body is in proportion to its distance, compared with its apparent size, the sun must, from this consideration alone, be more than 1200 miles in diameter, and must contain more than nine hundred millions of cubical miles. But how much greater his distance and magnitude are than what is now stated cannot be determined from such observations. The same idea may be illustrated as follows: Suppose a spectator at Edinburgh, which may be represented by the point A (Fig. 95,) and another at Capetown, in the southern extremity of Africa, about the time of our winter solstice, which position may be represented by the point E; both spectators might see the sun at the same moment, and he would appear exactly of the same size from both positions. Yet such spectators would be more than 4000 miles distant from each other in a straight line, and the observer at Capetown would be several thousands of miles nearer the sun than the one at Edinburgh. Now if the sun were only a few thousands of miles from the earth, he would appear of a very different magnitude to observers removed so far from each other, which is contrary to fact. Consequently, the sun must be at a very great distance from the earth, and his real size proportionable to that distance. For experience proves that objects which are of great magnitude may appear comparatively small when removed from us to a great distance. The lofty vessel, as it recedes from the coast towards the ocean, gradually diminishes in its apparent size, till at length it appears as a scarcely distinguishable speck on the verge of the horizon; and the aeronaut with his balloon, when they have ascended beyond the region of the clouds, appear only as a small dusky spot on the canopy of the sky, and sometimes entirely disappear. The following argument, which is level to the comprehension of every reflecting mind, proves that the sun is larger than the whole globe of the earth, and that the moon is considerably less. Previous to the application of the argument to which I allude, it may be proper to illustrate the law of shadows. The law by which the shadows of globes are projected is as follows: When the luminous body is larger in diameter than the opaque body, the shadow which it projects converges to a point which is the vertex of a cone, as in Fig. 96. When the luminous and the opaque body are of an equal size, the shadow is cylindrical, and passes on from the opaque body to an indefinite extent, as represented in Fig. 97. When the luminous body is less than the opaque, the shadow extends in breadth beyond the opaque body, and grows broader and broader in proportion to its distance from the opaque globe, as in Fig. 98. This may be illustrated by holding a ball three or four inches in diameter opposite to a candle, when the shadow of the ball will be seen to be larger in diameter in proportion to the distance of the wall or screen on which the shadow is projected. Now it is well known, and will readily be admitted, that an eclipse of the moon is caused by the shadow of the earth falling upon the moon, when the sun, earth, and moon are nearly in a straight line with respect to each other; and that an eclipse of the sun is caused by the shadow of the moon falling upon a certain portion of the earth. Let S (Fig. 99) represent the sun; E the earth; and M the moon, nearly in a straight line, which is the position of these three bodies in an eclipse of the moon. The shadow of the earth, at the distance of the moon, is found to be of a less diameter than the diameter of the earth. This is ascertained by the time which the moon takes in passing through the shadow. The real breadth of that shadow, at the moon's distance from the earth, is about 5900 miles, sometimes more and sometimes less, according as the moon is nearer to or further from the earth; but the diameter of the earth in its progress through space, and, by calculais nearly 8000 miles; therefore the shadow tion, it is found that it terminates in a point, of the earth gradually decreases in breadth as in Fig. 96, at the distance of about 850,000 miles. But when a luminous globe causes the shadow of an opaque globe to converge towards a point, as in Fig. 96, the luminous body must be larger in diameter than the opaque one. The sun is the luminous body which causes the earth to project a shadow on the moon; this shadow, at the moon, is less in breadth than the diameter of the earth; therefore it inevitably follows that the sun is larger than the earth; but how much larger cannot be determined from such considerations. From the same premises it necessarily fol lows that the moon is less than the earth. For the moon is sometimes completely covered by the shadow of the earth, although this shadow is less than the earth's diameter, and not only so, but sometimes takes an hour or two in passing through the shadow. If the sun were only equal to the earth in size, the earth's shadow would be projected to an indefinite extent, and be always of the same breadth, and might sometimes eclipse the planet Mars when in opposition to the sun. If the sun were less than the earth, the shadow of the earth would increase in bulk the further it extended through space (as repre. sented in Fig. 98,) and would eclipse the great planets Jupiter, Saturn, and Uranus, with all their moons, when they happened to be near their opposition to the sun; and in this case they would be deprived of the light of the sun for many days together. In such a case, too, the sun would sometimes be eclipsed, to the earth by the planet Venus, when in its inferior conjunction with that luminary: an eclipse which might cause a total darkness of several hours continuance. In short, if the sun were less than any one of the planets, the system would be thrown into confusion by the shadows of all these bodies increasing in proportion to their distance, and interrupting, periodically, for a length of time, the communications of light and heat. But as none of these things ever happen. it is evident that the sun is much larger than the whole terraqueous globe. All that requires to be taken for granted by the unlearned reader in this argument is, that the earth is a globular body; that an eclipse of the moon is caused by the shadow of the earth falling upon that orb; and that the shadow of the earth, at the distance of the moon, is of less breadth than the earth's diameter. The first two positions will readily be admitted; and the third position, respecting the breadth of the earth's shadow, may be received on the ground of what has been above stated, and on the authority of astronomers. For, if they were ignorant of this circumstance, they could not calculate eclipses with so much accuracy as they do, and predict the precise moment of the beginning and end of a lunat eclipse. If, then, any individual is convinced, from the consideration above stated, that the sun must be much larger than the earth, he has advanced one step in his conceptions of the magnificence of the heavenly bodies, and may rest with confidence on the assertions of astronomers in reference to the real distances and magnitudes of these orbs, although he may not be acquainted with the mathematical principles and investigations on which their calculations proceed. Before proceeding to the illustration of the trigonometrical principles on which astronomers proceed in determining the true distances of the heavenly bodies, it may be requisite, for the unlearned reader, to give a description of the nature of angles and the mode by which they are measured. An angle is the opening between any two lines which touch each other in a point; and the width of the opening determines the extent of the angle, or the number of degrees or minutes it contains. Thus if we open a pair of compasses, the legs of which may be represented by A B, B C, Fig. 100, an angle is formed of different dimensions, according as the extremities of the legs are removed further from or brought nearer to each other. If the legs are made to stand perpendicular to each other, as in Fig. 101, the angle is said to be a right angle, and contains ninety degrees, or the fourth part of a circle. The walls of a room generally stand at right angles to the floor. If the legs be separated more than a right angle, they form what is termed an obtuse angle, as in Fig. 102. When the angle is less than a right angle, it is called an acute angle, as in Fig. 100, and, consequently, contains a less number of degrees than ninety. All angles are measured by the arc of a circle described on the angular point; and every circle, whether great or small, is divided into 360 equal parts, called degrees. Thus, if I want to know the quantity of an angle at K, (Fig. 103) I place one point of the compasses at the angular point K, and describe the arc of a circle between the two sides L K, K M, and whatever number of degrees of a circle is contained between them is the quantity or measure of the angle. If, as in the present case, the angle contains the eighth part of a circle or half a right angle, it is said to be an angle of forty-five degrees. A triangle is a figure which contains three angles and three sides, as O PQ, Fig. 104. It is demonstrated by mathematicians, that the three angles of every triangle, whatever proportion these angles may bear to each other, are exactly equal to two right angles, or 180 degrees. Thus, in the triangle O P Q, the angle at Q is a right angle, or ninety degrees, and the other two angles, O and P, are together equal to ninety de Thus, if the angle at P be equal to thirty degrees, the angle at O will be equal to sixty degrees. Hence, if any two angles of a triangle be known, the third may be found by substracting the sum of the two known angles from 180 degrees, the remainder will be the number of degrees in the third angle. All the triangles have their greatest sides opposite to their greatest angles; and if all the angles of the triangle be equal, the sides will also be equal to each other. If any three of the six parts of a triangle be known (excepting the three angles,) all the other parts may