drawn with its circumference passing through four, or a greater number of, given points. Exhibit such a case. (12) In a given circle draw a chord which shall be both equal and parallel to a given chord in the same circle. (13) If an arc or a segment of a circle be given, complete the circle. (14) Through a given point within a given circle draw the least chord. (15) Through a given point within a given circle draw a chord which shall be equal to a given line not greater than the diameter of the circle. (16) If one circle intersect another, shew that the straight line joining the points of intersection is at right angles to the straight line joining their centres. (17) Shew that the two tangents, which can be drawn to a circle from a point without it, are equal to one another. (18) Shew that the straight line drawn through the middle point of an arc parallel to the chord of the arc is a tangent to the circle at that point. (19) Can two distinct straight lines touch a circle at the same point? (20) Divide a given arc into four equal parts. (21) Have equal circles equal circumferences or perimeters? Is this the case with equal squares, triangles, and other rectilineal equal plane figures? (22) If AB, CD be any two chords in a circle at right angles to each other, prove that the sum of the arcs AC, BD is equal to half the circumference. (23) From two given points draw two straight lines which shall meet in a given straight line, and be at right angles to each other. Within what limits only is this possible? (24) Apply Prop. VI. to draw a straight line at right angles to a given straight line from one extremity of it, when the given line cannot be produced'. (25) Construct a square, when the diagonal only is given. (26) If tangents be drawn to a circle from the extremities of a diameter, shew that any line intercepted between them, and touching the circle, subtends at the centre a right angle. (27) Shew that about a given circle a certain number of equal circles can be drawn touching it and each other; and find the number. (28) If two circles touch each other internally, and the radius of one of them be half that of the other, shew that every straight line, drawn from the point of contact to meet the outer circumference, is bisected by the inner one. (29) A straight line touches a circle, and from the point of contact A any chord AB is drawn ; BC is another chord parallel to the tangent, and BD a chord parallel to AC. Shew that the chords AB, AC, CD are equal to one another. (30) If the circumferences of two circles intersect each other, and through one of the points of intersection the diameters be drawn, shew that the other extremities of those diameters and the other point of intersection will be in one and the same straight line. (31) Three equal circles are given, in the same plane, of which no two intersect each other, find the point from which if tangents be drawn to each circle, those tangents shall be equal to one another. (32) What is the angle which the arc of a quadrant subtends at any point in the remaining portion of the circumference? Is it the same for all circles? (33) In any two circles which have the same centre*, if a chord be drawn to the outer one and intersecting the inner one, shew that the parts of this chord intercepted between the two circumferences are always equal. (34) If two circles touch each other, either externally or internally, and through the point of contact two straight lines be drawn forming four chords, two in each circle, shew that the straight lines joining the extremities of these chords in each circle are parallel to one another. Such circles are sometimes called 'concentric' circles. 64. PROPORTIONAL LINES AND AREAS. DEFINITION. Ratio is the relation which two or more things, or quantities of things, of the same kind bear to each other in respect of magnitude. And, for the purpose of this comparison, any two things are of the same kind only when the lesser of the two by multiplication can be made to exceed the other. Thus a lineal foot can be multiplied (22) until it exceed a lineal mile; therefore these are things of the same kind, and bear a certain ratio to each other. So likewise an oz. and a lb. in weight have a certain ratio; a quart and a gallon have a certain ratio; and so on. But an oz. and a mile are not things of the same kind. The one can never by multiplication be made to exceed the other; and consequently they bear no relation to each other in respect of magnitude, that is, they can have no ratio to each other. Similarly, a line may have ratio to a line, and an area to an area; but a line can have no ratio to an area, because by the multiplication of either we can never arrive at, or exceed, the other. 