at N; then, the intercepted arcs MAN and MBN are equal. For, drawing any secant AB parallel to the tangents, F M F and each of the intercepted arcs in this case is a semi-circumference. 30. Scholium 1. The straight line joining the points of contact of two parallel tangents is a diameter. 31. Scholium 2. According to the principle of (28), the tangent being regarded as a secant whose two points of intersection are coincident, the demonstration of the first case in the preceding theorem embraces that of the other two cases. RELATIVE POSITION OF TWO CIRCLES. 32. Definition. Two circles are concentric, when they have the same centre. 33. Definition. Two circumferences are tangent to each other, or touch each other, when they have but one point in common. The common point is called the point of contact, or the point of tangency. Two kinds of contact are distinguished: external contact, when each circle is outside the other; internal contact, when one circle is within the other. PROPOSITION XIII.-THEOREM. 34. When two circumferences intersect, the straight line joining their centres bisects their common chord at right angles. Let O and O' be the centres of two circumferences which intersect in the points A, B; then, the straight line 00' bisects their common chord AB at right angles. For, the perpendicular to AB erected A B 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 G 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' 00' > OA OA + O'A, and also A 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. We have the dif Let O, O', be the centres. ference of the radii OA - O'B = 00' + AB. Hence 00' is less than the difference of the radii by the distance AB. B 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. 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 from dividing one by the other; thus, the ratio of A to B is B 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. The ratio of two commensurable quantities can, therefore, be exactly expressed Br by a number whole or fractional (as in the preceding example and is called a commensurable ratio. by), 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 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 A B E CD 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 |