Let it be required to finda third proportional to the distances AB and CD. From any point E, *... Al............ |B and with the distance . . . . . . C. ---------.ID AB, describe aportion ' ' – of a circle, in which inflect FG equalto CD, and from G, with that distance, describe the semicircle FHI ; HI is the third proportional required. For the angles GEH and IGH are each of 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. 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. 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 LGR ; 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. 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 |