Imágenes de páginas

In the sight of our law the African slave-trader is a pirate and a with the voice suspended; but it should generally be read with telon; and in the sight of heaven, an offender far beyond the ordinary the falling intlection of the voice. depth of human guilt,

40. In reading, be careful to let the pause of the colon be a 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, indicated by a comma.

total cessation of the voice, and three times longer than that 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

Examples. densite to you?

The smile of gaiety is often assumed while the heart aches within : It is not the use of the innocent amusements of life which is dan.

though folly may laugh, guilt will sting. gerous, but the abuse of them; it is not when they are occasionally,

There is uo mortal truly wise and restless at the same time: wisdom but when they are constantly pursued; when the love of amusement is the repose of the mind. degenerates into a passion; and when, from being an occasional

Nature felt her inability to extricate herself from the consequences indulgence, it becomes an habitual desire.

of guilt: the gospel reveals the plan of Divine interposition and aid. The prevailing colour of the body of a tiger is a deep tawny, or Nature confessed some atonement to be necessary : the gospel dis. orange yellow; the face, throat, and lower part of the belly are covers that the atonement is made. Dearly white; and the whole is traversed by numerous long black

Law and order are forgotten : violence and rapine are abroad: the stripes.

golden cords of society are loosed. 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, of God: the marble pavement is trampled by iron hoofs : horses

The temples are profaned: the soldier's curse resounds in the house oving probably to its woolly fleece, is seldom molested.

neigh beside the altar. The horse is quick-sighted; he can see things in the night which

Blue wreaths of smoke ascend through the trees, and betray the his rider cannot perceive ; but when it is too dark for his sight, his

half-hidden cottage : the eye contemplates well-thatched ricks, and Kase of smelling is his guide.

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.

The necessaries of life are few, and industry secures them to every

man: it is the elegancies of life that empty the purse : the superExamples.

fluities of fashion, the gratification of pride, and the indulgence of Hast thou not set at defiance my authority ; violated the public luxury, make a man poor. peace, and passed thy life in injuring the persons and properties of thy

My dear children, I give you these trees : you see that they are in telor-subjects ?

good condition. They will thrive as much by your care as they will Oh, it was impious; it was unmanly; it was poor and pitiful !

decline by your negligence : their fruits will reward you in proportion Have not you too gone about the earth like an evil genius; blasting

to your labour. the fair fruits of peace and industry; plundering, ravaging, killing

A bee among the flowers in spring is one of the most cheerful objects

that can be looked upon. without law, without justice, merely to gratify an insatiable lust for

Its life appears to be all enjoyment: so dominion ?

busy and so pleased: yet it is only a specimen of insect life, with which, Art thou not, fatal vision, sensible to feeling as to sight? Or art by reason of the animal being half-domesticated, we happen to be thoa bat a dagger of the mind; a false creation, proceeding from the

better acquainted. beat-oppressed brain P

"Tis a picture in mnemory distinctly defined, with the strong and By such apologies shall man insult his Creator; and shall be hope upperishing colours of mind : a part of my being beyond my control, to flatter the ear of Omnipotence ? Think you that such excuses will

beheld on that cloud, and transcribed on my soul. Gain new importance in their ascent to the Majesty on high; and will

Yet such is the destiny of all on earth ; so flourishes and fades you trust the interests of eternity in the hands of these superficial majestic man. advocates ?

Let those deplore their doom whose hopes still grovel in this dark And shall not the Christian blush to repine ; the Christinn, from sojourn : but lofty souls, who look beyond the tomb, can smile at fate, before whom the veil is removed; to whose eyes are revealed the

and wonder why they mourn. giories of heaven?

If for my faded brow thy hand prepare some future wreath, let me Why, for so many a year, has the poet and the philosopher wandered

the gift resign : transfer the rosy garland : let it bloom around the ansidst the fragments of Athens or of Rome; and paused with strange temples of that friend beloved, on whose materual bosom, even now, and kindling feelings, amidst their broken columns, their mouldering

I lay my achirg head. temples, their deserted plains? It is because their day of glory is

Do not flatter yourselves with the hope of perfect happiness ; tliere past; it is because their name is obscured; their power is departed ;

is no such thing in the world. their influence is lost!

But when old age has on your temples shed her silver frost, there's Where are they who taught these stones to grieve; where are the

no returning sun: swift flies our summer, swift our autumn's fled, hands that hewed them; and the hearts that reared them ?

when youth, and spring, and golden joys are gone. Hope ye by these to avert oblivion's doom ; in grief ambitious, and

A divine legislator, uttering his roice from heaven; an almighty in asbes vain?

governor, stretching forth his arm to punish or reward : informning Can Do support be offered ; can no source of confidence be named ? us of perpetual rest prepared hereafter for the righteous, and of Is this the man that made the earth to tremble ; that shook the indignation and wrath awaiting the wicked: these are the considerakingdoms; that made the world like a desert; that destroyed the

tions which overawe the world, which support integrity, and check

guilt. Falsely luxurious, will not man awaken ; and, springing from the It is not only in the sacred fane that homage should be paid to the bed of sloth, cnjoy the cool, the fragrant, and the silent hour, to Most High: there is a temple, one not made with hands, the vaulted meditation due, and sacred song ?

firmament: far in the woods, almost beyond the sound of city-chime, But who shall speak before the king when he is troubled ; and who

at intervals heard through the breezeless air. eball boast of knowledge when he is distressed by doubt ?

As we perceive the shadow to have moved along the dial, but did Who would in such a gloomy state remain longer than nature

not perceive its moving; and it appears that the grass has grown, Traves; when every muse and every blooming pleasure wait without, though nobody ever saw it grow : so the advances we make in knowto bless the wildly devious morning walk ?

ledge, as they consist of such minute steps, are perceivable only by What a glorious monument of human invention, that has thus

the distance gone over. 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

MECHANICS.-IV. the light of knowledge and the charities of cultivated life; and has thus bound together those scattered portions of the human


PARALLEL FORCES. rece, between which nature seems to have thrown an insurmountable

The method given in the last lesson of finding the resultant of Who that bears a human bosom, hath not often felt how dear are

several forces holds good, whether they act all the same all those ties which bind our race in gentleness together; and how plane, or some of them upwards or downwards from it in difsweet their force, let fortune's wayward hand the while be kind or

ferent directions. For example, five forces, represented by the

lines 0 A, O B, OC, OD, O E, in Fig. 9, are thus applied to a point VI. THE COLON.

o of a body on the floor of a room; two of them, O A, OD, along :

the floor in two different directions ; another, O B, pointing to a 38. The Colon is composed of two periods, placed one above the 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 39. Sometimes the passage ending with a colon is to be read and last, o E, obliqnely downwards, pressing the body against

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the floor. On constructing, in such a case, the polygon of one; but, as their directions meet outside the body, it is neces. forces, we should have the figure as represented in perspective sary to show that their effect is the same as though the point below, one of whose sides, o A, is on the floor, while the others, of meeting was a real point of application. This, in a future A R, R R,, R, R., and R, Rg, are in the air. A figure of this kind lesson, can be demonstrated by a perfect proof; but, in the mean. is termed a twisted polygon, as though its sides had been all time, the following considerations will satisfy you that it is true. originally in the same plane, but, by a twist, some of them bad Let A P and B o be the two forces applied to the points A and

been pulled from it. You can B (as in Fig. 10), and o the outside point in which their direc.
see that, since such a polygon tions meet. Also, let or be the direction which their resultant
cannot be drawn on paper, so would take were the body extended to o and the forces there
as to have the magnitudes of applied. Suppose now that, in order to
its sides and angles there ac- extend it, a round bar of iron of uniform
curately represented, it can be thickness is firmly soldered to it, so as
of no practical use in finding to include the line or within its sub-
resultants. You might make stance. The body being thus extended,
one by fastening five rods of o may be considered a point of applica-
the proper lengths together at tion of both forces, which we may con.
the proper angles, but the ceive to be transferred to it by two

