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 ;

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 AD2=3AB2; and since AE=2AB, AE2=4AB2. In the right-angled triangles IBK and IBG, IK2=IB2+ BK2=4EB2+BK2=5AB2, IG2=IB2+BG2=4 AB2+ 2AB=6AB'; but (II. 23.) IC2=IB'+BC2+IB.2BO =4AB2+AB2+2AB2=7AB2. Again, GH being double

of BG, GH2=4. 2AB2=8AB2, and AI being the triple of AE, AI2=9AB'; and lastly, IAL being a right-angled triangle, ILIA+AL2=9AB+ AB=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= √8, 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.

DEFINITIONS.

1. The complement of an arc is its defect from a quadrant; its supplement is its defect from a semicircumference; and its explement is its defect from the whole circumference.

2. The sine of an arc is a perpendicular let fall from one of its extremities upon a diameter passing through the other.

3. The versed sine of an arc is that portion of a diameter intercepted between its sine and the circumference.

4. The tangent of an arc is a perpendicular drawn at one extremity to a diameter, and limited by a diameter extending through the other.

5. The secant of an arc is a straight line which joins the centre with the termination of the tangent.

In naming the sine, tangent, or secant, of the complement of an arc, it is usual to employ the abbreviated terms of cosine, cotangent and cosecant. A farther contraction is frequently made in noting the radius and other lines connected with the circle, by retaining only the first syllable of the word, or even the mere initial letter.

Let ACFE be a circle, of which the diameters AF and CE are at right angles; having taken any arc AB, produce the radius OB, and draw BD, AH perpendicular to AF, and BG,

CI perpendicular to CE. Of this assumed arc AB, the com

plement is BC, and the supplement

BCF; the sine is BD, the cosine BG or OD, the versed sine AD, the coversed sine CG, and the supplementary versed sine FD; the

tangent of AB is AH, and its cotangent CI; and the secant of the same arc is OH, and its cosecant OI.

F

D

(B

E

Several obvious consequences flow from these definitions:

1. Since the diameter which bisects an arc bisects also the chord at right angles, it follows that half the chord of any arc is equal to the sine of half that arc.

2. In the right-angled-triangle ODB, BD2+OD2 = OB2; and hence the squares of the sine and cosine of an arc are together equal to the square of the radius.

3. The triangle ODB being evidently similar to OAH, OD: DB: OA: AH; that is, the cosine of an arc is to the sine, as the radius to the tangent.

4. From the similar triangles ODB and OAH, OD: OB :: OA: OH; wherefore the radius is a mean proportional between the cosine and the secant of an arc.

5. Since BD=AD.FD, it is evident that the sine of an arc is a mean proportional between the versed sine and the