Abbildungen der Seite
PDF
EPUB

Let it be required to find a third proportional to the dis

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

them double the angle GFH or IFH at the circumference (III. 17. El.); whence the triangles GEH and IGH must also have the angles at the base equal, and are consequently similar: Wherefore (VI. 12. El.) EG: GH::GH: HI.

If the first term AB be less than half the second term CD, this construction, without some help, would evidently not succeed. But AB may be previously doubled, or assumed 4, 8, or 16 times greater, so that the circle FGH shall always cut FHI; and in that case, HI, being likewise doubled, or taken 4, 8, or 16 times greater, will give the true result.

PROP. XVII. PROB.

To find a fourth proportional to three given distances.

Let it be required to find a fourth proportional to the distances AB, CD, and EF.

From any point G, describe two concentric circles HI and KL with the distan

ces AB and EF; in the circumference of the first inflect HI equal to CD, assume any point K in the second 'circumference, and cut this in L by an arc described from I with the distance HK; the chord LK is the fourth proportional required.

[ocr errors]

B

C

D

-F

El

K

For the triangles ILG and HKG are equal, since their corresponding sides are evidently equal; whence the angle IGL is equal to HGK, and taking away HGL, the angle IGH remains equal to LGK; consequently the isoceles triangles GIH and GLK are similar, and GI: IH:: GL: LK, that is, AB: CD:: EF:LK.

If the third term EF be more than double the first AB, this construction, it is obvious, will not answer without some modification. It may, however, be made to suit all the variety of cases, by multiplying equally AB and the chord LK, as in the last proposition.

PROP. XVIII. PROB.

To find the linear expressions for the square roots of the natural numbers, from one to ten inelusive.

This problem is evidently the same as, to find the sides of squares which are equivalent to the successive multiples of the square constructed on the straight line representing the unit. Let AB, therefore, be that measure: And from B as a centre, describe a circle, in which inflect the radius four times, from A to C, D, E, and F; from the opposite points A and E, with the double chord AD, describe arcs intersecting in G and H,-with the same distance, and from the points D, F, describe arcs intersecting in 1,— and, with still the same distance and from E, cut the circumference in K ;

[merged small][ocr errors][ocr errors][merged small][merged small][merged small]

For, in the isosceles triangles ACB and BDE, the perpendiculars CO and DP must bisect the bases AB and BE; and the triangle ADI being likewise isosceles, IP= AP, and consequently IB=AE=2AB. But, from what has been formerly shown, it is evident that AK2=2AB2 and AD 3AB2; and since AE=2AB, AE2-4AB2. In the right-angled triangles IBK and IBG, IK2=IB2+ BK*=4EB*+BK2=5AB2, IG*IB*+BG2=4AB'+ 2AB2=6AB2; but (II. 23.) IC2=IB2+BC2+IB.2BO =4AB2+AB2+2AB2=7AB2. Again, GH being double

of BG, GH2=4.2AB=8AB', and AI being the triple of AE, AI2=9AB2; and lastly, IAL being a right-angled triangle, IL2=IA2+AL2=9AB+ AB2 = 10AB2.

If AB, therefore, denote the unit of any scale, it will follow, that AK= √2, AD=√3, AE= √4, IK=√5, IG=√6, IC=√7, GH=√§, IA= √9, and IL= √10.

ELEMENTS

OF

PLANE TRIGONOMETRY,

TRIGONOMETRY is the science of calculating the sides or angles of a triangle. It grounds its conclusions on the application of the principles of Geometry and Arithmetic.

The sides of a triangle are measured, by referring them to some definite portion of linear extent, which is fixed by convention. The mensuration of angles is effected, by means of that universal standard derived from the partition of a circuit. Since angles were shown to be proportional to the intercepted arcs of a circle described from their vertex, the subdivision of the circumference therefore determines their magnitude. A quadrant, or the fourth-part of the circumference, as it corresponds to a right angle, hence forms the basis of angular measures. But these measures depend on the relation of certain orders of lines connected with the circle, and which it is necessary previously to investigate.

« ZurückWeiter »