PROPOSITION VIII.-THEOREM. 22. A straight line cannot intersect a circumference in more than two points. For, if the line could intersect the circumference in three points, the radii drawn to these points would meet the line at unequal distances from the perpendicular let fall from the centre of the circle upon the line, and would be unequal, by Proposition XXI., Book I. PROPOSITION IX.-THEOREM. 23. A straight line tangent to a circle is perpendicular to the radius drawn to the point of contact. For any other point of the tangent, as D, must lie outside of the circle, and therefore the line OD, joining it with the centre, must be greater than the radius OC, drawn to the point of contact. OC is, then, the shortest line that can B D be drawn from O to the tangent AB, and is therefore perpendicular to AB, by Proposition XVII., Book I. 24. COROLLARY I. A perpendicular to a tangent line drawn through the point of contact must pass through the centre of the circle. 25. COROLLARY II. If two circumferences are tangent to each other, their centres and their point of contact lie in the same straight line. Suggestion. Through their point of contact draw a line perpendicular to the tangent at that point. (v. Corollary I.) PROPOSITION X.-THEOREM. 26. When two tangents to the same circle intersect, the distances from their point of intersection to their points of contact are equal. For the right triangles OAP and OBP (Proposition IX.) are equal, by Proposition X., Book I. A P B EXERCISES. 1. Theorem.-In any circumscribed quadrilateral, the sum of two opposite sides is equal to the sum of the other two opposite sides. 2. Theorem. If two circumferences are tangent, and from any point, P, of the tangent at their point of contact, tangents are drawn to the two circles, the points of contact of these tangents are equally distant from P. PROPOSITION XI.-THEOREM. 27. Two parallels intercept equal arcs on a circumference. We may have three cases: 1st. When the parallels AB, CD, are E both secants, then the intercepted arcs AC and BD are equal. For, let OM be the radius drawn perpendicular to the parallels. By Proposition VI., the point M is at once the middle of the arc AMB and of the arc CMD, and hence we have G M F B N H AM BM and CM 2d. When one of the parallels is a secant, as AB, and the other is a tangent, as EF at M, then the intercepted arcs AM and BM are equal. For the radius OM drawn to the point of contact is perpendicular to the tangent (Proposition IX.), and consequently perpendicular also to its parallel AB; therefore, by Proposition VI., AM = BM. 3d. When both the parallels are tangents, as EF at M, and GH at N, then the intercepted arcs MAN and MBN are equal. For, drawing any secant AB parallel to the tangents, we have, by the second case, and each of the intercepted arcs in this case is a semi-circumference. 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. 28. 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. 29. Definition. The ratio of two quantities is the quotient arising from dividing one by the other: thus, the ratio of A 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 (28). It is implied in the definition of ratio that the quantities compared are of the same kind. Hence, also, instead of the definition (28), 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 30. 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 Br quantities can, therefore, be exactly expressed by a number, whole or fractional (as in the preceding example by), is called a commensurable ratio. 31. 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 expressed by 32. 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 three A F B E 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 completed, and we have found FD as the required greatest common meas ure. |