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Katre of gravity, wo have the centre of gravity of as “azes the flat ends. Moreover, as all bodies so shaped may be con. E sarfa." Also let it be understood that the centre de sidered a collection of areas, one atom thick, piled on top of Comprar of a line, straight or curved, means that pou r ses each o-ber, either perpendicularly or with a slope, like cards, or
a pile of sovereigns, the centre of gravity of each must lie also
on the line joining the centres of gravity of the two areas which NO FIND CENTRES OF GRAVITY BY CONSTRUCT:03.
form their ends. The centre itself, therefore, is the point in Dis is done by the rule for finding the centre of parade Taich this line pierces the middle cross section, as at cand e, Suns giren in Lesson IV. (page 123). We al comence Fig. 23, in the cylinder and cube. But this requires us to be
with the most gene ate to find the centre of gravity of such areas, of which take ral case, Lamely :- first the triangle..
1. I end the cat 3. To find the Centre of Gravity of a Triangle. This we do by mon Centre of Gra- corsidering the triangle made up, as in the triangle a, in Fig. 30, rity of c. y r c of lines an atom thick, all parallel to the side A B. The centre of Bolus. the pas- of gravity of each line is at its middle point. If, therefore, rate Ce..fres and I can satisfy you that the middles of all the lines are on the Weights of ickich are line cX, which joins the vertex c with the middle m of A B, giren.-The masses tee centre for the whole triangle is somewhere on that line. may be anyhow I have, then, to prove that placed, but the ope cu bisects, or divides into ration is the same two equal parts, every line whether they are all parallel to A B. Suppose,
on the same plane, now, that I cut cu into Fig. 27.
as in the case of the three equal parts, cx, xy,
balls on the ground, 'yx, as in the triangle b, in Fig. 27 above, or are some in that plane, some above, and Fig. 30, and draw paral. see below. Let them be four in number and on the same ! lels to AB at the two
ase, their centres being A, B, C, D; then four parallel forces, points of section inside, the weights, act at these centres : what has to be done? Join meeting AC and BC each Erst A with B, and cut the joining line at x inversely as the in two points from which seeghts at these points. Next connect x and c, and cut cx parallels to cu are drawn, & I inversely as the two first weights to that at c. Lastly, ' meeting A B in four points, I being joined to D, divide D y at z inversely, as the weights of two on each side of m. the three balls already used are to that of the fourth, D. This Now, since cu is equally
divided, and the white iest point, z, is the required common centre of gravity.
Fig. 29. You observe that the joining and cutting of the lines is in no figures inside are parallelograms, it is evident that the line way influenced by, or dependent on, the bodies being on the parallel to cm marked a, b, on each side, are equal to each other, Eame or in different planes, or of their number. How many and to cx, the third part of cm. Hence the three small shaded soever they be, the operation is the same. Note, also, that a triangles next to Ac are equal to each other, and have equal angles. common centre of gravity can be outside the bodies of which it Their three sides parallel to A B are therefore equal, which shows is the centre.
that A u is cut by the parallels to c m into three equal parts. For 2. To find the Centre of Gravity of a Right Line.-A mechanical the same reason BM is cut into three equal parts; and since right line being, as we have agreed, a line of atoms of equalAx is equal to BM, the six parts into which AB is divided are size and weight, the case is that which we have considered in equal to each other. You thus see that the first parallel above Lesson IV., of a number of equal parallel forces acting at equal AB is made of parts, two on either side of cm, equal to the distances from each other, along a right line. The resultant parts below, and is therefore bisected by cm. The next above passes through the middle point of that line; hence the centre is also evidently bisected, being composed of two parts, one of gravity of a right ine is its middle point.
