elc h: CORRESPONDENCE. lighted to accept a commission in the St. Helena regiment, as a retirement some marches, and in the real tragedy of war on a grand scale, would be defrom the fatigues of a long and active life of toil and wounds. I, Sir, have an enemy Helena regiment. The operation of this feeling is, I appreserved thirty years, of which seventeen have been spent in the field against and I should be delighted to retire with my rank to the quiet and so peaceful, no officer will quit it to make room for other aspirants, like hend, precisely the cause of Miles's complaint: that the post is so pleasant Miles and Your most humble Servant, AN OLD LIEUTENANT-COLONEL OF BENGAL. London, 21st June 1831. infantry is singular, as Miles says; and it certainly prevails no where else; P.S. That the artillery at St. Helena receive no higher pay than the infantry and the artillery at St. Helena, being an anomalous corps altogebut on asking a cognoscent friend the cause of it, he informed me that the and four of infantry, of which regiment the Governor is Colonel. Unless, ther, form but one regiment between them; . e. four companies of artillery therefore, Miles can extend the boundaries of the island, or raise the revenue to the necessary mark, I fear his complaints in regard to rank and pay will not be redressed. The King's brevet has not been extended to St. Helena, for this plain reason I am told, that they are so isolated they cannot be called upon to do duty with H. M.'s forces, as is always the case in India. Trisection of an Angle. MR. EDITOR,-There appeared some time ago, in the United Service Journal, a notification extracted from a German periodical, announcing that the geometrical trisection of an angle had been effected by an Austrian officer at Vienna. This induced me to commence an investigation, of which the following demonstration is the result. I never saw the curve until I discovered it myself, nor can I learn that it has ever been published in any printed work; but I do not presume to call myself the original inventor; because I have since been assured that the curve and its properties, are known in the British Universities. However, as there may be many of your naval and military readers, who have never met with it before, I take the liberty of offering it to you, for their amusement. I have the honour to be, Sir, ALFRED BURTON. TRISECTION OF ANY ANGLE, BY MEANS OF A PECULIAR CURVE. Draw any right line ATC &c. (Fig. 1,) making AT=TC. With the centre T, and radius TC, describe the circular quadrantal arc, Cee B; meeting the perpendicular radius T B, in B. From A draw any number of right lines, AB, Ad, Ad &c. on which take always edeb, and edeb• &c. Then Bdd F is the curve; the property of which is, that all angles whose sides intersect thereon; and whose vertices are at A and C; are to each other, as 1 to 3. Therefore, to trisect any given angle d'CF, take on CF any point F. FC produced, take CT=CF=TA. On Erect TB, &c. &c. and describe the curve as above found, cutting d C in d. Join Ad, and the d·AF= Ld.CF For it has been already shown by Sir Isaac Newton, (therefore the proof need not be repeated here,) that if between any two diverging right lines, Ad and AF, there be taken AbbC=Cd; the d CF will always =3 / d AF. But on AC describe a semicircle; bisect it, by the TB, and draw the chord Ce. The AbC, is isosceles, ... any bisecting AC must cut the vertex b.. The AbCd, is also isosceles, ... any bisecting b'd, must cut the vertex C. The Ae C is an in a semicircle, . Ce is to, and bisects b'd'. Ld.CF. ··‚e'b'=e'd', and the ▲ d'AF= 3 Q. E. D. When the d·AF (Fig. 1,)=22° 30′, the sine or ordinate d'g is of like value with BT, and the rectangle d'g TB becomes the diagonal section of a cube, the only rectangle divisible in infinitum, into two similar and equal rectangles. The greater side being always to the less, as 2 to 1. For Ab is evidently the secant of LbAT, to radius AT. And A e', the chord of the supplement of double that angle. eb is the difference of these two quantities. And ed e' b⋅ . Hence, A d', and any absciss, or ordinate, are easily determined. Then, if AT, TB,=TC=d'g, be denoted by a, Ag will = a + √2a2 by the conditions. And as, Ad: Rada: Line 22° 30' Or as, a + √2a2: Rada: Tangent 22° 30'. The area of the curve BdFTB, (Fig. 2) is equal to the square ABCB, (or, which is the same thing, to the square of Tg, in Fig. 1.) For on TC, erect a square, and draw the diagonal BCA B. Draw also, Co Cb = C d. The triangles or semisquares BCh, and BCT, are equal and similar. Co= Cb, and Co is to, and bisects bd. ... LdCo= LoCb. The OCB= Z BCb, .'. L BCd=3 ▲ oC B. And as the like is true of all the infinitely numerous angles into which the angle BCh is divisible, it is evident that the curve Bd F, is generated by the triplication of the semisquare BCh, which may be conceived to expand like a fan, until the side Ch coincides with CF. When the angle BCF becomes 3 BCh. Therefore, as the sum of the differences of all these infinitely numerous angles, is tripled in the angle BCF, (while their common vertex C, and the values of their sides remain unaltered,) so likewise is the sum of all the infinitely small differences of their areas, tripled in the curvilinear area Bd FCB. Hence, the area BdFTB is equal to the square ABC B. Or if the curve be completed, by carrying it round on the opposite side of the line AF, then the entire area Bd FBT B, will be equal to the square of A C. A very simple instrument for mechanically trisecting an angle, may be made by cutting a piece of card, wood, or ivory, in the shape of the curve; distinguishing the point C by an indentation, thus A с Then, if dC F, be any given angle, apply the the side C F, and the mark at C to the vertex. and remove the instrument. Join Ad, and then with, and cutting C, will trisect the angle dC F. base of the instrument to Note the points A and d, a right line parallel there EDITOR'S PORTFOLIO; OR NAVAL AND MILITARY REGISTER. AFFAIRS AT HOME AND ABROAD. -The article on "Reform" at the commencement of our present Number, circumscribes our usual comments on passing events at home. The Reform question was again introduced, in an unmitigated