Min. The following table gives, with sufficient accuracy for any latitude in the United States south of Alaska, and for either the upper or the lower culminations of these bright stars, the value of the time interval, and the annual increase thereof in minutes, between the moment of vertical coincidence, and the moment of the culmination of Polaris. Min. For 5 Ursce Majoris in 1890 0.9 in 1900 + 2.6 Annual increase + 0.35 For 8 Cassiopeia in 1890 + 0.1 in 1900 + 3.4) Annual increase + 0.33 To establish the meridian, choose still weather, hang a plumb-bob from some high fixed object into a bucket of water, that it may be both free and steadfast, and select a place of observation so far southward that the plumb-line shall cover the breadth of sky between the reference star and the pole,-the farther the better. The point of observation may be an upright bodkin or compass-sight, fastened to a block movable horizontally eastward and westward. Watch for the moment when, from the point of observation, the plumb-line covers Polaris and the reference star. On the lapse of the tabular interval thereafter bring the plumb-line in range with Polaris by shifting the observation point laterally. That range will be the true meridian. Stakes may be set on it forthwith by means of candles. If the star in Cassiopeia be used within the coming two and a half years, attention is directed to the negative time interval. Its treatment hardly needs exposition. With a transit the plumb-line is not necessary, but special care should be taken to adjust the vertical thread of the telescope, and the horizontality of its transverse axis. This is best done by sighting up and down a fine cord or wire suspending a plummet in water. When making observations at night the cross hairs may be illuminated by reflecting light on the object glass from white paper. 13. By observation of the North Star at its extreme elongation. Find the time in Table II., and make the preparations above directed. Keep the plumb-line in range with the star until the star apparently ceases to move. Mark that range. Multi the distance in feet from the point of observation to the mark in the northern range just set. The product will be the distance from said northern range mark, square right or left, to a point in the true meridian passing through the point of observation. If the western elongation was observed, set off the calculated distance eastward from the northern range mark; if the eastern elongation was observed, set the distance off westward. If both the eastern and western elongations be observed, the true meridian will pass through the point of observation, bisecting the angle between the northern rauge marks. With a vernier instrument, the azimuth can be laid off directly, in degrees and minutes. 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, B I is perpendicular to BC. 49 2. Tangents drawn to a circle from the same point are equal. Thus, I B=E. 3. The angle DIE, at the intersection of tangents, is equal to the central angle BCE, 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 DI E. 5. Any ngle, BC E, 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 L FH 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 HN be made equal to HL, H K is called the tangential distance, and LN the deflection distance. 10. The exterior angle E H 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 equa! 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 |