on either side. Now, if I divide cu into five parts instead of This enables us to find the centre of gravity of a uniform rod. three, I have four other parallels also bisected by cm; if into 7 By * uniform," I mean such that the cross sections are of the or any other number, it is the same I can fill the whole triangle
same size with parallels to A B bisected each by the line c . The centre
As the cen- The centre of gravity of
tre of each a parallelogram can now
line is in its be shown to be the intermiddle, the centre of the whole bundle is in the cross section secting of its diagonals ut tibe rod's middle. And observe that this holds good of A C, B D (see c, Fig. 30); B till stuer bodies, besides were rods, which can be considered for, since the diagonals
Fig. 30. made up of equal parallel Lines, such as of a cylinder or uni- bisect each other, the line form pilier, or of a beam of taber, a eubical block of stone; B D is the bisector of the common side A c of both the triangles, the centres of gravity will be in the cross sections at their ABC and ACD. The centre of each, therefore, is on that siddle E. And it makes no difference whether the flat line, and therefore the common centre of both-that is, the
minder pdiar, beem, or block are perpendicular to centre of the parallelogram. But, by the same reason, conich it is supposed to be composed, as in cand e sidering the parallelogram made of the two triangles on BD, go to tiem, as at d and f (Fig. 29); tha the centre is on A C. Being thus on both diagonals, it is at
in the middle cross section parallel to their intersection.
4. To find the Centre of Gravity of a Polygon.—Let A B C D E weight, and equally distant from G, their common centre of
three areas of the three triangles. of the circle in which it is formed, for the ring may be con-
centre. If the polygon had more sidered as made up
Fig. 32. process is the same, and must be cles of atoms, lying continued until all the triangles into which it is necessary to one inside the other, and having the same centre, g, which, by divide the polygon have been gone over.
| the above, is therefore their common centre of gravity. 5. To find the Centre of Gravity of the Circumference of a The centre of gravity of a hollow sphere may, in like manner, Circle. Let the circumference be taken to be a curved line of be proved, by drawing lines through a to the atoms on its suratoms, as in a, in Fig. 32, to the right; and through the centre face, to be the centre of figure of the sphere; and a solid G of the circle let any line, A G B, be drawn passing through two sphere may be considered as consisting of a number of these of them, one on either side. Since these two are of equal ! hollow ones inside one another.
LESSONS IN PENMANSHIP.-XIV. the line c c, at which the up-stroke forming the loop or bow of
the letter e was commenced. IN Copy-slip No. 46 (page 196), an example was given of the Copy-slips Nos. 47 and 49, comprising the words tax and letter X. This letter is formed of the letter c twice repeated; axe, are given to show the learner how the letters x and e are the first, or the one to the left, being turned upside down, while connected with letters that precede or follow them. the second, or the one to the right, is formed in the ordinary way. In the last lesson it was said that the letters C, X, and e are The left half of the letter is commenced on the line c c with a | modifications of the letter o. The learner may prove this in a hair-line which is turned at the top to the right, and brought practical manner for his own satisfaction, if he will take the downwards without being thickened by pressure on the pen. trouble to make the letter o in pencil, on a piece of ruled paper, The hair-line is turned to the left as it approaches the line bb, and then trace the letter cor e over it in ink; or otherwise, by carried round, and terminated in a dot about midway between making the letters C and e, and then adding to them the fine the lines b b, cc. The right half is then added. It is made in hair-stroke on the right side that is required to form the comprecisely the same way as the letter C, the thick down-stroke plete oval of the letter 0. To show that x is a modification of o
touching the thin down-stroke of the turned c, and forming it will be necessary to make the letter o twice over, so that the y the thickened centre of the letter.
right side of the first touches the right side of the second, and In Copy-slip No. 48 the learner will find an example of the then trace the letter x over the double o thus formed; or, as in ' letter e, which is commenced on the central line, c c, by a hair- the case of c and e, the hair-stroke that is necessary to com
stroke carried up in a slanting direction to the right. This plete the oval of o may be added on the right and left of the hair-line is then turned at the top line, a a, and carried to the letter x. In the letters c, x, and e, the bottom-turn is carried left, and the letter is finished in the same manner as the letter to the right, beyond the limit of the bottom-turn of the letter o, C, or the right half of the letter x; but in making the thick in order to join them the more readily to any letter that may dowu-stroke care must be taken to let it pass over the point in follow them.
