PROBLEM VI.

Given the Latitude of a Place; to find the Time of the shortest Twilight, and its Duration.

RULE.

To the logarithmic tangent of the half of 18 degrees, add the logarithmic sine of the latitude; and the sum (abating 10 in the index,) will be the logarithmic sine of the sun's declination at the time of the shortest twilight, of a contrary name to the latitude: the day corresponding to this declination will be that required.

Again, to the logarithmic sine of the half of 18 degrees, add the logarithmic secant of the latitude; and the sum (abating 10 in the index,) will be the logarithmic sine of an arch, which, being doubled and converted into time, will be the duration of the shortest twilight.

Example.

Required the time of the shortest twilight, and its duration, in the year 1824, in latitude 50:48: N.?

which is south, of a contrary name to the latitude.

Duration of twilight 28:39:38", in time 15439:.

The days, in the Nautical Almanac, corresponding to the sun's declination 7:31 S., are March 2d and October 11th, which, therefore, are the days of the shortest twilight in the year 1824, in latitude 50:48: north; and the duration of the twilight, on those days, is 1:54.39.

PROBLEM VII.

Given the Latitude of a Place between 48:32 and 66:32 (the Limits of regular Twilight); to find when real Night or Darkness ceases, and when it commences.

RULE.

The complement of the latitude, diminished by 18 degrees, will be the declination of the sun, of the same name as the latitude, at the time when it ceases to be real night, and also when real night commences.

Example.

Required the interval of time, in the year 1824, during which there will be no real darkness or night, in latitude 50:48? north?

Solution. The complement of the latitude 39:12 N.-18:21:12'N. the sun's declination. Now, the days answering to 21:12 of north declination are, May 26th and July 17th. Upon the first of these days, therefore, real night ceases, and it commences upon the last. During this interval there is no real darkness, because the sun is less than 18 degrees below the horizon; and so on for any other latitude within the limits.

PROBLEM VIII.

Given the Sun's Declination and Semi-diameter; to find the Interval between the Instants of his lower and upper Limbs being in the Horizon of a known Place.

RULE.

Find the approximate time of the sun's rising or setting, by Problem I., page 124; to which time let the sun's declination be reduced, by Problem V., page 298.

To the logarithm of the sun's semi-diameter, expressed in seconds, add the constant logarithm 9. 124939, and call the sum a reserved logarithm ; then,

To the logarithmic co-sine of the sum of the latitude and declination, add the logarithmic co-sine of their difference: half the sum of these two logarithms, being subtracted from the reserved logarithm, will leave the logarithm of the interval of time, in seconds, between the instants of the sun's lower and upper limbs being in the horizon of the given place.

Example 1.

Required the interval between the instants of the sun's lower and upper limbs being in the horizon, at the time of its setting, July 13th, 1824, in latitude 50:48 N., and longitude 120: W.?

Apparent time of setting in Table L., to latitude 50:48. N.,

and declination 21:49:51 N. =

Longitude 120: west, in time =

7:57"12:

8. 0. 0

15:57 12:

Sun's declination at noon, July 13th, 1824, = 21:49:51′′N.
Correction of ditto for 15 5712!=

5.58

21:43.53′′N.

Sun's semi-diameter 15:45". S=945"..8 Log.=2.975799

Hence, the interval between the instants of the sun's limbs touching the horizon, is 4 minutes and 6 seconds.

Example 2.

Required the interval between the instants of the sun's upper and lower limbs touching the horizon, at the time of rising, October 1st, 1824, in latitude 40:30 N., and longitude 105? E.?

Apparent time of rising, in Table L., to latitude 40:30. N.,

Sun's declination at noon, Sept. 30th, 1824,= 2:52:46% S.
Correction of ditto for 11:10:48: =

+10.53

3: 3:39" S.

Sun's semi-diameter=16'1". 2=961". 2 Log.=2.982814

Hence, the interval between the instants of the sun's limbs touching the horizon, is 2 minutes and 49 seconds.

Note. The constant logarithm made use of in this problem is the arithmetical complement of the proportional logarithm of 24 hours esteemed as minutes. If the sun's diameter be used, instead of its semi-diameter, it must be expressed in minutes and decimal parts of a minute: in this case, the same result will be obtained by employing the constant logarithm 8.823909; viz., the arithmetical complement of the common logarithm of 15 degrees, or the motion corresponding to one hour of time.

SOLUTION OF PROBLEMS IN GNOMONICS OR DIALLING.

Dialling, or Gnomonics, is a branch of mixed mathematics, which depends partly on the principles of geometry and partly on those of astronomy; and it may be defined as being the method of projecting on the surface of any given body, whether plane or otherwise, a figure called a sun-dial,-the different lines of which indicate, by the shadow of a style or gnomon, when the sun shines thereon, the apparent time of the day.

The upper edge of the style or gnomon, which projects the sun's shadow on the plane of the dial, must be parallel to the earth's axis: hence, it is sometimes called the axis of the dial.

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 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 BC, 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.