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 points of the arc indicating the nuinber of degrees and minutes they ought to inake with the meridian. The stile KCL (fig. 6) must be 80 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 open. ngs 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 cvlindrical 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 froru 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 bere give a Table, calculated by the formula of ars 21, by which a horizontal dial may be constructed for any place is Great Britain.

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

39° 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-lides 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 Dials. 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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ways according to the view that is taken of the subject.

They may all, howover, be deduced from the formula investigated in art. 21, damely, that radius is to the sine of the latitude, as the tangent of the nora.y angle described by the sun between any hour and noon, is to the tangent of the angle which the hour-line on the dial makes with the meridian. From this formula we inmediately derive the following results.

Method 1. 27. Let CMO, C'M'O (fig. 8) be the meridian line on the dial, the space between CM, C'M' being left for the thickness of the stile, and CC its centre, and 6 C 6 the six o'clock hour-line.

Make a right-angled triangle cmo, fig. 9, of any magnitude, having one of its acute angles c equal to the latitude of the place.

In the meridian, take CM and CM' equal to cm, the hypothepuse of the triangle, and MO and M'O' equal to mo, the side opposite to the angle c.

Through M, M' draw PQ perpendicular to CO.

On O and O', as centres, with OM as a radius, describe quadrants MH, M'H'.

Divide each quadrantal arc into six equal parts.

Through the points of division draw the lines O1, O2, O3, &c. also, O' 11, O' io, O9, &c. meeting PQ in v, u, I, &c. and in r, s, t, &c.

From the points C, C' draw lines. Civ, C2u, C 3x, &c. to the points v, u, t, &c. and C 117, C 10 s, C9 t, &c. to the points T, So t, &c. and these will be the hour-lines of the dial, viz. C i and Cil will be the hour-lines of I in the afternoon and XI in the forenoon, and C 2, C 10 the hour-lines of II and X, and so on.

The hour-lines before six in the morning, and after six in the even. ing, are to be found from the adjoining intermediate hours, as directed in art. 22.

The demonstration of this construction is obvious ; for in the rightangled triangles OMv, CMv, we have

CM : Mv :: rad, : tan. MCV,

and Mv : MO :: tan. MOv : rad. Therefore, er æquo inv. CM : MO :: tan. MOv : tan. MCv, but CM : MO :: cm i mo :: rad. : sin. lat. ; hence, rad. : sin. lat. :: tan. 20v : tan. MCv.

Therefore, the angle MCU is rightly determined, and the demonstration applies alike to all the hour-lines.

This construction, although very simple, is rather inconvenient in practice, because the lines, 04, 05, and 08, 07, may go off the surface on which the dial is to be delineated, before they meet the line PQ. The next construction has not this defect.

Method 2. 28. Let CM, C'M'be the double meridian line (fig. 10), and 6C6. the six o'clock hour-line, and let cmo (fig. 9), be a right-angled trian. gie, constructed as directed in the first operation of Method i.

On C, C the centres of the dial, with a radius equal to cm, the hypothenuse of the triangle com (fig. 9), describe semicircles on opposite sides of the meridian.

On the same cevtres, with a radius equal to om, (the side opposite to the apple which is the latitude) describe other two semicircles on opposite sides of the meridian,

Divide each quadrant of the two semicircles into six equal parts, at the points of division 1, 2, 3, &c. 11, 10, 9, &c. and let the numbers be written at the points of division, in the same order, in respect to the meridian, as the characters for the hours are to be placed on the dial.

Then, to find the position of any hour-line, as, for example, that for thrce in the afternoon : let D be the third point of division on the inner circle, and E the third point of division on the outer circle, reckoned from the meridian on the quadrant through which the after. noon hour-lines are to pass. Draw EBA perpendicular to the meridian, and DB parallel to it, meeting the perpendicular in B.

Draw a straight line from C through B, and the line CB will be the hour-line for III in the afternoon, as required.

And in the very same way may all the other hour-lines be drawn on the dial.

To prove the truth of this construction, Ict EB meet the meridian in A, and join EC, which will evidently pass through D. Because BD is parallel to AC, CE: CD :: AE : AB ; but by the construction, CE : CD :: rad. ; sin. lat. ; and, by trigonometry, AE : AB :: tan. ACE : tan. ACB ; therefore, rad. : sin. lat. :: tan. ACE : tan. ACB; row ACE is equal to the horary angle which the sun describes in three hours ; therefore CB is the hour-line for three in the afternoon. (Art. 39.)

Construction of Dialling Scales. 29. There is another very elegant geometrical construction for the hour-lines, by which scales may be made for the construction of dials, which save the labour of dividing circles.

To construct these scales, divide AB (fig. 11), a quadrant of a circle, into six equal parts. Draw the line ba to touch the middle of the arc at G. Draw lines from the centre through A and B, the extremities of the arc, to meet the tangent in a and b, and also through the divisions, to meet the tangent in the points against wbich the numerals VI, V, IV, &c. are placed. Then the line between the extreme points a and b is the scale of hours.

Next, divide EF, a quadrant of the same circle, into 90 equal parts, (only every tenth division is marked in the figure). From the points of division draw perpendiculars to OF, the radius. Draw lines through E and the bottoms of the perpendiculars, and produce them, until they meet the circumference again in the points 10, 20, 30, sc. transfer the chords of the arcs D 10, D 20, D 30, &c. (also the chords of the intermediate arcs not distinguished in the figure) to a straight line df, numbering them as in the figure; and the line df will be the scale of latitudes.

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