BOOK SECOND.

PLANE GEOMETRY DEPENDING ON THE RIGHT LINE.

SECTION FIRST.

Comparison of Angles.

Def. 1. The doctrines of extension constitute the science of Geometry.

Def. 2. Solids have three dimensions, length, breadth, and thick

ness.

Def. 3. The boundaries of a solid are surfaces, the perimeter of a surface are lines, and the extremities of a line are points.

PROPOSITION I. [PRIMARY NOTION].

A Straight Line is such that it does not change its direc- (66) tion at any point in its whole extent.

This truth is not produced as a theorem, for it is incapable of demonstration; not as a problem, for there is nothing to be done; neither as a corollary, for it is the consequence of nothing; nor as an axiom, for it is hardly of the unconditional and absolute character of that enounced in the words, "the whole is equal to the sum of all its parts;" and it is not a definition, for we gain no new idea by the mere terms of the proposition: we only recognize by and in them one of those primary notions which we possess anterior to all instruction, and which, as they are necessary to, lie at the foundation of, every logical deduction. It would doubtless be out of place to enter here into any investigation in regard to the origin of our ideas; but I think it will be apparent, that the notion of continuity is coöriginal with that of personal identity, and therefore, antecedent to argumentation; and continuity measured out on

the one hand, in the lapse of events, as the periodic return of day and night, the revolutions of the celestial sphere, or the index of the chronometer, becomes time, and, on the other, attached to form by a personal passage over the surfaces of bodies, it becomes space; and, considered in regard to space and restricted by the notion of perfect sameness, continuity gives us the idea of direction, or that of the straight line.

Corollary 1. Two straight lines cannot intersect in more (67) points than one; for having crossed once, it is obvious from (66), that, in order to a second intersection, one of the lines, at least, must change its direction.

Cor. 2. Straight lines coinciding in two points, coincide (68) throughout, otherwise two straight lines would intersect in more points than one, which is contradictory to (67)...

Cor. 3. Straight lines coinciding in part, coincide through- (69) out, and form one and the same straight line (68). (Why?)

Cor. 4. Two straight lines cannot include a space (68). (70) (Why?)

Def. 4. A Plane is a surface with which a straight line may be made to coincide in any direction.

Def. 5. Two straight lines are said to be parallel when, situated in the same plane, they do not meet how far soever they may be produced.

Cor. 5. Through the same point, only one straight line can (71) be drawn parallel to a given line ;* for the directions of all the lines save one, drawn through the given point, will evidently (66) be such as to cause them to meet the given line if sufficiently produced. Scholium. The last corollary, though commonly given as an axiom, has been thought not sufficiently evident of itself.

It is not self evident, doubtless, if regarded as a consequence of any mere-definition that can be given of a straight line, but necessarily follows from that idea of a straight line, viz., continuity in sameness of direction, which we possess anterior to all definition. As such, it is, I believe, as well established as the primary truths in any department of human knowledge.

Application 1. To make a straightedge. Hav

ing formed a ruler as straight as possible and drawn a line with it upon a plane surface, turn the ruler

Fig. 2.

* When the word line is used alone, straight line is to be understood.

over upon the opposite side, and, with the same edge, repeat the line, coinciding with, or better, very nearly coinciding with the first; if the lines coincide, or appear equally distant throughout, the edge is sensibly straight.

Application 2. To test the parallelism of the edges of the ruler, place it against a second straightedge, and, having drawn a line, turn it end for end and draw a second. (Why ?)

Application 3. To test a plane, apply the straightedge to it in different directions.

Def. 6. When two lines intersect each other, or would intersect if sufficiently produced, the inclination of the one line to the other is called an Angle. It is obvious that angles are of different and comparable magnitudes, and, therefore, like other geometrical quantities, solids, surfaces, and lines, are capable of addition, subtraction, multiplication, and division, and, consequently, subject to mathematical investigation.

PROPOSITION II.

The sum of the adjacent angles formed by one line meet- (72) ing another, is always the same constant quantity.

B

A

A

A

Fig. 3.

