The Elements of Euclid, the parts read in the University of Cambridge [book 1-6 and parts of book 11,12] with geometrical problems, by J.W. Colenso1846 |
From inside the book
Results 6-10 of 34
Page 14
... angle ACB by the straight line CD ( 1.9 ) : the straight line AB is bisected in the point D. Because AC is equal to CB and CD is common to the two triangles ACD , BCD , the two sides AC , CD are equal to the two BC , CD , each to each ...
... angle ACB by the straight line CD ( 1.9 ) : the straight line AB is bisected in the point D. Because AC is equal to CB and CD is common to the two triangles ACD , BCD , the two sides AC , CD are equal to the two BC , CD , each to each ...
Page 20
... angles are together less than two right angles . Produce BC to D : Then , because ACD is the ex- terior angle of the triangle ABC , ACD is greater than the interior and opposite angle ABC ( 1. 16 ) : To each of these add the angle ACB ...
... angles are together less than two right angles . Produce BC to D : Then , because ACD is the ex- terior angle of the triangle ABC , ACD is greater than the interior and opposite angle ABC ( 1. 16 ) : To each of these add the angle ACB ...
Page 21
... angle ACB : much more then is the angle ABC greater than the angle ACB . Wherefore , The greater side & c . Q. E.D. PROP . XIX . THEOR . The greater angle of every triangle is subtended by the greater side , or has the greater side ...
... angle ACB : much more then is the angle ABC greater than the angle ACB . Wherefore , The greater side & c . Q. E.D. PROP . XIX . THEOR . The greater angle of every triangle is subtended by the greater side , or has the greater side ...
Page 26
... angle BAC shall be greater than the angle EDF . A B D C E F For , if it be not greater , it must be either equal to ... ACB , equal to the angles DEF , DFE , each to each , viz . ABC to DEF , and ACB to DFE , and have also one side equal ...
... angle BAC shall be greater than the angle EDF . A B D C E F For , if it be not greater , it must be either equal to ... ACB , equal to the angles DEF , DFE , each to each , viz . ABC to DEF , and ACB to DFE , and have also one side equal ...
Page 27
... angle BAC to the third angle EDF . For , if AB be not equal to DE , one of them must be greater than the other . Let ... ACB ( Hyp . ) ; therefore also the angle GCB is equal to the angle ACB , the less to the greater- which is absurd ...
... angle BAC to the third angle EDF . For , if AB be not equal to DE , one of them must be greater than the other . Let ... ACB ( Hyp . ) ; therefore also the angle GCB is equal to the angle ACB , the less to the greater- which is absurd ...
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The Elements of Euclid, the Parts Read in the University of Cambridge [Book ... Euclides No preview available - 2016 |
Common terms and phrases
ABCD adjacent angles angle ABC angle ACB angle BAC angle BCD angle EDF angle equal base BC BC is equal centre chord circle ABC circumference cuts the circle diameter double draw equal angles equal to F equiangular equilateral triangle equimultiples exterior angle fore given circle given line given point given straight line gnomon greater ratio inscribed intersection isosceles triangle less Let ABC Let the straight lines be drawn lines drawn meet multiple opposite angles opposite sides parallel to BC parallelogram pentagon perpendicular plane polygon PROB produced proportionals Q.E.D. PROP rectangle contained rectilineal figure remaining angle right angles right-angled triangle segment semicircle shew shewn square of AC straight line &c straight line AB THEOR touches the circle triangle ABC twice the rectangle Wherefore
Popular passages
Page 42 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.
Page 4 - Let it be granted that a straight line may be drawn from any one point to any other point.
Page 33 - F, which is the common vertex of the triangles: that is », together with four right angles. Therefore all the angles of the figure, together with four right angles are equal to twice as many right angles as the figure has sides.
Page 62 - 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 22 - If from the ends of a side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.
Page 58 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line.
Page 146 - ... may be demonstrated from what has been said of the pentagon : and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon.
Page 194 - 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 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another : 16.
Page 69 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.