PROPOSITION VIII. THEOREM 255 If two circles intersect, their line of centers bisects their common chord at right angles. ее Fig. 1 Fig. 2 HYPOTHESIS. O and O' (Fig. 1) are two circles whose circumferences intersect at A and B. CONCLUSION. OO' bisects AB at right angles. PROOF O and O' are two points each equidistant from A and B. § 221 ... OO' is the bisector of AB. § 188 Q. E. D. 256 COROLLARY 1. If two circles are tangent, the point of contact is in the line of centers. For since A and B are always equidistant from OO', if the © O and O' move apart, A and B will come together and coincide on OO' at T (Fig. 2). 257 COROLLARY 2. If two circles are tangent, they have a common tangent at the point of contact. For a perpendicular drawn to OO' at T is tangent to both circles. EXERCISES 257 What is the position of two circles whose line of centers is (1) greater than the sum of the radii ? (2) equal to the sum of the radii ? § 252 (3) less than the sum but greater than the difference of the radii ? (4) equal to the difference of the radii ? (5) less than the difference of the radii? EXERCISES 258 What is the position of two circles which can have (1) two common external and two common internal tangents ? (3) two common external tangents and no common internal tan- (4) one common external and no common internal tangent? (5) no common tangent? 259 The least chord which can be drawn through a fixed point within a circle is perpendicular to the diameter drawn through that point. [§ 249.] 260 What is the longest chord which can be drawn through a point within a circle? 261 If in a circle one chord bisects a second, both not being diameters, the first chord is greater than the second. 262 In two concentric circles, a chord of the outer circle which touches the inner is bisected at the point of contact. [§§ 251, 245.] 263 In two concentric circles all chords of the outer circle which touch the inner are equal. [§§ 251, 248.] 264 Radii of two circles drawn to the points of contact of a common tangent are parallel. 265 If two non-intersecting lines intercept equal arcs on a circumference, they are parallel. [Converse of Ex. 252.] 266 Two parallel chords drawn from the end-points of a diameter are equal. [By Ex. 252, arc b = arc b'. ... arc a .. the chords are equal by § 243.] = arc a'. b 267 If a quadrilateral is circumscribed about a circle, the sum of one pair of opposite sides is equal to the sum of the other pair. [§ 254.] 268 The median of a circumscribed trapezoid is equal to one-fourth of the perimeter. [Ex. 267, § 211.] 269 An inscribed trapezoid is isosceles. [Ex. 252, § 243.] 270 If two parallel tangents are drawn to a circle, the straight line which joins the points of contact is a diameter. 271 In the annexed figure, AP = AO. Prove that / BOC 3 Z P. B C MEASUREMENT DEFINITIONS 258 Ratio is the relation of two magnitudes of the same kind. This relation (ratio) is expressed by the number of times the first contains the second. Thus, the ratio of 6 ft. to 3 ft. is 2; the ratio of a to b is, or a b, and is read "the ratio of a to b." The number expressing the ratio of two magnitudes is always abstract. 259 To measure a magnitude is to find its ratio to another magnitude of the same kind, called the unit of measure. 260 Commensurable magnitudes are magnitudes which have a common measure. Thus, a foot and a yard are commensurable, for they have a common measure, the inch, which the first contains 12 times and the second 36 times. Incommensurable magnitudes are magnitudes which have no common measure. THE THEORY OF LIMITS 261 A constant is a quantity whose magnitude remains fixed. 262 A variable is a quantity whose magnitude may take an indefinite number of different values. 263 The limit of a variable is a constant, such that the difference between the variable and the constant can be made less than any assigned quantity, but not equal to zero. The variable is said to approach the constant as a limit. Thus, suppose a point P starting from A moves along the line AB, under the condition that it shall move half the distance from A to B the first second, half the remaining distance the second second, and so on indefinitely. Then 1. P can never reach B, for there is always half of some distance between them. 2. P can be made to approach as near to B as we choose, for by continuing the bisection, the distance between P and B can be made less than any assigned quantity. 3. The changing distance AP', AP", AP"", etc., is an increasing variable which approaches the constant AB as a limit; the changing distance P'B, P"B, P""B, etc., is a decreasing variable which approaches zero as a limit. Again. Consider the isosceles triangle ABC. Let the legs AB and AC constantly increase but always remain equal, and let the base BC remain constant. Then the angles B and C are two increasing variables, always equal and each approaching a right angle as a limit, which it can never become, for no triangle can have two right angles. Let it be clearly observed that the two variables, the angles B and C, always remain equal, and that the limits which they approach, two right angles, are equal. This fact is of great importance, and is formally set forth in the following 264 PRINCIPLE 1. If two variables are always equal, and each approaches a limit, their limits are equal. For since the variables are always equal, they are the same variable approaching a common limit. Their limits are, there fore, identical and equal. 265 PRINCIPLE 2. If a variable can be made less than any assigned quantity, the product of the variable by a constant or a decreasing quantity can be made less than any assigned quantity. Let x be the variable and c a constant or a decreasing quantity. Then cx is their product. Since x decreases indefinitely, cx decreases indefinitely, and therefore can be made less than any assigned quantity. 266 COROLLARY. If a variable can be made less than any assigned quantity, the quotient of the variable by a constant can be made less than any assigned quantity. Let x be the variable and c a constant. Then is their quotient. Now с x, which is the product of a variable (x) by a constant с x = (-1) x, х Therefore can be made less than any assigned quantity. с § 265 267 PRINCIPLE 3. If a variable x approaches a constant c as a limit, mx approaches mc as a limit, m being a constant. For cx can be made less than any assigned quantity. § 263 :.m(c − x) = (mc — mx) can be made less than any assigned quantity. 82 .. mx approaches mc as a limit. § 265 § 263 268 COROLLARY. If a variable x approaches a constant c as a limit, 269 PRINCIPLE 4. If each of two variables can be made less than any assigned quantity, their product can be made less than any assigned quantity. Let x and y be the variables and c a constant. Now cy can be made less than any assigned quantity. [§ 265.] But x can be made less than c. Hyp. ..xy can be made less than any assigned quantity. 270 COROLLARY. If a variable can be made less than any assigned quantity, the square of the variable can be made less than any assigned quantity. xy, which by § 269 can be made less than any 271 PRINCIPLE 5. If a variable x approaches a constant c as a limit, x2 approaches c2 as a limit, and √x approaches √c as a limit. For cnxnd; and since d can be made less than any assigned quantity by taking a large enough, x" approaches c" as a limit. If n = 2, x2 approaches c2 as a limit. If n = 1, √x approaches Vc as a limit. § 263 |