Elements of Geometry and Trigonometry from the Works of A.M. Legendre: Adapted to the Course of Mathematical Instruction in the United States |
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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
... , 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.
... , 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 the quan- tities themselves . B A Let A and B be any two quantities ; then wil ' denote their ratio . If we multiply both terms of this fraction by m , its value will not be changed ; and we shall have , mB B MA A ...
... proportional to the quan- tities themselves . B A Let A and B be any two quantities ; then wil ' denote their ratio . If we multiply both terms of this fraction by m , its value will not be changed ; and we shall have , mB B MA A ...
Page 56
... proportional to the quantities themselves . We have , Prop . VII . , A : B :: mA : mB . If we make m = 1 ± 2 , in which P is any fraction , q q we shall have , A : B :: A ± PA : B ± 2 B ; Չ 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 ± 2 , in which P is any fraction , q q we shall have , A : B :: A ± PA : B ± 2 B ; Չ 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 AC² altitude angle ACB apothem axis base and altitude base multiplied BC² bisect centre chord circumference coincide cone consequently convex surface corresponding cosec cosine Cotang cylinder denote diameter distance divided draw drawn edges equal bases equal in volume equal to AC equal to half equally distant Formula frustum given angle given line greater hence homologous hypothenuse included angle intersection less Let ABC logarithm lower base mantissa mean proportional measured by half number of sides opposite parallelogram perimeter perpendicular plane MN polyedral angle polyedron prism PROPOSITION XI proved pyramid quadrant radii radius rectangle regular polygons right-angled triangle Scholium segment semi-circumference side BC similar sine slant height sphere spherical angle spherical excess spherical polygon spherical triangle straight line tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence