Elements of Geometry: Containing the First Six Books of Euclid, with a Supplement on the Quadrature of the Circle, and the Geometry of Solids; to which are Added, Elements of Plane and Spherical TrigonometryB. & S. Collins; W. E. Dean, printer, 1836 - 311 páginas |
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Página 106
... multiple of C that B is of D , and therefore , as has been demonstrated , A : C :: B : D and inversely ( A. 5. ) C : A :: D ; B. : PROP . C. THEOR . If the first be to the second as the third to the fourth ; and if the first be a multiple ...
... multiple of C that B is of D , and therefore , as has been demonstrated , A : C :: B : D and inversely ( A. 5. ) C : A :: D ; B. : PROP . C. THEOR . If the first be to the second as the third to the fourth ; and if the first be a multiple ...
Página 107
... multiple of A ; and therefore , as is shewn above , D is the same multiple of C , and therefore C is the same part of D that A is of B. PROP . VII . THEOR . Equal magnitudes have the same ratio to the same magnitude ; and the same has ...
... multiple of A ; and therefore , as is shewn above , D is the same multiple of C , and therefore C is the same part of D that A is of B. PROP . VII . THEOR . Equal magnitudes have the same ratio to the same magnitude ; and the same has ...
Página 108
... = nF ; and if mAnB , mE / nF . Now , mA , mE are any equi- multiples whatever of A and E ; and nB , nF any whatever of B and F ; therefore A : B ; ; E ; F ( def . 5. 5. ) . PROP . XII . THEOR . If any number of 108 ELEMENTS.
... = nF ; and if mAnB , mE / nF . Now , mA , mE are any equi- multiples whatever of A and E ; and nB , nF any whatever of B and F ; therefore A : B ; ; E ; F ( def . 5. 5. ) . PROP . XII . THEOR . If any number of 108 ELEMENTS.
Página 110
... multiples of A and B , by the numbers m and n ; and first , let mA 7nB : to each of them add mB , then mA + mB7mB + nB . But mA + mB = m ( A + B ) ( Cor . 1. 5. ) , and mB + nB = ( m + n ) B ( 2. Cor . 2 . 5. ) , therefore m ( A + B ) 7 ...
... multiples of A and B , by the numbers m and n ; and first , let mA 7nB : to each of them add mB , then mA + mB7mB + nB . But mA + mB = m ( A + B ) ( Cor . 1. 5. ) , and mB + nB = ( m + n ) B ( 2. Cor . 2 . 5. ) , therefore m ( A + B ) 7 ...
Página 116
... multiple the base HC is of the base BC , the same multiple is the triangle AHC of the triangle ABC . For the same reason , whatever the base LC is of the base CD , the same mul- tiple is the triangle ALC of the triangle ADC . But if the ...
... multiple the base HC is of the base BC , the same multiple is the triangle AHC of the triangle ABC . For the same reason , whatever the base LC is of the base CD , the same mul- tiple is the triangle ALC of the triangle ADC . But if the ...
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ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC angles equal base BC bisected centre chord circle ABC circumference cosine cylinder demonstrated diameter divided equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given straight line greater Hence hypotenuse inscribed join less Let ABC Let the straight magnitudes meet multiple opposite angle parallel parallelogram parallelopiped perpendicular polygon prism PROP proposition quadrilateral radius ratio rectangle contained rectilineal figure remaining angle right angled triangle SCHOLIUM segment semicircle shewn side BC sine solid angle solid parallelopiped spherical angle spherical triangle square straight line BC THEOR third touches the circle triangle ABC triangle DEF wherefore