Fig. 10.
structure would probably break thin but strong wires, o A, O B, the mass
down before you arrived at of which is so small that it may be neglected in comparison
your resultant, and at the best with that of the body. The forces A P and B Q then evidently

the operation would be very become one force, acting along o R on rod and body together, Fig. 9.

troublesome. Calculation alone and producing the same effect on both as though they acted

can help in such cases; but at A and B. But the effect taken separately of the resultant on the “ twisted polygon has the educational value of giving the O R, and therefore of A P and B Q, is evidently the same student mechanical ideas.

namely, a pressure along its length. Their effects, therefore, EXAMPLES FOR PRACTICE.

on the body itself taken separately must be the same; and o,

although outside, may be considered a point of application. 1. Three forces act on a point o 1, equal to 3 pounds, o B to 5, and The two forces are reducible to one applied to the body at any oc to 7. The second lies between the other two, making with o A an angle of 30 degrees, and with o c 45 degrees. Find the pounds in the point on the line o e within the body. resultant, and the angle it makes with the least force o A.

TWO PARALLEL FORCES. 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

Third Case. The resultant single force can be determined in slope. Find the magnitude of this force and the pressure of the roller this case also by the parallelogram of forces, but the proof given on the plane.

by the greatest mechanician of antiquity-Archimedes of Syra. 3. From two points on a ceiling, five feet apart, a sixty-pound cuse-is, with a slight alteration, rauch preferable, on account weight is suspended by two strong cords, which meet at the point of of its simplicity. I shall first take suspension. The lengths of the cords are three and four feet respec- two equal parallel forces, which act

»D tively. Find the magnitudes of the forces by which they are strained in the same direction. Let A and B

4. Three weights of three, four, and five pounds are attached to three cords, which are knotted together at their other ends. The (Fig. 11) be the points of applicatwo cords bearing the lesser weights are thrown over two pulleys tion, and their directions those of fastened at a distance of 10 feet from each other, and at the same

the arrow-heads P and Q. Suppose, height, into a wall, the greatest weight hanging between them. Find moreover, that in magnitude they are

Fig. 11. the position in which the cords and weights will settle into equi- each one pound, or ounce, or ton-say librium.

one pound. Now, in the first place, the resultant, whatever You will observe that these problems are to be done by rule it be, must pass through the middle point of A B. The best and compass, etc. We have not yet come to the more effective reason I can give you for this is, that the resultant cannot, method of solving them by calculation. The geometric way, since the forces be equal, be nearer to one than to the other. however, of drawing and measuring is the best for giving you If it were a tenth of an inch nearer to a, it should be also a accurate ideas of the subject, and therefore indispensable tenth nearer to B. in the first stages. The lines you must carefully lay

Now, in order to find its magnitude and direction, let us supdown by a ruler, and the angles by a circular protractor, pose that two other forces, A C, B D, each equal to a pound, ara keeping in mind, as to the latter, that in every right applied to the body along the line A B in opposite directions. angle there are ninety degrees. The distances representing the These being equal, and therefore of themselves balancing each forces you must take from an ordinary scale; and observe, as other, can neither add to nor take from the effect of a P and to this, that you need not make in every case your drawings so

BQ, which may consequently be considered equivalent to the large that a whole inch be given to every pound of force. You four forces A P, BQ, A C, B D. Let the two at A be now may allow a quarter of an inch to each pound, or hundredweight, compounded into one, acting in some direction between them or ton, or even a tenth, if the numbers be large. All that is ne- (I care not which), and let the same be done with the two at B. cessary is to keep the proportion of your figures right, whether Now produce these resultant directions backwards, until they they be on a large or a small scale, as is done in mapping or meet at o, and transfer the resultants themselves to that point. drawing plans of buildings. For the above examples a scale Now resolve them back into their original components, and of a quarter of an inch for each pound will be quite sufficient. you have two pounds, o C, and 0 D,, acting against each other Perhaps for the third example tenths of an inch will best

answer. parallel to A B, and two separate pounds pulling from o down. In the next lesson the answers to these problems will be given. wards parallel to A P and B Q. The two former cancel each I now proceed to

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 Three cases present themselves for consideration.

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. The resultant is in magnitude equal to their sum, or to 2. When they meet without.

twice either force. 3. When the two forces are parallel to each other.

As an example to illustrate, take two equally strong horses First Case.—This is easily disposed of. When two forces pnlling a carriage; two equal forces are applied to the splinter meet within a body, the point of meeting may be taken as the bar, which give one force equal to double the strength of either point of application of both forces, which can there be com- horse acting at its middle point. When the carriage is backed, pounded into one; and the case thus becomes that of a single these forces are applied in the opposite direction directly to the force applied to a single point.

centre through the pole. Second Case.--Here also the two forces may be reduced to We are now in a position to find the resultant of any two



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parallel forces, the first step towards which is to determine the But count now the number of subdivisions on either side, resultant of any number of equal ones applied to a body at from a to A and B. There are four on the side of A and six on equal distances along a line. The number may be either odd or B's side—that is to say, the resultant cuts the line A B in the even. We shall consider each separately. First, take odd ; proportion of the numbers 4 and 6, with this peculiarity, howand let it be seven, as in Fig. 12. Now, supposing each to be ever, that the smaller number is on the side of the greater force. one pound, if we take the middle one, which is evidently at the This is what we might expect, for the resultant ought naturally

middle of the line A B, we find that to tend towards the greater, on account of its preponderance. there are three pounds on either When a line is cut in this way, the smaller portion being on the side of it acting in pairs at equal side of the greater number of pounds, it is said to be cut distances from m. The resultant inversely as the two numbers—that is, in the contrary order.

of the nearest pair gives, as proved 2. Now let us take the case of two odd numbers ; let them be Fig. 12.

above, two pounds at M; the next 9 and 7. It is evident that if we put another 9 pounds at A,

pair also give two, and so does the and 7 at B, the resultant of this second 9 and 7 should in every third. These make six pounds of resultant at m, which, with the respect agree and coincide with that of the first, and that the single one already there, are seven pounds—the sum of all the resultant of the four should be the sum of two nines and two forces for resultant. Were the number thirteen the conclusion sevens. But the double 9 at A is 18 pounds, and the double 7 would be the same. There would be six on either side of the at B 14 pounds. The case, therefore, becomes one of even num. middle one, and you would have a resultant of thirteen pounds; bers, and the line A B, as proved above, must be cut by the and the same holds good of any other odd number you select, resultant in the inverse proportion of 18 to 14. But to divide be it large or small.

a line so that there may be 18 parts one side and 14 on the Now, suppose we have an even number of such forces, say other becomes, by throwing every two of the subdivisions into six, as in Fig. 13, counting them from either end towards the one, the same thing as dividing it so that 9 may be on one side middle, there will be no middle pound; and the middle of the and 7 on the other. In this case then, also, A B is divided line A B will be in the middle of the space between the middle | inversely as the forces. pair of forces. What have we then? The inside pair gives two 3. When the numbers are one odd and the other even, say 4

pounds at m, so does the next and 7, the result is the same. By doubling each force you get outside, and so the next; and 8 and 14 pounds, both even numbers; the line A B is divided by there are evidently thus six the resultant inversely as 14 to 8, which is the same as 7 to 4 pounds of resultant at the cen. inversely as the forces.

tre of A B. Take any other even We have supposed in all these cases that the forces conFig. 13.

number, and the result is the tained an exact round number of pounds; but what should we

same; and thus, for both odd do if there were fractions of a pound in either or in both ? I and even numbers, we arrive at this conclusion :-The resultant say, reduce the forces to ounces, and work by round numbers in of any number of equal parallel forces acting on a body at equal ounces. If there were fractions of ounces, work in grains. distances along a line, is equal to their sum, and bisects the You can thus still secure round numbers, and the above proofs line joining the points of application of the extreme forces. will hold good. But what are

