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Prob. 14. To construct Mercator's Chart of the World.
1. Draw any indefinite line for an equator, marked o EQUATOR O in the plate.
2. Assume any point on this line for the position of the first meridian; that is to say, the Meridian of Greenwich.
3. Take any assumed distance in the compasses for 10° of longiade, or the unit of measure ; set this off on the equator from zero (0), or the meridian of Greenwich, eighteen times on both sides of
Then will 360° of longitude be set off; because 18+18=36, which, multiplied by 10', give 360°.
4. Through those 36 equal divisions on the equator draw straight and parallel lines at right angles to the equator : these straight and parallel lines will be the meridians 0, 10, 20, 30, &c. and which, on the projection, are continued across the chart at every 20th degree of longitude, east and west of zero, or Greenwich.
A TABLE OF MERIDIONAL PARTS, ,
For the Construction of Mercalor's Chart.
Deg. Mer. Pts. Deg. Mer. Pts. Deg. Mer. Pts. Deg. Mer. Pt: Deg. Mer. Ptsal
5. To lay down the parallels of this Mercator's chart, look in the foregoing Table of Meridional Parts for the number of parts corresponding to 10' of latitude. They are 603 ; divide this by 60, and the quotient 103 will be 10 degrees 3 minutes, because the 603 parts are minutes ; and the divisor 60 is also 60 minutes, equal to one degree. Take, then, 10° 3' from the degrees of longitude on the equator, and set them off, both north and south, from the zeros, 0,0, at the ex. tremities of the equatorial line ; and draw the parallels of 10° north and south latitude.
6. To lay down the parallel of 20° north or south latitude, divide by 60, and the meridional parts answer to 20",—viz. 1225,-and the quo. tient 20" 25', taken from the degrees of longitude on the equator, and
set cff both ways from the point zero, will give the exact position or the 20th degree of north and south latitude.
7. The latitude of 32° is 1888' + 60=31° 20' taken from uic equae tor, and set off both ways from the zeros.
8. That of 40° is 2623' +60'=43' 43' taken from the cquator, and set off both ways from the points zero.
9. That of 50°=3474'+60=57° 54' taken from the equator, and set off both ways from the points zero
10. That of 60°=4527' +60=75° 27'. &c.
11. From the points 120", 10°, 80, on the equator; draw those lines, which, radiating from a centre, represent the different points of the MARINER'S COMPASS. 96 chown in the following Table :
TABLE OF THE ANGLES, Which every Point and Quarter-Point of the Compass makes with
N. by E. N. by W.
N.E. by N.N.W. by N.
S.E. by S. S.W. by S.
2.48.45 5.37 30 8.26.15 11.15. 0 14. 3.45 16.52.30 19.41.15 22.30. 0 25.18.45 28. 7.30 30.56.15 93.45. O 36.33.45 39,22.30 42.11.15 45, 0.0 47.48.45 50.37.30 53.26.15 56.15. 0 59. S.45 61.52.SO 64.41.15 67.30. 0 70.18.45 73. 7.30 73.56.15 78.45. O 81.33 45 84.22.30 87.11.15 90, 0.0
12. At every 15th degree in the marginal lines of the chart, at top and bottom, set off the hours I. II. III. IV. V., &c. for the purpose of working the Problems on this map.
13. In fine, complete the chart by faithfully indicating all the chequered and indented sea-coast, it being the chief thing which in this chart claims attention : for, in all charts, the inland country is totally neglected, and, except particular towns, harbours, forts, light. houses, beacons, and rocks, directly on the margin of the sea, the interior country is left unadorned. But, on the water, the winds, currents, shoals, rocks, anchorages, and, in short, all things connected with nautical geography, are carefully and truly laid down, according to their direction, variation, bearing, and distance.
Though what is generally called the Globular Projection of the Sphere is preferable to the Stereographic in the construction of maps, 80 far as regards the division of lines, it has not the advantage of representing the angles on a map equal to their originals ; nor can the originals of the measures of lines and angles be obtained from it as from the stereographic ; since it depends upon no other principle than the whim of dividing the primitive and vertical circles equally, and drawing lines to form the meridians and parallels of latitude through every three points of division. When the eye is taken about three quarters of the radius out of the surface of the sphere, then all the circles, except those which pass through the eye, or parallel to the primitive, are projected into ellipses, those which pass through the eye are represented by straight lines, and those which are parallel to the primitive are projected also into circles. This is really a true projection, and the circles divide each other into equal parts very nearly; but it is difficult to project.
However, wherever accuracy is required it ought to be used, as being a more natural representation from which the original measures can be obtained.
