sarface" Also let it be understood that the centre of ty of a line, straight or curved, means that post for such

Fig. 27 above, or are some in that plane, some above, and e below. Let them be four in number and on the same pane, their centres being A, B, C, D; then four parallel forces, the weights, act at these centres: what has to be done? Join ft A with B, and cut the joining line at x inversely as the weghts at these points. Next connect x and c, and cut cx at inversely as the two first weights to that at c. Lastly, I being joined to D, divide D Y at z inversely, as the weights of the three balls already used are to that of the fourth, D. This iast point, z, is the required common centre of gravity.

You observe that the joining and cutting of the lines is in no way influenced by, or dependent on, the bodies being on the same or in different planes, or of their number. How many Foever they be, the operation is the same. Note, also, that a common centre of gravity can be outside the bodies of which it is the centre.

2. To find the Centre of Gravity of a Right Line.-A mechanical right line being, as we have agreed, a line of atoms of equal size and weight, the case is that which we have considered in Lesson IV., of a number of equal parallel forces acting at equal distances from each other, along a right line. The resultant passes through the middle point of that line; hence the centre of gravity of a right ine is its middle point.

This enables us to find the centre of gravity of a uniform rod. By "uniform," I mean such that the cross sections are of the same size and form throughout its length. Such a body may be considered a collection of

middle. the centre of the whole bundle is in the cross section at the rod's middle. And observe that this holds good of all other bodies, besides mere rods, which can be considered made up of equal para lues, such as of a cylinder or uniform piliar. or of a beam of timber, a eubical block of stone; the of gravity will be in the cross sections at their And it makes no difference whether the flat Serpillar, beam, or block are perpendicular to te supposed to be composed, as in c and e to them, as at d and f (Fig. 29); the in the middle cross section parallel to

adered a collection of areas, one atom thick, piled on top of each other, 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 form their ends. The centre itself, therefore, is the point in which this line pierces the middle cross section, as at e and e, Fig. 28, in the cylinder and cube. But this requires us to be able to find the centre of gravity of such areas, of which take first the triangle..

3. To find the Centre of Gravity of a Triangle.-This we do by ecnsidering the triangle made up, as in the triangle a, in Fig. 30, of lines an atom thick, all parallel to the side A B. The centre of gravity of each line is at its middle point. If, therefore, I can satisfy you that the middles of all the lines are on the line cx, which joins the vertex c with the middle м of A B, the centre for the whole triangle is somewhere on that line. I have, then, to prove that ca bisects, or divides into two equal parts, every line parallel to A B. Suppose, now, that I cut cм into three equal parts, cz, zy, y M, as in the triangle b, in Fig. 30, and draw parallels to A B at the two points of section inside, meeting AC and BC each in two points from which parallels to cм are drawn, meeting A B in four points, two on each side of M. Now, since cм is equally divided, and the white figures inside are parallelograms, it is evident that the line parallel to c M marked a, b, on each side, are equal to each other, and to cr, the third part of CM. Hence the three small shaded triangles next to AC are equal to each other, and have equal angles. Their three sides parallel to A B are therefore equal, which shows that A M is cut by the parallels to c M into three equal parts. For the same reason B M is cut into three equal parts; and since AM is equal to в M, the six parts into which A B is divided are equal to each other. You thus see that the first parallel above A B is made of parts, two on either side of c M, equal to the parts below, and is therefore bisected by c M. The next above is also evidently bisected, being composed of two parts, one on either side. Now, if I divide cM into five parts instead of three, I have four other parallels also bisected by CM; if into 7 or any other number, it is the same-I can fill the whole triangle with parallels to A B bisected each by the line c M. The centre of gravity of the triangle is therefore on C M.

Fig. 29.

But by a similar reasoning it can be shown that this centre of gravity must be in AL (in triangle a, Fig. 30) bisecting A c. Hence we have for rule that, in order to find the centre of gravity of a triangle, we

must join any two of its vertices with the middle

points of the sides opposite to them, and that the intersection G of the joining lines is the required point. This centre G is distant from м one-third of c M, and from L one-third of A L.

The centre of gravity of a parallelogram can now be shown to be the intersecting of its diagonals A C, BD (see c, Fig. 30); B for, since the diagonals bisect each other, the line

Fig. 30.

B D is the bisector of the common side A c of both the triangles, A B C and AC D. The centre of each, therefore, is on that line, and therefore the common centre of both-that is, the centre of the parallelogram. But, by the same reason, considering the parallelogram made of the two triangles on B D, the centre is on A C. Being thus on both diagonals, it is at their intersection.

