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In Geometry (B. 3, Prop. 17. Schol. 1, it is shown that angles are proportional to the arcs included between their sides, the arcs being described with equal radii, and it is also there stated that hence such arcs are properly the measures of angles.
So that if an arc included between two sides of one angle be double, or triple, or sextuple, an arc described with the same radius included between the sides of another angle, the first angle is double, triple, or sextuple the second.
The relative magnitude of angles may therefore be correctly expressed by means of the relative magnitudes of the arcs which measure them.
The relative magnitudes of quantities are commonly given by referring the quantities to be compared to some known standard of measure which must be always of the same kind with the quantities themselves.
This standard is called a unit. Thus a foot, a yard, &c. are units of length, and the idea of the relative lengths of two lines is obtained by its being said that one is seven feet or yards, and the other nine. Or the just conception of the length of a single line is had by being told how many feet, yards or miles it contains. The mind compares it with one of these well known units, which in imagination it repeats along its length.
Now the unit of measure, which is employed in a similar manner for giving the conception of the magnitude of an arc, is called a degree. A degree is the part of the circumference of a circle. The relation which any given arc bears to the whole circumference may be conveniently expressed by stating the number of degrees which the arc contains. Thus an arc of 90 degrees will be one fourth the whole circumference. An arc of 45 degrees will be one eighth. An arc of 30 degrees will be somewhat less. And it is plain that the length of the arc, as compared with the whole circumference, may be readily conceived, as soon as the number of degrees which it contains is mentioned.
So also the magnitude of the angles subtended by these arcs will, after a little familiarity, be rendered easily sensible to the mind. To speak of an angle of 10 degrees for instance, (A CB in the annexed diagram,) will suggest the image of a very acute angle, one of 60 degrees (A CD) a much larger acute angle, one of 140 degrees (ACE) an obtuse angle.
A degree being always the sto part of a circumference, a single degree will be larger in a larger circle than in a smaller, and this, so far from being inconvenient, is particularly advantageous in the measurement of angles; for since arcs described about the vertex of an angle as a centre with different radii, and included between the sides of the angle, bear the same relation to each other as the radii, (Geom. B. 5, Prop. 11, Cor.) and since the entire circumferences are also proportional to their radii, it follows that two concentric* arcs included between the sides of the same angle, and having the vertex of that angle for a centre, are the same aliquot parts of their respective circumferences. Consequently, two such arcs will contain the same number of degrees. Hence, to find the number of degrees contained in a given angle, the arc described for the purpose about the vertex, and extending from side to side of the angle, may be with any radius at pleasure.
* Having the same centre.
This may be distinctly seen in the following diagram.
Where the size of an angle is such that it does not embrace an exact even number of degrees of the circumference, smaller divisions called minutes, 60 of which make a degree, are employed. The angle is then said to contain as many degrees and minutes as there are degrees and parts of a degree each, sto over, between its sides. If the second side of the angle does not pass exactly through one of these smaller divisions, a still smaller kind termed seconds, 60 of which form a minute, or 360 a degree, must be introduced. More miuute divisions than these last are seldom used. When it becomes necessary to regard such, the same system is continued. The next denomination is thirds, 60 of which make a second, the next fourths, and so on.
The notation for these denominations is as follows. Degrees are written thuso; minutes thus'; seconds thus"; thirds
"' thus '', &c.; 30° 20' 10" is read thirty degrees, twenty minutes and ten seconds.
6. It is evident that the numbers used in the system of division for the circumference of the circle, are entirely arbitrary. Others might be employed with equal propriety, provided the same principle were observed. In fact the attempt has been made, and probably will be successful in France, to subvert the old system of division, and to adopt a decimal system in this as well as in every other sort of measurement. Thus a right angle, which is the unit of angles, is made to
contain 100° instead of 90; and the circumference will then contain 4000 instead of 360. 100' instead of 60=1° 100 =1'. The convenience of a decimal division we have experienced in this country in our system of Federal money. The French are likely, despite the despotism of custom, to enjoy the same advantage in all denominate numbers.
7. Another method of expressing the magnitude of angles is as follows.
A distance at pleasure is laid off from the vertex of the angle upon one of the sides, and a perpendicular there drawn to this side till it meets the other side of the angle. The ratio of this perpendicular to the distance from its foot to the vertex, serves to indicate the size of the angle.
For example, if the line BCDE be perpendicular to the line AB, and bc be one fourth AB, the angle bac is
, said to be an angle of 1. If BD be one half ab, the angle
AB BAD is said to be an angle of . If Be be equal to AB, BAE is said to be an angle of 1; and so on for other mag
AS nitudes. An angle of 1 is plainly half a right angle, or 450.
This kind of measurement is much used by engineers, to express the degree of slope in excavations and embankments.
8. The protractor which we are now prepared to describe, is an instrument for drawing upon paper an angle of any given number of degrees.
This instrument is made in a variety of forms; sometimes with a full circle divided into degrees, sometimes comprising only a semicircle, sometimes upon a rectangular rule having not the circumference but the radii drawn, as they would be through the divisions of the circumference if it were actually described. The first kind is made usually of brass, or of silver, which is less liable to corrosion, and communicates no unpleasant odor to the hands. It has a metallic radius move
able about the centre of the circle and extending beyond the circumference. This prolonged radius serves to point out the number of degrees, and is armed with a sharp pin under the outer extremity for the purpose of pricking the paper, so that when the instrument is removed a line may be drawn with pencil through this point, and that upon which the centre was placed ; which line shall be a radius corresponding to the number of degrees at which the instrument was set.
A which is the one most commonly seen, is a semi-circle of brass, (or other metal,) having the greater part of the interior cut out to render the instrument less heavy.
The semi-circumference is divided into degrees by marks made in the metal, and these are numbered from 0° to 180 (the number in a semi-circumference) both ways, in order that the counting may commence with convenience at either end.
The degrees are also sometimes divided into half degrees, and lines of different length are employed to mark more distinctly every five and every ten degrees."
The centre is marked by a notch in the straight side of the instrument, which side is a diameter of the semi-circle.t
* Such a division of instruments is termed graduation.
† This instrument may be made out of paper, and a large one so made is very accurate.