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The radius is also called a "Semi-diameter," because two radii in a straight line form a Diameter, or “measure through" the circle.
The difference of direction of any two radii, or lines drawn from the centre of a circle, is called an angle. And this difference of direction is measured on the circumference, which, for this purpose is divided into 360 equal parts, called degrees."
Degrees) are subdivided into sixtieths, called minutes ('). Minutes (') are subdivided into sixtieths, called seconds (").
When the measurement of any angle is given, the angular point is considered as at the centre of a circle. Thus, we say, 66 an angle of so many degrees, minutes, and seconds," referring to any circumference drawn from it; as 25° 16' 18'. Hence the magnitude of an angle does not consist in the length of the lines which include it, but in their difference of direction, or inclination to each other.
An angle is generally described by naming three letters, the middle one of which is standing where the lines meet, and the other two at their distant extremities.
M Every semi-circumference contains 180°, and every quadrant or quarter-circumference, 90°.
The angle of 90° is called a Right Angle, and the radii forming it are said to be Perpendicular to each other.
Angles are frequently referred to the right angle, and are termed Obtuse or Acute, as they contain a number of degrees greater or less than 90.
The deficiency of any acute angle when referred to 90°, is called its Complement: thus, 60° is the complement of 30°; 41° is the complement of 49°; and 66° 32', is the complement of 23° 28'.
P Every circumference measures exactly 44 seventh parts of its radius, or distance from the centre. Hence, the diameter or double radius is a little less than of the circumference; since the radius is a little less than . Circumferences therefore are in proportion to their radii or diameters so also are their 360th parts or degrees.
PLANE-INSIGNIFICANT QUANTITIES-RELATIVE VASTNESS AND MINUTENESS.
B Let writing-paper of the smoothest surface and finest texture, be pierced with a finely polished needle :—if the paper be held evenly, and the needle be inserted suddenly, and perpendicularly to the plane of the paper, the hole so made will be circular, not else.
Of this circular hole we have to remark :—
1. That, like any other circle, the edge of the hole, or circumference, is, in all its parts, equally distant from a certain place or point within the hole, called its centre.
2. That this said point is level with the edge of the hole, and therefore in the same plane with it, and with the surface of the paper: that is, (supposing our needle to be of perfectly cylindrical form,) it is just where the needle-point first touched the paper before it really pierced it. This central point therefore, "has position, but not magnitude:"—it is now in vacant space.
3. That every individual spot or part of the circular edge, although at an equal distance from the centre, has a different direction with regard to the centre, from that of any other spot or part of the circular edge. And that this difference of direction, which any two of these points in the circumference have with regard to the central point, is called their angle or "bearing."
4. That, as there are considered to be 360 exactly equal parts in the whole circumference of any circle, and, as the size of a circumference and of each of its degrees is in exact proportion to its distance from the centre, so each of the 360 degrees of our needle-hole circumference will be in exact accordance with the thickness of the needle which formed it.
© Place the paper flatly on the table, and on another portion of the same table, let another sheet of paper of the same thickness be placed, also with perfect flatness. If the table be even, the surfaces of these two sheets of paper
will be in the same plane; so also will our circular edge and its centre be, with the circumferences and centres of any needle-holes that may have been pierced in the other sheet of paper.
D The young reader will please to remark, that we have not said that these upper surfaces of the paper are in the same plane with the table surface; because they are raised above it by the thickness of the paper, which is not inconsiderable in this case.
But, there is another table in a distant part of the room, corresponding in height, and the floor is level;-the surface of this also is in the same plane with our first table; and our thickness of the paper being insignificant when compared with the distance between these tables, we may now safely say, our paper surface also is in the same plane. So, also, are the surfaces of our sheets of paper and of our two tables, in the same plane with any other tables of the same height in the adjoining rooms on the same floor of the house.
If the eye be brought to the edge of the table, so as to look along its surface and the surface of our paper, at any objects placed at a moderate distance, (say at an opposite house or window,) that portion of the window or house seen to coincide with the surfaces of our table and paper, will likewise be in the same plane with them, and with the centres and circumferences of our needle-holes, and with that part of space exactly intervening in the width of the street, &c.
