figures are said to be symmetrical with regard to a center when all their corresponding points are respectively symmetrical with regard to the center. (2) With respect to an axis. Two points are said to be symmetrical with regard to an axis when they are on opposite sides of a line (or axis) on a perpendicular to the axis, and equidistant from the axis. Two figures are said to be symmetrical with regard to an axis when any two corresponding points are symmetrical with regard to that axis. The placing of figures in symmetrical positions, as in Bk. I, Th. VII (§ 105), is quite common, as is the use of symmetrical points. Note also the figure of § 100, and exercise 120. 345. Positive and Negative Sects and Angles. There are many theorems in Geometry where different cases of the theorem require that sects or angles be added in one case, subtracted in a second, and in the third one of the things added or subtracted is zero. If these are arranged in one general statement, it can be shown that the sects, or angles, are all positive in the addition case, that one pair has become zero in the zero case, and then negative in the subtraction case. Examples of this can be found in the proof of Th. VII, Bk. I (§ 105), in §§ 207, 214, and 215, and in §§ 240 and 244. 346. Distance. Where the word "distance" is used, it always means the shortest possible distance. The distance between two points means the straight-line distance. That this is the shortest line has been partly proved in § 109, where the straight line between two points is shown to be less than any broken line between the points. That it is less than any curved line between the points has not been shown, but it can be proved quite easily by showing that the shortest line between two points must pass through any point on the straight line between those points, and therefore through every point on the straight line. The distance from a point to a line is the perpendicular from the point to the line (§ 113). The distance from a point to a circumference is the sect from the point to the circumference on the line through the center (§ 174). 347. Limits. There is some doubt as to the perfect accuracy of any of the proofs for the first limit theorem; the following proof is probably as free from objections as any. I. If two variables approaching limits are equal for all values, their limits are also equal. = vLx, where x and x' are quantities, either positive or negative, which can become indefinitely small, but cannot equal zero. II. v=v' (given), ... L-x= L'x' (eq. same), and L-L'x-x' (eq. +, - eq.). III. L, L' and L-L' are constants; x and x' can each be made less than any constant except each become less than (L-L'), and then IV. But this is impossible, since L-L' x − x'. · · . L — L' = 0. II. (1) If a variable approaches the limit zero, its quotient by a constant, and its product by a constant (other than zero), approaches zero. Given. To prove. Proof. v = 0; c. 2=0; vc = 0. с I. Let k be any constant quantity, however small. Then ck is a constant quantity, and v can become less than ck. v II. But if v<ck, then -<k (uneq. ÷ eq.). с IV. Since is less than k, a constant quantity, с however small, but is not zero, 2=0 (def.). Similarly ve 0. с (2) If a variable approaches any limit, its quotient by a constant, and its product by a constant (other than zero), will approach the quotient of its limit by the constant, and the product of its limit by the constant. III. If two variables are proportional to two constants, their limits are proportional to the same constants. 348. Incommensurable. Case. The incommensurable case of Th. V, Bk. II, will be taken as an example of the method to be applied to all theorems having the two cases. Let the central angles AOB and COD in circle o have no common divisor. Suppose an exact divisor of AOB to be applied to COD as often as possible, leaving a remainder XOD, which is therefore less than the divisor used. Then AOB and Cox have a common divisor, so ZAOB AB = by the commensurable case. But, if the divisor of AOB is taken smaller and smaller, that is, is made to approach the limit zero, the remainder XOD, being still smaller, will also approach the limit zero. Therefore Cox will approach the limit ▲ COD. Also its arc, CX, will approach the limit CD. But, since the variables ▲ cox and CX are proportional to the constants AOB and AB, their limits are also proZAOB AB portional to those constants; that is, Z COD = CD This method of proof will apply to Th. II, Bk. III, to Th. I, Bk. IV, and in fact to all proofs where the method of a common divisor is used. 349. Similar Figures. Similar figures have already been defined (§ 282) as those which have their corresponding angles equal, and their corresponding sides proportional. This, of course, applies only to polygons. The following definition is sometimes given, and while it is not as convenient for use in Plane Geometry, it has the merit of applying to all kinds of figures, including those of Solid Geometry. Two figures that may be placed in a pencil (or sheaf, in Solid Geometry) of lines, so that all pairs of corresponding points of the figures cut the respective lines of that pencil in the same ratio, are similar. These two definitions are identical in result, as far as polygons are concerned, as may be shown by proving the two following statements: 1. If two polygons that are mutually equiangular and have their corresponding sides proportional are placed with one pair of corresponding sides parallel, the polygons lying on the same side of those lines, the lines joining their corresponding vertices will form a pencil which is cut proportionally by the vertices of the polygons. 2. If two polygons lie in a pencil of lines, and their vertices cut the lines proportionally, the polygons are mutually equiangular, and their corresponding sides are proportional. 350. The Evaluation of Pi. The ratio of the circumference of a circle to its diameter is obtained by the use of the ratios of the perimeters of regular inscribed and circumscribed polygons of a large number of sides to the diameter. The circumference is longer than the |