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?

Hour line of 1, or 11 o'clock. 11:43:52 Log. tangent=9.317349

Proceeding in this manner, the several angles which the respective hour lines make with the meridian will be found to be as follows; viz.,

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, ab; 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 ik; 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 to 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 c6 I., c7 II., c8 III., c 9 IV., c0V.: produce a 5 VII. and a 4 VIII. for the hour lines of VII. and VIII. o'clock in the evening; and produce c9 IV. and c0 V. for the hour lines of IV. and V. in the morning. In the same manner may the quarter and half-hour 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 LP for the axis of the style or gnomon.

The style may have any shape the artist pleases, provided its edge a LP 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 ab and cd; 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 LP 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 I., page 415, 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?

Proceeding in this manner, the several angles which the respective hour lines make with the meridian will be found to be as follows; viz.,

CONSTRUCTION.

On the proposed plane draw the XII. o'clock hour line ab; parallel to which, at a distance equal to the intended thickness of the style, draw the line cd at right angles to the sub-style, or XII. o'clock line, 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 ik. Take the transverse distance 9:36:40% to 9:36:40% in the compasses, and set it off from g to 1, and, also, from i to 6. Take the transverse distance 20:2.44% to 20:2:44" 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. Then, from the centre a, draw the forenoon hour lines a 1 XI., a 2 X., &c. &c. ; and, from c as a centre, draw the afternoon hour lines c 6 I., c 7 II., &c. &c. Take the complement of the latitude in the compasses, viz., the transverse distance 39:11:45 to 39:11:45"; set it off from g to L, and draw the hypothenuse line a LP for the axis of the style or gnomon.

Now, when the dial is placed vertically, with its plane duly facing the south, the VI. o'clock hour line ef will be parallel to the plane of the horizon; and the style B a LP, directed downwards, making an angle with the sub-style or XII. o'clock hour line equal to the complement of the latitude, will be truly parallel to the earth's axis.

Since the sun cannot shine any longer on a dial of this description than from VI. in the morning until VI. in the evening, it is not necessary to describe hour lines upon it before or after those periods of time.

Note.-An erect direct north dial for a place in north latitude, is constructed exactly in the same manner as an erect direct south dial; but the position of the dial must be reversed: that is, the VI. o'clock hour line must be at the bottom instead of the top of the dial; and the style or gnomon must be directed upwards instead of downwards.

SOLUTION OF PROBLEMS RELATIVE TO THE MENSURATION OF HEIGHTS AND DISTANCES.

Since it is frequently of the greatest importance to the mariner, but at all times to the engineer or other military officer, to be able to ascertain the heights and distances of remote objects with precision, the following