69. The base, the angle at the vertex, and the sum of the sides given, to construct the triangle. When is the Problem impossible? 70. The base, the angle at the vertex, and the difference of the sides given, to construct the triangle. 71. On a given base to construct a triangle equivalent to a given triangle. 72. With a given altitude to construct a triangle equivalent to a given triangle. 73. Two sides of a triangle and the perpendicular to one of them from the opposite vertex given, to construct the triangle. 74. Two of the perpendiculars from the vertices to the opposite sides and a side given, to construct the triangle. 1st. When one of the perpendiculars falls on the given side. 75. An angle and two of the perpendiculars from the vertices to the opposite sides given, to construct the triangle. 1st. When one of the perpendiculars falls from the vertex of the given angle. 2d. When neither of the perpendiculars falls from the vertex of the given angle. 76. An angle and the segments of the opposite side made by a perpendicular from the vertex given, to construct the triangle. 77. Given an angle, the opposite side, and the line from the given vertex to the middle of the given side, to construct the triangle. When is the Problem impossible ? 78. An angle, a perpendicular from another angle to the opposite side, and the radius of the circumscribed circle given, to construct the triangle. When is the Problem impossible? 79. To divide a triangle into two parts in a given ratio, 80. To trisect a triangle by straight lines drawn from a point within to the vertices. 81. Parallel to the base of a triangle to draw a line equal to the sum of the lower segments of the two sides. 82. Parallel to the base of a triangle to draw a line equal to the difference of the lower segments of the two sides. 83. To inscribe in a given triangle a quadrilateral similar to a given quadrilateral. 84. To divide a given line so that the sum of the squares of the parts shall be equivalent to a given square. 85. To construct a parallelogram when there are given, 1st. Two adjacent sides and a diagonal. 2d. A side and two diagonals. 3d. The two diagonals and the angle between them. 86. To construct a square when the diagonal is given. 87. To construct a parallelogram equivalent to a given triangle and having a given angle. 88. To draw a quadrilateral, the order and magnitude of all the sides and one angle given. Show that sometimes there may be two different polygons satisfying the conditions. 89. To draw a quadrilateral, the order and magnitude of three sides and two angles given. 1st. The given angles included by the given sides. 2d. The two angles adjacent, and one adjacent to the unknown side. 3d. The two angles being opposite each other. 4th. The two angles being both adjacent to the unknown side. In any of these cases can more than one quadrilateral be drawn? 90. To draw a quadrilateral, the order and magnitude of two sides and three angles given. 1st. The given sides being adjacent. 2d. The given sides not being adjacent. 91. In a given circle to inscribe a triangle similar to a given triangle. 92. Through a given point to draw to a given circle a secant such that the part within the circle may be equal to a given line. 93. With a given radius to draw a circumference, 1st. Through two given points. 2d. Through a given point and tangent to a given line. 3d. Through a given point and tangent to a given circumference. 4th. Tangent to two given straight lines. 5th. Tangent to a given straight line and to a given circumference. 6th. Tangent to two given circumferences. State in each of these cases how many circles can be drawn, and when the construction is impossible. 94. To draw a circumference, 1st. Through two given points and with its centre in a given line. 2d. Through a given point and tangent to a given line at a given point. 3d. Tangent to a given line at a given point, and also tangent to a second given line. 4th. Tangent to three given lines. 5th. Through two given points and tangent to a given line. 6th. Through a given point and tangent to two given lines. 95. To draw a tangent to two circumferences. There can be drawn, 1st. When the circles are external to each other, four tangents. 2d. When the circles touch externally, three. 3d. When the circles cut, two. 4th. When the circles touch internally, one. 5th. When one circle is within the other, none. PLANE TRIGONOMETRY. CHAPTER I. PRELIMINARY. LOGARITHMS. 1. Logarithms are exponents of the powers of some number which is taken as a base. In the tables of Logarithms in common use, the number 10 is taken as the base, and all numbers are considered as powers of 10. And, since 10° 1, that is, since the Logarithm of 1 is 0, the Logarithm of any number between 1 and 10 is between 0 and 1, that is, is a fraction; the Logarithm of any number between 10 and 100 is between 1 and 2, that is, is 1 plus a fraction; and the Logarithm of any number between 100 and 1000 is 2 plus a fraction; and so on. And, as 10-10.1, that is, since the Logarithm of 0.1 is -1, the Logarithm of any number between 1 and 0.1 is between 0 and -1, that is, is -1 plus a fraction; the Logarithm of any number between 0.1 and 0.01 is between —1 and -2, that is, is -2 plus a fraction; and so on. The Logarithms of most numbers, therefore, consist of an integer, either positive or negative, and a fraction, which is always positive. The representation of the Logarithms of all numbers less than a unit by a negative integer and a positive fraction is merely a matter of convenience. 2. The integral part of a Logarithm is called the characteristic, and is not generally written in the tables, but can be found by the following RULE. The characteristic of the Logarithm of any number is equal to the number of places by which its first significant figure on the left is removed from units' place, the characteristic being positive when this figure is to the left and negative when it is to the right of units' place. E. g. The Logarithm of 59 is 1 plus a fraction; that is, the characteristic of the Logarithm of 59 is 1. The Logarithm of 5417.7 is 3 plus a fraction; that is, the characteristic of the Logarithm of 5417.7 is 3. The Logarithm of 0.3 is -1 plus a fraction; that is, the characteristic of the Logarithm of 0.3 is -1. The Logarithm of 0.00017 is -4 plus a fraction; that is, the characteristic of the Logarithm of 0.00017 is -4. 3. Since the base of this system of Logarithms is 10, if any number is multiplied by 10, its Logarithm will be increased by a unit; if divided by 10, diminished by a unit. |