An instance of this is the working of a hand fire-engine. you to do if there are fractions Suppose seven men at the lever on either side, that is, fourteen of grains ? Work them by hands on each lever; supposing the men to be equally arranged tenths, or hundredth, or thouand of equal strength, this makes fourteen equal forces applied sandth parts of grains, or by even at equal distances, the resultant of which is the muscular power far smaller fractions, and you of seven acting at the centre on either side.

will still have round numbers, Now we shall, without difficulty, find the resultant of two and you can say that the resulttrequal parallel forces. As before, let A and B be their points ant cuts A B inversely as these

Fig. 15. of application, and let us first suppose that they act in the same numbers, however great they be, direction. Measuring the forces by ds, or ounces, or even and therefore inversely as the forces. To trouble you about grains, there are three cases which may occur. The number, smaller fractions would only get you into a cloud of metaphysics day of pounds, in the forces may be both even, or both odd, or for no practical purpose. one odd and the other even. 1. We shall take “both even" first, I have proved this important principle only for particular and, for simplification sake, let them be six at A and four at B. even numbers, 6 and 4, but you will find that the reasoning will Divide now the line A B into ten eqnal parts, that is, into as many be the same whatever be the even numbers you choose. The parts as four and six together make. Extez also A B on either rule simply is to divide the line A B into as many equal parts as side, as represented (Fig 14) by the dotted lines, and measure there are pounds in both forces, and then to distribute all the off on the extensions any number of portions you please, each pounds at a in two batches on either side of that point, and to

equal to one of the do the same at B with the pounds there acting, observing to
subdivisions of A B. place the pounds as you go from A or B in any direction, at the
Beginning at A, sup- first, third, fifth, and so forth, points of division.
pose you apply a pound You are now in a position to find the resultant of three or
force at the end of the more parallel forces acting, say, at the

first subdivision to the points A, B, C, D, as in Fig. 15. First A
Fig. 14.

right, another pound join A with B, and cut it inversely as the

at the end of the third, forces which are there applied; next join the another at that of the fifth, and so on until you come to B. point x so found with c, and cut the joinYou will find then that there will be a pound at the end of the ing line at y inversely as the sum of the two first division from B. Put pounds now at the end of the first first forces to that at c; join this again division from a on the dotted line, on the third, and on the fifth, with D, and cut it inversely as the three and do the same on the dotted line from B, on the first and third first forces to that of D; and so proceed Count all the pounds you have; they are ten, five inside and five until you have exhausted all the forces. outside. Calling the points occupied by the extreme pounds The point z last found is that through Fig. 16. pand Q, the resultant of these ten, so distributed at equal which the resultant of all passes, and is distances, must pass through the middle, m, of P Q, and be ten called the centre of parallel forces. pounds, by the principle last established. But if we take Suppose, for example, that the centre was required in the case separately the three outside and the three inside a, they make six of parallel forces of 1, 2, 3, and 4 pounds applied to the four pounds acting at A.

Also the two pair on either side of B make corners of a square board, A, B, C, D (Fig. 16). First divide a e

The ten pounds at a must therefore produce into three parts, and take two next to A and one to B. The the same effect on the body as the six at a and the four at B, point x so found is the parallel centre for these two forces. and therefore most be the resultant of these forces; that is tó Join x now with c, and cut x c into six parts (the sum of 1, 2, say, the resultant is the sum of the components.

and 3), and take three next to c and three to x. The centre Y


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so found, which evidently will be the middle of c x, is the centre RULE.—Find the greatest common measure of two of them; of the three. Now join y with D, and divide y d into ten parts then find that of the common measure thus obtained and of the (the sum of 1, 2, 3, and 4), and take four next y and six next third ; then that of this common measure and the fourth, and so

This last point, z, is the centre of all the given forces. Try on. The last obtained will be the greatest commor measure of your own hands now on the following Examples, and in the next the given numbers. lesson we shall have for subject the centre of gravity, which is EXAMPLE.—Find the greatest common measure of 204, 357, a centre of parallel forces.

and 935. Examples.

