PLANE LOCI. If the position of a point is determined by certain conditions; also, if every point in some line, and no other point fulfil these conditions, the line is said to be a locus of the point. As a simple illustration of a locus, consider that of a point which is always equally distant from a given point. This is obviously a circle, whose radius is equal to that distance. So the locus of a point, which is always equally distant from a given straight line, is a line parallel to it, and at a distance from it equal to the given distance. EXERCISES. 1. Find the locus of the vertices of all the triangles that have the same base and one of their sides of a given length. 2. Find the locus of a point that is at equal distances from two given points. 3. Find the locus of a point that is equally distant from two given lines, either parallel, or inclined to one another. 4. To find the locus of the vertices of all the triangles that have the same base and equal altitudes. 5. To find the locus of the vertices of all the triangles that have the same base and one of the angles at the base the same. 6. To find the locus of the angular point opposite to the hypotenuse of all the right-angled triangles that have the same hypotenuse. 7. To find the locus of the vertices of all the triangles that have the same base and equal vertical angles. 8. To find the locus of the vertices of all the triangles that have the same base and equal areas. 9. To find the locus of the vertices of all triangles that have the same base and the sum of the squares of their sides equal to a given square. 10. If straight lines drawn from a given point to a given line be cut in a given ratio, to find the locus of the point of section. 11. To find the locus of the vertices of all the triangles that have the same base and the ratios of their sides equal. 12. If a straight line drawn from a given point, and terminating in the circumference of a given circle, be cut in a given ratio, the locus of the point of section is also the circumference of a given circle. 13. If one of the extremities of straight lines drawn through a given point be terminated in a given straight line, and their other extremities be determined, so that the rectangle under the segments of each line is equal to a given rectangle, the locus of these extremities will be the circumference of a circle; and if one of the extremities be terminated in the circumference of a circle, the locus of the other extremities is a straight line. 14. If the vertex of a triangle be fixed, and its altitude given, find the loci of the extremities of its base, so that the area may be constant. PORISMS. A porism is a proposition of an indeterminate nature, such that an indefinite number of quantities must fulfil the same conditions. As a simple example of a porism, let it be required to find a point such, that all straight lines drawn from it to the circumference of a given circle shall be equal. This point is evidently the centre of the circle. Problems of loci and porisms are in many cases mutually convertible. The preceding problem becomes a problem of the latter kind when the centre is given, and it is required to find the locus of all the points that are at a given distance from it. 091 · 9ail 1. Two points being given, to find a third, through which any straight line being drawn, the perpendiculars upon it from the two given points shall be equal. 2. Three points being given, to find a fourth, through which any straight line being drawn, the sum of the perpendiculars upon it from two of the given points on one side of it, shall be equal to the perpendicular on it from the third. 2 There are similar porisms when four, five, or any number of points are given, by which another point is to be found such, that any straight line being drawn through it, the sum of the perpendiculars from the points on one side of it shall be equal to the sum of those from the points on the other side of it. The point so found is called the point of mean distance; or if the points be considered to be equal, physical, or material points, it is called the centre of gravity. 3. A straight line and a circle being given, to find a point such, that the rectangle under the segments of any straight line drawn through it, and limited by these, shall be equal to the rectangle contained by the external segment of a diameter perpendicular to the line, and produced to meet it, and the diameter itself. eq This problem is the 13th exercise on loci converted into a porism. Many other problems of loci may be similarly converted; and conversely. PLANE TRIGONOMETRY. Plane Trigonometry treats of those relations that subsist between the sides and angles of triangles, by which their numerical values may be computed. These relations are established by means of certain lines connected with the angles, called trigonometrical lines. When a sufficient number of the parts of a triangle are known, it may be constructed, and then the unknown parts can be measured. This method, however, which is called construction, or the graphic method, would only give a moderate approximation; but, by trigonometrical computation, the values may be found to any required degree of accuracy. The sides are measured by some line of a determinate length, chosen as the unit of measure-as a foot, a yard, a mile, &c.; and the unit of measure of angles is the 90th part of a right angle, or the 360th part of four right angles. As the angles at the centre of a circle are proportional to the arcs subtending them, these arcs may be taken as their measures. The circumference of a circle is accordingly supposed to be divided into 360 equal parts, called degrees; each of these into 60 equal parts, called minutes; each of these into 60 equal parts, called seconds; and so on. If, therefore, a circle be described from the angular point as a centre, the number of degrees, minutes, &c., contained in the arc intercepted by the lines containing the angle, is the measure of that angle. An angle is also sometimes measured by the length of the intercepted arc of a circle, divided by the radius, whose centre is the angular point in this case, 3.1415926 = 180°. DEFINITIONS OF TRIGONOMETRICAL LINES. 1. The complement of an arc is its difference from a quadrant; and that of an angle its difference from a right angle. 2. The supplement of an arc is its defect from a semicircle; and that of an angle is its defect from two right angles. 13. The sine of an arc is a line drawn from one of its extremities, perpendicular to the radius passing through its other extremity. 4. The tangent of an arc is a line touching it at one extremity, and limited by the radius produced through its other extremity. 5. The secant of an arc is the length of the line, which is intercepted between the extremity of the tangent and the centre. 6. The versed sine of an arc is that portion of the radius intercepted between the sine and the extremity of the arc. 7. The supplemental versed sine, or suversed sine, is the difference between the versed sine and the diameter. 8. The sine, tangent, secant, &c., of the complement of an arc, are concisely termed the cosine, cotangent, cosecant, &c., of that arc. These terms, for conciseness, are usually contracted into sin., tan., sec., vers., suvers., cos., cot., cosec., covers., and F that of AOB; BE is the sine of AB; AE its versed sine; DE its suversed of BC, sine; also BG, CH, and OH are the sine, tangent, and secant or the cosine, cotangent, and cosecant of AB. COROLLARIES FROM THE DEFINITIONS. 1. The sine of a quadrant, or of a right angle, is equal to the radius. 2. The tangent of half a right angle is equal to the radius. For if angle AOF were half a right angle, so would AFO (I. 32), and therefore AF would be equal to AO. 3. The sine, tangent, &c., of an arc are equal in length to those of its supplement. For BE is the sine of DCB; AF is the tangent of AKL, and therefore of the equal arc DCB; and OF is its secant. 4. The radius is a mean proportional between the tangent and cotangent of an arc. For FAO, OCH are similar triangles, and therefore AF: AO = OC: CH. If the arc AB be called A, and the radius OA be called R, then tan. A: R=R: cot. A, or R2 tan. A cot. A. 5. The radius is a mean proportional between the sine and cosecant of an arc. |