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Ajouterai-je à ces raisons, une idee qui m'est venue plusieurs fois. Suppose que le même triangle, dont vous vous occupez, soit mis sous les yeux d'un ëtre intelligent, dontla stature et celle des objets qui l'environment soient cent fois plus grandes que celle des objets environmans—mon raisonne- ment sera toujours le même, et ne perdra rien de A C la force. Croiez-vous, cependant, qu’il fút possible que crestät dans l'equation, C = p : (A, B, c): Et si c restoit, les géans dont je parle deduiroient-ils de cette equation la même valeur que vous? Il faudroit que cela fät, car l'objet a les mêmes dimensions dans les deux cas.”
‘I am sorry that, on reconsidering the subject maturely, I cannot assent to the force of this reasoning, however clearly and neatly it is here developed. The whole stress of the argument, it may be perceived, lies in the distinction which M. Legendre endeavours to establish between angles and lines, —a distinction which I hold at bottom to be merely arbitrary. Angles and lines are both equally real quantities, though of different kinds; they are capable of being measured, and consequently represented by numbers, by referring each of them to some determinate measure or unit of its own denomination. Angles are measured or expressed numerically by angles, and lines by lines. It is true, that the mensuration of angles is facilitated by a reference to the subdivision of the circuit or entire revolution; yet even this mode of denoting angular magnitude is evidently only conventional. As standards for measuring straight lines, nature has furnished the limbs of the human body, and the extent of our globe itself. Such units of mensuration are not indeed very definite or readily attainable; but they are not therefore the less real or prominent. Nor is there any essential difference in principle between the expressing of an angle by degrees, of which 360 or 400 are contained in a complete revolution, and the denoting of a straight line on the French system, for instance, by the number of metres it includes, each of which is the forty millionth part of the entire circumference of the earth. Angles and lines hence present to the mind no radical or absolute
discrimination, and therefore the argument grounded on such a distinction must lose all its efficacy.
Admitting, however, what the slightest inspection readily confirms, that the third angle is merely derived from the other two, M. Legendre demonstrates with great elegance, the property that the three angles of a triangle are equal to two right angles. Letting fall from the right angle a perpendicular on the hypotenuse, he divides any right-angled triangle into two subordinate triangles, which have each of them two angles equal to those of the original triangle; whence the acute angles of that triangle are alternately equal to the angles which compose the right angle. But every triangle may be divided into two right-angled triangles, by letting fall a perpendicular from the vertex on the base, and consequently the acute angles of both these triangles, and which form the angles at the base, and the vertical angle of the primary triangle,_ are together equal to two right angles.
This theorem may be proved somewhat more directly. In the triangle ABC, let the angle CBA be greater than ACB, and draw BD, and then DE, making the angles ABD and BDE each equal to ACB. The triangles ABC and ADB having the common angle BAC and the angle ACB equal to ABD, their third angles ABC and ADB must be equal. But the triangles BCD and BDE have also a common angle CBD, and equal angles DCB and BDE : whence the third angle BDC is equal to BED, and therefore the supplementary angle ADB, equal to ABC, is equal to DEC. Again, the triangles ABC and DEC having two common or equal angles, their third angles BAC and EDC are equal; wherefore the three angles ABC, BCA and BAC of the original triangle, are respectively equal to BDA, BDE and EDC, and hence equal to two right angles.—If the triangle ABC be equiangular, divide it into two scalene triangles ABD and CBD, the angles of which, or the angles of the original triangle, together with the
adjoining angles ADB and BDC, must be equal to four right angles, and consequently the angles of that triangle are equal to two right angles.
But the proposition is easily derived from another view of the subject. If we suppose a ruler turning about the point A, to change its direction AC into AB, then opening at B till it gains the direction BC, and finally wearing about the point C till it acquires the opposite position CA; thus changing its direction with respect to a remote object, by three successive openings all to the same side, the ruler, being now reversed, must have performed half a circuit; that is, the three angles of a triangle, which constitute those openings, are equal to two right angles.
The profound geometer already quoted, pursuing his refined argument, has, from the consideration of homogeneous quantities, likewise attempted to deduce the proportionality of the sides of equiangular triangles. But in this abstruse research, assumptions are still disguised and mixed up in the progress of induction. Such indeed must be the case with every kind of reasoning on mathematical or physical ob. jects, which proceeds a priori, without appealing, at least in the first instance, to external observation. Of this kind are some of those ingenious analytical investigations respecting the laws of motion and the composition of forces. In fact, no elementary physical truth can ever be discovered by any process of calculation, which merely combines or embodies the various assumptions that have been tacitly made into a general result. The principle of sufficient reason, introduced by Leibnitz, appears to be nothing but an artificial mode of dress. ing out an hypothesis, which the celebrated Boscovich has well exposed in his excellent notes to a didactic poem by Stay, entitled Philosophia Recentior.
14. Proposition twenty-second. The subject of parallel lines has exercised the ingenuity of modern geometers; for Euclid had only endeavoured to evade the difficulty, by styling the fundamental proposition an axiom. The investigation now given seems to be one of the best adapted to the natural progress of discovery. It is almost ridiculous to scruple about admitting the idea of motion, which I have employed for the sake of clearness. But even that futile objection might be obviated, by considering merely the successive positions of the straight line extending through the given point.
15. Proposition thirtieth. That invaluable instrument, Hadley’s quadrant, is founded on the second corollary, annexed as an obvious consequence of the proposition. A ray of light SA, from the sun, impinging against the mirror at A, is reflected at an angle equal to its incidence; and now striking the halfsilvered glass at C, it is again reflected to E, where the eye likewise receives, through the transparent part of that glass, a direct ray from the boundary of the horizon: Hence the triangle AEC has its exterior angle ECD and one of its interior angles CAE, respectively double of the exterior angle BCD and the interior angle CAB, of the triangle ABC ; wherefore the remaining interior angle AEC, or SEZ, is double of ABC; that is, the altitude of the sun above the horizon is double of the inclination of the tyo mirrors. But the glass at C remaining fixed, the mirror at A is attached to a moveable index, which marks their inclination.
The same instrument, in its most improved state, and fitted with a telescope, forms the sextant, which, being admirably calculated for measuring angles in general, has rendered the most important services to geography and navigation.
16. Proposition thirty-fourth. This problem is generally constructed somewhat differently.
In AB take any point C, and on BC (I. 1. cor.) describe an equilateral triangle CDB, on its side DB, another DEB; and F on DE the side of this, a third equilateral triangle DFE; join the last vertex F with the point B ; and BF is the perpendicular required.
Because the triangles CDB and DBE are equilateral, the angles CBD and DBE are each of them equal to two-third parts of a right angle (I. 30. cor. 1.); and the triangles BDF, BEF, having the sides BD, DF equal to BE, EF, and the side BF common, are (I. 2.) equal, and consequently the angles FBD and FBE are equal, and each of them the half of DBE. The angle FBD, being therefore one-third part of a right angle, and the angle DBA two-third parts, the whole angle FBC must be an entire right angle, or the straight line BF is perpendicular to AB.
A C 13
BOOK II. *
1. A simple proposition might be here introduced. A straight line bisecting two sides of a triangle, is parallel to the base. The straight line DE which joins the middle points of the sides AB and BC, is parallel to the base AC of the triangle ABC. For join AE and CD. Because the triangles ADC, BCD stand on equal bases AD, DB, and have the same vertex or altitude, they are (II. 2.) equivalent, and therefore ADC is