First, we find the greatest common 204 ) 357 (1

measure of 204 and 357 to be 51, by the 1. Three equal parallel forces act at the corners of a triangle; find rule given for two numbers. the centre through which their resultant passes.

153 ) 201 (1 2. A force of a pound is applied to one end of a beam, of three at

153 the other, and of two at the middle; find the centre of these forces, they being parallel to each other.

51 ) 1533 3. A weight of one pound and three-quarters hangs from one end of

153 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. 4. A door is seven feet high and three feet wide, and the centres of Next, we find the greatest common measure of 51 51) 935 ( 18

51 its hinges are distant one foot from its ends. A force of twenty-three and 935, which we see to be 17. pounds is applied along its upper edge, pulling it off its hinges, and

425 one of thirty-seven along the lower. Find the strains on the hinges. Hence, according to the rule, 17 is the greatest

408 common measure of 204, 357, and 935.

17) 51 (3

We do not give the reasons for the truth of the LESSONS IN ARITHMETIC.-VIII.

foregoing rules, as they cannot be satisfactorily GREATEST COMMON MEASURE.

established without the aid of algebra. 1. A composite number, as already defined (seo Lesson VI.,

4. The above rules are infallible methods for finding the Art. 2), is one which is produced by multiplying two or more greatest common measure of two or more numbers. In practice, numbers or factors together.

however, we can frequently dispense with these operations, and A prime number is one which cannot be produced by multi-determine the greatest common measure by inspection, or by plying two or more numbers together; it cannot, therefore, be splitting up the numbers into their elementary or prime exactly divided by any whole number except unity and itself. | factors. Thus 1, 2, 3, 5, 17, 31, etc., are frime numbers, or primes, as It is evident that if two or more numbers have a common they are sometimes called.

measure at all, they must be composite numbers, i.e., capable of A measure of any given number is a number which will divide being separated into factors. If any given numbers be sepathe given number exactly without a remainder. Thus, 3 is a rated into prime factors, the greatest common measure will measure of 9, 25 is a measure of 75.

evidently be the product of all the factors which are common to A common measure of two or more numbers is a number which each of the given numbers. will divide each of them without a remainder. Thus, 2 is a Thus, 75, 135, and 300, when separated into their primo common measure of 6, 8, 12, 18, 30, etc.

factors, are respectively The greatest common measure of two or more numbers is

3 * 5 * 5,3 * 5 * 9, and 2 * 2 * 3 * 5 * 5 the greatest number which will divide them all without a remainder. Thus, 9 is the greatest common measure (or, as it Now, the factors which are common to all of these are 3 and 5, is sometimes written for shortness, the G. C. M.) of 18, 27, 36, and therefore 3 x 5—that is, 15—is the greatest common and 45.

measure of 75, 135, and 300. 2. To find the greatest common measure of two given numbers.

5. We subjoin a

Rule for dividing a composite number into its prime factors. RULE.-Divide the greater by the less, then the preced- Divide the given number by the smaller number, which will ing divisor by the remainder, and so on, until there is no divide it without a remainder ; then divide the quotient in the remainder. The last divisor will be the greatest common mea- same way, and continue the operation until the quotient is sure required.

unity. The divisors will be the prime factors of the given EXAMPLE.—To find the greatest common measure of 532 and number. 1274. Arrange the process thus :

The reason of the truth of the above rule may be thus ex.

plained :532 ) 1274 (2

Every division of a number, where there is no remainder, 1064

resolves it into two factors-namely, the divisor and quotient.

But in the above rule the divisors in each case are the smallest 210) 532 (2

numbers which will divide the given number and the successive

quotients without a remainder : hence they are all prime num112) 210 (1

bers, and the division is continued until the quotient is nnity. Hence, clearly, the product of all these divisors (which are all

primes) will be equal to the original number. In other words, 98 ) 112 (1

these divisors are the prime factors of the given composite number.

EXAMPLE.— Resolve 16170 into its prime factors. Arrange 14) 98 (7 the process thus :

2) 16170




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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, 16. How often could 43046721 be subtracted from 7, 7, il; or, 16170 = 2 X 3 X 5 X 7 X 7 X 11.

