PART IV. SECTION I. GEOMETRIC EXTENSIONS. 220°. Let two lines L and M passing through the fixed points A and B meet at P. When P moves in the direction of the arrow, L and M M B A L approach towards parallelism, and the angle APB diminishes. Since the lines are unlimited (21°, 3) P may re cede from A along L until the segment AP becomes greater than any conceivable length, and the angle APB becomes less than any conceivable angle. And as this process may be supposed to go on endlessly, P is said to "go to infinity" or to "be at infinity," and the LAPB is said to vanish. But lines which make no angle with one another are parallel, .. Parallel lines meet at infinity, and lines which meet at infinity are parallel. The symbol for "infinity" is ∞. The phrases "to go to infinity," "to be at infinity," must not be misunderstood. Infinity is not a place but a property. Lines which meet at ∞ are lines so situated that, having the same direction they cannot meet at any finite point, and therefore cannot meet at all, within our apprehension, since every point that can be conceived of is finite. The convenience of the expressions will appear throughout the sequel. Cor. Any two lines in the same plane meet at a finite point if the lines are not parallel, at infinity if the lines are parallel. N" 221°. L and M are lines intersecting in O, and P is any point from which PB and PA are || respectively to L and M. A third and variable line N turns about P in the direction of the arrow. B Y M When N comes to parallelism with L, AX becomes infinite and BY becomes zero. ...o is indefinite since U may have any value we please. 2. The motion continuing, let N come into the position N'. Then AX' is opposite in sense to AX, and BY' to BY. But AX increased to ∞, changed sign and then decreased absolutely, until it reached its present value AX', while BY decreased to zero and then changed sign. .. a magnitude changes sign when it passes through zero or infinity. 3. It is readily seen that, as the rotation continues, BY' increases negatively and AX' decreases, as represented in one of the stages of change at X" and Y". After this Y" goes off to ∞ as X" comes to A. Both magnitudes then change sign again, this time BY" by passing through ∞ and AX" by passing through zero. Since both segments change sign together the product or rectangle remains always positive and always equal to the constant area U. 222°. A line in the plane admits of one kind of variation, rotation. When it rotates about a fixed finite point it describes angles about that point. But since all the lines of a system of parallels meet at the same point at infinity, rotation about that point is equivalent to translation, without rotation, in a direction orthogonal to that of the line. Hence any line can be brought into coincidence with any other line in its plane by rotation about the point of intersection. 223°. If a line rotates about a finite point while the point simultaneously moves along the line, the point traces a curve to which the line is at all times a tangent. The line is then said to envelope the curve, and the curve is called the envelope of the line. The algebraic equation which gives the relation between the rate of rotation of the line about the point and the rate of translation of the point along the line is the intrinsic equation to the curve. 224°. A line-segment in the plane admits of two kinds of variation, viz., variation in length, and rotation. If one end-point be fixed the other describes some locus depending for its character upon the nature of the variations. The algebraic equation which gives the relation between the rate of rotation and the rate of increase in length of the segment, or radius vector, is the polar equation of the locus. When the segment is invariable in length the locus is a circle. 225°. A line which, by rotation, describes an angle may rotate in the direction of the hands of a clock or in the contrary direction. If we call an angle described by one rotation positive we must call that described by the other negative. Unless convenience requires otherwise, the direction of rotation of the hands of a clock is taken as negative. An angle is thus counted from zero to a circumangle either positively or negatively. The angle between AB and A'B' is the rotation which brings AB to A'B', and is either a or -ẞ, and the sum of these A two angles irrespective of sign is a cir cumangle. B' a When an angle exceeds a circumangle the excess is taken in Geometry as the angle. Ex. QA and QB bisect the angles CAB and ABP externally; to prove that ✩P=24Q. The rotation which brings CP to AB is 2a, AB to BP is +26, .. LP=2(ß− a). Also, the rotation which brings AQ to AB is -a, and AB to BQ is +ẞ, . 4Q=ß-a. 4(CP. BP)=24(AQ. BQ). P B a This property is employed in the working of the sextant. 2 226°. Let AB and CD be two diameters at right angles. The rectangular sections of the plane taken in order of positive rotation and starting from A are called respectively the first, second, third, and fourth quadrants, the first being AOC, the B second COB, etc. The radius vector starting from coincidence with OA may describe the positive LAOP, or the negative LAOP'. 3 4 D A M Let these angles be equal in absolute value, so that the AMOP=AMOP', PM being ▲ on OA. Then PM - P'M, since in passing from P to P', PM passes through zero. sin AOP’= P'M - PM = and cos AOP' OM_OM = OP' OP cos AOP. . the sine of an angle changes sign when the angle does, but the cosine does not. 227°. As the angle AOP increases, OP passes through the several quadrants in succession. When OP lies in the 1st Q., sin AOP and cos AOP are both positive; when OP lies in the 2nd Q., sin AOP is positive and cos AOP is negative; when OP lies in the 3rd Q., the sine and cosine are both negative; and, lastly, when OP lies in the 4th Q., sin AOP is negative and the cosine positive. Again, when P is at A, LAOP=o, and PM=0, while OM=OP. .. sin o=0 and cos o=1. When P comes to C, PM-OP and OM=0, and denoting a right angle by π sin=1, and cos=0. When P comes to B, PM=0 and OM= - I, sin =0, and cos = - I. (207°, Cor. 4) Finally when P comes to D, PM - OP and OM=0. These variations of the sine and cosine for the several quadrants are collected in the following table : |