« 이전계속 »
as into an te in coma ruler on ■esponding e into the e into the come into 'the three , shewing, a quantity ingles, lgle of any first, after four right
hesitation he interior They have ry as any ting like a. '• to the n "under aj "would 8 "coincide "finite sp "to this i "does not
A defini intended t
If we w of the uti then use neglect in
Geomet tion be coi to introdu parcel of t by these 1 formed b] the idea i definition.
Since tl commensu should be (or some c
As the i lines as pi face of a 1 as the thii plane suri being unli
The definition has other faults: instead of saying "bounded in one direction, unlimited in the other," he should have said bounded in two directions, unbounded otherwise, (or elsewhere): for surely the thing meant is bounded by two straight lines, and therefore in two directions. Moreover, not unlimited, but unbounded, is the opposite of bounded. If any where, surely in geometrical definitions, it is indespensible that words should be used with strict propriety, so as to avoid confusion. With equal propriety it might be said, that the angle ABC + the angle BCD = the corner A BD.
As I hold that the definition of angle here proposed is a failure, so likewise is the demonstration of the property in Prop. VI., that the sum of the three angles of the triangle are equal j) to two right angles; and for the same reason. The space E F below the line D B C G may belong to the angle A, or to any thing else, as in the annexed figure.
Instead of systematizing and refining till we get our ideas into an atmosphere too sublime for them to be of any use, we may take in common sense view of the subject. In the triangle ABC lay a ruler on the line A B, marking the ends towards A and B with the corresponding letters. Turn the ruler about A, till the end marked B come into the direction A C; then let it turn about C, till the end A come into the direction C B; and finally let it turn about B, till the end B come into the direction B A. The ruler has thus turned about each of the three angles, and the ends marked A and B have changed places, shewing, that the sum of the three angles of the triangle are equal to quantity formed by turning a straight line half round, or to two right angles.
In like manner, if we measure the exterior angles of the triangle of any polygon, the ruler at last will have the same direction as at first, after having gone completely once round; or after having described four right angles.
If I were to write a Treatise on Geometry, I should without hesitation introduce these as demonstrations of the theorems regarding the interior angles of a triangle, and the exterior angles of any polygon. They have long appeared to me to be quite as evident and satisfactory as any principles in Geometry. A good treatise should be something like a. good map, shewing not merely one high road through the country, but also the principal cross-roads connecting the different parts of the country with each other. In point of fact, no one is considered to be master of the subject, till he be pretty fully acquainted with these cross-connections.
We may chop logic as long as we please, but after all there is precisely the same difficulty in conceiving straight lines to be lengthened by producing them, as in conceiving angles to be increased by the continued revolution of one of the sides about a point, or by the lengthening of the circular arc measuring them. Each is accomplished by motion. The straight line is produced by another straight line laid partly upon it, and partly beyond it, or by conceiving the line to move along itself, all points between the fore-end of the old line, and the rear end of the new being common to both lines. In like manner, an angle or circular arc is increased either by a line revolving about a point, or by conceiving the arc to move upon itself, so as to have all the points between the fore-end of the old arc, and the rear end of the new arc, common to both ares. In this way the idea of a fixed centre is unnecessary for any but the first part of the arc, just as the idea of a fixed direction is unnecessary for any but the first part of the straight line.
Notes on the Recent Earthquakes on the North-Western Frontier. Bj Lieutenant R. Baird Smith, Bengal Engineers.
On the forenoon of Saturday, the 19th of February 1842, a seven shock of an Earthquake was experienced at different points in the countries on our North-Western Frontier, and extending thence i affected, although with much reduced intensity, several of the district of the North-Western Provinces.
The remotest point at which its devastating influence was experi enced, and relative to which any authentic intelligence has yet reacha us, was the city of Jellalabad, where extensive injury was done to th fortifications and to the buildings throughout the place. The motioi of the earth is described as having been of an undulating charactei producing symptoms similar to those of sea-sickness in many of th persons who felt it; and in one account it is asserted, that the groum opened and closed again with loud noise in several places. Such
phenomenon is a very common accompaniment of a severe Earthquake, and by the extent to which it occasionally reaches, has proved one of the most fatal causes of destruction to life and property. The details at the effect of the Earthquake at Jellalabad are very brief and imperfect; this is, however, simply what might have been anticipated from the circumstances undeT which the gallant force now there are placed, but we shall probably at a future time obtain information of a more definite and satisfactory character. Three bastions, with, I presume, their connecting curtains, are said to have been levelled with the ground, and a painful interest is attached to this particular effect of the shock, from its having thrown open the defences of the small but resolute body of troops then occupying the city, and exposing them to an assault from the Affghans, at a time when they must necessarily hare had much internal confusion to contend against. In darker times, superstition would have tended to unnerve still more our brave friends, but on this occasion their courage appears to have risen even above the level of their difficulties, and brilliant success in repelling the assault was no more than the well-merited reward of their devotedness and energy.
From Jellalabad the shock affecting a portion of the Suffied Koh range of mountains, with the numerous subordinate ranges that diverge from these, reached Peshawur. From the circumstance of General Pollock's force being encamped at Kawulsur, about eight miles from Peshawur, and the communication being uninterrupted, our details are much fuller, and more satisfactory, than would otherwise have been the case.
The following extracts from letters published in the Delhi Gazette, pre the most perfect account of the different effects of the Earthquake that I have been able to find, although it is much to be regretted, that on the most important point, that namely, of the exact time of the occurrence of the shock, much discrepancy exists.
Extract from a Letter, dated Kawulsur, 20th February, 1842.
"Yesterday a fearful Earthquake visited this part of the world. The shock, which came on between 10 and 11 o'clock was long continued, and men, horses, tents, even the ground under us, and the hills in the