at its middle point C, passes through both centres (16); and there can be but one straight line drawn between the two points O and O'. 35. Corollary. When two circumferences are tangent to each other, their point of contact is in the straight line joining their centres. It has just been proved that when two circumferences intersect, the two points of intersection lie at equal distances from the line joining the centres and on opposite sides of this line. Now let the circles be supposed to be moved so as to cause the points of intersection to approach each other; these points will ultimately come together on the line joining the centres, and be blended in a single point C, common to the two circumferences, which will then be their point of contact. The perpendicular to 00' erected at C will then be a common B tangent to the two circumferences and take the place of the common chord. PROPOSITION XIV.-THEOREM. 36. When two circumferences are wholly exterior to each other, the distance of their centres is greater than the sum of their radii. Let O, O' be the centres. Their distance 00' is greater than the sum of the radii OA, O'B, by the portion AB interposed between the circles. A B PROPOSITION XV.-THEOREM. 37. When two circumferences are tangent to each other externally, the distance of their centres is equal to the sum of their radii. Let O, O', be the centres, and C the point of contact. The point C being in the line joining the centres (35), we have 00' = OC+ O'C. PROPOSITION XVI.-THEOREM. 38. When two circumferences intersect, the distance of their centres is less than the sum of their radii and greater than the difference of their radii. Let O and O' be their centres, and A one of their points of intersection. The point A is not in the line joining the centres (34); and consequently there is formed the triangle A00', in which we have 00' < OA + O'A, and also 00' > OA - O'A (I. 67). Α PROPOSITION XVII-THEOREM. 39. When two circumferences are tangent to each other internally, the distance of their centres is equal to the difference of their radii. Let O, O', be the centres, and C' the point of contact. The point C being in the line joining the centres (35), we have 00' = OC — O'C. PROPOSITION XVIII.-THEOREM. 40. When one circumference is wholly within another, the distance of their centres is less than the difference of their radii. Hence 00' is less than the difference of the radii by the distance AB. 41. Corollary. The converse of each of the preceding five propositions is also true: namely 1st. When the distance of the centres is greater than the sum of the radii, the circumferences are wholly exterior to each other. 2d. When the distance of the centres is equal to the sum of the radii, the circumferences touch each other externally. 3d. When the distance of the centres is less than the sum of the radii, but greater than their difference, the circumferences intersect. 4th. When the distance of the centres is equal to the difference of the radii, the circumferences touch each other internally. 5th. When the distance of the centres is less than the difference of the radii, one circumference is wholly within the other. ست MEASURE OF ANGLES. As the measurement of magnitude is one of the principal objects of geometry, it will be proper to premise here some principles in regard to the measurement of quantity in general. 42. Definition. To measure a quantity of any kind is to find how many times it contains another quantity of the same kind called the unit. Thus, to measure a line is to find the number expressing how many times it contains another line called the unit of length, or the linear unit. The number which expresses how many times a quantity contains the unit is called the numerical measure of that quantity. 43. Definition. The ratio of two quantities is the quotient arising A B from dividing one by the other; thus, the ratio of A to B is To find the ratio of one quantity to another is, then, to find how many times the first contains the second; therefore, it is the same thing as to measure the first by the second taken as the unit (42). It is implied in the definition of ratio, that the quantities compared are of the same kind. Hence, also, instead of the definition (42), we may say that to measure a quantity is to find its ratio to the unit. The ratio of two quantities is the same as the ratio of their numerical measures. Thus, if P denotes the unit, and if P is contained m times in A and n times in B, then, 44. Definition. Two quantities are commensurable when there is some third quantity of the same kind which is contained a whole number of times in each. This third quantity is called the common measure of the proposed quantities. Thus, the two lines, A and B, are commensurable, if there is some line, C, which is contained a whole number of times in each, as, for example, 7 times in A, and 4 times in B. B C The ratio of two commensurable quantities can, therefore, be exactly expressed by a number whole or fractional (as in the preceding example by and is called a commensurable ratio. 45. Definition. Two quantities are incommensurable when they have no common measure. The ratio of two such quantities is called an incommensurable ratio. If A and B are two incommensurable quantities, their ratio is still Ar MB E CD 46. Problem. To find the greatest common measure of two quantities. The well-known arithmetical process may be extended to quantities of all kinds. Thus, suppose AB and CD are two straight lines whose common measure is required. Their greatest common measure cannot be greater than the less line CD. Therefore, let CD be applied to AB as many times as possible, suppose 3 times, with a remainder EB less than CD. Any common measure of AB and CD must also be a common measure of CD and EB; for it will be contained a whole number of times in CD, and in AE, which is a multiple of CD, and therefore to measure AB it must also measure the part EB. Hence, the greatest common measure of AB and CD must also be the greatest common measure of CD and EB. This greatest common measure of CD and EB cannot be greater than the less line EB; therefore, let EB be applied as many times as possible to CD, suppose twice, with a remainder FD. Then, by the same reasoning, the greatest common measure of CD and EB, and consequently also that of AB and CD, is the greatest common measure of EB and FD. Therefore, let FD be applied to EB as many times as possible: suppose it is contained exactly twice in EB without remainder; the process is then com pleted, and we have found FD as the required greatest common measure. The measure of each line, referred to FD as the unit, will then be as follows: we have The proposed lines are therefore numerically expressed, in terms of the unit FD, by the numbers 17 and 5; and their ratio is 17 47. When the preceding process is applied to two quantities and no remainder can be found which is exactly contained in a preceding remainder, however far the process be continued, the two quantities have no common measure; that is, they are incommensurable, and their ratio cannot be exactly expressed by any number whole or fractional. 48. But although an incommensurable ratio cannot be exactly expressed by a number, it may be approximately expressed by a number within any assigned measure of precision. A Suppose denotes the incommensurable ratio of two quantities B A and B; and let it be proposed to obtain an approximate numerical expression of this ratio that shall be correct within an assigned 1 measure of precision, say Let B be divided into 100 equal 100 parts, and suppose A is found to contain 314 of these parts with a remainder less than one of the parts; then, evidently, we have 314 that is, is an approximate value of the ratio within the as signed measure of precision. A Α B To generalize this, denoting as before the incommensurable B ratio of the two quantities A and B, let B be divided into n equal |