LESSONS IN ARITHMETIC.—XIV. colonies—nay, empires-are made ; and the object of the people
in going was to establish a settlement where politics and religion, DECIMALS (continued).
which were discouraged at home, might have freedom to live, 9. Multenecation of Decimals.
and liberty to grow. An embargo was laid upon the ships, and Io mnltiply 6-34 by 2-149.
for the time their departure was delayed. Some of the would-be
634 x 2149
voyagers never pursued their journey; they refused to give the
guarantees which were required of thern before they could get Now the numerator shows us that we must multiply the
licence to go; they returned to their homes and their duty, and figures together as in whole numbers, and the denominator
made themselves names in English history for ever. Among shows is that the result will have as many decimal places as
them were John Hampden, who first tried conclusions with the There are decimal places in the multiplier and multiplicand
king by refusing to pay a tax levied by the royal authority only; together
Sir Arthur Hazelrig, one of the most determined enemies the 634 x 2149 = 1362 465,
kingly power ever had ; John Pym, the fature leader of the ud the required result must have 5 decimal places. Hence the House of Commons, and promoter of all the constitutional Eswar is. 13-62466.
resistance which Parliament subsequently offered to the king's Hemce we see the truth of the
illegal pretensions; and last, not least, Oliver Cromwell! These Eule for the Multiplication of Decimals.
and many kindred spirits were flying from tyranny and oppres. Multiply the two numbers together, as in whole numbers, and sion at home, going with their worldly wealth to follow in the out off from the resulting product as many decimal places as the footsteps of the Pilgrim Fathers, who, a few years before, had
um of the number of decimal places in the multiplier and sailed and founded in the wild regions of the West a colony multiplicand.
where freedom was to flourish till it grew up and overshadowed Ole-When the number of significant figures in the product the land. is not as great as the sum of the number of decimal places in Certainly fate was cruel. Had these eight ships sailed! the multiplier and multiplicand, we must prefix ciphers.
Had Cromwell, and Pym, and Hampden, and the rest, been EXAMPLE.—Multiply .013 by .02.
suffered to depart, how might not English history have been Multiplying as in whole numbers, we get 26 ; but since there written differently? None, of course, can tell whether, among are 5 decimal places in the multiplier and multiplicand together, the noble army of patriots who at that time thronged Parliawe prefix 3 ciphers to 26, and the required result is by the rule ment, there might not have been found another Hampden, *00026.
another Pym to impeach Lord Strafford, another “Cromwell, The reason of this may also be seen analytically thus : guiltless of his country's blood;" but taking the men as they •013 X •02 = 1 x 1 = 3 = •00026 (Arts, 5, 6). were at the time, and considering what they afterwards became,
it is excusable to speculate upon what different scenes would EXERCISE 32.
have presented themselves, had not the "unlucky order of em1. Find the products of the following numbers, and point bargo been issued from the privy council. them according to rule :-
But why were these men going ? England had been the 1. .96 -5
15. 213.02 x 4.318.
home, not of themselves only, but of their forefathers for gene2. •358 * *096.
16. 10.2016 ~ 38.26.
rations. Cromwell's family counted among its recent members, 3. 1.0013 * .25.
17. 164.023 x 1.678.
as poor Charles afterwards found, and tried to use the knowledge 4. 3.6051 X 4.1.
18. 9:40061 X 15.812.
in bribing his enemy-that same Henry Cromwell who was 5. .1003 6:12.
19. 731042 x 10.021.
secretary to Cardinal Wolsey, and who, after that statesman's 6. 8.0001 X *004.
20, 40-4368 x 1.2904. 7. 0006 * *00012.
| fall in 1530, had risen in King Henry's service, till he became
1. 75.35060 X 62.3906. 8. •3005 X 0035.
2. 3150301 x 17.0352.
Earl of Essex, and was finally promoted to the honour of being 9. 100.0008 X 000306.
3. 0000713 X 2.30561.
executed, by order of the master he had served too well-the 10. 25067823 X 0000001.
21. 42.10062 X 3821013. master “whose commands," as Mr. Hallam tersely observes, 11. 394.20023 * *00000003.
25. 1'0112034 X *0620034. " were crimes." The other emigrants were no less illustrious, 12. 2561-21035 X 4.300506.
26. 64'301257 x 1.000402. no less bound by the strongest ties to the land of their birth. 13. 44'016 X 43.
27. 840003:1709 x 112:10371.
| What motive could they have for voluntarily forsaking all that 14. 35*601 x 1.032.
