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 55 II .
... 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 55 II .
Page 57
... proportional to the quan- tities themselves . B Let A and B be any two quantities ; then denote their ratio . will A If we multiply each term of this fraction by m , its value will not be changed ; and we shall have , mB B MA A whence ...
... proportional to the quan- tities themselves . B Let A and B be any two quantities ; then denote their ratio . will A If we multiply each term of this fraction by m , its value will not be changed ; and we shall have , mB B MA A whence ...
Page 58
... proportional to the quantities themselves . We have , Prop . VII . , A : B :: MA : mB . If we make m = 1 ± p in which p is " any fraction , զ q we have , Р A : B :: A ± A : B ± զ PB ; q which was to be proved . PROPOSITION X. THEOREM ...
... proportional to the quantities themselves . We have , Prop . VII . , A : B :: MA : mB . If we make m = 1 ± p in which p is " any fraction , զ q we have , Р A : B :: A ± A : B ± զ PB ; q which was to be proved . PROPOSITION X. THEOREM ...
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
ABCD AC² adjacent angles altitude apothem Applying logarithms bisects centre chord circle circumference circumscribed cone consequently convex surface cosec cosine Cotang cylinder denote diagonals diameter difference distance divided draw drawn edges equilateral feet find the area formula frustum given angle given line given point greater hence homologous hypothenuse included angle inscribed intersection isosceles less Let ABC log sin lower base lune mantissa number of sides parallel parallelogram parallelopipedon perimeter perpendicular plane angles plane MN polyedral angle polyedron principle demonstrated prism PROBLEM.-In PROPOSITION proved pyramid quadrant radii radius rectangle regular polygon right angles right-angled triangle Scholium secant segment semi-circumference similar sine slant height solution sphere spherical angle spherical polygon spherical triangle square straight line subtracting supplement Tang tangent THEOREM THEOREM.-Show tri-rectangular triangle ABC triangular prism triedral angle upper base vertex vertices volume whence
Popular passages
Page 28 - If two triangles have two sides of the one equal to two sides of the...
Page 90 - A cos 6 = cos a cos c + sin a sin c cos B cos c = cos a cos 6 + sin a sin 6 cos C Law of Cosines for Angles cos A = — cos B...
Page 18 - Things which are equal to the same thing, are equal to each other. 2. If equals are added to equals, the sums are equal. 3. If equals are subtracted from equals, the remainders are equal. 4. If equals are added to unequals, the sums are unequal.
Page 4 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.
Page 253 - For, the point A being the pole of the arc EF, the distance AE is a 'quadrant ; the point C being the pole of the arc DE, the distance CE is likewise a quadrant : hence the point E is...
Page 61 - A Circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center.
Page 94 - A'B'C', and applying the law of cosines, we have cos a' = cos b' cos c' + sin b' sin c' cos A'. Remembering the relations a' = 180° -A, b' = 180° - B, etc. (this expression becomes cos A = — cos B cos C + sin B sin C cos a.
Page 126 - If two polygons are composed of the same number of triangles, similar each to each and similarly placed, the polygons are similar.
Page 46 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 125 - THEOREM. Similar triangles are to each other as the squares of their homologous sides.