65. DEF. The measure of the ratio between any two magnitudes is, (not their difference, but) the number of times the one contains, or is contained in, the other. Thus, if the line AB, upon being multiplied three times (22), becomes equal to the line CD, that is, if CD contains AB exactly three times, then the measure of the ratio of CD to AB is 3, that is, CD bears the same relation to AB in magnitude which 3 does to 1. But in order that two magnitudes of the same kind may have a ratio to each other, it is not necessary that one should contain the other an exact integral number of times. Thus, for example, let A be a magnitude which contains another magnitude taken as the unit of measurement, whatever that may be, 5 times; and let B be another magnitude, of the same kind, containing the same unit 3 times; then the ratio of A to B will be that of 5 to 3. In this case A may be said to contain B once and two-thirds of a time; and the measure of the ratio of A to B is 13, or the fraction §. Similarly in other cases. In the case here supposed, a certain multiple of A is equal to another certain multiple of B, that is, three times A five times B. Thus, if A be a line which contains a lineal foot 5 times, and B another line which contains it 3 times, then A = 5 feet, and B = 3 feet; the ratio of A to B is that of 5 feet to 3 feet, that is, 5 to 3; and 3 times A 15 feet 5 times B*. = = 66. DEF. PROPORTION is the equality of ratios. Thus, if the ratio of A to B be equal to the ratio of C to D, then A, B, C, D are said to be proportionals, or in proportion. Observe, A, B, C, D, in order to be proportionals, need not be all of the same kind. It is only necessary that A and B be of the same kind, and likewise C and D of the same kind; but the one pair of magnitudes may be different from the other pair. Thus, one line, A, may have the same ratio to another line, B, that one area, C, has to another area, D, in which case A, B, C, D are proportionals. The ratio of two magnitudes is often expressed by placing the symbol: between them; thus A: B signifies the ratio of A to B. So then, if A, B, C, D are proportionals, A: BC: D; but this is generally written thus, A: B :: C: D; and is read 'A is to B as C to D', which means that A has the same ratio to B which C has to D. 67. PROP. I. If two straight lines be intersected by any number of parallel lines, so that the parts of one of them intercepted between the parallels are equal to one another, the parts also, of the other line between the same parallels shall be equal to one another. Let ABCD be any straight line, such that AB=BC = CD; or similarly, whatever the number of parts may be of which it is composed. Through the points A, B, C draw parallel lines Aat, Bb, Cc, Dd, meeting another straight line in the points a, b, c, d; then also ab = bc = cd. Through the point a draw ae D parallel to AB, meeting Bb in e; and through b draw bf parallel C B d * In this section single letters will often be used to denote lines and other magnitudes, to avoid superfluous writing, where it may be done without risk of error. + This is read' A little a,' 'B little b', &c. to BC meeting Cc in f. Then since A Bea, and BCfb are parallelograms, ae = AB, and bf=BC (40); but AB =BC, :. ae=bf. Again, because Bb is parallel to Cc, LA Be = 4 BCf; and because ae is parallel to AB, LABe: = 4aeb; also because bf is parallel to BC, BCf=4bfc, .. 4aebbfc. And since ae, bf are parallel to the same line ABC, they are parallel to each other, and ... < eab4fbc. Hence in the two triangles aeb, bfc, there are two angles in the one equal to two angles in the other, each to each, and the side common to those angles in the one equal to the side which is common to the two angles, equal to them, in the other, .. the triangles are equal in all respects, and the side ab the side bc (39). Similarly it may be shewn, by drawing cg parallel to CD, that bc=cd; .. ab=bc=cd. And the proof may be extended to any any number of parts. COR. 1. The proof here given is independent of the length of the line A a, and will therefore hold when A and a coincide in the same point, that is, when the two given lines meet in A. COR. 2. Conversely, if two straight lines be composed of the same number of equal parts, the straight lines Aa, Bb, Cc, &c., joining corresponding points in them, will be parallel. 68. PROP. II. To divide a given straight line into any number of equal parts. Let AB be the given straight line, which is to be divided into any proposed number of equal parts, three suppose, as the process is the same whatever the number may be. b. A F From A draw any other indefinite line Ab, forming an angle with AB; in Ab take any point D conveniently near to A, and with centre D and radius DA |