186. A straight line perpendicular to a radius at its extremity is a tangent to the circle. Let BA be the radius, and MO the straight line perpendicular to BA at 4. We are to prove MO tangent to the circle. From B draw any other line to M 0, as BC H. BH > BA, § 52 (a I measures the shortest distance from a point to a straight line). .. point H is without the circumference. But B H is any other line than B A, .. every point of the line M O is without the circumference, except A. .. MO is a tangent to the circle at A. $ 171 Q. E. D. 187. COROLLARY. When a straight line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact, and therefore a perpendicular to a tangent at the point of contact passes through the centre of the circle. 188. When two circumferences intersect each other, the line which joins their centres is perpendicular to their common chord at its middle point. Let C and C be the centres of two circumferences which intersect at A and B. Let: A B be their common chord, and CC' join their centres. § 184 We are to prove CC' I to A B at its middle point. A I drawn through the middle of the chord A B passes through the centres C and C', (a I erected at the middle of a chord passes through the centre of the O). .. the line C C', having two points in common with this I, must coincide with it. ..CC is I to A B at its middle point. Q. E. D. Ex. 1. Show that, of all straight lines drawn from a point without a circle to the circumference, the least is that which, when produced, passes through the centre. Ex. 2. Show that, of all straight lines drawn from a point within or without a circle to the circumference, the greatest is that which meets the circumference after passing through the centre. 189. When two circumferences are tangent to each other their point of contact is in the straight line joining their centres. A 0 B Let the two circumferences, whose centres are C and C', touch each other at 0, in the straight line A B, and let CC' be the straight line joining their centres. We are to prove O is in the straight line C C'. AI to A B, drawn through the point 0, passes through the centres C and C', $ 187 (a I to a tangent at the point of contact passes through the centre of the O). .. the line C C", having two points in common with this I, must coincide with it. ..O is in the straight line C C". Q. E. D. Ex. A B, a chord of a circle, is the base of an isosceles triangle whose vertex C is without the circle, and whose equal sides meet the circle in D and E. Show that C D is equal to CE, ON MEASUREMENT. 190. DEF. To measure a quantity of any kind is to find how many times it contains another known quantity of the same kind. Thus, to measure a line is to find how many times it contains another known line, called the linear unit. 191. DEF. The number which expresses how many times a quantity contains the unit, prefixed to the name of the unit, is called the numerical measure of that quantity; as 5 yards, etc. 192. DEF. Two quantities are commensurable if there be some third quantity of the same kind which is contained an exact number of times in each. This third quantity is called the common measure of these quantities, and each of the given quantities is called a multiple of this common measure. 193. DEF. Two quantities are incommensurable if they have no common measure. 194. DEF. The magnitude of a quantity is always relative to the magnitude of another quantity of the same kind. No quantity is great or small except by comparison. This relative magnitude is called their Ratio, and this ratio is always an abstract number. When two quantities of the same kind are measured by the same unit, their ratio is the ratio of their numerical measures. a 195. The ratio of a to 6 is written or a :b, and by this is meant : b' times 6 is contained in a; or, what part a is of b. I. If b be contained an exact number of times in a their ratio is a whole number. If b be not contained an exact number of times in a, but if there be a common measure which is contained m times in a and n times in b, their ratio is the fraction II. If a and b be incommensurable, their ratio cannot be exactly expressed in figures. But if 6 be divided into n. equal parts, and one of these parts be contained m times in a with 1 a remainder less than part of b, then is an approximate 1 value of the ratio correct within õ' 22 a m n т n n a a n? Again, if each of these equal parts of b be divided into n equal parts; that is, if b be divided into na equal parts, and if one of these parts be contained m' times in a with a remainder 1 m' less than part of b, then is a nearer approximate value na 1 of the ratio correct within 7' By continuing this process, a series of variable values, m' m etc., will be obtained, which will differ less and n a less from the exact value of We may thus find a fraction b which shall differ from this exact value by as little as we please, that is, by less than any assigned quantity, Hence, an incommensurable ratio is the limit toward which its successive approximate values are constantly tending. m n? . ON THE THEORY OF LIMITS. 196. DEF. When a quantity is regarded as having a fixed value, it is called a Constant ; but, when it is regarded, under the conditions imposed upon it, as having an indefinite number of different values, it is called a Variable. 197. DEF. When it can be shown that the value of a variable, measured at a series of definite intervals, can by indefinite continuation of the series be made to differ from a given constant by less than any assigned quantity, however small, but cannot be made absolutely equal to the constant, that constant is called the Limit of the variable, and the variable is said to approach indefinitely to its limit. If the variable be increasing, its limit is called a superior limit; if decreasing, an inferior limit. 198. Suppose a point 4 to move from A toward B, under the conditions that the first second it shall move one-half the distance from A to B, that is, to M; the next second, one-half the remaining distance, that is, to M'; the next second, one-half the remaining distance, that is, to M", and so on indefinitely. Then it is evident that the moving point may approach as near to B as we please, but will never arrive at B. For, however M M M" B |