21st of June and December, take the tangent of 23° 28′, the sun's declination at that time, and it will be 434, if the radius were A Cor 1000; but as the radius is the cosine of the latitude, which is 559, we must say as 1000: 559 = 434 : 243, the length of A D and A C. On the 21st of February, April, August, and October, the sun's declination is nearly 11° 19′, the tangent of which for a radius of 1000 is 200; but for a radius of 559, the cosine of the latitude, it will be 112, which is the distance of the stile from 4 on both sides on the 21st of the months already mentioned. On the 21st of January, May, July, and November, the sun's declination is nearly 20° 8' the tangent of which, for the radius 1000, is 367; but for the radius 559 it will be 205, which is the distance of the stile from A, on both sides, on the 21st of these months, the names of the months being inserted beside the points, as in Fig. 32. The horary points are now to be determined in the manner already mentioned, and the dial will be finished. In order to place the dial, we have only to turn it round till the stile of the analemmatic dial indicates the same hour with that of the horizontal one, and it will then be properly placed.

Description of a New Dial in which the Hours are at Equal Distances in the Circumference of a Circle.3

With any radius describe the circle FXIIB (Plate XI, Fig. 36), draw A XII for the meridian, and divide the quadrants FXII, BXII, each into six equal parts for hours. To the latitude of the place add the half of its complement, or the height of the equator, and the sum will be the inclination of the stile, or the angle D A C. Thus, at Edinburgh, the latitude is 55° 58′, the complement of which, or the altitude of the equator, is 34° 2′; the half of which, 17° 1′, being added to 55° 58', gives 72° 59′ for the inclination of the stile or the angle DAC. The position of the stile in the figure is that which it must have on the 21st of March and September, when the sun crosses the equator; but when the sun has north declination, the point A must move towards D, and when he is south of the equator, it must move in the opposite direction. In order to find the position of the point A for any declination of the sun, multiply together the radius of the dial, the tangent

3 This dial was invented by M. Lambert, and is described and demonstrated in the Ephemerides of Berlin, 1777, p. 200.

of half the height of the equator at the place for which the dial is constructed, and the tangent of the sun's declination, and the product of these three quantities, divided by the square of the radius of the tables, will give the distance of the moveable point A from the centre of the circle FXIIB.

Let it be required, for example, to find the position of the point 4 on the 21st of December and June, when the declination of the sun is a maximum, or 23° 28', the radius A B of the dial being divided into 100 equal parts.

Sum 21.1234013 Log. of product.

From this logarithm substract 20, the logarithm of the square of the radius, and the remainder will be 1.1234013 Log. 13.29. Take 13 parts, therefore, in your compasses, and having set them both ways from A, the limits of the moveable stile will be marked out.

For any other declination, the position of the point A may be found in a similar manner. It will be sufficient in general to determine it for the declination of the sun when he enters each sign, and place these positions on the dial, as represented in Fig 32.

The length of the stile AC, or its perpendicular height HC, must always be of such a size that its shadow may reach the hours in the circle FXII B. For any declination of the sun, its length AC may be determined by plain trigonometry. A XII is always given, the inclination of the stile DAC is also known, the angle A XIIC is equal to the sun's meridian altitude, and therefore the whole triangle may be easily found in the common way, or by the following trigonometrical formula :

AC the length of the stile =

AXIIX Sin. Merid. Alt.

Improve

Sin. (180° Angle of Stile + Merid. Alt.) Notwithstanding the simplicity in the construction of this dial, the motion of the stile is troublesome, ment upon and should if possible be avoided. For this purpose it by La Grange. the idea first suggested by the celebrated La Grange will be of essential utility. He allows the stile to be fixed in the centre A, and describes with the radius AB, circles upon

the different points where the stile is to be placed between A and D, and on the other side of A, which is not marked in the figure. All these circles must be divided equally into hours like the circle FXIIB, and when the sun is in the summer solstice, the divisions on the circle nearest the stile are to be used; when he is in the winter solstice, the circle farthest from A must be employed, and the intermediate circles must be used when the sun is in the intermediate points. This advice of La Grange may be adopted also in analemmatic dials.

CHAPTER XXI.

ON THE CAUSE OF THE TIDES ON THE SIDE OF THE EARTH OPPOSITE TO THE MOON.

T

Ir has always been reckoned difficult for those unacquainted with physical astronomy, to understand why the sea ebbs and flows on the side of the globe opposite to the moon. This fact, indeed, has frequently been regarded, and sometimes adduced, by the ignorant, as an unsurmountable objection to the Newtonian theory of the tides, in which the rise of the waters is referred to the attraction of the sun and moon. From an anxiety to give a popular explanation of this subject, Mr. Ferguson has been led into an error of considerable importance, in so far as he ascribes the tides on the side of the earth opposite the moon, to the excess of the centrifugal force above the earth's attraction.' It cannot be questioned, indeed, that the earth revolves round the common centre of gravity of the earth and moon, at the distance of nearly 6000 miles from that centre; and that the side of the earth opposite the moon has a greater velocity, and consequently a greater centrifugal force than the side next the moon; but as the side of the earth farthest from the moon, is only 10,000 miles from the centre of gravity, it will describe an orbit of 31,415 miles in the space of 27 days 8 hours, or 656 hours, which gives only a velocity of 47 miles an hour, which is too small to create a centrifugal force, capable of raising the waters of the ocean. The true cause of the rise of the sea may be understood

1 See Vol. I, p. 35.

from Plate XI, Fig. 38, where A B C is the earth, O the common centre of gravity of the earth and moon, round which the earth will revolve in the same manner as if it were acted upon by another body placed in that centre. Let AM, BN, CP, be the directions in which the points A, B, C, would move, if not acted upon by the central body; and let B b n be the orbit into which the centre B of the earth is deflected from its tangential direction BN. Then since the waters at A are acted upon by a force, as much less than that which influences the centre of the earth, as the square O B is less than the square of O A, they cannot possibly be deflected as much from their tangential direction AM, as the centre B of the earth; that is, instead of describing the orbit Am, they will describe the orbit e a. In the same manner, the waters at c being acted upon by a force as much greater than that which influences the centre B of the earth, as the square of O B exceeds the square of O C, will be deflected farther from their tangential direction than the centre of the earth, and instead of describing the orbit cp, will describe the orbit hci.

As the earth, therefore, when revolving round the centre of gravity O, will be acted upon by the moon in the same way as by another body placed in that centre, it will assume an oblate spheroidal form abc; so that the waters at c will rise towards the moon, and the waters at a will be left behind, or will be less deflected than the other parts of the earth, by the lunar attraction, from that rectilineal direction in which all revolving bodies, if influenced only by a projectile force, would naturally move.