(Use § 174, 5, on the first equation before clearing it of fractions.) Ratio plays a very important part in science, though the ratio idea is often disguised to such an extent by the scientific notation that the pupil thinks in other terms than those of ratio or measurement. For example, the mysteries of Specific Gravity disappear when one feels that the specific gravity is simply the ratio of a volume of some substance to an equal volume of some substance taken as a standard. The standard for liquids and solids is water. One cubic centimeter (c.c.) of water weighs 1 gram, or 1 cu. ft. weighs 621 pounds. For gases the standard is usually hydrogen; sometimes air, which is 14.44 times as heavy as hydrogen, is used. Ex. A cubic foot of steel weighs 490 lb. Find specific gravity of steel. Specific gravity of steel = 490 62.5 = 7.84. It is customary to write specific gravity in a decimal form, not as a common fraction. 1 liter = 1000 cubic centimeters (c.c.). 1 kilogram = 1000 grams. 1 c.c. water weighs 1 gram. 1 liter hydrogen weighs 0.09 gram. Specific gravity air (hydrogen standard) is 14.44. 22. Ice weighs 57.5 pounds to the cubic foot. Find its specific gravity. 23. The specific gravity of oak is 0.8. Find the weight of 1 cubic foot. 24. A cubic foot of lead weighs 706 pounds. Find its specific gravity. 25. A cubic foot of copper weighs 550 pounds. Find its specific gravity. 26. The specific gravity of aluminum is 2.6. Steel is how many times as heavy ? LIST OF CONSTRUCTIONS 63. To draw a straight line equal to a given straight line PAGE 49 67. To construct an angle equal to a given angle 72. To bisect a given angle 53 80. To construct a triangle when three sides are given 92. To draw a perpendicular bisector of a straight line 64 93. To draw a perpendicular to a line from any point in the line 65 96. To draw a perpendicular to a line from a given point with out the line : 67 119. To draw a straight line parallel to a given straight line 97 175 LIST OF THEOREMS Lines THEOREM VII 91. If two straight lines intersect, the vertical angles are equal THEOREM VIII 94. If a perpendicular is erected at the middle point of a line, I. Any point in the perpendicular is equidistant from the extremities of the line II. Any point not in the perpendicular is unequally distant from the extremities of the line THEOREM ІХ 97. From a point without a line but one perpendicular can be drawn to the line THEOREM XII 100. If two unequal oblique lines drawn from a point in a perpendicular to the line, cut off unequal distances from the foot of the perpendicular, the more remote is the greater THEOREM ХІІІ 101. If oblique lines are drawn from a point to a straight line and a perpendicular is drawn from the point to the line, I. Two equal oblique lines cut off equal distances from the foot of the perpendicular II. The greater of two unequal oblique lines cuts off the greater distance from the foot of the perpendicular THEOREM XIV PAGE 64 65 67 70 71 120. Two lines parallel to the same line are parallel to each other 97 THEOREM XV 121. A line perpendicular to one of two parallels is perpendicu lar to the other 98 THEOREM XVI 124. If two parallel lines are cut by a transversal, the alternate interior angles are equal PAGE 100 THEOREM XVII 125. If two lines are cut by a transversal, and the alternateinterior angles are equal, the lines are parallel 101 THEOREM XXI 131. Any point in the bisector of an angle is equidistant from the sides of the angle THEOREM XXVI 137. If a series of parallels intercept equal parts on one transver sal, they intercept equal parts on every transversal THEOREM XXVII 138. The line joining the middle points of two sides of a triangle is parallel to the third side and equal to one half of it THEOREM ХХVIII 139. The line joining the middle points of the non-parallel sides of a trapezoid is parallel to the bases and equal to one half of their sum Lines which Meet in a Point THEOREM ХХІХ 140. The bisectors of two of the angles of a triangle intersect on the bisector of the third angle THEOREM ХХХ 106 111 112 113 113 • 142. The perpendiculars erected at the middle points of the sides of a triangle meet in a point which lies in the perpendicular bisector of the third side 115 THEOREM ХХХІ 144. The three altitudes of a triangle meet in a common point THEOREM XXXII 145. Two medians of a triangle meet in a point of the third median 116 116 Triangles THEOREM I 70. Two triangles are equal when two sides and the included angle of one are equal respectively to two sides and the included angle of the other THEOREM ІІ 71. Two triangles are equal when a side and two adjacent angles of the one are equal respectively to a side and two adjacent angles of the other THEOREM ІІІ 78. In an isosceles triangle the angles opposite the equal sides are equal. THEOREM IV 81. Two triangles are equal when three sides of one are equal respectively to three sides of the other THEOREM V 89. Any side of a triangle is greater than the difference of the other two sides THEOREM VI 90. The sum of two sides of a triangle is greater than the sum of two lines drawn from any point within the triangle to the extremities of the third side of the triangle THEOREM Х 98. Two right triangles are equal when the hypotenuse and an acute angle of the one are equal, respectively, to the hypotenuse and acute angle of the other PAGE 51 52 55 57 62 63 68 THEOREM ХІ 99. Two right triangles are equal when the hypotenuse and leg of the one are equal, respectively, to the hypotenuse and leg of the other 69 THEOREM XVIII 126. The sum of the angles of a triangle is equal to two right angles 103 |