shape, by Lord John Russell, on Friday the 24th inst., His Majesty having opened the Parliament by a Speech from the Throne on the previous Tuesday. The Bill was brought in, and read on Saturday, and was ordered to be read a second time on Monday the 4th July. In its present form it will be warmly opposed. Serious riots have occurred at Merthyr Tidvil, the centre of an extensive mining district in Wales, in which little, if any, distress is stated to exist. These daring excesses were suppressed, with some loss of life, by the steadiness and resolution of the few troops who could be brought to the spot. We give the details in another place. Ireland continues disturbed by partial insurrection, while in other districts it is wasted by famine. A sanguinary affray recently took place at Newtownbarry between the country people, and the police supported by the yeomanry. The former had fire-arms, were apparently organised, and attacked the Yeomanry, who retaliated, as it would appear, in self-defence, after one of their number had been shot, and several wounded. What have not the ruthless Agitators of that unhappy country to answer for! Extraordinary vicissitudes, though unattended with decisive results, U. S. JOURN. No. 32. JULY 1831. have marked the operations of the belligerents in POLAND, since the occurrences included in our last notice. The Polish commander, suddenly breaking through the Russian line by one of those bold, though rash movements which have distinguished his active tactics, advanced rapidly to the north by the Narew, driving the feeble links of the Russian chain before him; and having occupied Lomza, a fortified post, and reached Tycozin, within a short distance of Bialystock on the Wilna Road, garrisoned by the Russian Guards, he succeeded in throwing the respective corps of Generals Chlapowski and Gielgud into Lithuania, to aid and stimulate the insurgents of that province. This was the ostensible object of his irruption. The consequences of a movement so hazardous, in a military sense, did not, however, escape the vigilance of Count Diebitsch, who, rallying from the torpor and depression of baffled schemes and thwarted pretensions, made an effort, though without adequate promptitude and vigour, to cut off the main Polish army from Praga. Retiring at first from the Bug to concentrate a sufficient force, he made a flank attack on Skrzynecki, then in full retreat. The encounter took place at Ostrolenka, on the Narew, to the advantage of the Russians, though the gallant Pole cut his way through, and precipitately retreated on Praga. Here he remained for some time to refresh his troops and repair his losses. On the 14th June, the Po 2 E lish Commander again advanced, with the avowed determination of striking a decisive blow. Meanwhile the several corps thrown into Lithuania are stated to have gained advantages in the field, and to have succeeded in rousing and sustaining the spirit of insurrection. There has been a fatality throughout this war, favourable to the Poles. Amongst its signs we may enumerate the frozen bridges of the Vistula gliding suddenly away-the cholera waging an exterminating war upon their ill-provided opponentsdisunion and fatigue paralysing the hostile army-and, in fine, the "Balkan-passer" checked, distracted by difficulties, and undermined, at length succumbing to his altered destiny by taking refuge in the grave. Count Diebitsch died, as variously stated, of apoplexy or cholera, at his head-quarters near Pultusk, on the 10th of June. Count Paskewitsch Erivanski, his victorious co-operator by the line of the Caucasus in the war of Turkey, is expected to succeed him. Of FRANCE there is little new to be recorded. The march of anarchy proceeds, denoted by those periodical outbreaks of popular licentiousness, which ever and anon convulse society, and shake to its foundation the flimsy fabric of government in that bankrupt and besotted country. Paris has been the scene of renewed disorders-the King has been insulted in the course of his tour through the Provinces of the Rhine -Russia, it is stated, has been equally uncivil, for which summary reprisals are threatened-a mighty expedition is destined to chastise the "insolence" of Don Miguel, at which Spain very naturally waxes wroth; and, in truth, it will be a miracle if our Gallic neighbours do not find some outlet for their martial spirit and overgrown army. The BELGIANS have formally elected Prince Leopold as their King by a large majority of the Congress ; but there being two parties to this election, it is not yet confirmed, although it is expected His Royal Highness will accept the proffered Crown, upon suitable conditions. The deputation sent to offer it has returned to Brussels with his ultimatum, which is supposed to be in the affirmative. Gen. Chassé has been compelled to restrain the Rabble of Antwerp by the only arguments of any weight with "les Braves Belges." A revolution has been enacted in the enlightened empire of Brazil. A revolt was got up at Rio by a handful of ignorant conspirators, which so disgusted Don Pedro, who deemed himself par excellence the Constitutional Emperor of either world, that he abdicated the sceptre of that magnificent country and its semi-savage population in favour of his son, an unhappy infant. The Ex-Emperor has been rusticating, after the fashion of Dioclesian, on the soil of Liberal France, having come to Europe in H. M. Ship Volage, Lord Colchester. His Ex-Majesty has arrived in London. In TURKEY a serious insurrection of the Albanians, under the Pasha of Scutari, has been defeated by the Grand Vizier; by which success, the power of Sultan Mahmoud appears to be still further confirmed. GREECE is the theatre of renewed convulsions, and appears as far removed as ever from the blessings of peace, or the security of life or property. Without established Institutions, or a Government entitled to respect, what does the nominal Freedom of these wayward Greeks avail them? It cannot even protect them from themselves. In the VALAIS, there have been also disturbances. Who that knows that sterile Region of Cretins and Imbecility will ascribe this movement to the March of Intellect? |