centre of parallel forces.

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4. A door is seven feet high and three feet wide, and the centres of its hinges are distant one foot from its ends. A force of twenty-three pounds is applied along its upper edge, pulling it off its hinges, and one of thirty-seven along the lower. Find the strains on the hinges.

LESSONS IN ARITHMETIC.-VIII.

GREATEST COMMON MEASURE.

1. A composite number, as already defined (see Lesson VI., Art. 2), is one which is produced by multiplying two or more numbers or factors together.

A prime number is one which cannot be produced by multiplying two or more numbers together; it cannot, therefore, be exactly divided by any whole number except unity and itself. Thus 1, 2, 3, 5, 17, 31, etc., are prime numbers, or primes, as they are sometimes called.

A measure of any given number is a number which will divide the given number exactly without a remainder. Thus, 3 is a measure of 9, 25 is a measure of 75.

A common measure of two or more numbers is a number which will divide each of them without a remainder. Thus, 2 is a common measure of 6, 8, 12, 18, 30, etc.

The greatest common measure of two or more numbers is the greatest number which will divide them all without a remainder. Thus, 9 is the greatest common measure (or, as it is sometimes written for shortness, the G. C. M.) of 18, 27, 36, and 45.

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4. The above rules are infallible methods for finding the greatest common measure of two or more numbers. In practice, however, we can frequently dispense with these operations, and determine the greatest common measure by inspection, or by splitting up the numbers into their elementary or prime factors.

It is evident that if two or more numbers have a common measure at all, they must be composite numbers, i.e., capable of being separated into factors. If any given numbers be sepa rated into prime factors, the greatest common measure will evidently be the product of all the factors which are common to each of the given numbers.

Thus, 75, 135, and 300, when separated into their prime factors, are respectively

3 x 5 x 5, 3 × 5 × 9, and 2 × 2 × 3 × 5 × 5

Now, the factors which are common to all of these are 3 and 5, and therefore 3 x 5-that is, 15-is the greatest common measure of 75, 135, and 300.

5. We subjoin a

Rule for dividing a composite number into its prime factors. Divide the given number by the smaller number, which will divide it without a remainder; then divide the quotient in the same way, and continue the operation until the quotient is unity. The divisors will be the prime factors of the given

2. To find the greatest common measure of two given numbers. RULE. Divide the greater by the less, then the preceding divisor by the remainder. and so on, until there is no remainder. The last divisor will be the greatest common measure required. EXAMPLE.-To find the greatest common measure of 532 and number. 1274. Arrange the process thus:

532) 1274 (2

1064

210) 532 (2

420

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112) 210 (1

112

98) 112 (1 98

14) 98 (7 98

The reason of the truth of the above rule may be thus explained:

Every division of a number, where there is no remainder, resolves it into two factors-namely, the divisor and quotient. But in the above rule the divisors in each case are the smallest numbers which will divide the given number and the successive quotients without a remainder: hence they are all prime numbers, and the division is continued until the quotient is unity. Hence, clearly, the product of all these divisors (which are all primes) will be equal to the original number. In other words, these divisors are the prime factors of the given composite number.

EXAMPLE.-Resolve 16170 into its prime factors. Arrange the process thus:

.1

Here, in accordance with the rule, we divide 1274 by 532, which gives a remainder 210; then 532 (the preceding divisor) by 210, giving a remainder 112; again 210 (the preceding divisor) by 112, which gives a remainder 98; then 112 (the preceding divisor) by 98, which leaves a remainder 14; and lastly, 98 by 14, which gives no remainder. 14, therefore, according to the rule, is the greatest common measure of 532 and 1274. 3. To find the greatest common measure of three or more given numbers.

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Hence the prime factors of which 16170 is composed are 2, 3, 5, 7, 7, 11; or, 16170 = 2 × 3 × 5 × 7 x 7 x 11.

EXERCISE 19.

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16. How often could 43046721 be subtracted from 22876792454961, and. at last leave no remainder ?

