Area of H> area of R; i.e. area of s > area of R. Q.E.D. 584. Cor. Of all polygons having a given number of sides and a given area, that which has a minimum perimeter is regular. Ex. 1149. Among the triangles inscribed in a given circle, the one that has a maximum perimeter is equilateral. Ex. 1150. Of all polygons having a given number of sides and inscribed in a given circle, the one that has a maximum perimeter is regular. VARIABLES AND LIMITS. THEOREMS PROPOSITION I. THEOREM 585. If a variable can be made less than any assigned value, the product of the variable and any constant can be made less than any assigned value. Given a variable V, which can be made less than any previously assigned value, however small, and let K be any constant. To prove that V. K may be made as small as we please, i.e. less than any assigned value. Assign any value, as a, no matter how small. Now a value for v may be found as small as we please. a Take V < Then V. K<a; i.e. V · K may be made less K than any assigned value. Q.E.D. 586. Cor. I. If a variable can be made less than any assigned value, the quotient of the variable by any constant, except zero, can be made less than any assigned value. HINT. = V 1 587. Cor. II. If a variable can be made less than any assigned value, the product of that variable and a de creasing value may be made less than any assigned value. HINT. Apply the preceding theorem, using as K a value greater than any value of the decreasing multiplier. 588. Cor. III. The product of a variable and a variable may be a constant or a variable. 589. Cor. IV. If a variable can be made less than any assigned value, the square of that variable can be made less than any assigned value. (Apply Cor. II.) Ex. 1151. Which of the corollaries under Prop. I is illustrated by the theorem: "The product of the segments of a chord drawn through a fixed point within a circle is constant " ? 590. The limit of the product of a variable and a constant, not zero, is the limit of the variable multiplied by the constant. Given any variable v which approaches the finite limit Z, and let K be any constant not zero. To prove the limit of K. V = K • L. Let R=L-V; then V=L— R. .'. K · V = K. L-K· R. But the limit of K. R = 0. 591. Cor. The limit of the quotient of a variable by a constant is the limit of the variable divided by the constant. V, which is the product of the variable and a constant. PROPOSITION III. THEOREM 592. If two variables approach finite limits, not zero, then the limit of their product is equal to the product of their limits. Given variables and v' which approach the finite limits L Then VL-R and V'= L' — R'. .. v. v' L. L' (L'. R+L. R' - R. R'). = -- But the limit of (L' .. the limit of V. V' the limit of [L · L' — (L' · R + L · R' · Q.E.D. 593. Cor. If each of any finite number of variables approaches a finite limit, not zero, then the limit of their product is equal to the product of their limits. PROPOSITION IV. THEOREM 594. If two related variables are such that one is always greater than the other, and if the greater continually decreases while the less continually increases, so that the difference between the two may be made as small as we please, then the two variables have a common limit which lies between them. Given the two related variables AP and AP', AP' greater than AP, and let AP and AP' be such that as AP increases AP' shall decrease, so that the difference between AP and AP' shall approach zero as a limit. To prove that AP and AP' have a common limit, as AL, which lies between AP and AP'. Denote successive values of AP by AQ, AR, etc., and denote the corresponding values of AP' by AQ', AR', etc. Since every value which AP assumes is less than any value which AP assumes (Hyp.) .. AP < AR'. But AP is continually increasing. Hence AP has some limit. (By def. of a limit, § 349.) Since any value which AP' assumes is greater than every value which AP assumes (Hyp.) .. AP'> AR. But AP' is continually decreasing. Hence AP' has some limit. Suppose the limit of AP (By def. of a limit, § 349.) the limit of AP'. Then let the limit of AP be AK, while that of AP' is AK'. Then AK and AK' have some finite values, as m and m', and their difference is a finite value, as d. But the difference between some value of AP and the corresponding value of AP' cannot be less than the difference of the two limits AK and AK'. This contradicts the hypothesis that the difference between AP and AP' shall approach zero as a limit. .. the limit of AP = the limit of AP' and lies between AP and AP', as AL. Q.E.D. 595. THEOREM. With every straight line segment there is associated a number which may be called its measurenumber. a u For line segments commensurable with the unit this theorem was considered in §§ 335 and 336; we shall now consider the case where the segment is incommensurable with the chosen unit. Given the straight line segment a and the unit segment u; to express a in terms of u. Apply u (as a measure) to a as many times as possible, suppose t times, then tu<a < (t+1) u. Now apply some fractional part of u, say, Ρ to a, and sup Then apply smaller and smaller fractional parts of u to a, say и и и and suppose them to be contained to, tз, tå, times p2 p3 respectively, then to • u < a <t2+1 p2 p2 ... Now the infinite series of increasing numbers t, t to p' p2 none of which exceeds the finite number t + 1, defines a numbern (the limit of this series) which we shall call the measurenumber of a with respect to u. Moreover, this number n is unique, i.e. independent of p (the number of parts into which the unit was divided), for if m is any number such that m<n, then m • u < a, and if m>n, then m u>a; we are therefore justified in associating the number n with a, and in saying that nu = a. 596. Note. Manifestly, the above procedure may be applied to any geometric magnitude whatever, i.e. every geometric magnitude has a unique measure-number. 597. Cor. If a magnitude is variable and approaches a limit, then, as the magnitude varies, the successive measure-numbers of the variable approach as their limit the measure-number of the limit of the magnitude. 598. Discussion of the problem: To determine whether two given lines are commensurable or not; and if they are commensurable, to find their common measure and their ratio (§ 345). Moreover, GD is the greatest common measure of AB and CD. For every measure of AB is a measure of its multiple CE. Hence, every common measure of AB and CD is a common measure of CE and CD and therefore a measure of their differ |