SECTION III.

ON PARALLELS.
DEFINITIONS.

"Geometers have usually preferred to define parallel lines by the property of being in the same plane and never meeting. This, however, has rendered it necessary for them to assume, as an additional axiom, some other property of parallel lines; and the unsatisfactory manner in which properties for that purpose have been selected by Euclid and others has always been deemed the opprobrium of elementary geometry."

MILL.

When two straight lines coincide with each other, they have the same direction, but are not distinct; they may however be conceived to be separated from one another, but kept in the same position relatively to each of their points, that is, with the same direction; they are then said to be parallel to one another. Whence

20. Straight lines parallel to one another, or parallels, are straight lines having the same direction.

It has been seen that two straight lines which coincide are said to have either the same direction or opposite directions, according to the manner they are supposed to have been made to coincide; so also two straight lines which do not coincide, but have the same direction, are said to have either the same direction or opposite directions, according to the manner they are supposed to have been brought into their respective positions; thus, two straight lines having opposite directions may also be said indifferently to have the same direction or opposite directions; consequently they are parallel.

It follows from this definition that two parallels cannot have one common point without coinciding throughout their extension; whence, straight lines cutting each other are not parallel, which is already evident from the fact that they have not the same position relatively to the point of section, and consequently, have not the same direction. It follows also that two distinct parallels remain throughout on the same side of each other.

It is obvious that when two straight lines contained in the same plane are neither perpendicular nor parallel to one another, they are oblique. Perpendicularity, obliquity, and parallelism are thus the only three positions that two straight lines may assume relatively to one another: of these, perpendicularity and parallelism are the most remarkable.

21. The like extremities of two parallels are those which

are on the same side of a transversal; the unlike extremities are those which are not on the same side of a transversal.

It is evident that straight lines joining the unlike extremities of two parallels cut each other between these parallels; the straight lines which join their like extremities, although their prolongations may cut each other, have no point in common between the parallels.

THEOREM 16.

Two perpendiculars on the same straight line are parallel to one another.

Let the straight lines A B and C D be both perpendicular to the indefinite straight line E F.

E

C

B

D

F

Let the intercept A C be reduced until the points A and C coincide, then the straight line EF will not have changed in position; but the perpendiculars A B and C D will coincide with each other [9]: therefore they have the same direction; that is, they are parallel to each other, W. W. T. B. D. Inversely

If one of two straight lines oblique to each other be perpendicular to a third, the other is not perpendicular to the same.

COROLLARY I. A perpendicular and an oblique to the same straight line are oblique to each other.

COROLLARY II. From any point without a straight line a parallel may be drawn to the latter [Post. VIII].

THEOREM 17.

A perpendicular to a straight line, is perpendicular to its parallel.

Let A B and C D be two straight lines parallel to each other, and let OM be a perpendicular to A B and cut CD in N.

If O M and C D were not perpendicular, then CD and AB would not be parallel to each other [16i]: which is impossible [Hyp.]; therefore O M and C D are perpendicular to one another, W. W. T. B. D. Inversely

An oblique to a straight line is oblique to its parallel.

COROLLARY I. Two parallels, however far produced, do not meet each other (10).

COROLLARY II. If one of two parallels be perpendicular to a straight line, the other is perpendicular to the same.

COROLLARY III. Two parallels to the same straight line are parallel to each other. (Eucl. I. 30.)

COROLLARY IV. No more than one parallel can be drawn to a straight line through a point without it (9).

Scholium. A parallel to a straight line is determined when one of its points is known.

THEOREM 18.

(Eucl. Ax. 12.)

A perpendicular and an oblique to the same straight line, when indefinitely produced, cut each other.1

Let A B be a perpendicular to MN, and CD an oblique to the

same.

IN

D'

Let CD be conceived to turn on the point N so that it inclines more and more on MN. When CD coincides with NM, it cuts A B in M. Let CD, still turning on the point N, move back to a position C'D' between NM and its first position; it will then cut A B in a point E distinct from M.

Now let CD move back to its first position. As long as its direction is different from that of A B, that is as long as it is not perpendicular to MN, it will cut A B; for at whatever point CD should be conceived to leave A B, these two straight lines would have one point in common and differ in direction: therefore they would still cut each other [5]. But CD is not perpendicular to M N [Hyp.]; therefore, if indefinitely produced, it will cut A B, which W.T.B.D. Inversely

A straight line is perpendicular to another if, however far produced, it cannot meet another perpendicular to the same.

1 The student must here, as throughout Plane Geometry, supply the condition that these lines be contained in the same plane.

COROLLARY I. Two straight lines not parallel to one another, when indefinitely produced, cut each other.

COROLLARY II. The perpendiculars to straight lines oblique to one another, when indefinitely produced, cut each other.

COROLLARY III. If a straight line cut another, the first, when indefinitely produced, cuts every parallel to the second. Scholium. Two parallels meet at an infinite distance.

From any point C and D of C D let there be drawn to AB two perpendiculars CA and D B: the lengths of these two lines will be the distances from the points C and D to AB [11]. Let also M N be the middle perpendicular on the intercept AB, cutting CD at the point N [18 iii]: it will also be perpendicular on CD [16].

Now let the plane containing the figure be folded upon MN. Then M B will coincide with MA [9 i] and A with B, also N D with NC [91]. Because the straight lines MA and MB coincide with each other and become one and the same line, the perpendiculars D B and C A coincide with each other [9], and the point D, already coinciding with some point in N C, coincides with the point C.