4. To find the Centre of Gravity of a Polygon.-Let A B C DE (Fig. 31) be the polygon, and from the angle A draw the dotted lines A C, A D to the remote angles c and D. The polygon is thus cut up into three triangles. Let G, H, and к be the centres of gravity of these latter figures; there are thus three bodies whose centres, G, H, and K, are known, and whose masses are the

B

three areas of the three triangles. Suppose now that you had calculated these areas, and had them written down in numbers. Then join G with H and cut G H at x inversely as the numbers expressing the areas of the triangles A B C, A D C. Connect x now with K, Dand cut K X at y inversely, as the quadrilateral A B C D to the triangle AED; the point y is the required centre. If the polygon had more Fig. 31. sides than are in Fig. 31, the process is the same, and must be continued until all the triangles into which it is necessary to divide the polygon have been gone over.

E

5. To find the Centre of Gravity of the Circumference of a Circle. Let the circumference be taken to be a curved line of atoms, as in a, in Fig. 32, to the right; and through the centre G of the circle let any line, A G B, be drawn passing through two of them, one on either side. Since these two are of equal

weight, and equally distant from G, their common centre of gravity is the middle of A B, that is, the point G. So, likewise, going round the figure, the centre of gravity of every opposite pair of atoms is G, and therefore G is the common centre of all, or of the circumference.

The centre of gravity of a ring is thus seen to be the centre of the circle in which it is formed, for the ring may be considered a bundle of circles an atom thick, bound together, one above and around the other, so as to have for common centre of gravity the centre of the central circle. The centre of gravity of the area of a circle is also the centre of figure of the circle, for the area may be considered as made up of a number of circles of atoms, lying

由

Ө

Fig. 32.

one inside the other, and having the same centre, G, which, by the above, is therefore their common centre of gravity.

The centre of gravity of a hollow sphere may, in like manner, be proved, by drawing lines through a to the atoms on its surface, to be the centre of figure of the sphere; and a solid sphere may be considered as consisting of a number of these hollow ones inside one another.

LESSONS IN PENMANSHIP.-XIV.

IN Copy-slip No. 46 (page 196), an example was given of the letter X. This letter is formed of the letter C twice repeated; the first, or the one to the left, being turned upside down, while the second, or the one to the right, is formed in the ordinary way. The left half of the letter is commenced on the line c c with a hair-line which is turned at the top to the right, and brought downwards without being thickened by pressure on the pen. The hair-line is turned to the left as it approaches the line bb, carried round, and terminated in a dot about midway between the lines b b, c c. The right half is then added. It is made in precisely the same way as the letter C, the thick down-stroke touching the thin down-stroke of the turned c, and forming the thickened centre of the letter.

In Copy-slip No. 48 the learner will find an example of the letter e, which is commenced on the central line, c c, by a hairstroke carried up in a slanting direction to the right. This hair-line is then turned at the top line, a a, and carried to the left, and the letter is finished in the same manner as the letter C, or the right half of the letter X; but in making the thick down-stroke care must be taken to let it pass over the point in

the line cc, at which the up-stroke forming the loop or bow of the letter e was commenced.

Copy-slips Nos. 47 and 49, comprising the words tax and axe, are given to show the learner how the letters X and e are connected with letters that precede or follow them.

In the last lesson it was said that the letters C, X, and e are modifications of the letter o. The learner may prove this in a practical manner for his own satisfaction, if he will take the trouble to make the letter O in pencil, on a piece of ruled paper, and then trace the letter C or e over it in ink; or otherwise, by making the letters C and e, and then adding to them the fine hair-stroke on the right side that is required to form the complete oval of the letter o. To show that x is a modification of o it will be necessary to make the letter O twice over, so that the right side of the first touches the right side of the second, and then trace the letter X over the double ✪ thus formed; or, as in the case of C and e, the hair-stroke that is necessary to complete the oval of o may be added on the right and left of the letter X. In the letters C, X, and e, the bottom-turn is carried to the right, beyond the limit of the bottom-turn of the letter O, in order to join them the more readily to any letter that may follow them.

the liberty of the subject. The wisdom of Elizabeth's advisers had used these instruments sparingly, and had kept them as much as possible out of sight. They were now to fall into hands which knew not how to use them wisely-hands which clutched the blade instead of the hilt of the weapon, and got themselves badly cut accordingly.

