RELATIVE POSITION OF Two CIRCLES. Proposition 11. Theorem. 220. If two circumferences intersect each other, the right line joining their centres bisects their common chord at right angles. Hyp. Let O, O' be the centres of two Oces which intersect each other; and A, B their pts. of intersection. To prove that the line 00' bisects AB at rt. s. A Proof. Because O and O' are each equally distant from A and B, .. the line 00′ bisects AB at rt ≤ s. (179) (67) 221. COR. 1. Conversely, the perpendicular bisector of a common chord passes through the centres of both circles. 222. COR. 2. If we suppose the circles to be moved so that the point A approaches the line 00', the pt. B will also approach the line; and since the line 00' is always perpendicular to the middle of AB (220), the two points. C A D A and B will ultimately come together on the line 00', and be united in a single point common to the two circles. The common chord AB will then be a common tangent to the two circumferences at their point of contact. Hence, when two circumferences are tangent to each other, their point of contact is in the straight line joining their centres; and the perpendicular at this point is a common tangent to the two circumferences. Proposition 12. Theorem. 223. If two circumferences intersect each other, the distance between their centres is less than the sum and greater than the difference of the radii. Hyp. Let the Os with centres O, O' intersect at A. Join OO', AO, AO’. To prove 00' < OA+ AO', and > OA- AO'. 00' < 0A + AO', and 00′ > OA — AO'. Either side of a ▲ < the sum and > the difference of the other two 224. COR. 1. If the distance of the centres of two circles is greater than the sum of their radii, they are wholly exterior to each other. 225. COR. 2. If the distance of the centres of two circles is equal to the sum of the radii, they are tangent externally. 226. COR. 3. If the distance of the centres is less than the sum and greater than the difference of the radii, the circles intersect. 227. Cor. 4. If the distance of the centres is equal to the difference of the radii, the circles are tangent internally. 228. COR. 5. If the distance of the centres is less than the difference of the radii, one circle is wholly within the other. SCH. If two circles intersect and the radius of either circle drawn to a point of section touches the other circle, the circles intersect orthogonally, i.e., at right angles. EXERCISES. 1. Through a given point P either inside or outside a given circle whose centre is 0, two straight lines PAB, PCD are drawn making equal angles with OP, and cutting the circle in A, B, C, D: prove that AB = CD, and PA PC. 2. P is a point inside a circle whose centre is 0: prove that the chord which is at right angles to OP is the shortest chord that can be drawn through P. Let APB be to OP, CPD any other chord through P; draw OE L to CD, etc. 3. If two circles cut each other, any two parallel straight lines drawn through the points of intersection to cut the circles are equal. 4. Two circles whose centres are A and B intersect at C; through C two chords DCE, FCG are drawn equally inclined to AB and terminated by the circles: show that DE= = FG. 5. Prove that the two tangents drawn to a circle from an external point are equal and equally inclined to the straight line joining the point to the centre of the circle. 6. A is a point outside a given circle whose centre is 0; with centre A and radius AO a circle is described, and with centre O and radius equal to the diameter of the given circle another circle is described, cutting the last in B; OB is joined, cutting the given circle in E: prove that AE is tangent to the given circle. THE MEASUREMENT OF ANGLES. 229. To measure a quantity is to find how many times it contains another quantity of the same kind taken as a standard of comparison. This standard is called the unit. Thus, if we wish to measure a line, we must take a unit of length, and see how many times it is contained in the line to be measured. The number which shows how many times a quantity contains the unit, is called the numerical measure of that quantity. 230. The relative magnitude of two quantities, measured by the number of times which the first contains the second, is called their ratio. Thus, the ratio of A to B is or A : B. A B' Since the ratio of two quantities is found by dividing the first by the second, therefore the ratio of two quantities is the number which would express the measure of the first, if the second were taken as unity. The ratio of two quantities is the same as the ratio of their numerical measures. Thus, if A contains the unit m 28 times, and B contains A 28m it 9 times, we have = 28 9 231. When a quantity is contained an exact number of times in two quantities of its kind, it is called their com mon measure. Two quantities are commensurable when they have a common measure. The ratio of two commensurable quantities can be expressed by a whole number or by a fraction. Thus, if each of the two lines A and B contains some line C an exact number of times, A the two lines A and B are com- C mensurable, and the line C is their common measure. Their ratio is expressed by the fraction §. If C is not contained an exact number of times in A and B, but if there be a common measure which is contained, say, 25 times in A and 19 times in B, then the ratio of A to B is the fraction. Generally, if there be a common measure which is contained m times in A and n times in B, their ratio will be 232. Two quantities are incommensurable when they have no common measure. The ratio of such quantities is called an incommensurable ratio. This ratio cannot be exactly expressed in figures; but its numerical value can be obtained approximately as near as we please. Thus, suppose A and B are two lines whose ratio is √2. We cannot find any fraction which is exactly equal to 12; but by taking a sufficient number of decimals, we may find √ to any required degree of approximation. 1000000 √2> 1.414213 and < 1.414214. 1000000 That is, the ratio of A to B lies between 1414818 and 1414214, and therefore differs from either of these ratios by less than one-millionth. And since the decimals may be continued without end in extracting the square root of 2, it is evident that this ratio can be expressed as a fraction with an error less than any assignable quantity. In general, when A and B are incommensurable, divide B into n equal parts each equal to x, so that B = nx, where n is an integer. Also let A > mx but <(m +1)x; then |