This ambiguity is avoided by the following conventions:

1. We assign to each line a positive and a negative direction. The positive direction is that from the beginning to the end of a finite line. The angle they form is then that between their positive directions. This is the same as the angle between two lines going out from the same point in the respective positive directions of the lines.

2. We consider one side of the angle as that from which we measure, and we measure the angle to the other side in a positive direction, as explained in §§ 3 and 8.

The four values are thus reduced to one.

If two lines are parallel, their angle is 0° or 180° according as they are similarly directed or oppositely directed.

Projections of Lines.

66. Def. The projection of a finite straight line AB upon an indefinite line XY is the distance A'B' between the feet of the perpendiculars from A and B upon the indefinite line.

B.

X

B

Y

To find the length of a projection. Through one end of the line, as A, draw AC parallel to XY, meeting BY in C. Then

The projection is equal to the length of the line projected into the cosine of the angle which the two lines form with each other.

Algebraic sign of a projection. Let the positive direction of the line OB be from 0 to B, and let the line turn on O into the successive positions OC and OD.

If on the line of projection we regard directions toward the right as positive, the projection A'B' will be positive. The whole line OC will be projected at the point C; the projection will therefore be zero, a result given by the formula (1), because cos 90° = 0.

C

Α'

B

The projection A'D' of OD will be negative because it falls in the negative direction. This also corresponds to the formula, because the angle between the two directions OD and A'B' is obtuse.

If we suppose OB to perform a complete revolution around 0, we readily see that its projection goes through the same series of changes as the cosine of the angle which it forms with the line of projection.

67. Projection of sides of a polygon. Let ABCDE be any polygon the positive directions of whose sides correspond with the circuit we should form in going round the polygon, so as to reach its vertices in alphabetical order. We shall then have, for the projections on the line X,

Proj. of AB A'B', which is +;

=

D

66

The positive direction being arbitrary, we might equally take the directions AE, ED, DC, etc., as positive. Each of the projections would then have the opposite algebraic sign from that just given.

The student will remark that the projection of the line is positive or negative according as the projection of its end is on the positive or negative side of the projection of its beginning. We wish now to determine the sum of the projections, and for this purpose must understand the algebraic addition of lines.

68. Algebraic addition of lines. In geometry the sum of two segments AB and BC is defined

A

B

Ꭰ C F

as the segment AC, formed by put-
ting AB and BC end to end in the same straight line.

E

In trigonometry and modern geometry we distinguish between the beginning and the end of each segment, and between the positive and negative directions upon the segment; the positive direction being from the beginning toward the end; the negative, from the end toward the beginning.

When this distinction is attended to we must, in designating a segment by letters at its termini, write that letter first which is at the beginning of the segment, so that the letters shall follow each other in the positive order. The segment BA will then be the negative of the segment AB.

We now generalize the definition of the addition of lines as follows:

Def. The algebraic sum of several lines is formed by placing the beginning of each line after the first at the end of the line next preceding. This sum is then the segment from the beginning of the first line to the end of the last one.

Example.-In the preceding figure we have

as in geometry, because both segments are positive.

But if we consider the segment CD as beginning at Cand ending at D, then, by the above definition, the algebraic sum of the segments AC and CD will be the segment AD, from the beginning of AC to the end of CD. That is, a negative segment will be subtractive in the same way that in algebra the addition of a negative to a positive quantity implies sub

traction.

In general, whenever A, B, C, D, E, F represent points upon a straight line, we have

AB+BC+ CD+DE+ EF = AF,

however these points may be situated.

If the end of the last line coincides with the beginning of the first, the sum will be zero, by definition. Hence, however the points A, B, and C may be situated, we have

69. Let us now return to the projected polygon. On the preceding system, the sum of the projections of the several sides upon the line X is

A'B'B'C' + C'D' + D'E' + E'A' = 0.

The same thing would be true if we took any other straight line as the line of projection. Hence:

THEOREM I. The algebraic sum of the projections of the sides of a polygon upon any straight line is zero.

Since the projection of each side is equal to its length into the cosine of the angle it makes with the line of projection, this theorem may be expressed in the following form:

If the sides of a polygon be a, b, c, etc., and the angles which these sides make with any straight line be a, ß, y, etc., we shall have

a cos a + b cos B+ c cos y + etc. = 0.

C

We may imagine the sides of the polygon all taken up and placed with their beginnings at the same point, their length and direction remaining unchanged.

Their several projections will then have the same values as before, and in consequence the algebraic sum of the projections will still be zero.

70. We now have the following

E

DE C

A

E

B

theorem, the demonstration of which is left as an exercise for the

student:

THEOREM II. If the algebraic sums of the projections of three or more straight lines upon any two non-parallel lines are each zero, these lines when placed end to end without changing their directions will form a polygon, the end of the last line falling upon the beginning of the first:

NOTE. To fix the ideas the student may suppose the lines as first given to all emanate from one point, as in the last figure.

The demonstration is begun by showing that in case the sum of the projections upon a straight line is zero, then, when the lines are placed end to end, the end of the last line and the beginning of the first must lie on the same perpendicular to the line of

Ni

projection. Thus, in the figure, the sum of the projections of the four unbroken lines is zero, although they do not form a polygon. But, with such a figure, the projections will not be zero on any other non-parallel line.

71. THEOREM III. If a, b, c, etc., be the sides of a polygon, and a, b, y, etc., the angles which these sides form with any straight line, we shall have

a sin a+b sin ẞ+c sin y + etc. = 0.

Proof. Let OX be the base line from which the angles a, ß, y, etc., are counted; AB, any side

of the polygon; a, its length; a, the angle which AB makes with X.

Draw OY perpendicular to OX, and let PQ be the projection of AB upon OY. AB will then

P

make with OY an angle 90° a, and we shall have

Projection PQ a cos (90°
= a cos (90° — a) = a sin a.

B

-X

Treating all the other sides in the same way, the algebraic sum

of their projections upon OY is found to be

a sin absin 6+ c sin y + etc.,

which sum is zero by Theorem I.

THEOREM IV. If the sum of the projections of a series of straight lines upon any two non-parallel lines be zero, the sum of their projections upon any third line will be zero.