The ugly instruments in question were the Star Chamber and High Commission, tribunals unknown to the common law of the land, exercising a jurisdiction quite incompatible with the existence of liberty, and apt to become the means of all sorts of oppression. It would take too much space to examine here the whole history of these courts. With regard to the former of them, the Star Chamber, much ignorance prevails, and advantage has been taken to throw a sentimental and false colour upon its actions, with a view to making it an element in the composition of historical romances. It will be sufficient to say that it was a court composed of the king himself, and such members of his privy council as he chose to summon; that it took cognizance of certain offences not then noticed as such by the ordinary law courts, such as libel and slander, and also assumed a right to take any case it chose from the consideration of the regular courts of law, and especially the criminal courts, and deprived a man in this way of the right of trial by his peers, which had been secured for him by Magna Charta. The lords of the council were at once judges and jury, even in cases where the crown was concerned; there was not any appeal from their sentence, and the sentences of the court were often most ruinous (notwithstanding the clause of the Great Charter which forbade any man to be fined to such an extent as would prevent his getting a livelihood), even where they did not condemn a man to imprisonment, and sometimes to torture. Any punishment short of death-and many of the punishments came only just short of it-the court of Star Chamber asserted its power to inflict; and the claim having been put forward in action at a time when men were not able to question it, came at length to be looked on almost as a matter of course, except by those who suffered by it, and by those faithful guardians of the liberties of England who only bided their time to announce that the court itself was an illegal thing, and ought to be abolished.

The High Commission was a tribunal invented under Queen Elizabeth, a sort of ecclesiastical Star Chamber, composed of ecclesiastics, who made it their business to "sniff out moral taints, and to be down on any one who worshipped God in any other way than that prescribed by the Church of England. It was armed with power to fine and imprison, and this power it used till resistance became so strong, even under Elizabeth, that it was deemed prudent to admonish it from above. It was a sort of Protestant Inquisition; but Englishmen were not Spaniards, and the seeds of priestly tyranny were crushed ere they could grow into a plant. Still it existed, in company with the Star Chamber, which ever waxed more and more intolerable in its administration under the successors of Elizabeth.

Men had endured much from the Tudor princes, as they always will endure at the hands of rulers whose strong personal character makes them respected, even though feared; but from princes of the House of Stuart, they were by no means ready to put up with insult and oppression, so that when members of Parliament were cited to appear in the Star Chamber to answer, as to a crime, for language spoken by them in their place in Parliament, they resisted, and remonstrated with the king, and declared what he had done to be a breach of privilege of Parliament. Against other acts of the Star Chamber, and of the government, the Houses also protested, and Puritans in politics, as well as in religion, who had been trained up in Elizabeth's parliaments, and who sat in the parliaments of James, uttered their words of remonstrance and warning, not fearing even the dismal dungeons in the Tower, which the chances were would be their reward for their boldness.

The king was despicable, his government was weak; the Parliament men were for the most part noble, and unquestionably they were strong; so all through the reign of James I., 1603-1625, there were perpetual conflicts between the sovereign and the people, and though when the king died the Crown had not given up any of its so-called prerogatives, there had been conjured up a deep spirit of resistance to them, a spirit which found expression in the reign of James's successor, his ill-fated son, Charles I.

Englishmen could be induced to rise up and say, "This thing shall not be." With a government as weak, or weaker than James's, Charles pretended even greater claims than his father, and exercised his prerogative even more annoyingly and more tyrannically. He levied certain taxes on the people, not only without the consent of Parliament, but in direct contravention of several statutes; he issued proclamations, and required them to be obeyed as laws; he resented the offer of advice as unwarrantable interference; and he refused finally to summon the counsellors, whose advice was always so unpalatable. Brought up in the notion that kings are appointed directly by God, and that the Church of England was also of Divine institution, he put forward offensively his own claims on the one hand, and backed with all his might the claims of the Church on the other. In order to do this he was necessitated to employ very extensively, in the face of increasing opposition, the two courts of which mention has been made.

Two members of Parliament, Sir John Eliot and Sir Dudley Digges, were imprisoned by order of the Star Chamber, for "seditious" words used by them, as members, when the Duke of Buckingham was impeached; and when the House refused to vote supplies till its members were released, the king threatened them, but gave way about his prisoners. Then came a series of attacks on the constitution by the king and his ministers, which were repelled with more or less damage to the good-will between him and his people; the king tried to govern without Parliament, and Parliament was resolved there should be no peace for him if he did. With the Earl of Strafford as chief adviser in state affairs, and Archbishop Laud as head of the Church, Charles strove to make himself an absolute king, caring little apparently how rough-shod he rode over the feelings and affections of his people. The honour of the nation was forgotten by a disgraceful foreign policy, pirates from Morocco were allowed to prey upon ships in the English Channel, the influence of England abroad had sunk to zero, and at home all power and statesmanship were directed to the one object of laying the nation, bound hand and foot, at the feet of the king.

