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on the earth's surface which have their horizons perpendicular to one another, must have their zeniths 90 degrees asunder. Hence it folows, that a vertical south or north dial at any place, would be a borizontal dial to a place 90° south or north from that place, and on the same meridian. A vertical south or north dial at Edinburgh, for instance, the latitude of which is 56°, would be a horizontal dial at place on the same meridian, and in 34° of south latitude, and i would show the same bour of the day at both places, because the time of noon happens at both at the same instant.
37. Hence, if we put E for the horary angle from noon, and C for the angle which the corresponding hour-line makes with the meridian of a north or south dial, and L for the latitude of the place we get immediately, from the formula of art. 21, Tan. C=Cos. L tan. E
(2) a general formula for constructing a north or south dial; and by this, the angles which the hour-lines make with the meridian may be computed. The other three methods of constructing a horizontal dial apply equally to vertical south or north dials, only substituting the complement of the latitude, or what it wants of goo for the latitude, observing that the axis must make with the plane of the dial an angle equal to the complement of the latitude, and must be in the meridian, and then it will point to the pole of the world, as must be the case in all dials.
38. In north latitudes, a north dial is only illuminated when the sun is on the north side of the equator ; and the nearest times of the day to noon that can be shown by it, are those at which the sun passes the prime vertical on the day of the summer solstice. A south dial can never be illuminated before six in the morning, nor after six in the evening ; because, when the sun rises earlier, and sets later, he does not pass the prime vertical so early as six in the morning, and he crosses it again before six in the evening. It will be unnecessary, therefore, to describe upon either more hour-lines than can be wanted.
Vertical East and West Dials 39. These dials are traced upon vertical planes, facing directly east and west ; their planes, therefore, coincide wiib the plane of the meridian, and pass through the poles of the world.
To explain the nature of these dials, let us suppose that NS, fig. 1ĉ is a straight line traced upon their planes in the direction of the earth axis, and that it is crossed at right angles by a straight line EQ, whics will be the intersection of the planes of the equinoctial and the meridian Let us also suppose, that AB is a thin cylindrical rod, held directly over the line ab, and parallel to it, by two supports Aa, BL; and that this rod passes through C the centre of a circle, which lies „n the plane of the equinoctial circle, and which touches the plane of the meridian in c, the bottom of a perpendicular Cc : this circle will evidently be an equinoctial dial, of which AB is the axis.
Let CK be the shadow which the axis projects on the plane of the ciscle, and let it be produced to meet the vertical plane in k; then a
line drawn through k, perpendicular to EQ, will evidently be the direction of the shadow which the rod AB projects on the vertical plaane, at the same instant of time that it projects on the equinoctial dial the shadow CK ; and as the hours are indicated on the equinoctial dial by the position of the revolving shadow CK, they will also be shown on the vertical plane EQNS, by the successive positions of the rectilineal shadow FkF, which wil always be parallel to NS.
Now, as the plane a ABb is perpendicular to the plane of the meridian, and passes through the poles, it must be the plane of the six o'clock hour circle, or that circle in the heavens passing through the poles of the world, in which the sun is always seen at six in the morning and six in the evening. Therefore the arc cK of the equinoctial dial, intercepted between the perpendicular Cc and Ck, the position of the shadow at any time will be the measure of the horary angle described by the sun in the heavens, between six o'clock and that time ; and the straight line ck, the distance of the shadow of the rod AB from the line ab immediately under it, will be the tangent of that arc to the radius Cc.
40. Let the horary angle from six o'clock be denoted by E', let ck, the distance of the hour-line from ab, be x ; also let Cc, the height of the rod above the plane of the dial, be denoted by d, then becanse rad. : tan. E! :: :x, the general formula expressing the position of the hour-Anes on an cast or west dial, in respect of the line ab, will be (supposing radius=1) x=d tan. El
(3) from which it appears that, in these dials, the position of the honrlines in respeot of each other is altogether independent of the latitude of the place. Indeed, the same thing might have been inferred from what has been said in art. 33 and 34, for a vertical east or west dial, for any place whatever, would manifestly be an horizontal dial at the equator.
MENSURATION is the art of computing the extension, superficies, a solidity of lines, surfaces, or solids, from given data.
Def. Every quantity is measured by some other quantity of the same kind ; as a line by a line, a surface by a surface, and a solid by a solid ; and the number which shows how often the lesser, called the measuring unit, is contained in the greater, or quantity measured, is called the content of the quantity so measured. Thus, if the quantity to be measured be the
D rectangle ABCD, and the little square E, whose side is one inch, be the measuring unit, then, as often as the said little square is contained in the rectangle, 80 many square inches the rectangle is mid to contain : so that if the length DC be supposed 5 inches, and the breadth AD 3 inches, the content of the rectangle will be 3 times 5, or 15 square inches : because, if lines be drawn parallel to the sides, at an inch distance one from another, they will divide the whole rectangle ABCD into 3 times 5, or 15 equal parts, of one inch each. And, generally, whatever the meapures of the two sides may be, it is evident that the rectangle will contain the square E, as many times as the base AB contains the base of the square, repeated as often as the altitude AD contains the altititude of the square.
Hence we have the following rule for any parallelogram whatever.
Prob. 1. To find the area of a parallelogram, whether it be a square, a rectangle, a rhombus, or a rhomboides.
Multiply the length by the perpendicular height, and the product will be the area.
Ex. 1. What is the area of a square ABCD, whose side AB or BC is 2f. 3 in. ? By duodecimals.
By decimals. 2 3i
2.25 2 3i