17. How many times does 310314420 contain 39390 ?
18. What number is that which divided by 123456 would

measure of the following give a quotient of 826451, and a remainder of 70404 ?
19. Work the following examples in multiplication :—

5. 1879 and 2125.

6. 75, 125, and 60.

7. 183, 3996, and 108.

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1. 42634 x 63. 2. 50035 x 56. 3. 72156 x 1000. 4. 42000 x 40000. 5. 80000 × 25000. 6. 2567345 x 17. 7. 4300450 x 19. 8. 9803404 × 41. 9. 6710045 x 71.

4. Find the greatest common measure of the following 10. 3156701 x 18. numbers by resolving them into factors :

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9. Divide one thousand billions by 81 and 729. 10. Divide a thousand thousand millions by 111. 11. Divide a thousand millions of millions by 1111. 12. Divide 908070605040302010 by 654321. 13. Divide 4678179387300 by the following divisors, separately, 2100, 36500, 8760, 957000, 87700, 1360000, and 87000. 14. If the annual revenue of a nobleman be £37960, how much is that per day, the year being supposed to be exactly 365 days.

15. What is the nearest number to one thousand billions that can be divided by 11111 without a remainder?

11. 7000541 × 91. 12. 4102034 × 99. 13. 42304 × 999.

14. 50421 x 9999. 15. 67243 × 99999. 16. 78563 × 93. 17. 34054 x 639. 18. 52156 x 756. 19. 41907 × 54486. 20. 26397 x 21648 21. 12900 × 14000 22. 64172 × 42432. 23. 26815678 × 81 24. 85 x 85. 25. 256 X 256. 26. 322 x 325.

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27. 5234 × 2435. 28. 48743000 × 637. 29. 31890420 × 85672. 30. 80460000 × 2763. 31. 2364793 × 8485672. 32. 1256702 x 999999. 33. 6840005 x 91 x 61. 81, 45067034 x 17 x 51. 35. 788031245 x 81 x 16. 36. 61800000 × 23000. 87. 12563000 × 4800000. 38. 91300233 x 1000000. 89. 680040000 x 1000000.

17. 3562189 ÷ 225. 18. 685726 + 32000. 19. 723564 ÷ 175. 20. 892565 225. 21. 456212 + 275. 22. 925673 +125. 23. 763421 ÷ 175. 24. 876240 ÷ 275.

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ALTHOUGH leaves have a great variety of uses, yet the principal is that of respiration or breathing. In this manner they become the representatives of lungs in animal beings. But though plants breathe, the vegetable function of respiration in them is not to be considered as similar to that function in animals. On the contrary, it is directly the reverse: the very gas which animals expel from their lungs as useless or injurious, plants receive through the medium of their leaves, take out of it that which is suitable to their wants, then exhale the portion which is refuse to them, but which is necessary to the existence of animals. What a train of reflections does the contemplation of this beautiful provision call forth! Not only are vegetables useful in supplying us with food and timber, not only do they beautify the landscape with their waving branches and picturesque forms, but they are absolutely necessary to the exist ence of animal life as a means of purifying the atmosphere!

The breathing function of leaves is far too important to admit of being lightly passed over with these few remarks, yet a difficulty occurs in pursuing it further, inasmuch as to understand the precise theory of vegetable respiration the reader must be acquainted with certain facts in chemistry. readers, doubtless, are acquainted with these chemical facts, others are not; consequently, the best plan will be to present a slight outline of these facts at once.

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Some

To begin, then did the reader ever set fire to a bit of stick or a little charcoal? No doubt he has. What does the reader think becomes of this stick or charcoal? Is it lost, destroyed?

Oh no, there is no such thing as destruction in all nature; substances, even when they appear to be destroyed, only change their form. What, then, becomes of a piece of stick or a piece of charcoal when we burn either in the fire? Now, whenever philosophers desire to study the conditions of an experiment, and the choice of more than one set of conditions stands before them, they very properly take the simplest. We have here two sets of conditions; the burning of a stick is one, the burning of a piece of charcoal is the other. The latter being the simpler of the two, we will take it, and use it for our purposes; moreover, we

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