The Star Chamber was set in motion against the opponents of the kingly power, and indeed against all who ventured to criticise the actions of government. Sir David Foulis was fined £5,000 for dissuading a friend from paying an unlawful tax; Prynne, a barrister of Lincoln's Inn, for an abusive book he had written against some of the practices in the king's household, and against the ultra-High Church practices of the primate, was sentenced to be disbarred, to be put in the pillory at Cheapside and at Westminster, to have both ears cut off, to be fined £5,000, and to be imprisoned for life! People were ruinously fined for turning their arable land into pasture, in contravention of some obscure law of Henry VII.; for refusing to lend money to the king; and for encroaching on the royal forests. One man, Morley, was fined £1,000 for reviling and striking one of the king's servants at Whitehall; another, named Allison, was fined £1,000, imprisoned, and pilloried at Westminster, for having said falsely that the Archbishop of York had incurred the king's displeasure. For calling the Earl of Suffolk "a base lord," Sir Richard Granville was ordered to pay £4,000 to the earl and £4,000 to the king; Sir G. Markham having thrashed Lord Darcy's huntsman for abusing him, and having promised to do the like by Lord Darcy, should he approve his servant's conduct, was fined £10,000.* Landed proprietors being ordered by the king's proclamation not to live idly in London, but to go to their estates, were fined in the Star Chamber for non-compliance. In 1637 Burton, a divine, and Bastwick, a physician, were condemned for sedition and schism to the same punishment as had been inflicted on Prynne, and that unfortunate man having again offended, was further mutilated and fined another £5,000. Williams, Bishop of Lincoln, was fined £10,000, and sent to the Tower, for some trumpery offence against Laud; Osbaldistone, the master of Westminster School, for having nicknamed Laud in a letter to Williams, was ordered to be pilloried before all his school, and to pay £5,000, but he saved himself by flight. Lilburne, charged with distributing seditious pamphlets, was whipped by the hangman, pilloried, and imprisoned with irons on him.

It was under circumstances like these, when despair seemed

to have seized the minds of men; when the king was hurrying forward headlong in a career of violent misgovernment, and no one was found to stand in his way and stop his mad course; when oppression seemed to be triumphant, and right and justice were openly trodden under foot; when honour had gone from England, and the homes of her people were no longer pleasant places, that Hampden, and Pym, and Hazelrig, and Cromwell proposed to quit her shores and begin life anew in America. The royal order, arbitrarily issued, prevented them as we have seen. They returned to their homes and their duties, and when, compelled as a last resource to summon Parliament, whose advice he had not sought for eleven years, the king again addressed the House of Commons, these men were in their places, resolved to do their duty to the uttermost. even to exceed it

Earl of Strafford, the supporter of the impeachment of Laud, the life and soul of all the constitutional opposition which the parliament made to the king. His name is not to the warrant for the execution of Charles I. (January 30, 1648-49), though with Hampden, Hazelrig, and two more, he was one of those five members whose arrest the king in 1641 endeavoured to effect in person (see "Historic Sketches," IV., page 120); but his name stands out brilliantly among those advanced patriots and purely disinterested men who in 1641, immediately after the execution of Lord Strafford, wrung from the king a consent to the abolition by statute of the courts of Star Chamber and High Commission.

Of Oliver Cromwell, the fourth man among the detained, it is unnecessary now to write. Much has been said for him,

some will say. Be that as it may, of the men whom Charles's order stopped from emigrating, Hampden in the same year brought forward the question of the king's right to levy taxes, when he resisted even to trial the demand which was made on him for ship-money; and he fell subsequently, mortally wounded, at Chalgrove, early in the war between the king and the parliament. Sir Arthur Hazelrig was foremost among the more intemperate enemies of the king in all the subsequent troubles, but he did not identify himself remarkably with any of the great questions upon which the sword had finally to pronounce judgment. Of Pym much, but scarcely enough, has been written. Unselfish, truly persuaded as to the course he was pursuing, unswerving in his fidelity to that course, incorruptible, calm amidst tumults, a fountain of wisdom in a sea of folly, he was eminently fitted for the post which he a long while filled, that of leader of the popular party in the House of Commons. He was the framer of the articles of impeachment against the

much more, but less weighty, has been said against him; but his name and his character have brightened since the light of honest, critical inquiry was turned upon him. Some there are who cannot admire him enough for his policy, which raised the foreign influence of England to a height it had not attained since Henry the Fifth was crowned in France, and which at home brought order, albeit by a stern method, out of the chaos into which the Great Rebellion had thrown all things. Others there are who seem to think that nothing can atone for a usurpation which nevertheless declined to perpetuate itself by esta blishing a dynasty, and who can never forgive or forget the fact that Cromwell's name appears among the first signatures on Charles's death-warrant, and that but for him that death-warrant would never have been written.*

* For Synopsis of Events in the Life and Reign of Charles I., and List of Contemporary Sovereigns, see page 122.