22876792454961, last leave no remainder ? EXERCISE 19.

17. How many times does 310314420 contain 39390 ?

18. What number is that which divided by 123456 would 1. Find the greatest common measure of the following give a quotient of 826451, and a remainder of 70404 ? numbers:

19. Work the following examples in multiplication :1. 285 and 465. 5. 1879 and 2:25.

1. 42631 X 63. 14. 50421 x 9999, 2. 532 and 1274.

27. 5234 x 2435. 6. 75, 125, and 60.

2. 50035 X 56. 15. 67213 X 99999. 28. 48743000 X 637. S. 683 and 2775. 7. 183, 3996, and 108. 3. 72156 X 1000. 16, 78563 X 93,

29. 31890 120 X 85672. 4. 2145 and 3171. 8. 672, 1440, and 3172. 4. 42000 X 40000. 17. 31054 X 639.

30. 80160000 x 2763. 2. Resolve all the composite numbers from 9 to 108 into their

5. 80000 X 25000. 18. 52156 X 756,

31. 2364793 X 8185672.

6. 2567345 X 17. 19. 41907 X 54486. prime factors.

32. 1256702 X 999999.

7. 4300 150 X 19. 20. 26397 X 21618 33. 68 10005 X 91 X 61. 3. Resolve into their prime factors 180, 420, 714, 836, 2898,

8. 9803104 X 41. 21. 12900 X 14000 34. 45067034 X 17 X 51. 11492, 1728, 1492, 8032, 71640, 92352, 81660.

9. 67100 15 x 71. 22. 64172 X 42132. 35. 788031245 x 81 x 16. 4. Find the greatest common

measure of the following 10. 3156701 * 18. 23. 26815678 X 81 36. 61800000 X 23000. numbers by resolving them into factors :

11. 7000541 X 91. 24. 85 X 85.

37. 12563000 X 4800000. 12. 4102034 X 99. 25. 256 X 256.

38. 91300233 X 1000000. 1. 36, 60, and 108.

2. 56, 84, 140, and 168.
13, 42304 x 999. | 26. 322 X 325.

89. 680040000 x 1000000. 3. 5355, 6545, 17017, 36465, 91385. 5. Find the greatest common measure of the following

20. Work the following examples :onmbers :

1. 1188 + 33.

9. 31256726 • 15. 17. 3562189 225, 1. 105 and 165. 3. 140, 210, and 315.

2. 3128 + 86.

10, 42367581 45. 18. 685726 + 32000. 2. 108, 126, and 162. 4. 24, 42, 54, and 60.

3. 2516 i 37.

11. 16753672 + 35. 19. 723564 – 175. 4. 7125 95. 12. 3256385 * 55.

20. 892565 225. 6. Find all the divisors common to the following numbers:

5. 568210 + 42. 13. 45672400 – 25. 21. 456212 • 275. 1. 15, 18, 24, and 36.

4. 82, 118, and 146.

6. 785372 ^ 63. 14, 6245634 45, 22. 925673 + 125, 2. 11, 29, 12, and 35.

5. 42 and 66.

7. 896736 72.

15. 8245623 + 125. 23. 763121 + 175. S. 10, 35, 50, 75, and 60.

8. 67231568 5. 16. 462156 + 75.

21. 876240 + 275, 7. Resolve the following numbers into their prime factors :- 21. How long would it take a vessel sailing 100 miles per 1. 120 and 144.

7. 1492 and 8032.

day to circumnavigate the earth, whose circumference is 25000 2. 180 and 420.

8. 4604 and 16806.

miles ? 3. 711 and 836.

9. 71640 and 20780,

22. The distance of the earth from the sun is 95000000 of 4. 574 and 2898.

10. 8-1570 and 65480.

miles : how long would it take a balloon, going at the rate of 5. 11-192 and 980.

11, 92352 and 1660.

100000 miles a year, to reach the sun ? 6. 650 and 1728.

