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 Geometry |
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Page 32
... consequently , the op posite sides are parallel ; namely , the side AB parallel to CD , and BD pa rallel to AC ; and , therefore , the figure is a parallelogram . COR . 3. Hence , also , if the opposite angles of a quadrilateral be ...
... consequently , the op posite sides are parallel ; namely , the side AB parallel to CD , and BD pa rallel to AC ; and , therefore , the figure is a parallelogram . COR . 3. Hence , also , if the opposite angles of a quadrilateral be ...
Page 42
... consequently , the angle ACD is greater than the angle ADC ; and , since the greater side is opposite to the greater angle ( 19. 1. ) ; therefore the side AD is greater than the side AC . COR . 1. The perpendicular measures the true ...
... consequently , the angle ACD is greater than the angle ADC ; and , since the greater side is opposite to the greater angle ( 19. 1. ) ; therefore the side AD is greater than the side AC . COR . 1. The perpendicular measures the true ...
Page 43
... consequently AH = DF . Now ( by hyp . ) , we have DF AC ; and therefore , AH AC . But by the last prop- osition , the oblique line AC can- not be equal to the oblique line AH , which lies nearer to the per- pendicular AB ; therefore it ...
... consequently AH = DF . Now ( by hyp . ) , we have DF AC ; and therefore , AH AC . But by the last prop- osition , the oblique line AC can- not be equal to the oblique line AH , which lies nearer to the per- pendicular AB ; therefore it ...
Page 86
... consequently divides the circle into two equal parts ; hence the portion AEF is equal to the portion AEFB , which is absurd . COR . The arc of a circle whose chord is a diameter , is a semicircum- ference , and the included segment is a ...
... consequently divides the circle into two equal parts ; hence the portion AEF is equal to the portion AEFB , which is absurd . COR . The arc of a circle whose chord is a diameter , is a semicircum- ference , and the included segment is a ...
Page 87
... consequently lie in the circumference of the circle , whose centre is F , and radius FA . It is obvious , that besides this , no other circumference can pass through the same points ; for the centre , lying in the perpen- B F E dicular ...
... consequently lie in the circumference of the circle , whose centre is F , and radius FA . It is obvious , that besides this , no other circumference can pass through the same points ; for the centre , lying in the perpen- B F E dicular ...
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Common terms and phrases
ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC angle BCD base BC bisected centre chord circle ABC circumference cosine cylinder demonstrated described diameter divided draw equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given rectilineal given straight line greater Hence hypotenuse inscribed join less Let ABC Let the straight magnitudes meet opposite angle parallel parallelogram parallelopiped perpendicular polygon prism PROB PROP proportional proposition quadrilateral radius ratio rectangle contained rectilineal figure remaining angle right angled triangle SCHOLIUM segment semicircle shewn side BC sine solid angle spherical angle spherical triangle square straight line BC THEOR touches the circle triangle ABC triangle DEF wherefore
Popular passages
Page 49 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.
Page 9 - When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle ; and the straight line which stands on the other is called a perpendicular to it.
Page 27 - Straight lines which are parallel to the same straight line are parallel to one another. Triangles and Rectilinear Figures. The sum of the angles of a triangle is equal to two right angles.
Page 17 - The angles which one straight line makes with another upon one side of it, are either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of it, the angles CBA, ABD : these shall either be two right angles, or shall together be equal to two right angles. For...
Page 13 - UPON the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity.
Page 294 - 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 55 - In every triangle, the square on the side subtending either of the acute angles, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the...
Page 24 - ... sides be equal, each to each; and also the third angle of the one to the third angle of the other. Let ABC, DEF be two triangles which have the angles ABC, BCA equal to the angles DEF, EFD, viz.
Page 125 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Page 78 - The angle in a semicircle is a right angle ; the angle in a segment greater than a semicircle is less than a right angle ; and the angle in a segment less than a semicircle is greater than a right angle.