## Elements of Geometry and Trigonometry: From the Works of A.M. Legendre |

### From inside the book

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Page 28

... B4 G A E D GI + IC > GC , and BI + IA > AB ;

... B4 G A E D GI + IC > GC , and BI + IA > AB ;

**whence**, by addition , recollecting that the sum of BI and IC is equal to BC , and the sum of GI and IA , to GA , we have , AG + BC > AB + GC . Or , since AG = AB , and GC = 28 GEOMETRY . Page 52

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**whence**, A = a ; clearing of fractions , we have , BC = AD ;、 which was to be proved . Cor . If B is equal to C , there will be but three pro- portional quantities ; in this case , the square of the mean is equal to the product of the ... Page 53

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**whence**, = A C ; B and , A : B B : F : G ;**whence**, = A G F • From Axiom 1 , we have , D G = F ;**whence**, C D :: F : G ; which was to be proved . Cor . If the antecedents , in two proportions , are the same the consequents will be ... Page 54

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**whence**, B D = · A If we take the reciprocals of both members ( A. 7 ) , we have , A C = ;**whence**, B : A B Ꭰ which was to be proved . • • D : C ; PROPOSITION VI . THEOREM . If four quantities are in proportion , they will be in pro ... Page 55

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**whence**, = Α 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 nD n C ;**whence**, mA : mB :: nC : nD ; MA = which was to be proved . PROPOSITION IX . THEOREM . If two ...### Other editions - View all

### Common terms and phrases

ABē ABCD ACē adjacent angles altitude 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 difference distance divided draw drawn edges equally distant feet find the area Formula frustum given angle given straight line greater hence homologous hypothenuse included angle interior angles intersection less Let ABC log sin lower base mantissa measured by half number of sides opposite parallel parallelogram parallelopipedon perimeter perpendicular plane MN polyedral angle polyedron prism PROPOSITION proved pyramid quadrant radii radius rectangle regular polygons right angles right-angled triangle Scholium secant segment semi-circumference side BC similar sine slant height sphere spherical polygon spherical triangle square subtracted Tang tangent THEOREM triangle ABC triangular prisms triedral angle upper base vertex vertices whence

### Popular passages

Page 126 - The square of the hypothenuse is equal to the sum of the squares of the other two sides ; as, 5033 402+302.

Page 59 - 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 18 - The circumference of every circle is supposed to be divided into 360 equal parts called degrees, and each degree into 60 equal parts called minutes, and each minute into 60 equal parts called seconds, and these into thirds, fourths, &c.

Page 104 - The square described on the hypothenuse of a rightangled triangle is equal to the sum of the squares described on the other two sides.

Page 6 - The logarithm of any power of a number is equal to the logarithm of the number multiplied by the exponent of the power.

Page 28 - If two triangles have two sides of the one equal to two sides of the...

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 99 - The area of a parallelogram is equal to the product of its base and altitude.

Page 172 - If two planes are perpendicular to 'each other, a straight line drawn in one of them, perpendicular to their intersection, will be perpendicular to the other.

Page 214 - A sphere is a solid bounded by a curved surface, every point of which is equally distant from a point within called the center.