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DEFINITIONS, ILLUSTRATIONS, ETC.
solidity eight times; a three-feet globe twenty-seven times; and a four-feet globe sixty-four times that of a one-foot globe.
Circles (circular spaces) must, in like manner, be compared by considering the squares of their radii, or of their diameters.
Illustration. The three following are Quadrants, (quarters of circular areas) described with the radii 1, 2, and 3, respectively, and included in squares constructed on those radii.
H The quantity of light received from an object into each of two telescopes, will, therefore, be in the ratio of the squares of the diameters of the tubes, or of the glasses called the "object-glasses" at their ends; or, if the telescopes be of the reflecting construction, in the ratio of the squares of the diameters of the reflectors; and the internal arrangements being equally good, their powers of revealing the features of an object, will be in that ratio.
B The earth reflects the sun's rays upon the moon (as she upon the earth); so that, with the unassisted sight, we can frequently perceive the darker portion of her disk. Dr. Herschel's ten feet reflector, which has, perhaps, an aperture of fifteen inches diameter, collected sufficient of this earth-light, to show him even the most obscure of her spots on that portion.
C It was formerly usual to form deep well-holes in observatories, for the purpose of seeing the stars in the day time; because in the old uncorrected object-glass of telescopes, all the more oblique rays, that is, those admitted into the object-glass at a considerable distance
from its centre, as those beyond a and p, (see fig. below,) were decomposed in passing through it, into their parts or colours, (see "Refraction,") and, causing distortion and confusion, were necessarily "stopped out."
The improvements introduced by Dollond and others,* in the construction of the object glasses of telescopes, now allow of their apertures being so enlarged, (see A P,) as to admit the light from a star in quantity sufficient (even when mixed with strong daylight,) to reveal the star to the eye in the focus of the instrument. A large star may be thus seen, even when near the sun; and Dr. Dick informs us, in his "Celestial Scenery," that he has frequently observed the planet Venus, even when at her superior conjunction; i. e., just northward or southward of the sun, and at her greatest distance from us.
D The planet Jupiter, at its more than fivefold distance from the sun, receives, on any portion of its surface, less than one twenty-fifth of the light which a similar portion of our own planet's surface receives. (C on. p. 13.) If imagination can enable us to furnish its inhabitants with eyes, the pupils of which shall have a six-fold width, and consequently present a thirty-six fold surface for the reception of light, we may conceive of them as admitting to nerves, fitted to receive and enjoy it, a daily splendour far exceeding our own.
K Let the line A B be the diameter of a circle, and C the centre: DK then will the arc D B be the measure of the angle formed at C by the radii C D and C B.
If we now drop a perpendicular, D E, from D, the extremity of the measuring arc, upon C B, (the radius with which it was formed),
Mr. Ross of Regent Street has very recently made material improvements.
DEFINITIONS, ILLUSTRATIONS, ETC.
such perpendicular will be the Sine of the angle measured by that arc.
In like manner, if we turn the figure, so as to make G C horizontal, and D F be let fall from the extremity D of the arc D G upon the radius G C, D F is the Sine of the angle measured by the arc D G.
Now, the Arc D G is the complement (N p. 2,) of the arc D B; for G B contains 90°.
For the same reason D B is the complement of D G. The Sine of a complement, is, for shortness' sake, called its Cosine.
Hence, whilst D E is the sine of the angle measured by the arc D B, the line D F, or its equal E C, is its Cosine. So, D E. or its equal F C, is the cosine of the angle expressed by the arc D G.
The arc A D, or what D B wants of being a semicircumference, is called its supplement: in like manner D B is the supplement of A D. Now D E is the sine of both of these
that is, "Every sine is common to two arcs, which are supplements to each other, or make, together, 180°"
If a line, as B T, be drawn from B, the extremity of the radius, and at right angles to it, B T is called a Tangent. If the radius CD be lengthened so as to reach the tangent at K, C K is called the Secant of the arc D B, &c.
F The ancient Greeks employed Trigonometry (or "the measuring of triangles.") Hipparchus, one of their most celebrated astronomers, is supposed to have been the originator of the science. The Chord of the Arc (answering to the string of a bow,) was their means of solving problems. The Moors or Arabians, during the sojourn of astronomy with them, and when Europe was in darkness, introduced the Sine or half-chord, an improvement of great utility. The Latin word sinus (a fold), is supposed to be a translation of an Arabic word referring to the two halves of the chord being conceived to be folded together. The Europeans have introduced the secant and tangent since the 15th century.
↳ Parallax is that apparent angular motion of an object which is occasioned by changing the point from which we view it.
See Chronology of Ancient Observations, (Index).
Illustration. Whilst the boy A at the bottom of the hill, at a certain instant, sees his kite and the distant balloon in the same line, and opposite the point B, (in the sky), M, who is half-way up the hill, sees the kite opposite k, and the balloon opposite b. But S, who is on the summit, sees the kite opposite the point k', (on the ground,) and the balloon opposite B'.
M The amount of this parallactic displacement, is less as the distance of the object is greater. The difference of apparent positions of the distant balloon, is much less to A and S, than the difference of those of the kite. The moon having the same position, viz., close to the Evening Star from both stations, on account of the comparatively insignificant remoteness of those stations.
N Parallax, used astronomically, is the difference of apparent position of a heavenly body, viewed from the surface of the earth and from the earth's centre.
G An observer, who has the object in his zenith, sees it as if he were situated at the centre of the earth. Parallax, therefore, which is greatest when the object is in the horizon, is less as the altitude of that object is greater, and is nothing when it is vertical.
Illustration. Let A be an individual standing on the brow of a cliff, and G and H the stations of two inhabitants of a building. We will suppose it to be starlight, and G to record the star or point opposite to which the individual at A appears to be, from his station at the top of the tower, at the same instant that H records the star opposite to which A appears to be from his station at its base. It is evident that the arc, in the concave of the sky between such two stars, will have the same number of degrees as that which the building intercepts in the opposite direction from the view of A. (See note to L on p. 2.) If, with the distance A G, we draw an arc G K, then G H, the distance between the stations of the two inhabitants, will be the sine of that arc, i. e., the sine of the angle of parallax.
In like manner, if G be the centre of the earth, M the moon, would appear from that station, (or from Z, the station of an observer who
DEFINITIONS, ILLUSTRATIONS, ETC.
has her in his zenith,) to be opposite to the point g; whilst to H, who has her in his horizon, she appears at h. Now the distance between G and H being known, (see Q p. 20,) the distance M G of the moon from the earth's centre, can readily be found from her parallax.
P The sine of any arc used to measure an angle has always a known ratio to the radius with which that arc was drawn. Thus, if the radius (as A G) be one mile or one inch, the sine of 14° will be a little more than a quarter of a mile, or quarter of an inch: sine of 30° will be exactly half a mile, or half an inch: sine of 49° will be a little more than three quarters of a mile, &c. The sine of the arc (h g,) or (H G,) the parallax of the moon, is found to be about of radius M G. Hence (H G) the semi-diameter of our earth, or 4000 miles, (which is in this case the sine,) is of the distance of the moon, (in this case our radius,) and 4000 × 60=240000 miles.
other in south latitude, to observe, on the same day, the zenith-distance of the sun's centre at his station, when it is at the highest, that is, on their meridian and at their common noon.
Then, as will be seen by the figure above, these two zenith-distances