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in which radius is supposed 1. The logarithmic formula, deduced from it, may be expressed in words at length, thus :

To the logarithmic tangent of the horary angle described by the sun between noon and the given time, add the log. sine of the latitude, and the sum, alating 10, (the log. of rad.) is the logarithmic tangent of the angle which the hour-line on the dial makes with the meridian line.

23. Er. Let it be required to calculate the angles which the hourlines on a horizontal dial, for Edinburgh, make with the meridian or 12 o'clock line the latitude of Edinburgh being about fifty-six degrees, a calculation for the hour-lines of XI in the forenoon, and I in the afternoon, would be as follows:

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Hence it appears, that the hour-lines for XI in the forenoon, and I in the afternoon, must each make with the meridian an angle of 12° 32′. The angles which the remaining hour-lines make with the meridian may be found in the same way, and will be as follows:

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The hour-lines of V in the morning, and VII in the evening, make the same angles with the meridian as the hour-lines of VII in the morning and V in the afternoon; but they lie on opposite sides of the VI o'clock hour-lines. In like manner, the hour-lines of IV in the morning, and VIII in the evening, make the same angles with the meridian as the hour-lines of VIII in the morning and IV in the afternoon, and so on.

The construction of the dial is now very easy, as it requires nothing more than to make an angle of a given number of degrees. Thus, draw the meridian line CM (fig. 7), and cross it at right angles by the six o'clock hour-line CG; and as the style of the dial must have some thickness, it will be proper to draw two parallel lines CM, C'M' for the meridian line, so that the distance between them may be equal to that thickness.

From the points C, C', draw the lines CI, C'XI on opposite sides of the meridian, so that the angles MCI, M'C'XI may be each 12° 32'; and these lines will be the hour-lines of I in the afternoon, and XI in the forenoon; the former lying on the east and the latter on the west side of the meridian, when the dial is placed in its proper position. In the same way, all the other hour-lines may be laid down on the plane of the dial, using a scale of chords, or a protractor, such as is commonly sold by mathematical instrument makers. Or a qua

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ant of a circle pq may be described on C as a centre, and divided nto 90 equal parts, and the hour-lines drawn at once through the Doints of the arc indicating the nuinber of degrees and minutes they ought to make with the meridian. The stile KCL (fig. 6) must be so constructed that the angle contained by CK and CL, the edges of one of its planes, may be 56°, the latitude of the place, and it may be fixed into the plane of the dial by two tenons at C and L let into openngs made to receive them. The edge CK must stand directly over the meridian line CM, and then the afternoon hours will be shown by the limit of the shadow of the triangular plane KCL.

The stile may have any shape, provided its edge CK be a straight line. It may even be a cylindrical rod, but in that case the hourlines ought to be tangents to its section with the plane of the dial. The angles they make with the meridian will, however, be the same.

24. Instead of an axis directed to the pole, we may substitute vertical pin; for if, from any point K in the axis, a perpendicular KI be let fall on the meridian line, and the axis be removed, leaving the vertical line KL, it is evident that the shadow of its top K will come to any hour-line at the same instant that the edge of the shadow of the axis CK would have fallen on that line.

To form this stile, we must, at any point L in the meridian, erect a vertical pin of such a height, that a line drawn from its top to the centre of the dial, may make with the meridian an angle equal to the latitude. In this case the meridian may be a single line if the stile have a sharp point, and then the extremity of the shadow will point out the hour of the day. This kind of stile, however, cannot indicate the hour for some time after sun-rise and before sun-set, because of the shadow extending beyond the limits of the dial.

The hours may also be indicated by the shadow of any point whatever, provided a line drawn from it to the centre of the dial pass through the pole of the world. Hence the stile may be any ornamental or emblematical figure; for example, Time, and the hour may be shown by the shadow of the point of his scythe, &c.

25. We shall here give a Table, calculated by the formula of art 21, by which a horizontal dial may be constructed for any place in · Great Britain.

A Table of the angles which the Hour-lines form with the Meridian on a Horizontal Dial for every half Degree of Latitude, from 50° to 59° 30'.

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In this Table, the angles formed by the lines for V in the morning and VII in the evening, IV in the morning and VIII in the evening, &c. are not marked, because, it has been already observed, they are the same as those for VII in the morning and V in the evening, VIII in the morning and IV in the evening, only they lie on opposite sides of the VI o'clock hour-lines.

The use of the Table may be easily comprehended: if the place for which a horizontal dial is to be made, corresponds with any latitude in the Table, the angles which the hour-lines make with the meridian may be seen at once. For example, it appears that the hourlines of XI and I must, in the latitude of 56°, make angles of 12° 32' with the meridian. If the latitude be not contained in the Table, proportional parts may be taken without any sensible error. Thus, if the latitude be 54° 15', and the angles made by the hour-lines of XI or I be required; as it appears from the Table that the increase of 30 in the latitude, viz. from 54° to 54° 30', corresponds to an increase of 4' in the hour angle at the centre of the dial, we may infer, that an increase of 15' will require an increase of 2′ nearly; and, therefore, that the angle required will be 12° 16′.

Geometrical Construction of Horizontal Diais.

26. As every geometrical problem admits of various constructions, so the hour-lines on a horizontal dial may be determined in various

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