The plane of the gnomon must be perpendicular to that of the dial. The plane on which it is erected is called the sub-style: in horizontal dials it may be called the meridian, or 12 o'clock line. The angle comprehended between the style and the sub-style, is called the elevation of the style: this angle, in horizontal dials, is always equal to the elevation of the pole, or the latitude of the place for which it is computed; but, in erect direct north or south dials, it is equal to the complement of the latitude of such place. Those dials whose planes are parallel to the plane of the horizon, are called horizontal dials; but such as have their planes perpendicular to the plane of the horizon, are called vertical or erect dials. Those vertical dials whose planes are either parallel or perpendicular to the plane of the meridian, are called direct erect dials. One of these must always face one of the cardinal points of the horizon, according as it may be a north, south, east, or west, erect dial. All other erect dials are called declining dials. Those dials whose planes are neither parallel nor perpendicular to the plane of the horizon, are called reclining dials. In this place, however, we shall only show the method of constructing a horizontal dial, and, also, that of a north or south erect direct dial; these being by far the most useful, and, indeed, the most common of all the varieties in dialling. PROBLEM I. Given the Latitude of a Place; to find the Angles which the Hour Lines make with the Sub-Style or Meridian Line of a Horizontal Sun-Dial. GENERAL PROPOSITION. In every right angled spherical triangle, radius is to the sine of one of the legs containing the right angle, as the tangent of the angle adjacent to that side is to the tangent of the other containing side of the triangle. This is merely a variation of the equation for finding the leg B C, in Problem IV., page 189: hence, the following RULE. To the logarithmic sine of the latitude, add the logarithmic tangent of the sun's horary angle from noon; and the sum (abating 10 in the index), will be the logarithmic tangent of the angle comprehended between the corresponding hour line and the sub-style, at the centre of the dial. Note. Since the sun's apparent motion in the ecliptic is at the rate of 15 degrees to an hour, therefore, at one hour from noon the sun's horary angle is 15; at two hours from noon it is 30:; and so on. Example. Required the angles which the hour lines make with the sub-style, or meridian line of a horizontal dial, in a place situated in 50:48:15: north latitude? To find the Angle at one Hour from Noon. 9.889296 Latitude of the place = . . 50:48:15 Log. sine = Hour line of 2, or 10 o'clock = . 24: 6'20 Log. tangent=9. 650735 11:43:52" 53. 18.53 90. 0. 0 The hour lines of VII. in the evening and V. in the morning, make the same angles with the meridian, on the opposite side of the VI. o'clock hour line, as the hour lines of VII. in the morning and V. in the evening. In the same manner the hour lines of VIII. in the evening and IV. in the morning make the same angles with the meridian as the hour lines of VIII. in the forenoon and IV. in the afternoon; and so on. The angles for the halves, quarters, or other subdivisions of the hours, are to be determined in the above manner. The angles which the different hour lines, &c. make with the meridian, being thus determined, the dial may then be very readily constructed, by means of a pair of compasses, and the line of chords on a common Gunter's scale, or of that on a Sector: the latter, however, should be preferred, because the degrees thereon are generally divided into halves, and sometimes quarters, which gives it a decided advantage, in point of accuracy, over that on Gunter's scale. : CONSTRUCTION. On the proposed plane draw the meridian, or XII. o'clock hour line, a b; parallel to which, at a distance equal to the intended thickness. of the gnomon or style, draw the line cd: perpendicularly to these draw the VI. o'clock hour line ef. Open the Sector to any convenient extent, and take the transverse distance 60 to 60: (on the line of chords) as a radius in the compasses, and, from a as a centre, describe the arc gh with the same radius, and from c as a centre, describe the arc i k; and, since the hour lines are less distant from each other about noon than in any other part of the day, it is advisable to have the centres of those quadrants or arcs at a little distance from the centre of the plane of the dial, on the side opposite to XII., so as tɔ allow of the hour distances being enlarged near the meridian under the same angles in the plane of the dial: thus, the centre of the plane is at A; but the centres of the quadrants or arcs are taken a little below it, at the points a and c. Take the transverse distance 11:43:52% to 11:43:52", in the compasses, from the line of chords, and set it off from g to 1, and, also, from i to 6: take the transverse distance 24:6:20%, in the compasses, and set it off from g to 2, and from i to 7; and proceed in the same manner with the remaining horary angles. Now, from the centre a draw the forenoon hour lines a 1 XI., a 2 X., a 3 IX., a 4 VIII., a 5 VII.; and, from c as a centre, draw the afternoon hour lines c 6 I., c 7 II., c 8 III., c 9 IV., c 0 V.: produce a 5 VII. and a 4 VIII. for the hour lines of VII, and VIII. o'clock in the R R evening; and produce c 9 IV. and c 0 V. for the hour lines of IV. and V. in the morning. In the same manner may the quarter and halfhour lines be drawn (and minutes if necessary), by setting off the computed corresponding angles from the meridian: these, however, have been omitted in the above diagram, with the view of preventing embarrassment. Take the latitude 50:48:15" in the compasses, viz., the transverse distance 50:48:15% to 50:48:15", and set it off from g to L, and draw the hypothenuse line a L P for the axis of the style or gnomon. The style may have any shape the artist pleases, provided its edge a L P be a perfectly straight line. It should be a metallic substance, and must be of an equal thickness with the breadth of the space comprehended between the two parallel straight lines a b and c d; in which space it must be erected truly perpendicular to the plane of the dial: then, since the angle B a P is equal to the latitude, the straight edge of the style a L P will be directed to the elevated pole of the world, and, hence, parallel to the earth's axis when the dial is truly set; the shadow of which, when the sun shines, will indicate the hour of the day. Note. Since the hour of the day indicated by a sun-dial is expressed in apparent solar time, it must be reduced to mean time, by Problem XIX., page 369, so as to make it correspond with that shown by a well-regulated watch or clock. PROBLEM II. To find the Angles on the Plane of an erect direct south Dial for any proposed north Latitude, or on that of an erect direct north Dial for any proposed south Latitude. RULE. To the logarithmic co-sine of the latitude, add the logarithmic tangent of the sun's horary angle from noon; and the sum (abating 10 in the index), will be the logarithmic tangent of the angle comprehended between the corresponding hour line and the sub-style, at the centre of the dial. Example. Required the angles which the hour lines on an erect direct south dial make with the sub-style or 12 o'clock line, in latitude 50:48:15" north? To find the Angle at one Hour from Noon. Latitude of the place= 50:48:15 Log. co-sine-9.800699 Sun's horary ang. at 1 from noon-15. 0. 0 Log. tangent-9.428053 Hour angle of 1, or 11 o'clock = 9:36:40" Log. tangent-9. 228752 To find the Angle at two Hours from Noon. Latitude of the place= 50:48:15 Log. co-sine 9.800699 Sun's horary ang. at 2 from noon-30. 0. 0 Log. tangent-9.761439 Hour angle of 2, or 10 o'clock 20: 2:44" Log. tangent 9. 562138 Proceeding in this manner, the several angles which the respective hour lines make with the meridian will be found to be as follows; viz., |