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 distances 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 pro-portional required. I AB E G H 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 inclusive. 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; and from A and K, describe arcs inter secting in L: Then 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=2AB and AD=3AB2; and since AE=2AB, AE=4AB2. In the right-angled triangles IBK and IBG, IK*=IB*+ BK2=4EB2+BK2=5AB2, IG* = IB2 + BG2 = 4 AB2 + 2AB=6AB2; but (II. 23.) IC2 =IB2 +BC2 + IB.2BO =4AB2+AB2+2AB2=7AB2. Again, GH being double of BG, GH2=4.2AB=8AB2, and AI being the triple of AE, AI=9AB2; and lastly, IAL being a right-angled triangle, IL=IA+AL2=9AB2 + 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= √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 mea sures 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. 1 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, |