DIALLING, or the method of constructing sun-dials, is a branch of mixed mathematics, which depends partly on the principles of geometry, and partly on those of astronomy.
The General Principles of Dialling. 1. The principles of astronomy teach us that the earth moves in an orbit about the sun, and completes a revolution in a year ; while, at the same time, it revolves uniformily from west to east on its axis, which, although it changes its place, is yet always parallel to a fixed imaginary line, called the axis of the world. By the first of these motions, the sun appears to move round the heavens, completing a revolution in the course of a year; and by the second, the sun, and all the heavenly bodies, have an apparent diurnal motion about the earth from east to west,
2. The motion of the earth in its orbit is not equable; and hence it happens, that the apparent motion of the sun in the heavens is not quite uniform : besides, the plane of that motion does not coincide with the plane of the diurnal motion. On these two accounts, the apparent diurnal motion of the sun differs a little from uniformity.
3. In the theory of dialling, however, we are to suppose that the sun's diurnal motion is always perfectly uniform, and that it moves throughout the day in a circle parallel to the equator ; but as neither of these hypotheses is strictly true, the time of the day shown by a dial will in general differ from that shown by an accurate clock. However, the difference admits of exact estimation, and tables bave beea calculated which show its amount for every day throughout the year.
4. In constructing dials, it is also usual to leave the effect of refraction out of consideration ; its effect might, indeed, be exactly appreciated, and tables formed, by which the time indicated by the dial might be corrected; or the dial might even be so constructed as to give the time cleared from the error. But this would be a degree of refinement which may very well be overlooked in the practice of what, since the invention of clocks and watches, is now little more than a scientific recreation.
3. If the earth's radius had any sensible proportion to its distance from the sun, that ought to be taken into account in the construction of dials But the earth is almost a mere point, as seen from the sun ; and herce it happens that the diurnal motion of the sun about any
line on the earth's surface, which is parallel to its axis, may be accounted uniform, exactly as if it were performed about the axis itself.
6. To understand the nature of a dial, let us suppose that eEF (fig. 1) is a straight rod or wire, parallel to the axis of the earth ; or which, if produced, woald pass through the pole of the heavens ; and let us suppose that one of its extremities terminates at e in a plane, abcd having any position whatever. Let us farther suppose, that the wire passes through E, the centre of a circle ABCD, described on some solid substance, and that it is perpendicular to the plane of that circle: then, as the wire passes through the poles of the heavens, the circle ABCD will be parallel to the terrestrial equator, and it will be in the plane of the equinoctial circle in the heavens, because on the earth's surface any plane whatever, parallel to the equator, may be considered as coincident with it, when produced to the celestial sphere.
Now, because the axis of the earth perpendicular to the plane of the circle which the sun appears to describe in the heavens by his diurnal motion, and passes through its centre, and that the same is alm.ost exactly true of every line parallel to the earth's axis ; when the circle ABCD is illuminated by the sun, the wire EP will project a shadow upon it, which will revolve about E as a centre, passing over equal arcs of the circumference in equal intervals of time. If, therefore, we suppose the circumference of the circle to be divided into twenty-four equal parts, and the points of division to be numbered 1, 2, 3, 4, &c. to 12, and again 1, 2, 3, 4, &c. to 12, as in the figure, and the circle to have such a position, that the shadow falis upon E 12 at noon ; then, at one o'clock, it will have the position El; at two o'clock, it will have the position E 2; at three, the position E 3; and so on. In short, the hour of the day, froni sunrise to sunset, will be indicated by the shadow, just as it is shown upon a watch by the motion of the hour hand. And as we suppose the motion of the sun to be quite uniform, the shadow will always have the same position at the same hour every day throughout the year.
7. If the two planes ABCD, alcd are illuminated at once by the sun, the rod eEF will project a shadow on them both. Let us suppose that at the instants the line EF projects its shadow in the directions of the lines E 12, E 1, E 2, &c, on the upper plane, the shadow of eE falls in the lines e 12, e 1, e 2, &c. respectively on the lower plane ; and let other cotemporaneous positions of the shadows be found for every hour the sun can shine on the planes ; then, as the shadow will always come to the same position on each plane at the same bour of the day, the hours will be indicated also by the shadow on the plane abcde.
8. Each of the planes ABCD, abcd is a dial: we have supposed the upper plane to be perpendicular to the axis of the world ; and in this particular position, the shadow will describe equal angles on it in equal times. The plane of the dial may, however, have any position; but if it is not perpendicular to the earth's axis, the motion of the