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would, were it not for parallax, be exactly equal to their difference of latitude; and whatever discrepancy there might be found on comparing their observations, supposing them to be exactly made, and with proper allowance for the refraction of the atmosphere, must be equal to this parallax
"This once determined, the Horizontal Parallax," (and from it the distance of the sun,) "is easily found by dividing the angle of parallax so found by the sum of the sines of the two latitudes." (See Sir John Herschel's Ast. p. 191.)
The distance of the sun is such, as to cause his horizontal parallax, as found by this process and by others more elaborate, to be eight seconds of a degree: a quantity equal only to about one four-hundredth of that of the moon, and showing his distance to be about 400 times greater. (See M p. 18.)
? But it remains to introduce the pupil to a method by which may be ascertained (although, from circumstances, not with great precision,) the diameter of the globe on which we live.
point in the circumference, and another line be drawn from the same point E to H, touching the circle; the whole length of this line D E, multiplied by its part E F, (or the rectangle D EXE F,) will be exactly equal to the length E H multiplied by itself (or, to "the square of E H.")
So that, supposing I find that this line E H is 6, and therefore its square (Q) 36; if I likewise know that E F is 2, E D must, of course, be 18; since it will take 18 to produce the rectangle T, (36) by multiplication by 2.
By referring to the illustration of the rectangle and the equal square, (page 12,) it will be understood that the length, D E, bears the same ratio to E H, the length of the square Q, that the width of that square bears to E f, the width of the narrow rectangle T.
Suppose a pole 10 feet high to be placed on the sea-shore, and a telescope or theodolite, to be fixed horizontally to the top of it: it is found that an individual looking through the telescope, (as fig. above,) can just discern the top of another pole of the same height, similarly placed on an opposite shore, at the distance of eight miles from him. That is, the top of the other pole, or some signal (as a brilliant lamp,) upon it, will be seen just peeping over the horizon at A.
Now it is plain that the line of sight B A, must just touch the earth in the horizon at A, half-way between the two poles; or at the distance of four miles from either of them. And the top of the upright pole B, is 10 feet above the circle of the earth,' and would, if it were to grow far enough downward, pass through the earth's centre, and become a Diameter.
Apply this experiment to the theorem on the page we have just attended to, and we shall see that 10 feet multiplied by the diameter of our earth, must, if the observation be correct, be equal to the square of 21,120 feet, or 4 miles, viz., 446,054,400 feet.† So that, dividing this square number by 10 feet, the width of our rectangle, (see p. 11,) we get 44,605,440 feet, or 8448 miles for our diameter; which, although not quite correct, is so only in consequence of the want of exactness in ascertaining the distance between the telescopes, and of the refraction of the atmosphere, which throws out the straightness of our line of sight; and, varying with the weather, enables the observer to look farther round the curvature of the earth's surface at one time than at another.
L An eye placed lower than B, or 10 feet, could not, of course, see this object. The horizon of a man six feet high, being distant not more than about three miles, and a boy of five feet seeing to a distance
* Sir J. W. Herschel's Astronomy, p. 21.
+ It may help the memory of the pupil, if we say that a field four miles square, would contain the same quantity of grass or corn, as a field of a length equal to the diameter of the earth, and ten feet wide only. See Index " Light."
of not more than about two miles and a quarter. That is, the rate of curvature of our earth is in a similar proportion; viz., about ten feet in four miles; six feet in three miles; five feet in two and a quarter miles, &c.
ELLIPSE, PARABOLA, HYPERBOLA.
R These three figures, as well as the circle and triangle, may, all of them, be produced by the cutting of a Cone. They are therefore denominated "Conic Sections," and may be briefly described thus:
If the cutting plane be carried through both sides of the cone, otherwise than parallel to the base, (which is circular,) the figure will differ from a Circle, (C)
and be elongated into an Ellipse, (as E).
If the cutting plane pass through one side of the cone, and through its base, in a direction parallel to the other side, the figure will be a Parabola, (as P).
If the cutting plane pass through a side and the base in a direction less inclined to the base than the other side is, the figure will be an Hyperbola, (as H).
s If two pins be stuck in a smooth board, and the ends of a thread, considerably longer than their distance, be attached to them, the point of a pencil which keeps this thread stretched, if moved as limited by it, will mark out an Ellipse, which will be more or less like a circle, according to the degree by which the length of the thread exceeds the distance of the pins. The places of the pins will be the two focuses (foci). If we take the Ellipse and draw a line A B through the foci, and
DEFINITIONS, ILLUSTRATIONS, ETC.
take c, the middle point between them, and draw a line M m, crossing it at right angles to A B.
Then A B is the Major Axis; Mm the Minor Axis; c is the Centre, and c F or cf, the excentricity of the Ellipse.
The difference of distance of F from A and from B, will, of course, vary with the shape of the ellipse, i. e., with its excentricity; but a line drawn from M to F, will always give the medium or mean distance.
Each of the "Conic Sections" has its peculiar properties, which are well known to mathematicians.
"They are naturally allied to each other, and one curve is changed into the other perpetually, when it is either increased or diminished. Thus, the curvature of a circle, being ever so little increased or diminished, passes into an ellipse; again, the centre of an ellipse going off infinitely, and the curvature being thereby diminished, it is changed into a parabola; and lastly, the curvature of a parabola being ever so little changed, there ariseth the first of the hyperbolas, the innumerable species of which will, all of them, arise orderly by a gradual diminution of the curvature; till this quite vanishing, the last hyperbole ends in a right line."-Dr. Hutton's Mathematical Dictionary.
V The Planets in their courses around the sun, describe Ellipses of very little excentricity, of which the sun is a focus.
"Three or four comets describe very long Ellipses, and nearly all the others that have been observed, are found to move in curves which cannot be distinguished from Parabolas. There is reason to think that two or three comets, which have been observed, move in Hyperbolas.
Every thing that is said respecting the motion of a planet, or body of any kind, round the sun, in consequence of the sun's attraction, applies equally well to the motion of a satellite about a planet; since the planet attracts with a force following the same law, though smaller, as the attraction of the sun. Thus, the moon describes an Ellipse round the earth, the earth being the focus of the Ellipse: Jupiter's satellites describe each an Ellipse about Jupiter, and Jupiter is in one focus of each of these Ellipses: the same is true of the satellites of Saturn and Uranus."Professor Airy on Gravitation.
METHODS OF MARKING THE FIXED STARS.
The most ancient method of distinguishing a star, is that which gives its place in the picture of its constellation; as "the first," (or westernmost,) "in the belt of Orion :" "the first in the left wing of Virgo," &c. The Arabians affixed names to the principal ones; and some of these names are still retained, as Aldebaran, Rigel, Fomalhaut, &c.
Bayer, a lawyer and astronomer of Germany, in the beginning of the 17th century, introduced the method of indicating the relative brilliancy or apparent magnitudes of the stars, in any constellation, in the order of the letters of the Greek alphabet. When the Greek letters are exhausted in this way, the stars next in brilliance are similarly marked by the small letters of the Roman alphabet, a, b, c, &c. Flamsteed, and the more modern astronomers, have agreed in using the letters of Bayer. There is also a numbering of the stars, which refers to their places in the several published catalogues of different astronomers.
It will be well for those who are unacquainted with the sounds of the Greek letters, to spell out their names as given below; as well as those words which are given in the next class of definitions. The capitals are seldom, if ever, used to the stars.