28. 0·831567834 x .00000008.
was dear to them in nationality, and turning their backs upon 2. In 1 rod there are 165 feet: how many feet are there in the country they loved ? Disgust at things as they were in the 41.3 rods?
country, and despair of ever seeing them become better. Shortly 3. In 1 degree of the earth's circumference there are 69.05 stated, these were the causes which drove such men away. British miles : how many miles are there in 360 degrees ? 4. In 1 barrel there are 31.5 gallons : how many gallons in
“We strove for honours—'twas in vain : for freedom-'tis no more," 65-25 barrels ?
they might have said with the indignant Roman citizens. 5. In 1 inch there are 2-25 nails : how many nails are there Henry the Eighth had begun that system of ruling by virtue in 60-5 inches ?
of his own strong will, which the nation afterwards, for national 6. In 1 square rod there are 30-25 square yards : how many purposes and under circumstances of national danger, allowed square yards are there in 26-05 rods ?
his daughter Elizabeth also to exercise. But even under her, 7. In 1 square rd there are 272-25 square feet: how many beneficent and nationally glorious as her reign was, the people, square feet are there in 160 rods?
by their representatives in Parliament, were perpetually striving to put a bridle on that sovereign power which the queen was so
fond of wielding. They loved her much, but they loved their HISTORIC SKETCHES.-VII.
children more, and they would not suffer her to forge chains for
freeborn limbs, nor permit that they and theirs should breathe KING CHABLE' VETO ON EMIGRATION.
by royal permission. When the dangers which caused the FATE was almost cruel to King Charles the First. One act of people for a while to submit themselves wholly to her, had his, or rather let us call it one act of his government, recoiled passed away, no time was lost in winning back rights and more upon his bed than ever fonl cannon recoiled upon its privileges which Elizabeth and her high-handed father had gonder. Biglat vennels were lying in the Thames in the early taken into their own hands. In the re-conquest it was inevitable part of the year 1637, bound for the plantations” in America. that collisions should take place between the queen and the When they were about to sail, an order came from the king in Parliament, and collisions did actually take place; but owing Souncil forbidding the masters of them to go. Obedience was to the perseverance of the House of Commons, and to the great erated by the royal officers from the all-unwilling masters, and good sense of Elizabeth, who always knew when to loosen the the intending poemsengers were compelled to land again, to dis. reins which were being held too short, the result of these disembark thuir bruggage, and to renounce the object of their putes was always favourable to right and liberty, and never cost yogare. The ships were emigrant ships, laden with colony. The queen a whit of her people's affection. But when she died, der store, and intended for colonists' use; the people in 1603, and was succeeded by James of Scotland, there were
taken passage in them were of the stuff from which / still some ugly instruments at the disposal of the crown against the liberty of the subject. The wisdom of Elizabeth's advisers Englishmen could be induced to rise up and say, “This thing had used these instruments sparingly, and had kept them as shall not be.” With a government as weak, or weaker than much as possible out of sight. They were now to fall into James's, Charles pretended even greater claims than his father, hands which knew not how to use them wisely-hands which and exercised his prerogative even more annoyingly and more clutched the blade instead of the hilt of the weapon, and got tyrannically. He levied certain taxes on the people, not only themselves badly cut accordingly.
without the consent of Parliament, but in direct contravention The ugly instruments in question were the Star Chamber and of several statutes; he issued proclamations, and required them High Commission, tribunals unknown to the common law of the to be obeyed as laws; he resented the offer of advice as un. land, exercising a jurisdiction quite incompatible with the exist warrantable interference; and he refused finally to summon the ence of liberty, and apt to become the means of all sorts of counsellors, whose advice was always so unpalatable. Brought oppression. It would take too much space to examine here the up in the notion that kings are appointed directly by God, and whole history of these courts. With regard to the former of that the Church of England was also of Divine institution, he them, the Star Chamber, much ignorance prevails, and advantage put forward offensively his own claims on the one hand, and has been taken to throw a sentimental and false colour upon its backed with all his might the claims of the Church on the other. actions, with a view to making it an element in the composition In order to do this he was necessitated to employ very exten. of historical romances. It will be sufficient to say that it was sively, in the face of increasing opposition, the two courts of a court composed of the king himself, and such members of his which mention has been made. privy council as he chose to summon; that it took cognizance Two members of Parliament, Sir John Eliot and Sir Dudley of certain offences not then noticed as such by the ordinary law Digges, were imprisoned by order of the Star Chamber, for courts, such as libel and slander, and also assumed a right to "seditious" words used by them, as members, when the Duke of take any case it chose from the consideration of the regular Buckingham