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AE: FK, and consequently (V. 19.) as one of the antecedents AB, BC, CD, DE or AE, is to its consequent FG, GH, HI, IK or FK, so is the amount of all those antecedents, or the perimeter ABCDE, to the amount of all the consequents, or the perimeter FGHIK. Again, the triangle CAB is to the triangle HFG (VI. 21. cor. 1.) in the duplicate ratio of AB to FG,-the triangle DAC is to the triangle IFH in the duplicate ratio of AC to FH, or of AB to FG, and the triangle EAD is to KFI in the duplicate ratio of AD to FI or of AB to FG ; wherefore (V. 19.) the aggregate of the triangles CAB, DAC, and EAD, or the area of the polygon ABCDE, is to the aggregate of the triangles HFG, IFH, and KFI, or the area of the polygon FGHIK, in the duplicate ratio of AB to FG, of BC to GH, of CD to HI, or of DE to IK. Cor. Hence also the perimeter ABCDE is to the perimeter FGHIK, as any diagonal AD to the corresponding diagonal FI, and the area ABCDE is to the area FGHIK in the duplicate ratio of AD to FI.

PROP, XXV. PROB.

. To construct a rectilineal figure that shall be. similar to one, and equivalent to another, given rectilineal figure.

Let it be required to describe a rectilineal figure similar to A, and equivalent to B.

On CD, a side of A, describe (II. 8.) the rectangle CDFE, equivalent to that figure, and on DF describe the

rectangle DGHF equivalent to the figure B ; find (VI.16.) IKamean

proportional be

tween CD and

DG, and on IK A-

construct, in the C - a-w /xx Ur - -

same position, a fi- D I JK
gure X similar to
the rectilineal fi- IC T. H.
gure A; this will be likewise equivalent to B.

For the figures A and X, being similar, must (VI. 24.) be in the duplicate ratio of their homologous sides CD and IK ; and since IK is a mean proportional between CD and DG, the duplicate ratio of CD to IK is the same as the ratio of CD to DG (V.24.); consequently the figure A is to the figure X as CD to DG, or (V. 25. cor. 2.) as the rectangle CF to the rectangle DH ; but the figure A is equivalent to the rectangle CF, and therefore (W. 4.) the figure X is equivalent to the rectangle DH, that is, to the figure B.

PROP. XXVI. THEOR.

. A rectilineal figure described on the hypotenuse of a right-angled triangle, is equivalent to similar figures described on the two sides.

Let ABC be a right-angled triangle; the figure ACFE described on the hypotenuse is equivalent to the similar

figures AGHB and BIKC, described on the sides AB and BC.

For draw BD perpendicular to the hypotenuse. And since (VI. 15. cor. 1.) AC; AB : : AB : AD, therefore AC is to AD in the duplicate ratio of AC to AB, that is, (VI. 24.), as the figure on AC to the figure on AB. For the same reason, AC is to CD in the duplicate ratio of AC to BC, or as the figure on AC to the figure on BC. Whence (V. 19. cor. 2.) AC is to the two segments AD and CD taken together, as the figure on AC to both the figures on AB and BC; and the first term of the analogy being thus equal to the second, the third must be equal to the fourth (V. 4.), or the figure described on the hypotenuse is equivalent to the similar figures described on the two sides.

PROP. xxvii. THEOR.

The arcs of a circle are proportional to the angles which they subtend at the centre.

Let the radii CA, CB, and CD intercept arcs AB and BD; the arc AB is to BD, as the angle ACB to BCD.

For (I. 5.) bisect the angle ACB, bisect again each of its halves, and repeat the operation indefinitely. An angle ACa will be thus obtained less than any assignable angle. Let this angle ACa or BCB (I. 4.) be repeatedly applied about the point C, from BC towards DC; it must hence, by its multiplication, fill up the angle BCD, nearer than any possible difference. But the elementary angle ACa being equal to BCb, the corresponding arc Aa is (III, 12.) equal to Bb. Consequently this arc Aa and its angle ACa, are like measures of the ič, arc AB and the angle ACB, and they are both contained equally in the arc BD and its corresponding angle BCD. Wherefore AB : BD: : ACB : BCD. Cor. Hence the arc AB is also to BD, as the sector ACB to the sector BCD ; for these sectors may be viewed as alike composed of the elementary sector ACa.

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PROP. XXVIII. THEOR.

The circumference of a circle is proportional to the diameter, and its area to the square of that diameter.

Let AB and CD be the diameters of two circles;–the circumference AFG is to the circumference CKL, as AB to CD; and the area contained by AFG is to the area contained by CKL, as the square of AB to the square of CD. For inscribe the regular hexagons AEFBGH and ClKDLM. Because these polygons are equilateral and equiangular, they are similar; and consequently (VI. 24. cor.) the diagonal AB is to the corresponding diagonal CD, as the perimeter AEFBGH to the perimeter CIKDLM. But this proportion must subsist, whatever be the number of chords inscribed in either circumference. Insert a dodecagon in each circle between the hexagon and the circumference, and its perimeter will evidently ap

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proach nearer to the length of that circumference. Proceeding thus, by repeated duplications,—the perimeters of the series of polygons that arise in succession, will continually approximate to the curvilineal boundary, which forms their ultimate limit. Wherefore this extreme term, or the circumference AEFBGH, is to the circumference CIKDLM, as the diameter AB to the diameter CD.

Again, the hexagon AEFBGH (VI. 24. cor.) is to the hexagon CIKDLM in the duplicate ratio of the diagonal AB to the corresponding diagonal CD, or (V. 24.) as the square of AB to the square of CD. Wherefore the successive polygons which arise from a repeated bisection of the intermediate arcs, and which approach continually to the areas of their containing circles, must have still that same ratio. Consequently the limiting space, or the circle AEFBGH, is to the circle CIKDLM, as the square of AB to the square of CD.

Cor. 1. It hence follows, that if semicircles be described

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