be known from them. Thus, if the side PQ, and the angles at P and Q be known, we can find the length of the sides PO and O Q. It is on this general principle that the distances and magnitudes of the ho venly bodies are determined. In order to understand and apply this principle, it is necessary that we explain the nature of a parallax. A parallax denotes the change of the apparent place of any heavenly body, caused by being seen from different points of view. This may be illustrated by terrestrial objects as follows: Suppose a tree 40 or 50 yards distant from two spectators, who are 15 or 20 yards distant from each other; the one will perceive the tree in a line with certain objects near the horizon, which are considerably distant from those which appear in the direction of the tree, as viewed from the station occupied by the other spectator. The difference between the two points near the horizon where the tree appears to coincide to the two different spectators is the parallax of the object. If the tree were only 20 or 25 yards distant, the parallax would be twice as large; or, in other words, the points in the horizon where it was seen by the two spectators would be double the distance, as in the former case; and if the tree were two or three hundred yards distant, the parallax would be proportionally small. Or, suppose two persons sitting near each other at one side of a room, and a candle placed on a table in the middle of the room, the points on the opposite wall where the candle would appear to each of the two persons would be considerably distant from each other; and this distance may be called the parallax of the candle as viewed by the two observers. This may be illustrated by Fig. 105, where R and S may represent the positions of the observers; a the candle or tree; and T and U the points on the opposite wall or in the horizon where the candle or the tree appears to the respective observers. The observer at R sees the intermediate object at U; and the one at S sees it in the direction S T. The angle R a S, which is equal to the angle Ta U, is called the angle of parallax, which is the difference of position in which the object is seen by the two observers. If, then, the distance between the observers R $ be known, and the quantity of the angle R a S, the distance between the observers and the object can also be known by calculation. Let us now apply this principle to the heavenly bodies. In Fig. 106 let the semicircle S, T, A, R, S, represent a section of the concave of the heavens; the middle circle, E C, the earth; M the moon; C the centre of the earth; and E H the sensible horizon of a spectator at E. It is evident that if the moon be viewed from the earth at the point E, she will be seen in the horizon at the point H; but were she viewed at the same time from C, the centre of the earth, she would appear among the stars at the point K, in a more elevated position than when seen from the surface of the earth at E. The difference between those two apparent positions of the moon, or the angle K M H, is called the moon's horizontal parallax. Astronomers know from calculation in what point of the heavens the moon would appear as viewed from the earth's centre; and they know from actual observation where she appears as viewed from the surface; and, therefore, can find the difference of the two positions, or the angle of parallax. This angle might likewise be found by supposing two spectators on different parts of the earth's surface viewing the moon at the same time. Suppose a spectator at E, who sees the moon in the horizon at H; and another observer, on the same meridian, at B, who sees her in his zenith at K; the parallax, as formerly, will be K H. The parallax of a heavenly body decreases in proportion to its altitude above the horizon, and at the zenith (A) it is nothing, for the line from the centre of the earth coincides with that from the surface, as C E A. Thus, the parallax of the moon at N (a b) is less than the horizontal parallax, KH; but from the parallax observed at any altitude, the hori zontal parallax can be deduced; and it is from this parallax that the distance of the moon or any other heavenly body is determined. The greater the distance of any body from the earth, the less is its parallax. Thus the heavenly body G, which is further from the earth than the moon, has a less parallax (cd) than that of the moon, K H. Now the parallax of the moon being known, it is easy to find the distance of that orb from the earth; for in every triangle, if one side and two angles be known, the other angle and the other two sides can also be found. In the present case, we have a triangle E M C, in which the side E C, or the semidiameter of the earth, is known. The angle M E C is a right angle, or ninety degrees; and the parallactic angle E M C is supposed to be found by observation. From these data, by an easy trigonometrical calculation, the length of the side C M, or the distance of the moon from the centre of the earth, can be determined with the utmost