8. If a common tangent be drawn to any number of circles which touch each other internally, and, from any point in this tangent as a centre, a circle be described cutting the others, and from this centre lines be drawn through the intersections of the circles respectively; the segments of them within each circle will be equal. 9. If from any point without a circle lines be drawn touching it ; the angle contained by the tangents is double the angle contained by the line joining the points of contact, and the diameter drawn through one of them. 10. If from any two points in the circumference of a circle there be drawn two straight lines to a point in a tangent to that circle, they will make the greatest angle when drawn to the point of contact. 11. From a given point within a given circle to draw a straight line which shall make with the circumference an angle less than the angle made by any other line drawn from that point. 12. If one chord in a circle bisect another, and tangents drawn from the extremities of each be produced to meet, the line joining their points of intersection will be parallel to the bisected chord. 13. If two circles cut each other, the greatest line that can be drawn through the point of intersection, is that which is parallel to the line joining their centres. 14. If from any point within an equilateral triangle perpendiculars be drawn to the sides, they are, together, equal to a perpendicular drawn from any of the angles to the opposite side. 15. If the points of bisection of the sides of a given triangle be joined, the triangle, so formed, will be one-fourth of the given triangle. 16. The difference of the angles at the base of any triangle, is double the angle contained by a line drawn from the vertex perpendicular to the base, and another bisecting the angle at the vertex. 17. If the tangents drawn to every two, of three unequal circles, be produced till they meet, the points of intersection will be in a straight line. 18. If the three sides of a triangle be bisected, the perpendiculars drawn to the sides, at the three points of bisection, will meet in the same point. 19. If from the three angles of a triangle lines be drawn to the points of bisection of the opposite sides, these lines intersect each other in the same point. 20. The three straight lines which bisect the three angles of a triangle meet in the same point. 21. The two triangles, formed by drawing straight lines from any point within a parallelogram to the extremities of two opposite sides, are together, half the parallelogram. 22. The figure formed by joining the points of bisection of the 23. If from the three angles of a triangle lines be drawn to the points of bisection of the opposite sides, the squares of the distances, between the angles and the common intersection, are, together, one-third of the squares of the sides of the triangle. 24. If squares be described on the three sides of a right-angled triangle, and the extremities of the adjacent sides be joined; the triangles so formed, are equal to the given triangle, and to each other. 25. If squares be described on the hypothenuse and sides of a rightangled triangle, and the extremities of the sides of the former and the adjacent sides of the others be joined, the sum of the squares of the lines joining them, will be equal to five times the square of the hypothenuse. 26. The vertical angle of an oblique-angled triangle inscribed in a circle, is greater or less than a right angle, by the angle contained between the base, and the diameter drawn from the extremity of the base. 27. If the base of any triangle be bisected by the diameter of its circumscribing circle, and, from the extremity of that diameter, a perpendicular be let fall upon the longer side, it will divide that side into segments, one of which will be equal to half the sum, and the other to half the difference of the sides. 28. A straight line drawn from the vertex of an equilateral triangle, inscribed in a circle to any point in the opposite circumference, is equal to the two lines together, which are drawn from the extremities of the base to the same point. 29. The straight line bisecting any angle of a triangle inscribed in a given circle, cuts the circumference in a point, which is equidistant from the extremities of the sides opposite to the bisected angle, and from the centre of a circle inscribed in the triangle. 30. In any triangle, if perpendiculars be drawn from the angles to the opposite sides, they will all meet in a point. 31. If from the three angles of any triangle three straight lines be drawn, to the points where the inscribed circle touches the sides, these lines shall intersect each other in the same point. 32. If from the centre of a circle a line be drawn to any point in the chord of an arc, the square of that line, together with the rectangle contained by the segments of the chord, will be equal to the square described on the radius. 33. If two straight lines in a circle cut each other at right angles, the sums of the squares of the two lines joining their extremities will be equal. 34. If two points be taken in the diameter of a circle, equidistant from the centre, the sum of the squares of the two lines drawn from these points to any point in the circumference, will be always the same. 35. If in the diameter of a circle two points be taken at equal distances from the centre, the sum of the squares of two lines drawn from these points, to any point in the circumference of the circle, is a constant quantity. 