was impeached ; and when the House refused to courts of law, and especially the criminal courts, and deprived a vote supplies till its members were released, the king threatened man in this way of the right of trial by his peers, which had them, but gave way about his prisoners. Then came a series of been secured for him by Magna Charta. The lords of the attacks on the constitution by the king and his ministers, which council were at once judges and jury, even in cases where the were repelled with more or less damage to the good-will between crown was concerned; there was not any appeal from their him and his people; the king tried to govern without Parliasentence, and the sentences of the court were often most ruinous ment, and Parliament was resolved there should be no peace for (notwithstanding the clause of the Great Charter which forbade him if he did. With the Earl of Strafford as chief adviser in any man to be fined to such an extent as would prevent his state affairs, and Archbishop Laud as head of the Church, Charles getting a livelihood), even where they did not condemn a man strove to make himself an absolute king, caring little apparently to imprisonment, and sometimes to torture. Any punishment how rough-shod he rode over the feelings and affections of his short of death-and many of the punishments came only just people. The honour of the nation was forgotten by a disgraceful short of it—the court of Star Chamber asserted its power to foreign policy, pirates from Morocco were allowed to prey upon inflict; and the claim having been put forward in aotion at a ships in the English Channel, the influence of England abroad time when men were not able to question it, came at length to had sunk to zero, and at home all power and statesmanship were be looked on almost as a matter of course, except by those who directed to the one object of laying the nation, bound hand and sufferec by it, and by those faithful guardians of the liberties of foot, at the feet of the king. England who only bided their time to announce that the court The Star Chamber was set in motion against the opponents of itself was an illegal thing, and ought to be abolished.
the kingly power, and indeed against all who ventured to The High Commission was a tribunal invented under Queen criticise the actions of government. Sir David Foulis was fined Elizabeth, a sort of ecclesiastical Star Chamber, composed of £5,000 for dissuading a friend from paying an unlawful tax; ecclesiastics, who made it their business to “sniff out moral Prynne, a barrister of Lincoln's Inn, for an abusive book he had taints," and to be down on any one who worshipped God in any written against some of the practices in the king's household, other way than that prescribed by the Church of England. It and against the ultra-High Church practices of the primate, was was armed with power to fine and imprison, and this power it sentenced to be disbarred, to be put in the pillory at Cheapside ned till resistance became so strong, even under Elizabeth, and at Westminster, to have both ears cut off, to be fined that it was deemed prudent to admonish it from above. It was £5,000, and to be imprisoned for life! People were ruinously a sort of Protestant Inquisition; but Englishmen were not fined for turning their arable land into pasture, in contravention Spaniards, and the seeds of priestly tyranny were crushed ere of some obscure law of Henry VII.; for refusing to lend money they could grow into a plant. Still it existed, in company with to the king; and for encroaching on the royal forests. One the Star Chamber, which ever waxed more and more intolerable man, Morley, was fined £1,000 for reviling and striking one of in its administration under the successors of Elizabeth.
the king's servants at Whitehall; another, named Allison, was Men had endured much from the Tudor princes, as they fined £1,000, imprisoned, and pilloried at Westminster, for always will endure at the hands of rulers whose strong personal having said falsely that the Archbishop of York had incurred character makes them respected, even though feared ; but from the king's displeasure. For calling the Earl of Suffolk “a base princes of the House of Stuart, they were by no means ready to lord," Sir Richard Granville was ordered to pay £4,000 to the put up with insult and oppression, so that when members of earl and £4,000 to the king ; Sir G. Markham having thrashed Parliament were cited to appear in the Star Chamber to answer, Lord Darcy's huntsman for abusing him, and having promised as to a crime, for language spoken by them in their place in Par- to do the like by Lord Darcy, should he approve his servant's liament, they resisted, and remonstrated with the king, and conduct, was fined £10,000.* Landed proprietors being ordered declared what he had done to be a breach of privilege of by the king's proclamation not to live idly in London, but to go Parliament. Against other acts of the Star Chamber, and of to their estates, were fined in the Star Chamber for non-comthe government, the Houses also protested, and Puritans in pliance. In 1637 Burton, a divine, and Bastwick, a physician, politics, as well as in religion, who had been trained up in were condemned for sedition and schism to the same punishment Elizabeth's parliaments, and who sat in the parliaments of as had been inflicted on Prynne, and that unfortunate man James, uttered their words of remonstrance and warning, not having again offended, was further mutilated and fined another ieering even the dismal dungeons in the Tower, which the £5,000. Williams, Bishop of Lincoln, was fined £10,000, and chances were would be their reward for their boldness.
sent to the Tower, for some trumpery offence against Laud; The king was despicable, his government was weak; the Osbaldistone, the master of Westminster School, for having Parliament men were for the most part noble, and unquestion. nicknamed Laud in a letter to Williams, was ordered to be ably they were strong; 60 all through the reign of James I., | pilloried before all his school, and to pay £5,000, but he saved 1603-1625, there were perpetual conflicts between the sovereign himself by flight. Lilburne, charged with distributing seditious and the people, and though when the king died the Crown had pamphlets, was whipped by the hangman, pilloried, and imnot given up any of its so-called prerogatives, there had been prisoned with irons on him. conjured up a deep spirit of resistance to them, a spirit which It was under circumstances like these, when despair seemed found expression in the reign of James's successor, his ill-fated son, Charles I.
* This case occurred in the previous reign, but it shows the tension But much had yet to be borne before order-loving, law-fearing to which the power of the court could be strung.
to have seized the minds of men ; when the king was hurrying Earl of Strafford, the supporter of the impeachment of Land. forward headlong in a career of violent misgovernment, and no the life and soul of all the constitutional opposition which the one was found to stand in his way and stop his mad course ; parliament made to the king. His name is not to the warrant when oppression seemed to be triumphant, and right and justice for the execution of Charles I. (January 30, 1648-49), thongh were openly trodden under foot; when honour had gone from, with Hampden, Hazelrig, and two more, he was one of those England, and the homes of her people were no longer pleasant five members whose arrest the king in 1641 endeavoured to places, that Hampden, and Pym, and Hazelrig, and Cromwell effect in person (see “Historic Sketches,” IV., page 120): but proposed to quit her shores and begin life anew in America. his name stands out brilliantly among those advanced patriots The royal order, arbitrarily issued, prevented them as we have and purely disinterested men who in 1641, immediately after seen. They returned to their homes and their duties, and when, the execution of Lord Strafford, wrung from the king a consent compelled as a last resource to summon Parliament, whose to the abolition by statute of the courts of Star Chamber and advice he had not sought for eleven years, the king again ad. | High Commission. dressed the House of Commons, these men were in their places, Of Oliver Cromwell, the fourth man among the detained, resolved to do their duty to the uttermost, even to exceed it it is unnecessary now to write. Much has been said for him,
bome will say. Be that as it may, of the men whom Charles's much more, but less weighty, has been said against him; but order stopped from emigrating, Hampden in the same year his name and his character have brightened since the light of brought forward the question of the king's right to levy taxes, honest, critical inquiry was turned upon him. Some there are when he resisted even to trial the demand which was made who cannot admire him enough for his policy, which raised the on him for ship-money; and he fell subsequently, mortally foreign influence of England to a height it had not attained wounded, at Chalgrove, early in the war between the king and since Henry the Fifth was crowned in France, and which at the parliament. Sir Arthur Hazelrig was foremost among the home brought order, albeit by a stern method, ont of the chaos more intemperate enemies of the king in all the subsequent into which the Great Rebellion had thrown all things. Others troubles, but he did not identify himself remarkably with any of there are who seem to think that nothing can atone for a usurthe great questions upon which the sword had finally to pro- pation which nevertheless declined to perpetuate itself by estenounce judgment. Of Pym much, but scarcely enough, has been blishing a dynasty, and who can never forgive or forget the fact written. Unselfish, truly persuaded as to the course he was that Cromwell's name appears among the first signatures on pursuing, unswerving in his fidelity to that course, incorruptible, Charles's death-warrant, and that but for him that death-warrant calm amidst tumults, a fountain of wisdom in a sea of folly, he | would never have been written.* was eminently fitted for the post which he a long while filled, that of leader of the popular party in the House of Commons. | For Synopsis of Events in the Life and Reign of Charles I., and He was the framer of the articles of impeachment against the | List of Contemporary Sovereigns, see page 122.