23. Divide 467000000000 by 25000000000.

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That our readers may have sufficient practice in multiplication and division, we give in this lesson upwards of one hundred

LESSONS IN BOTANY.-IV. examples in these rules. The operations should be contracted SECTION VI.-LEAVES CONSIDERED AS TO THEIR when practicable, and the correctness of every result should be

FUNCTIONS. tested by the methods given in our Lessons on Multiplication ALTHOUGH leaves have a great variety of uses, yet the principal and Division.

is that of respiration or breathing. In this manner they become EXERCISE 20.

the representatives of lungs in animal beings. But though 1. Find the product 678954 X 72, by multiplying by succes- plants breathe, the vegetable function of respiration in them is Eive factors.

not to be considered as similar to that function in animals. On 2. Find in the same way the product 78530700 x 1250.

the contrary, it is directly the reverse : the very gas which 3. Find the product of the following by dividing by succes.

animals expel from their lungs as useless or injurious, plants sive factors :

receive through the medium of their leaves, take out of it that

which is suitable to their wants, then exhale the portion which 1. 16128 • 24, 3. 91080 + 72.

5. 142857 • 112. |

is refuse to them, but which is necessary to the existence of 2, 25760 4 56. 4. 123156 168.

animals. What a train of reflections does the contemplation of 4. Divide 9643 by 30, by 300, and by 3000.

this beautiful provision call forth! Not only are vegetables 5. Divide 3360000 by 17000.

useful in supplying us with food and timber, not only do they 6. Divide 123456789 by 290000.,

beautify the landscape with their waving branches and pic7. Multiply and also divide :

turesque forms, but they are absolutely necessary to the exist1. 98734 by 5. 4. 103561203 by 15, 7. 25426 by 125.

ence of animal life as a means of purifying the atmosphere! 2. 53960201 by 5. 5. 1125 by 75.

8, 237135 by 75.

The breathing function of leaves is far too important to 3. 1256 by 15. 6. 5093123 by 75. 9. 3929764 by 125. admit of being lightly passed over with these few remarks, yet

a difficulty occurs in pursuing it further, inasmuch as to under8. Work the following examples in multiplication

stand the precise theory of vegetable respiration the reader 1 856783 X 999. 7. 39567 X 85, 13. 107206 X 486819. must be acquainted with certain facts in chemistry. Some 2. 337-065 X 99999, 8. 3567 X 284, 14. 59634281 5432. readers, doubtless, are acquainted with these chemical facts, 8. 3-1567 22. 9. 293621 X 546, 15, 62327453 x 90091.

others are not; consequently, the best plan will be to present a 4, 93200 X 38. 10. 149628 X 246. 16. 49532816 X 58673.

slight outline of these facts at once. 5. 210354 X 46. 11. 274032 X 9612. 17. 101299867 X 14059,

To begin, then: did the reader ever set fire to a bit of stick 6. 149681 x 52. 12. 1429 461 X 10812. 18. 637589931 x 98765.

or a little charcoal ? No doubt he has. What does the reader 9. Divide one thousand billions by 81 and 729.

think becomes of this stick or charcoal? Is it lost, destroyed ? 10. Divide a thousand thousand millions by 111.

Oh no, there is no such thing as destruction in all nature; 11. Divide a thousand millions of millions by 1111.

substances, even when they appear to be destroyed, only change 12. Divide 908070605040302010 by 654321.

their form. What, then, becomes of a piece of stick or a piece 13. Divide 4678179387300 by tho following divisors, sepa- of charcoal when we burn either in the fire ? Now, whenever rately, 2100, 36500, 8760, 957000, 87700, 1360000, and 87000. philosophers desire to study the conditions of an experiment,

14. If the annual revenue of a nobleman be £37960, how and the choice of more than one set of conditions stands before auch is that per day, the year being supposed to be exactly them, they very properly take the simplest. We have here two

sets of conditions; the burning of a stick is one, the burning of 15. What is the nearest number to one thousand billions that a piece of charcoal is the other. The latter being the simpler of can be divided by 11111 without a remainder ?

the two, we will take it, and use it for our purposes; moreover, we

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365 days.

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