Let the line AO meet the line BOC in O, making the angle AOB any whatever, and, in like manner, let the line A'O meet the line B'OC' in O, making the angle A'OB' any whatever. Placing the first figure upon the second, let the point ' O coincide with the point O and the line OB take the direction OB' then will OC take the direction OC' (69) and the line BOC will coincide with B'OC', also OA will take a certain direction OA", which we are at liberty to suppose between OA' and OC'; whence the angle AOB will be equal to the angle A"OB', the two becoming identical, and AOC to A′′OC'; but the angle A"OB' is, by hypothesis, the sum of the angles A'OB' and A'OA", and therefore greater than A'OB' by A'OA" (12), while the angle A"OC' is less than A'OC' by the same quantity; therefore the sum of the angles A"OB', A"OC' is equal to the sum of the angles A'OB', A'OC' (1), whence the sum of the angles AOB, AOC, is equal to the sum of the angles A'OB', A'OC′ (17) which was to be proved. The same in the language of symbols;

Let BOC coincide with B'OC', and OA take the direction OA";

and

< AOB = A"OB' = A'OB' + A'OA"

AOC A′′OC' A'OC' — A'OA";

=

=

ZAOBAOC = A'OB' + A'OC'. Q. E. D.*

The method employed in the preceding demonstration is obviously that of superposition, and the principles upon which it is grounded are the nature of the straight line, the whole is equal to the sum of its parts, and, equals added to equals, the sums are equal.

Def. 7. If the angle AOBAOC, then AOB and AOC are called Right Angles; hence, AOB or AOC being the half of AOBAOC: = a constant quantity (72), is also a constant quantity, or,

B

Fig. 32.

(73)

Cor. 1. All right angles are equal to each other. Cor. 2. The sum of the adjacent angles formed by one (74) line meeting another is equivalent to two right angles.

Cor. 3. Conversely; two lines met by a third, so as to (75) make the sum of the adjacent angles equal to two right angles, form one and the same straight line.

For if not, let BOX be a straight line, while

therefore BOC does not differ from the straight line BOX.

Cor. 4. The vertical angles formed by intersecting lines (76)

are equal.

If a and b be vertical angles, while c is adjacent to

Cor. 5. The sum of all the angles that can be formed at a (77) given point and on the same side of a straight line, is equal

to two right angles (2).

Cor. 6. The sum of all the angles that can be formed round a given point, is equivalent to four right angles.

Fig. 35.

(78)

Fig. 36.

* "Quod erat demonstrandum," which was to be proved.

+ We shall use the symbol for two right angles, and for four, while L will signify a single right angle.

Application 1. To make a Rightangle for drawing perpendiculars. Form a triangle, having one angle as nearly right as possible; set its base upon a straight edge and draw a line by its perpendicular, turn the triangle over upon the opposite face, and repeat the line. The instrument may be three inches in base and six in altitude, and cut from pasteboard, brass, or ivory, or framed of wood.

Fig. 4.

Application 2. To make a right angle in the field. Take a piece of board about 10 inches square, bore a hole through the centre, and fit to it a staff 4 feet long, so that, when the staff is stuck vertically in the ground, the board shall turn freely upon its top and continue during a complete revolution in the same plane, which will be determined by sighting its upper surface at a given object; draw two lines at right angles to each other through the centre and from corner to corner. In these lines at the corners may be stuck four needles-and to know whether they are accurately at right angles, we have only to place the board as nearly as possible in a horizontal position, and then to sight at two marks in range with the needles, in the lines drawn at right angles, and to see if we hit the same marks when the board is turned a quarter round. Why?

The SURVEYOR'S CROSS is the same instrument, only better finished.

Fig. 5.

The sight vanes (fig. 5) are hairs opposite slits,

and vice versâ; and the piece that turns upon the staff is represented in fig. 53.

Fig. 52.

To survey with the cross, it is necessary to be provided with two straight flag-staffs, which should Fig. 53. be wound with a red flag; or they may be square, and the adjoining faces painted red and white. In running a line, one staff is to go before and the other is to be left at the last position of the cross for a back sight. A measuring rod, tape line, or Gunter's chain is also to be provided. The chain is 4 rods or 66 feet in length, and centesimally divided by a hundred links, each, consequently, equal to 7'92 inches. It is a maxim in surveying land that all instruments, whether for measuring lines or angles, must be kept in a horizontal position; for it is the base, or the projection of the field upon the same horizontal plane that is required.