SECTION III. DIALLING. DIALLING is founded on the first motion of the heavenly bodies or the diurnal motion of the earth on its axis-and its theory depends on the elements of spherical trigonometry. But in an elementary work like this, we shall confine our instructions to what is practicable by a manual operation alone. The plane of every dial represents the plane of some great circle on the earth, and the gnomon of the earth's axis; the vertex of a right gnomon, the centre of the earth or visible heavens; and the plane of the dial is just as far from this centre as from the vertex of this style. The earth itself, compared with its distance from the sun, is considered as a point; and, therefore, if a small sphere of glass be placed upon any part of the earth's surface, so that its axis be parallel to the axis of the earth, and the sphere have such lines upon it, and such planes within it as are required, it will show the hours of the day as truly as if it were placed at the centre of the earth, and the shell of the earth were as transparent as glass. To construct an Equatorial Dial. Take a piece of oak plank a foot square, and 1 to 2 inches thick, with two sunk cross bars on the back to prevent warping in the sun. Divide it into a circle of 24 equal parts, beginning at 1 and going on to 12, and again repeated to complete the circle. The two lines of 6 and 12 must be at right angles; the board painted white and the lines black. Set a wire pin in the centre of the circle of the same thickness as the black lines, quite perpendicular, and at right angles with the lines 6 and 12. Find the latitude of the place, say 53°; subtract from 90° 37° or co-latitude; make two boards to this angle; place the dial level on a post, and the face northwards, elevated by the angle board behind, and the hours will be pointed correctly, by the shadow of the wire on the lines, for six months, when the sun is north of the line. In places south of the line, the face of the dial will be reversed, and stand to the south. 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'. 223 288 52 26 13 39 59 55 27 72 17 90 00 56 26 20 40 36 72 22222 55 37 72 22 90 00 33 90 CO 39 90 00 58 30 12 59 00 12 59 30 13 00 26 27 40 45 56 10 72 44 90 00 In this table the angles formed by the lines for the hour of v. in the morning and that of VII. in the evening, — IV. in the morning and VIII. in the evening, &c. are not marked, because 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. For the edge of the plane, by which the time of day is marked, is the style of the dial, and it is parallel to the axis of the earth. Hence the line on which this plane is erected is the substyle; and the angle included between the substyle and style is the elevation or height of the style or gnomon; moreover, this angle is always equal to the elevation of the pole, or latitude of the place. From this explanation it is clear that an erect south dial in 511° north latitude would be a horizontal dial on the same meridian 90° southward, which falls in with 3810 south latitude. The use of the foregoing table may be easily comprehended: thus, 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 are found by inspection. For example, the hour lines 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′, which would agree with Milton in Westmoreland, or Kirby Wiske, 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, namely, 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'. If these instructions be followed, we think every young gardener may be able to make a dial for the division of his own time, and the government of his duties at stated periods of the day. If by a good chronometer he could mark the shade of a tall tree at noon, by tracing a circle of some radius within the clear shadow of the stem, he might, on that circle, set off the angles corresponding to the hour lines in the foregoing table, and, by planting a flower or a shrub in the intersection of the hour line with the circle, exhibit the dial of paradise. 133 CHAP. VI. MECHANICS. MECHANICS, in the enlarged sense of the word, includes the whole range of natural philosophy or physics, and developes innumerable principles of the pure and mixed mathematics. But in a limited sense we confine mechanics to those general principles that govern forces as combined with matter. Hence, the sciences of statics and dynamics; upon which, however, this work cannot enter further than to explain a very small number of the principles that may be serviceable to those for whom we write. Force either produces or destroys motion. The unit of force may be taken at 1lb. troy, equal to 22-815 cubic inches of distilled water, which, divided into 5760 equal parts, the weight of each is a grain troy, and 7000 such grains make 1lb. avoirdupois. Hence 15 lb. avoirdupois represent 15lb. a force of 15 units. When a body is held at rest by two forces, these are said to be equal to one another. Here the forces act in opposite directions, and in the same straight line. We may multiply the forces, but that which counteracts its antagonists exercises double, treble, &c., their intensity if it preserve the equilibrium. Fig. 46. B Lines may represent forces in magnitude, and also in direction. When a third force is required to constitute an equilibrium, if lines be measured from this point in the direction of the forces, so as to contain each a given unit of length as many times as there are units in each force, then these lines will form the adjacent sides and dia A * This standard is fixed by act of parliament dated June 24th, 1824: temperature 62° Fahrenheit; barometer to stand at 30 inches. gonal of a parallelogram. Thus, if B A, C A, in fig. 46., be two forces acting upon the point A, we determine the magnitude and direction of the force which will hold them at rest by completing the parallelogram A B D C, and drawing the diagonal A D, which represents both the magnitude and direction, of the force that will keep the others in equilibrium. Thus we see that equilibrium results from the simultaneous action of several forces on a body, or a material point, when they reciprocally destroy each other's action, and the body, though free to move, remains at rest. Similar pressure, as noticed above, exhibits force counteracting force, and entire absence of motion. We thence infer that all bodies present themselves to us in a state of rest or of motion. But this last is the change of rectilinear distance between two points. And as the force producing motion may act upwards or downwards, &c., we trace motion in a straight line, bent or curvilinear, accelerated or retarded. Hence inertia is a contingent condition of rest or motion, yet the earth, in her orbicular and annual motions, and all the rest of the planets, wheel round the sun, in unresisting space, without effort and without cessation. The listless savage at the equator is carried along with a velocity of 101,200 feet per second of time, or 69,042 miles an hour: and the trees there, moving with this velocity, have not a leaf disturbed. Let but a hurricane sweep the same region with a force of 120 miles an hour, and all is desolation! The two revolutions of the earth, on its axis daily, and round the sun in a year, are employed as the standards of motion. The equatorial diameter is 26 miles longer than the polar, causing a variation in the gravitating force of one part in 194, by which means a mass of 195 lb. at the pole, weighs but 194 lb. at the equa tor. Bodies near the surface of the earth fall through about sixteen feet in a second of time. Hence we discover that force and velocity, time and space, are quantities in a continual flux, passing through a series of proximate stages. Thus it is, that the velocity of falling bodies accumulates, |