at those bases; or it must make an angle greater than this; or it must never fall in with any of the angular points at all, which can only be in consequence of the series having met the axis. In the first case, it is plain that the angles opposite to the bases are not right angles; for if so, there would be two straight lines perpendicular to a third straight line, and which yet meet. Still less could the lines in question meet if the angles were less than right angles; therefore they must be greater. In the second case, it has been shown that the series being continued must meet the axis; and consequently the sides opposite to the bases must form an interior polygon, and meet the axis also. From which it is clear that the angles cannot be right angles, for then the sides would be in one straight line, and two straight lines would inclose a space; and still less could the meeting be effected if the angles were less than right angles; therefore they are greater. And the same inferences hold good in the third case; in consequence of its having been proved that in this case also the exterior series must have met the axis. In a note it is attempted to show, that this third case might be dispensed with, by proving that there must be made at least one pair of cusps after the moving line has passed beyond the extremity of the side of the quadrilateral figure which was prolonged to make the axis, without the possibility of this being prevented by the series meeting the axis. 6,?-, ། ༽

If all this can be established, it is an easy inference that if the angles at the base of the quadrilateral figure are right angles, the angles opposite to the base must be so also. From which it readily follows, that the side opposite to the base is equal to the base. That the angles of any right-angled triangle are equal to two right angles, may be proved by completing the quadrilateral figure And by drawing a perpendicular to a side that lies between two acute angles, from the angular point opposite, any triangle may be divided into two right-angled triangles; from which it may be inferred that the three angles of any triangle are equal to two right angles. After this, the proposition conveyed in what is commonly called the 12th Axiom, may be proved in the case where one of the angles is a right angle, by taking a point in the line which makes an acute angle with the line that intersects the two others, drawing a perpendicular from this point to the intersecting line, and then constructing ranks of quadrilateral figures which from the previous data it is easy to prove rectangular and equal in all respects, and showing that the line from a point in which the perpendicular was drawn, must coincide with the diagonals of a succession of these rectangular figures, and consequently must

t

at length meet the other line. And when this is proved in the case where one of the angles is a right angle, it is easy to prove the same in the case where neither is a right angle.

precise resting

If this should happen to be true, it appears to be no reasonable objection to say that it is long. If it can be shortened, till it is as short as the First Proposition of the First Book, it will be well; but if it cannot, it is not a reasonable objection to say that it is long. It be to know, of any individual geometer, at what point recommend the introduction of bad reasoning for the sake of brevity. There are, on the common calculation, forty-seven mortal Propositions, before arriving at the demonstration of the square of the hypothenuse. Why should not two-thirds of these be cut down, by an effort of our royalists in Geometry?"

ART. XVIII.—Considérations Politiques sur le temps présent. Paris. 1830.

THERE is a prodigious difference between the day before a great victory, and the day after. And manifold are the occupations, the interests, the engagements, which start into existence in the first moments of the felicitous decision. Some collect the facts, some crown the victors, some honour the lost, some assist the disabled; or more strictly they do all and each of these at once, to the greatest extent which the nature of things admits. Among all these employments, there is one more, which is not incompatible with any of the rest; and that is to endeavour to extend the influence of the results,-an occupation peculiarly acceptable to those, who though removed from any direct operation on the scene of action, have laboured for months and years, in the department that was open to them, to clear some roads for possible advances, to provide some checks for possible efforts of the adversary, and in their remote sphere to organize or to forward some branch and portion of the general aim. All men cannot be every where; if somebody is to be nearest, somebody must be further off; but it is a pleasant thing to have ridden private gentleman on the right and the victorious side, though the service went no further than cheering on certain individuals to throw their caps up for the good old cause, or helping to have a wounded drum-boy laid snugly in a captain's quarter.

It is done. The Stuarts are beaten over again. Our forefathers are in the right in August; though they were in the wrong in June. The rubbish that we like fools and milksops had

allowed to be heaped upon their graves, has all been shovelled away by the sapeurs of the National Guard of Paris. It will be long before we are told again in the House of Commons, that it would be absurd for Englishmen to think of resisting tyranny, in the existing state of the continent. The horn blowers of arbitrary power in England, have bethought themselves of asking, what the French people have got by their revolution. This is what we have got by it; and the French people are probably as well acquainted with their own side of the account.

And what have the French people got; and what is each individual's share? Ask what each Englishman would have got, if the sovereign and form of government supported by the British people had been removed by foreign arms, and supplanted by a race whom every village in England had sacrificed twenty of its children to keep out; and if this worst of sufferings and deepest of misfortunes, had by one heroic effort of the metropolis been cast off, and the intrusive usurpers sent to eat their discreditable bread in foreign lands. Inquire accurately, what portion of satisfaction would have been the particular rever sion of each individual Englishman, if, in about the time that a birthday ox takes in roasting, his country had risen from being an object of commiseration to the benevolent and of scorn to the haughty, to set itself at the head of all previously existing combinations of human kind, and stand forth a lode-star and a guide, with brightness that history cannot parallel nor imagination surpass. Fancy a man yesterday nothing, to-day every thing. Calculate the difference between bearing about an iron in the soul, and walking in the consciousness of being pursued and pressed upon by the gratitude of human kind. Exhaust all images of such as have lain down in sorrow and risen in joy; and then tell the English absolutist, what each individual Frenchman has gained by his Revolution. La belle France! whose men were so gallant, and whose women were so beautiful; and that had suffered so much in the cause of all mankind! If there be gratitude to heaven for raising up some people to be a light and a guide to Europe, there shall be yet a subsidiary thanksgiving, that this people should have

been hers.

But this is diverging; but who can help it? The proposal was, to forward the results. And what are the results? In England, these. That fifty years of the labour of bad governors to bury and depress the principles of our own Revolution, have been swept away at a blow; and we, the friends of that Revolution and of all revolutions that the defence of civil liberty makes.

necessary, are uppermost. Heaven forbid an atom of the honour should be taken from those to whom honour is due. As far as Englishmen were concerned (saving always the exertions of some half score of our countrymen who fought at Paris) it was pretty nearly an act of God. But here we are. The men who have overwhelmed us with debt to pay for helping despotism in all corners of the earth, who charged us with the American and French wars, and then told us to look at the continent, and see what chance there was of resisting tyranny at home, the men who made peace only to give us the Corn Laws and the Six Acts, and bestowed on us the last twigs of that birch within twenty-four hours of the deliverance of Paris, the men who made the Manchester massacres (military light has been thrown upon Manchester massacres since then; there will be no more ;)-all these are just now rolling in the gulph, like their prototypes of poetry. They are in the condition of the wolf that is taken in a pitfall; who, say the naturalists, is so alarmed, that he may be gone in upon and muzzled, without offering to resist. They know their strength; and we know ours. Nine tenths of the British people are at this moment united in one strong bond of attachment and zeal for the recovered principles of their forefathers, and anxious to prove that they are not unworthy to weave a portion of the same web. They have only to show themselves and to come forward, in all those constitutional ways with which they happily are provided; and their adversaries will surely take advice will act like sensible men, and lie down quietly when they are down, and save a great mass of trouble to themselves and every body. Let them bethink themselves of their own argument,-a Daniel come to judgment!-If the people of England had no physical chance for resistance then, what ultimate chance would there be for the refusal of its just claims now? But they will be wiser; they will know better than to run needlessly into harm's way. There will always, under the very best state of things, be a modicum of scraps and cheese-parings for Tory man to live by; and they will be contented with what heaven sends them, without attempting to increase it by paying the double of its worth.

This then is the first result to England;-that her people, like the French, have risen in a week, from the state of conquest in which they were placed by the unfortunate successes of the Tory arms. They are no longer the sub jacti, the thralls of the sword and of the bow, of the Holy Alliance, nor of the English branch of it. They have walked forth, by the blessing of Providence and many strange combined interferences, into a

state of light and liberty, to which there is nothing comparable in their history, except the change produced by the accession of the other William. What our forefathers were then, we are now. There is no use in being diffuse, in carrying on the parallel.

1

In this new state of things, the manifest duty and interest of the British people, is to begin, support, and promote every measure, at home and abroad, by which the stamp of final ill success and defeat may most effectually and prominently be put upon the measures of their enemies. It is to do every thing, which may help to undo what the others did. To think of staying their hand, because the things to be undone were the deeds of British misgoverners and British bad ministers, would be as absurd as to have carried on the war with Holland after the disappearance of James the 2nd. It is true that the war had been carried on with English treasure and with English blood; but it is precisely because it had been so carried on and in the teeth of the interests of the English people, that war, if there is to be war, must be on the other side when things have changed. The Jacobites, no doubt, looked back with poignant regret to the times when Englishmen cut the throats of Dutch republicans, and Dutch republicans cut theirs, for the special end and purpose of maintaining a Holy-Alliance pressure upon the liberties of both ;-but it would have been a most strange result, if Englishmen, after their liberation from the Holy Alliance influence, had allowed themselves to be misled by appeals to the glories of Dutch wars, and the successes of Holy-Alliance battles. If Englishmen loved military glory, there might be glory on the right side, as well as on the wrong; and it was not long before they had a Marlborough, whose glories were as good as any that had preceded. It would have been a most vain and bootless process, that should have tried to stay the progress of events, by getting up dinners to anti-Dutch commanders; and those commanders themselves, would have been weaker men than they were ever taken for, if they had not the genius to put themselves at the head of the new movement, instead of harping upon the old. Russell in the Downs, was as mute as Wellington at Manchester; and where there was no use in grumbling, those were wisest who said least. We are all content, from the throne downwards; with exceptions in about the same proportion, as the white blackbirds and prodigious gooseberries in a country newspaper. The whole game is on our side of the board, king, queen, and all; if the others can produce a bishop or a knight, it is as much as they can show. Every body is in great good humour with