cumference through the three points fixed. Pupil should note the segment and give the geometrical proof. Prove that any inscribed in this segment equals b.]

EXERCISES.

109. To construct a when the base, altitude, and

of vertex is given. [Use the above problem.]

110. To construct a when two sides and the opposite one side are given. Show solution when the is (1) acute, (2) right, (3) obtuse.

111. (1) To construct a ▲ when the base, the vertical ▲,

112.

and the median from base are given. When is this impossible?

(2) What is the locus of the vertex of a ▲ when the base and the vertical are given?

Problem.

To draw a common tangent to two given

circles.

R

How many common tangents may be drawn to these Os? How many common tangents cross the line of centers?

[Join O to O'. With O for center, draw a with radius R-r. From O' draw a tangent to this. Draw its radius to the point of tangency and continue to the circumference of large . At this point draw a tangent to large O. Can you prove this line also a tangent to O? Try to invent a way to draw the tangents which cross the line of centers.]

EXERCISES.

113. Can you find the center of a without bisecting any straight line?

114. Inscribe a rectangle in a using diameters only to fix points.

115. Draw a O and take any point in it. What is the longest possible line that can be drawn from this point to a point in the circumference?

116.

Draw a

and fix any point without it.

Draw the shortest possible line from the point fixed to the circumference of the circle.

117. What is the locus of the mid-point of a chord of a given length in a given circle.

118. Can a tangent of any kind be drawn to a point within it?

from a

119. Can a tangent be drawn to a point in the circumference when the center is not known?

120. Describe a circle on one of the sides of an isosceles A and show how the circumference cuts the base.

121. Find the locus of the mid-point of a ladder as the foot of the ladder is pulled away from a vertical wall.

122. Through a given point within an angle draw a line intercepted between the sides and bisected at the given point. [See Ex. 107 and Ex. 108 for method; and, if you fail, see $118.]

123. Construct an equilateral when the altitude is

given.

124. Trisect a given rt. Z.

125. A circle is wholly without or within another circle, according as their central distance is greater than the sum, or less than the difference, of their radii.

[From Ex. 125 can you show how the central distance is related to the radii if two Os intersect?]

BOOK III.

MEASUREMENT.

172.

Plane Geometry deals with lines, angles, surfaces, and areas of surfaces. In comparing these magnitudes hitherto it has been deemed sufficient to prove their equality, or to prove that one is greater.

[When it is known that two s of a are unequal, what follows? If in the same or in equal Os chords are unequally distant from the center, what relation do these chords have? Can you think of other propositions wherein lines have been proved unequal? Think of propositions in which s have been proved unequal, etc., etc]

But it now becomes necessary for us to find the exact relation of the size of two or more magnitudes.

Case I.--If in comparing the lengths of two lines, a and b, we find that b contains, or measures, a an exact number of times, say three times, what is the relation of b to a? of a to b?

Case II.—If in comparing the lines a and b we find that one is not an exact measure of the other, but that a third line, c, is exactly contained in a four times and in 6 exactly nine times, what is the relation of a to b? of b to a? What is their common measure? Can you tell the common measure in Case I.? When is one line a common measure of another? Suppose a and b are given and c is required. Can you find c?

[How did you find the highest common factor of two or more quantities in Arithmetic and Algebra when you were unable to factor the quantities?]

We find a is contained three times in 6 with a remainder,

We find r is contained in a once with a remainder, r'.
Apply r' to r.

We find r' is contained in r twice and there is no remainder. What must be added to r to produce a? many times? b is also how many times r'?

a is then how

What then is the common measure of a and b? What is the relation of b to a? of a to b? What is the relation of r to a? to b? to r'?

Case III-Let us try to find a common measure of the diagonal and a side of a square.

Let A B C D be the required square. Draw the diagonal CA. Apply C B to CA (C B < C A; [Auth.] CA is not twice C B; [Auth.]. CB is contained in C A once with a

remainder. Lay off on C A, C P equal to C B; then the remainder is A P. Erect a 1 to CA at P, which cuts A B at N. then BNN P; [Auth.] also N P = A P, and A PN is a rt. isos., and A P N M is a square. [Give proof of each statement.] When, then, we apply the remainder, A P, to a side, C B, or to its equal, A B, A P is contained twice with the remainder N P'. Show that N P' (second remainder) is contained in A P (first remainder) twice with a remainder, and consequently, show there will always be a remainder and that there is no common measure of the diagonal and a side of a square.

Such lines are said to be incommensurable.

Do you know of other lines that are incommensurable? While it is impossible to get the exact relation of two incommensurable magnitudes, we may approximate that relation to any required degree of accuracy. (1) Thus, when C B is divided into 10 equal parts, [Review § 117.] A C contains more than 14 of those parts and less than 15. (2) If C B is divided into 100 parts, A C will contain more than 141 and less than 142 of them, etc.

In the first instance A C is more than 14 and less than 15 times B C.

Tell the relation of A C to B C in the second instance. Tell the relation of B C to A C in each instance.

Extract the square root of 2 true to six decimal places and further approximate the relation of A C to B C. Can you in each instance find two lines, one a little less than A C and one a little more than A C, to which B C does bear an exact relation? Find the difference of these two lines in each instance. Do these lines get closer and closer to A C, and does their difference grow less and less? What is the difference when you use all of the six decimal places required above?