36. If on the diameter of a semicircle, two equal circles be described, and in the space included by the three circumferences á circle be inscribed, its diameter will be half the diameter of either of the equal circles. ELEMENTARY PRINCIPLES OF PLANE TRIGONOMETRY. PLANE TRIGONOMETRY is that branch of Mathematics by which, if certain parts of a plane triangle be given, the other parts may be computed. As the sides and angles of triangles are quantities of different kinds, they cannot be compared with each other; but the relation between the sides and the magnitude of the angles may be discovered by comparing the sides with certain lines drawn in and about a circle, on which lines the arcs of the circle which measure the angles of the triangles depend. DEFINITIONS. 1. If two lines meet in the centre of a circle, the arc of the circumference intercepted between them is called the measure of the angle which they contain. 2. If the circumference of a circle be divided into 360 equal parts, each of these parts is called a degree ; if a degree be divided into 60 equal parts, each of these parts is called a minute; and if a minute be divided into 60 equal parts, each of these parts is called a second, &c. And whatever number of degrees, minutes, seconds, &c. are contained in any arc, the angle at the centre, which that arc measures, contains the same number of degrees, minutes, seconds, &c. Cor. 1. Any arc is to the whole circumference of which it is a part as the number of degrees, &c. contained in the arc, is to 360 degrees; and any angle is to four right angles in the same proportion. Cor. 2. Whatever be the radii with which the arcs are described which measure any angle, the arcs contain the same number of degrees, &c. For the arcs are like parts of 360 degrees. marks : °,!,", "I, &c. Thus 37° 14' 20' 38'' is 37 degrees, 14 minutes, 20 seconds, and 38 thirds. Cor. 4. Two arcs, whose sum is equal to a semicircle, or two angles, whose sum is equal to two right angles, are called supplements of each other. Cor. 5. The difference between an arc and a quadrant, or between an angle and a right angle, is called the complement of the arc or the angle. Cor. 6. The sine of an arc is a perpendicular let fall from one of its extremities upon the diameter which passes through the other extremity. Cor. 7. The versed sine of an arc is that portion of the diameter intercepted between the sine and the circumference. Cor. 8. The suversed sine of an arc is the versed sine of its supplement. Cor. 9. The tangent of an arc is a perpendicular to the diameter at one extremity of the arc, and terminated by the diameter produced, which passes through the other extremity. Cor. 10. The secant of an arc is the straight line drawn from the centre to the termination of the tangent. Remark. The sine, tangent, secant, &c. of the complement of an arc, are usually termed the cosine, cotangent, and cosecant of that arc. To illustrate the above definitions, let A E be the diameter of a circle AIEK; CI a perpendicular ra I dius from the centre C; AG and IH perpendiculars to AE and CI, meeting F G CH drawn from the centre in G and H. E A с D From B, where CH meets the circumference, draw B D and B F perpendicular to AC and CI. Then the arcs AB K к and BI E, and the angles A C B and BC E are supplements of each other. The arcs A B and BI, the arcs EIB and BI, the angles ACB and BCI, and the angles EC B and BCI, are respectively complements of each other. B D is the sine of the arc A B, or of the arc EIB; and it is also considered as the sine of either of the supplemental angles, ACB or BCE. A G is the tangent, and C G the secant of the arcs or the angles of which В D is the sine. AD is the versed sine of the arc A B, and E D its suversed sine. BF or C D is the sine of I B, or the cosine of AB; IH is the tangent, and C H the secant of I B; or I H is the cotangent, and CH the cosecant of A B. IF is the versed sine of I B, or the coversed sine of A B. And in all cases the sine, cosine, tangent, &c. of any arc, is called also the sine, cosine, tangent, &c. of the angle measured by that arc. For the sake of brevity, these technical terms in trigonometry are usually contracted as follows: H Thus for Sine of an arc, as AB, is put Sin. A B. do. Cos. A B. do. Tan. A B. Cot. A B. do. Sec. A B. do. Cosec. A B. Vers. A B. Suvers. A B. Covers. A B. From the above definitions we deduce the following obvious consequences : viz. 1st, When an arc is evanescent, or becomes nothing, its sine, tangent, and versed sine, are evanescent also ; and its secant and cosine are each equal to the radius. 2d, The sine and the versed sine of a quadrant are each equal to the radius, its cosine is evanescent, and its secant and tangent are infinite. 3d, The chord of an arc is twice the sine of half the arc. 4th, The radius B C, the sine BD, and the cosine C D, form a right-angled triangle C BD. The secant CG, the tangent G A, and the radius CA, form also a right-angled triangle CGA, similar to CBD; and the cosecant H C, the radius C I, and the cotangent I H, form another right-angled similar to CBD or CGA. Hence C D + D B? = C B? ; or cos? + sin? = rad?. CA? + A Go = CG'; or rad? + tano = sec?. CI? + IH2 = CH”; or rad? + cot? = cosec?. And CD: DB::CA : A G; or cos : sin :: rad : tan; or tan = rad. sin sin ; or tan = if rad be unity. COS > COS rad. cos COS COS CF : FB ::CI:IH; or sin : cos : : rad : cot; or cot = COS ; or cot = if rad be unity. sin sin CD:CB::CA:CG; or cos : rad : : rad : sec; or sec = rad 1 į or sec = if rad be unity. CF: CB ::CI:CH; or sin : rad :: rad : cosec ; or cosec rado 1 ; or cosec = if rad be unity. sin sin' GA:AC ::CI:IH; or tan : rad :: rad : cot; or tan, cot = rado rad ; ; or tan = sin cot 1 |