observed, the true meridian will pass through the point of observation, bisecting the angle between the northern range marks. With a vernier instrument, the azimuth can be laid off directly, in degrees and minutes. PROPOSITIONS AND PROBLEMS RELATING TO THE CIRCLE. XVI. PROPOSITIONS RELATING TO THE CIRCLE. The following propositions, demonstrable by simple processes of geometrical reasoning, may be regarded as axiomatic. 1. In any circle a tangent is perpendicular to radius at the tangent point. Thus, BI is perpendicular to BC. 49 2. Tangents drawn to a circle from the same point are equal. Thus, IB=I E. 3. The angle DIE, at the intersection of tangents, is equal 10 the central angle BC E, included between radii to the tangent points. 4. If a chord BE connect the tangent points, the angles I BE, IEB, are eqnal: each of them is equal to half of the central angle BCE, or of the intersection angle DIE. 5. Any angle, BCE, at the centre, subtended by the chord BE, is double the angle BFE, at the circumference, on the same side of the chord B E. 6. Acute angles at the circumference, subtended by equal chords, are equal. 7. An acute angle, KFH, between a tangent and a chord, is called a tangential angle, and is equal to the peripheral angle LFH subtended by an equal chord; each is equal to half the central angles FCH, or HCL, subdivided by the same chords. 8. The exterior angle LHN at the circumference, between two equal chords, is called a deflection angle: it is equal to the central angle, or to twice the tangential angle, subtended by either chord. 9. If FK be made equal to FH, and HV be made equal to HL, H K is called the tangential distance, and LN the deflection distance. 10. The exterior angle EH N at the circumference, between two unequal chords, is equal to the sum of their tangential angles, or to half the sum of their central angles. XVII. CIRCULAR CURVES ON RAILROADS. 1. The circle is divided, for convenience, into 360 equal parts, called degrees. A circle 36,000 feet in circumference would be cut by such subdivision into 360 parts, each 100 feet long, and subtending an angle of one degree at the centre; its |