with the same three given parts, the result will be always a repetition of the same triangle. The corresponding sides of the successive ones will not differ in length, and the angles will not differ in magnitude.

There is one exception to this principle, pointed out in B. 3, Prob. 11, Geom., where two sides and the angle opposite one of them are given, in which case two triangles can be constructed with the given parts.

3. Three parts of a plane triangle being given then, (except they be the three angles,) it ought to be possible to find the other three, since these are fixed by their dependance upon the three given.

This may be accomplished with sufficient accuracy for many purposes, by means of constructions, such as are exhibited at problems 8, 9 and 10 of the 3d book of Geometry.

We shall repeat one of these constructions, enunciating the problem somewhat differently.

The two sides and included angle of a triangle being given, let it be required to find the remaining side and the other two angles.

Let A and B be the two given sides, and c the given included angle. Draw two lines DH and G of indefinite length, making with each other an angle equal to the given angle c. Lay off on the first of these the given line A

A B

D

C

F

B

-H

E

from D to E, and on the second the given line в from D to F. Join EF. The only possible triangle DEF will thus be formed with the three given parts, in which EF will be the required side, and E and F the required angles.

The finding the unknown parts of a triangle by means of those which are given, is called its solution.

4. The method of solution just exhibited is rendered more practically useful by the employment of scales of equal parts and protractors.

The most simple form of a scale of equal parts, is shown in the annexed figure.

It is a straight rule divided into any number of equal parts: in this example ten and one of these again into ten, so that the smallest division is one hundredth of the whole length of the rule.

The following is the manner of usingit.

Suppose that it is required to draw upon paper a line equal in length to 56.

Place one foot of a pair of dividers at the line of division marked 5, and extend them till the other foot reaches exactly to the sixth smaller division mark on the right of 0; the feet of

the dividers will then be at a distance of 56 apart. To draw now the required line upon paper, let a be the point from which it is to be drawn. Placing one foot of the dividers at A, extended the distance 56 obtained from the scale, describe with the other an arc of a circle on the side towards which the line is to be drawn; then from a draw the line in the proper direction, terminating it at the arc before described, and it will be the line required.

Another line of 42 being measured from the scale and laid down upon the paper, the two lines will be in the ratio of 56 to 42. If they are lines upon a map, and the first corresponds to a line of 56 feet upon the ground, the second will correspond to a line of 42 feet.

If the first represent 56 yards, or chains, or miles, the second will represent 42 yards, or chains, or miles. And in general lines upon the same drawing which are measured in parts of the same scale must be understood to be expressed in units of the same kind.

5. Before describing the protractor which is an instrument for laying off angles, it will be necessary to explain the method of estimating the magnitude of angles.

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.

1 360

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.

angle, one of 60 degrees (A C D) a much larger acute angle,

one of 140 degrees (A C E) an obtuse angle.

A degree being always the 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, 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 thus°; 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