G Let the pupil endeavour to conceive of the centre of the needle-hole as the position of our sun, and of the circumference of the hole as representing the earth's annual path of 600 millions of miles around it; then, perhaps, the dotted pattern of the paper on the wall of the room, (if a considerably large one,) or points taken in the lofty ceiling, may represent the position of some of the nearest fixed stars, such as Sirius," Arcturus," &c., on the same scale of distance. It will be allowed, that no change of position in this tiny circumference of our needle-hole, can be considered as nearer to the wall, or more remote from it; than another.
We have said the nearest fixed stars: what then shall
represent the most distant! Each improvement in the
DEFINITIONS, ILLUSTRATIONS, ETC.
penetrating power of telescopes, has revealed new wonders in the infinity around us. Can the pupil's imagination furnish us with a room, the walls and ceiling of which shall be 35,000 times as distant as those of the present room are? Herschel has calculated such to be the relative remoteness of some of the wondrous objects he has viewed! But these estimates cannot be said so much to give us any notion of the power of the Deity, as to correct the errors we should fall into, by supposing His powers to have any limits like those which belong to our faculties by supposing that numbers, and spaces, and forces, and combinations, which overwhelm us, are any obstacle to the arrangements which His plan requires.
The difficulty which appears to reside in numbers, and magnitudes, and stages of subordination, is a difficulty produced by judging from ourselves-by measuring with our own sounding line; when that reaches no bottom, the ocean appears unfathomable. Yet, in fact, how is a hundred millions of miles a great distance? how is a hundred millions of times a great ratio? not in itself It is clear that the greatness of these expressions of measure, has reference to our faculties only. Our astonishment and embarrassment take for granted the limits of our own nature."*
To return to our needle-hole :-We have supposed its circumference to represent the orbitual path of our Earth around the Sun, placed at the centre. Uranus, the planet of comparatively late discovery by Dr. Herschel, the six moons of which have peculiarities of motion, "which seem to occur at the limits of our system, as if to prepare us for farther departure from all its analogies, in other systems which may yet be disclosed to us," has an orbit nineteen times as remote from the sun as our own. If our needle-hole then, were enlarged, until it could be measured by a shilling; we should have the orbit oft his remote wanderer on the same scale! The Comets, vanishing into space and re-appearing
Whewell's Bridgewater Treatise, pp. 276-7, which the young reader is strongly recommended to procure for perusal.
+ Sr. J. F. W. Herschel, Cab. Cyclo. p. 299.
after lengthy periods, may, perhaps, extend their most eccentric courses to a distance double or treble of this.
* * * *
K We see, then, that the fixed stars "have no immediate relation to our system. The obvious supposition is, that they are of the nature and order of our sun If, then, these are suns, they may,* like our sun, have planets revolving around them; and these may, like our planet, be the seats of vegetable, and animal, and natural life: : we may thus have in the universe, worlds, no one knows how many, no one can guess how varied; but however many, however varied, they are still but so many provinces in the same empire, subject to common rules, governed by a common power!†
L But the stars which we see with the naked eye, are but a very small portion of those which the telescope unveils to us. * The number of stars which crowd some parts of the heavens, is truly marvellous; Dr. Herschel calculated that a portion of the milky way, about ten degrees long and two and a half broad, contained 258,000. In a sky so occupied, the moon would eclipse 2000 of such stars at once?‡
If the first flash of that view of the universe which science reveals to us, does sometimes dazzle and bewilder men, a more attentive examination of the prospect, shows how unfounded is the despair of our being the objects of Divine Providence, how absurd the persuasion that we have discovered the universe to be too large for its Ruler."§
M We will tear off from the sheet of writing-paper, the part which contains the hole pierced by the needle, and wetting it with water procured from a pond in which the vegetable matter has lately decayed, we will press it between two pieces of thin glass, and direct a high power of
See Index," Double Stars."
+ Whewell's Bridgewater Treatise, p. 270. § p. 288.
The hole partly closes when the paper is soaked; it is better on this account to pierce a similar hole in a very thin piece of whalebone or ivory.