Elements of Geometry and Trigonometry |
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Page 51
... proportional to the other three . When the second term is equal to the third , it is said to be a mean proportional between the extremes . In this case , there are but three different quantities in the proportion , and the last is said ...
... proportional to the other three . When the second term is equal to the third , it is said to be a mean proportional between the extremes . In this case , there are but three different quantities in the proportion , and the last is said ...
Page 53
... same the consequents will be proportional . For , the antecedents of the second couplets may be made the consequents of the first , by alternation ( P. III . ) . PROPOSITION V. THEOREM . If four quantities are in proportion BOOK II . 53.
... same the consequents will be proportional . For , the antecedents of the second couplets may be made the consequents of the first , by alternation ( P. III . ) . PROPOSITION V. THEOREM . If four quantities are in proportion BOOK II . 53.
Page 55
... proportional to any equimultiples of the second couplet . Assume the proportion , B D A B :: CD ; whence , = A C If we multiply both terms of the first member by m , and both terms of the second member by n , we shall have , mB MA = nD ...
... proportional to any equimultiples of the second couplet . Assume the proportion , B D A B :: CD ; whence , = A C If we multiply both terms of the first member by m , and both terms of the second member by n , we shall have , mB MA = nD ...
Page 56
... proportional to the quantities themselves . We have , Prop . VII . , A : B :: mA : mB . If we make m = ± in which p is any fraction , 9 9 we shall have , Ρ A B :: : A ± A : BB ; 9 which was to be proved . PROPOSITION X. THEOREM . If ...
... proportional to the quantities themselves . We have , Prop . VII . , A : B :: mA : mB . If we make m = ± in which p is any fraction , 9 9 we shall have , Ρ A B :: : A ± A : BB ; 9 which was to be proved . PROPOSITION X. THEOREM . If ...
Page 58
... proportional , their squares will be proportional . Cor . 2. If the principle of the proposition be extended to three or more proportions , and the corresponding terms of each be supposed equal , it will follow that , like powers of ...
... proportional , their squares will be proportional . Cor . 2. If the principle of the proposition be extended to three or more proportions , and the corresponding terms of each be supposed equal , it will follow that , like powers of ...
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Common terms and phrases
ABCD AC² adjacent angles altitude angle ACB apothem Applying logarithms base and altitude bisect centre chord circle circumference circumscribed coincide cone consequently convex surface corresponding cosec cosine Cotang cylinder denote diagonals diameter distance divided draw drawn edges equally distant feet find the area Formula frustum given angle given line given point greater hence homologous hypothenuse included angle interior angles intersection less Let ABC log sin logarithm lower base mantissa mean proportional measured by half middle point number of sides opposite parallel parallelogram parallelopipedon perimeter perpendicular plane MN polyedral angle polyedron prism PROBLEM PROPOSITION proved pyramid quadrant radii radius rectangle regular polygons right angles right-angled triangle Scholium secant segment side BC similar sine slant height sphere spherical polygon spherical triangle square Tang tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence