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.

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2. Tangents drawn to a circle from the same point are equal. Thus, IBI E.

3. The angle DIE, at the intersection of tangents, is equal to the central angle BC E, included between radii to the tangent points.

4. If a chord BE connect the tangent points, the angles IBE, IEB, are equal: 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 BE.

6. Acute angles at the circumference, subtended by equal chords, are equal.

7. An acute angle, KF H, 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 F K be made equal to FH, and HN be made equal to HL, HK is called the tangential distance, and LN the deflection distance.

10. The exterior angle E HN 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.

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

chain 100 feet long being the unit generally adopted by American engineers for field measurements, any circular arc having that radius, of 5,730 feet, is called a one-degree curve, for the reason that one chain is equivalent to an arc of one degree at the circumference.

2. The circumferences of circles vary directly as their radii: hence, in any circular arc struck with half that radius, or 2,865 feet, one hundred feet at the circumference would subtend an angle of two degrees at the centre. Such an arc is called a two-degree curve. If one-third of the primary radius of 5,730 feet, or 1,910 feet, be used, the arc is called a threedegree curve; and so on.

3. It should be borne in mind, however, that these measurements are supposed to be made around the arc itself, and not on lines of chords. Since field measurements with the chain are always made on the lines of the chords, which are shorter between given points at the circumference than the lines of the arcs, as a bowstring is shorter than the bow, it is plain that, in advancing towards the centre of the one-degree curve by a series of concentric circles having radii equal to one-half, one-third, &c., of the primary radius, the chord 100 feet long differs more and more in length from the arc subtended by it, the bow being more and more arched in relation to the string. Thus, in the circle having a radius equal to one-twentieth of the primary radius, the chord 100 feet long subtends an angle of 20° 06', at the centre, instead of 20°, and the arc is 100.5 feet in length, instead of 100 feet. In order, therefore, that the chord of 100 feet may subtend arcs of 10, 20, 30, &c., in regular succession, the radii of these successive arcs must be somewhat greater than the above method by subdivision of the primary radius would make them; though, as might be inferred from the extreme case given by way of illustration, the difference is not appreciable in ordinary field practice, and radii, together with all the functions dependent on them, may usually be held to vary as the degree of curvature, or central angle per 100 feet chord, varies,

XVIII.

TO FIND THE RADIUS, THE APEX DISTANCE, THE LENGTH, THE DEGREE, ETC., OF A CURVE.

1. Let DB, AL be two straight lines intersecting at D. Lay off equal distances, D A, D B; erect perpendiculars at A and

B, meeting at G, and connect A B, D G. From the centre G, with radius G A, draw the curve A H B.

The point D will be the P. I.; A and B, tangent points; DA, D B, the tangents, or apex distances, which denote by AD; DH, the external secant, or S; HN, the middle ord, or O. Let the long chord A B, connecting the tangent points, be called C, Call the deflection angle to a

2. By XVI. 3 and 4, angle D A B =DBA=AGD=DGB = + I.

3. GIVEN THE INTERSECTION ANGLE I AND RADIUS R, TO FIND THE APEX DIST. A D.

Then ADR tan. I=1,910.1 × 0.3191 = 609.5.

4. Measure from the P. I. equal distances, D M, D F, along the tangents. Measure, also, M F and D K, the distance from D to the middle point of M F. Then, by reason of similarity in the triangles M D K, DA G,

MK:DK::AG: AD::R:T