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X.

7. If the sun be in the zenith at noon, and at LECT. the same time has no declination, you are then under the equinoctial, and so have no latitude.

If the sun be in the zenith at noon, and has declination, the declination is equal to the latitude, north or south. These two cases are so plain, that they require no examples.3

3 The latitude of a place may be found with equal facility and accuracy, by taking the meridian altitude of the planets and fixed stars, and observing the same directions which are given by our author in the case of the sun. When fixed stars, however, are employed, their altitude must be corrected by refraction only, as their parallax is not sensible.-ED.

LECTURE XI.

LECT.

XI.

OF DIALING.

HAVING shewn in the preceding Lecture how to make sun-dials by the assistance of a good globe, or of a dialing scale, we shall now proceed to the method of constructing dials arithmetically; which will be more agreeable to those who have learned the elements of trigonometry, because globes and scales can never be so accurate as logarithms, in finding the angular distances of the hours. Yet, as a globe may be found exact enough for some other requisites in dialing, we shall take it in occasionally.

The construction of sun-dials on all planes whatever, may be included in one general rule: intelligible, if that of a horizontal dial for any given latitude be well understood. For there is no plane, however obliquely situated with respect to any given place, but what is parallel to the horizon of some other place; and therefore, if we can find that other place by a problem on the terrestrial globe, or by a trigonometrical calculation, and construct a horizontal dial for it; that dial, applied to the plane where it is to

3

XI.

serve, will be a true dial for that place. Thus, LECT. an erect direct south dial in 51 degrees north latitude, would be a horizontal dial on the same meridian, 90 degrees southward of 51 degrees north latitude; which falls in with 38 degrees of south latitude but if the upright plane declines from facing the south at the given place, it would still be a horizontal plane 90 degrees from that place, but for a different longitude; which would alter the reckoning of the hours accordingly.

CASE I.

1. Let us suppose that an upright plane at London declines 36 degrees westward from facing the south; and that it is required to find a place on the globe, to whose horizon the said plane is parallel; and also the difference of longitude between London and that place.

Rectify the globe to the latitude of London, and bring London to the zenith under the brass meridian, then that point of the globe which lies in the horizon at the given degree of declination (counted westward from the south point of the horizon) is the place at which the above-mentioned plane would be horizontal.-Now, to find the latitude and longitude of that place, keep your eye upon the place, and turn the globe eastward until it comes under the graduated edge of the brass meridian; then the degree of the brass meridian that stands directly over the place, is its latitude; and the number of degrees in the equator, which are intercepted between the meridian of London and the brass meridian, is the place's difference of longitude.

Thus, as the latitude of London is 51 de

XI.

LECT. grees north, and the declination of the place is 36 degrees west; I elevate the north pole 51 degrees above the horizon, and turn the globe until London comes to the zenith, or under the graduated edge of the meridian; then, I count 36 degrees on the horizon westward from the south point, and make a mark on that place of the globe over which the reckoning ends, and bringing the mark under the graduated edge of the brass meridian, I find it to be under 304 degrees in south latitude: keeping it there, I count in the equator the number of degrees between the meridian of London and the brasen meridian (which now becomes the meridian of the requir ed place) and find it to be 42. Therefore an upright plane at London, declining 36 degrees westward from the south, would be a horizontal plane at that place; whose latitude is 30 degrees south of the equator, and longitude 423 degrees west of the meridian of London.

PLATE
XXIII.
Fig. 1.

Which difference of longitude being converted into time, is 2 hours 51 minutes.

The vertical dial declining westward 36 degrees at London, is therefore to be drawn in all respects as a horizontal dial for south latitude 30 degrees; save only, that the reckoning of the hours is to anticipate the reckoning on the horizontal dial, by 2 hours 51 minutes: for so much sooner will the sun come to the meridian of London, than to the meridian of any place whose longitude is 423 degrees west from London.

2. But to be more exact than the globe will shew us, we shall use a little trigonometry.

Let NE SW be the horizon of London, whose zenith is Z, and P the north pole of the sphere; and let Zh be the position of a vertical plane at

XI.

Z, declining westward from S (the south) by LECT an angle of 36 degrees; on which plane an erect dial for London at Z is to be described. Make the semidiameter Z D perpendicular to Z h, and it will cut the horizon in D, 36 degrees west of the south S. Then, a plane in the tangent HD, touching the sphere in D, will be parallel to the plane Zh; and the axis of the sphere will be equally inclined to both these planes.

Let WQE be the equinoctial, whose elevation above the horizon of Z (London) is 38 degrees; and PRD be the meridian of the place D, cutting the equinoctial in R. Then, it is evident, that the arc RD is the latitude of the place D (where the plane Zh would be horizontal) and the arc RQ is the difference of longitude of the planes Zh and DH.

In the spherical triangle WDR, the arc WD is given, for it is the complement of the plane's declination from S the south; which complement is 54° (viz. 90°-36°): the angle at R, in which the meridian of the place D cuts the equator, is a right angle; and the angle RWD measures the elevation of the equinoctial above the horizon of Z, namely 38 degrees. Say, therefore, as radius is to the co-sine of the plane's declination from the south, so is the co-sine of the latitude of Z to the sign of RD the latitude of D; which is of a different denomination from the latitude of Z, because Z and D are on dif ferent sides of the equator.

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See Playfair's Elements of Geometry. Spher. Trig.

Prop. XIX. ED.

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