precision, provided the angle of parallax has been accurately ascertained. Before proceeding to illustrate by examples the method of calculating the distances of the heavenly bodies when the parallax is found, I shall present an example or two of the mode of computing the heights and distances of terrestrial objects, the principle on which we proceed being the same in both cases. Suppose it were required to find the height of the A 107 tower C B (Fig. 107,) we first measure the distance from the bottom of the tower, B, to a station at the point A, which suppose to be one hundred feet. From this station, by a quadrant or other angular instrument, we take the angle of elevation of the top of the tower, or the angle CA B, which suppose to be fortya B seven and a half degrees. Here we have a triangle in which we have one side, A B, and two angles; namely, the angle at A=474°, and the angle at B, which is a right angle, or 90°, as the tower is supposed to stand perpendicular to the ground; therefore the side C B, which is the height of the tower, can be found, and likewise the other side, A C, if required. To find C B, the height of the tower, we make A B the radius of the circle, a portion of which measures the angle A; and the side B C, or the height of the tower, becomes the ta gent of that angle. And as there is a certain known proportion between the radius of every circle and the tangent, the height of the tower will be found by the following proportion: As the radius: is to the tangent of the angle A, 473°: so is the side A B, 100 feet: to CB, the height of the tower-109 feet. The following is the calculation by logarithms: Logarithm of the 2d termTangent of 4710 Logarithm of A B=100 feet— :: 3d term Logarithm of radius-1st term Logarithm of C B, 4th term109 feet= 10.0379475 2.0000000 12.0379475 10.0000000 2.0379475 Suppose the angle at E to be seventy three degrees and the angle at F sixty-eight degrees. As all the angles of a triangle are equal to two right anges, or 180°, if we add these two angles and subtract their sum from 180°, the remainder, 39°, will be the measure of the angle at D. It is a demonstrated proposition in trigonometry, that in any plane triangle, the sides are in the same proportion as the sines of the opposite angles. A sine is a line drawn through one extremity of an arc perpendicular upon the diameter or radius passing through the other extremity, as a d (Fig. 107.) In order, then, to find the distance (E D) between the tree and the house on the other side of the river, we state the following proportion: As the sine of D, 38°, the angle opposite to E F, the known side: is to the sine of the angle F, 68°, opposite the side sought, E D:: so is the length of the line E F 200 yards: to the distance, E D, between the tree and the house-2944 yards. The following is the operation by logarithms: 2d term-Sine of angle, F-68° 9.9671659 3d term-E F=200 yards. Log. 2.3010300 12.2681959 place as seen from the surface at E; or, in diameter 3d term-Radius 1st term-Sine of angle, D=39° 9.7988718 1st term-Sine of 57 minutes, 5 4th term-D E-2943 yards- 2.4693241 In these examples the logarithms of the second and third terms of the proportion are added, and from their sum the logarithm of the first term is subtracted, which leaves the logarithm of the fourth term; as in common numbers, the second and third terms are multiplied together, and their product divided by the first term; addition of logarithms corresponding to multiplication of whole numbers, and subtraction to division. The logarithms of common numbers, and of sines and tangents, are found in tables prepared for the purposes of calculation. I shall now state an example or two in reference to the celestial bodies. Suppose it is required to find the distance of the moon from the earth. In Fig. 109, let E C represent the earth; M the moon; E the place of 109 seconds 3.598243 10.000000 13.598243 8.220215 5.378028 According to this calculation, the moon is two hundred and thirty-eight thousand, eight hundred miles from the earth. In round numbers we generally say that the moon is 240,000 miles distant; and, in point of fact, she is sometimes considerably more than 240,000 miles distant, and sometimes less than the number above stated, as she moves in an elliptical orbit, her horizontal parallax varying from 54 to above 60 minutes. To find the Diameter of the Moon.-In Fig. 110 let A G B represent the moon, and an observer at the earth. The apparent diameter of the moon at its mean distance, as measured by a micrometer, is 31 minutes, 26 seconds, represented by the angle A CB; the 110 B a spectator observing the moon in his sensible horizon; E Mb and C M a the direction of the moon as seen from the centre of the earth at C, or from its surface at B; a the place or the moon as seen from the centre, and b its half of us, or the angle formed by the semidiauer of the moon, A C G, is 15 minutes, seconds. The distance of the moon, G C, supposed to be found as above stated, namely, 238,800 miles. Here, then, we have is |