Similarly the external pt. P' may be found, thus:

ME' = MB (const. 3), and a ☐ is constructed on AE' (const. 4). The proof is essentially the same.

NOTE. This problem was known to the Greeks, probably as early as Pythagoras. So important has been considered this section (cutting) of the line, that it is often called the Golden Section. Its importance in Geometry will be seen as the student proceeds. The expression "median section" is also applied to this division.

COROLLARIES. 1. CE, in the above figure, is divided at A in Golden Section.

Step 4 of proof, ... CA2 = AB2.

2. If a line is divided in Golden Section, and the less segment is laid off on the greater, then the greater segment is divided in Golden Section.

For consider CE, which is divided at A in Golden Section (cor. 1). Lay off on CA, from A, a segment YA AE, or = AP, then CA = AB, and YA AP, it follows that Y divides CA (the greater segment of CE) in Golden Section.

=

EXERCISES. 290. In the figure of pr. 6, ME2 = 5 MA2. (Eudoxus.) 291. In the figure of pr. 6, (PB + † AP)2 = 5 (AP)2.

(Eudoxus.)

Section 3. Practical Mensuration of Surfaces.

For practical purposes a surface is measured as follows: 1. A square unit is defined as a square which is one linear unit long and one linear unit wide.

That is, a square inch is a square that is 1 in. long and 1 in. wide; a square meter is a square that is 1 m. long and 1 m. wide, etc. In the figure the shaded square is considered as a square unit.

2. If two sides of a rectangle are 3 in. and 5 in. respectively, then, in the figure, the area of the

strip AB is 5 X 1 sq. in., and the total area is 3 X5 X1 sq. in. or 15 sq. in.

Theoretically, a rectangle rarely has sides

both of which exactly contain any linear

B

unit, however small. Such cases are discussed in Book IV. But for practical purposes the above method is approximate to any required degree.

At present it is necessary for the student to learn that geometry gives him an instrument for practical work. It will accordingly be assumed that the measurements can be made to any degree of approximation, and that the expressions area, measure, etc., are understood in their ordinary sense. It has already been explained that the rectangle of two lines corresponds to the product of two numbers; hence, in practice, lines are represented by numbers and their rectangles by the products of those numbers. This practical measurement will be exemplified hereafter, as it has already been to some extent; in the numerical exercises.

ABSCISSAS AND ORDINATES. If,

in the annexed figure, XX'IYY' at O, and y1 OX, then x, is called the abscissa of point P1, and y1 is called the ordinate of that point.

Similarly x2, y2, are the abscissa and ordinate of P2, and so on for P8, P4, etc.

PP
Y2

PRACTICAL MENSURATION OF SURFACES.

101

To find the area of the field P1 P2 P3 P4, the area of the trapezoid between y1 and y4 may be found, and from this may be subtracted the areas of the trapezoids between y1 and y2, y2 and yз, yз and у4; this plan is sometimes used by surveyors, OY representing the north line and OX the east line. It is more convenient, however, to let OX pass through the most southern point På, or OY pass through the most western point P1. Surveyors usually take the latter plan and find the areas of the trapezoids between x3 and ×4, X3 and x2, etc.

EXERCISES. 292. The abscissas of the several corners of a triangular field are 3, 5, 8, and the corresponding ordinates are 2, 8, 5. Draw the figure and find the area. (Th. 3.)

293. The abscissas of the several corners of a convex field are 2, 3, 5.5, 7 chains, and the corresponding ordinates are 4.2, 2.1, 3, 8 chains. Draw the figure and find the area of the field in acres. (Th. 3.)

294. The abscissas of the several corners of a convex field are 0, 100, 150, 250, 350, and the corresponding ordinates are 0, 125, - 75, 120, 0; draw the figure and find the area. (The minus sign before the 75 means that that ordinate extends below the line OX; it does not affect the sign of the corresponding area.)

295. The abscissas of the several corners of a triangular field are 0, 5.8, 8.96, and the corresponding ordinates are 0, 4.86, 0. Draw the figure and find the area.

296. A field is in the form of a rhombus, the obtuse angle being twice the acute angle; the shorter diagonal is 300 feet. Find the area of the field, in square feet.

297. A railroad embankment extends through a farm 1 mile long, its rails being in straight lines perpendicular to the two parallel sides of the farm; the embankment is 80 ft. wide at the bottom at one end, and 60 ft. at the other. How much land was taken for railroad purposes?

298. The hypotenuse of a right-angled triangle is 277, and one side is 115. Find the other side and the area.

299. A surveyor, wishing to erect a perpendicular to a line on the ground, drives two stakes, A, B, 12 links apart; to these he fastens the ends of a 24-link segment, and stretches the chain, at the end of the 9th link from A, to C. Show that AC LAB. (This method of erecting perpendiculars was known to the temple and pyramid builders, and surveyors employed for this purpose were called "rope stretchers." The method is still used in practical field work.)

12

DEFINITIONS. A circle is the finite portion of a plane bounded by a curve, which is called the circumference, and is such that all points on that line are equidistant from a point within the figure called the center of the circle.

For corollaries to the definition of a circle, see p. 59.

Certain definitions are here repeated for convenience.

If two equal figures are necessarily congruent, as in the case of circles, angles, squares, and line-segments, the word equal is ordinarily used to express congruence. Hence congruent circles (see p. 59, cor. 3) are ordinarily called simply equal.

A straight line terminated by the center and the circumference is called a radius.

A straight line through the center terminated both ways by the circumference is called a diameter.

The straight line joining any

E

two points on a circumference is called a chord.

C

B

Hence a diameter is a chord passing through the center. figure, AE and BD are chords.

In the

The expressions center, radius, diameter, chord, of a circumference are sometimes used instead of center, etc., of a circle, where their use avoids the needless repetition of the word circle.

The line of which a chord is a segment is called a secant.

In the figure, XY is a secant.

A part of a circumference is called an arc.

In the figure, BCD is an arc. In naming an arc, as in naming an angle, the counter-clockwise order of the letters is followed, and arcs so named are considered positive.

One-half of a circumference is called a semicircumference. In the figure, ẨỀ and ỀÀ are semicircumferences.

A fourth part of a circumference is called a quadrant.

If the sum of two arcs is a quadrant, each arc is called the complement of the other; if the sum equals a semicircumference, each is called the supplement of the other; if the sum equals a circumference, each is called the conjugate of the other.

In the figure, AB is the supplement of BE and the conjugate of BA.
An arc greater than a semicircumference is called a major
arc; one less than a semicircumference is called a minor arc.
In the figure, AB, BD, DE are minor arcs; DEA is a major arc.

An angle formed by two radii is called a central angle.
In the figure, AOB, BOE are central angles.

A central angle is said to stand upon the arc which lies within the angle and is cut off by the arms.

AOB, BOE stand upon AB, BE, respectively.

The arc upon which the sum of two central angles stands is called the sum of the arcs upon which those angles stand. Similarly for the difference of two arcs.

Thus, AE AB + BE, and AB

=

AD – BD.

Conjugate arcs are said to be subtended by their common chord.

In the figure, BD and DB are each said to be subtended by chord BD. The word subtend is variously used in geometry. It means to extend under or to be opposite to. Hence, in a triangle a side is said to subtend an opposite angle, a chord is said to subtend an arc, an arc is said to subtend a central angle, etc.

A portion of a circle cut off by an arc and two radii drawn to its extremities is called a sector, and the central angle which stands upon that arc is called the angle of the sector. In the figure, OAB is a sector, and angle AOB is its angle; similarly, OBCDEA is a sector, and BOA is its angle.