Elements of Geometry and Trigonometry: From the Works of A. M. Legendre |
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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 antecedent 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 antecedent 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 the quan- tities themselves . Let A and B be any two quantities ; then denote their ratio . B will A If we multiply both terms of this fraction by m , its value will not be changed ; and we shall have , mB B = ; MA Α ...
... proportional to the quan- tities themselves . Let A and B be any two quantities ; then denote their ratio . B will A If we multiply both terms of this fraction by m , its value will not be changed ; and we shall have , mB B = ; MA Α ...
Page 56
... proportional to the quantities themselves . We have , Prop . VII . , A : B :: mA : mB . If we make m = 1 ± p > 9 in which P is any fraction , we shall have , A : B :: A ± 2oA : B ± p -B ; q q which was to be proved . PROPOSITION X ...
... proportional to the quantities themselves . We have , Prop . VII . , A : B :: mA : mB . If we make m = 1 ± p > 9 in which P is any fraction , we shall have , A : B :: A ± 2oA : B ± p -B ; q q which was to be proved . PROPOSITION X ...
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
AB² ABCD altitude apothem Applying logarithms centre chord circle circumference cone consequently convex surface cosec Cosine Cotang cylinder demonstrated in Book denote diameter distance divided draw edges Equation feet find the area Find the logarithmic following RULE frustum given angle greater hence homologous hypothenuse included angle inscribed intersection less Let ABC linear units log cot log sin lower base lune mantissa multiplied number of sides opposite parallel parallelogram parallelopipedon perpendicular plane MN polar triangle polyedral angle polyedron principle demonstrated prism proportional PROPOSITION proved pyramid quadrant radii radius rectangle regular polygon right angles right-angled triangle Scholium segment similar six right slant height solution sphere spherical angle spherical excess spherical polygon spherical triangle square straight line subtracting Tang tangent THEOREM triangle ABC triangular prism upper base vertex volume whence write the following