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 53
... proportional to the three others . 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 three others . 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 55
... same , the consequents are proportional . For , the ante- cedents of the second couplets may be made the conse- quents of the first , by alternation ( P. III . ) . PROPOSITION V. THEOREM . If four quantities are in proportion BOOK II . 55.
... same , the consequents are proportional . For , the ante- cedents of the second couplets may be made the conse- quents of the first , by alternation ( P. III . ) . PROPOSITION V. THEOREM . If four quantities are in proportion BOOK II . 55.
Page 57
... proportional to the quan- tities themselves . B A Let A and B be any two quantities ; then will denote their ratio . If we multiply each term of this fraction by m , its value will not be changed ; and we shall have , mB B = ; whence ...
... proportional to the quan- tities themselves . B A Let A and B be any two quantities ; then will denote their ratio . If we multiply each term of this fraction by m , its value will not be changed ; and we shall have , mB B = ; whence ...
Page 58
... proportional to the quantities themselves . We have , Prop . VII . , A B mA : mB . If we make m = 1 ± we have , P p q in which is any fraction , q A : B :: A ± A ± 2 A P B + B ; q q 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 = 1 ± we have , P p q in which is any fraction , q A : B :: A ± A ± 2 A P B + B ; q q which was to be proved . PROPOSITION X. THEOREM . If ...
Page 60
... proportional , their squares are 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 are 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 AC² adjacent angles altitude angles is equal apothem base and altitude bisects centre chord circle circumference circumscribed cone consequently convex surface corresponding Cosine Cotang cylinder denote diagonals diameter distance divided draw drawn edges equally distant equilateral feet find the area formula frustum given angle given line given point greater hence homologous hypothenuse included angle interior angles intersection less Let ABC logarithm lower base mantissa mean proportional measured by half number of sides parallel parallelogram parallelopipedon perimeter perpendicular plane angles plane MN polyedral angle polyedron prism PROPOSITION proved pyramid quadrant radii radius rectangle regular polygon right angles right-angled triangle Scholium secant segment side BC similar sine slant height solution sphere spherical angle spherical polygon spherical triangle square Tang tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence