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 Sperical Trigonometry |
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Page 12
... the remainder ( 3 . K H D B E F Ax . ) BG : But it has been shewn that BC is equal to BG ; wherefore AL and BC are each of them equal to BG ; and things that are equal to the same are equal to one another ; therefore 12 ELEMENTS.
... the remainder ( 3 . K H D B E F Ax . ) BG : But it has been shewn that BC is equal to BG ; wherefore AL and BC are each of them equal to BG ; and things that are equal to the same are equal to one another ; therefore 12 ELEMENTS.
Page 22
... shewn that it is not equal to AB ; therefore AC is greater than AB . B PROP . XX . THEOR . Any two sides of a triangle are together greater than the third side . Let ABC be a triangle ; any two sides of it together are greater than the ...
... shewn that it is not equal to AB ; therefore AC is greater than AB . B PROP . XX . THEOR . Any two sides of a triangle are together greater than the third side . Let ABC be a triangle ; any two sides of it together are greater than the ...
Page 23
... shewn that BA , AC are greater than BE , EC ; much more then are BA , AC great- er than BD , DC . B D A E Again , because the exterior angle of a triangle ( 16. 1. ) is greater than the interior and opposite angle , the exte- rior angle ...
... shewn that BA , AC are greater than BE , EC ; much more then are BA , AC great- er than BD , DC . B D A E Again , because the exterior angle of a triangle ( 16. 1. ) is greater than the interior and opposite angle , the exte- rior angle ...
Page 25
... shewn that it is not equal to it : therefore the angle BAC is greater than the angle EDF . B : E D 26 PROP . XXVI . THEOR . If two triangles 4 OF GEOMETRY . BOOK I. 25 DF; but the angle BAC greater than the angle ...
... shewn that it is not equal to it : therefore the angle BAC is greater than the angle EDF . B : E D 26 PROP . XXVI . THEOR . If two triangles 4 OF GEOMETRY . BOOK I. 25 DF; but the angle BAC greater than the angle ...
Page 29
... shewn that the angle AGK is equal to the angle GHF ; therefore also AGK is equal to GKD ; and they are alter- nate angles ; therefore AB is parallel C ( 27. 1. ) to CD . E PROP . XXXI . PROB . G B H -F K D To draw a straight line ...
... shewn that the angle AGK is equal to the angle GHF ; therefore also AGK is equal to GKD ; and they are alter- nate angles ; therefore AB is parallel C ( 27. 1. ) to CD . E PROP . XXXI . PROB . G B H -F K D To draw a straight line ...
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Common terms and phrases
ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC 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 magnitudes meet multiple 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 solid parallelopiped spherical angle spherical triangle square straight line BC THEOR touches the circle triangle ABC triangle DEF wherefore
Popular passages
Page 95 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...
Page 68 - THE angles in the same segment of a circle are equal to one another...
Page 23 - 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 74 - 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.
Page 78 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.
Page 9 - 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 75 - If a straight line touch a circle, and from the point of contact a...
Page 18 - AT a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle. Let AB be the given straight line, and A...
Page 134 - EQUIANGULAR parallelograms have to one another the ratio which is compounded of the ratios of their sides.
Page 136 - AB is (7. 5.) to AD, as AE to AG ; and DC to CB, as GF to FE; and also CD to DA, as FG to GA ; therefore the sides of the parallelograms ABCD, AEFG about the equal angles are proportionals; and they are therefore similar to one another (1.