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 Trigonometry |
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Page 27
... interior , and opposite an- gles . Let ABC be a triangle , and let its side BC be produced to D , the exterior angle ACD is greater than either of the interior opposite angles CBA , BAĆ . Bisect ( 10. 1. ) AC in E , join BE and produce ...
... interior , and opposite an- gles . Let ABC be a triangle , and let its side BC be produced to D , the exterior angle ACD is greater than either of the interior opposite angles CBA , BAĆ . Bisect ( 10. 1. ) AC in E , join BE and produce ...
Page 28
... interior and opposite angle ABC ; to each of these add the angle ACB ; therefore the angles ACD , ACB are greater than the angles ABC , B ACB ; but ACD , ACB are to- A C D gether equal ( 13. 1. ) to two right angles ; therefore the ...
... interior and opposite angle ABC ; to each of these add the angle ACB ; therefore the angles ACD , ACB are greater than the angles ABC , B ACB ; but ACD , ACB are to- A C D gether equal ( 13. 1. ) to two right angles ; therefore the ...
Page 30
... interior and opposite angle , the exterior angle BDC of the triangle CDE is greater than CED ; for the same reason , the exterior angle CEB of the triangle ABE is greater than BAC ; and it has been demonstrated that the angle BDC is ...
... interior and opposite angle , the exterior angle BDC of the triangle CDE is greater than CED ; for the same reason , the exterior angle CEB of the triangle ABE is greater than BAC ; and it has been demonstrated that the angle BDC is ...
Page 34
... interior and opposite angle BCA : which is im- possible ( 16.1 . ) wherefore BC is not unequal to EF , that is , it is equal to it ; and AB is equal to DE ; therefore the two AB , BC , are equal to the two DE , EF , each to each ; and ...
... interior and opposite angle BCA : which is im- possible ( 16.1 . ) wherefore BC is not unequal to EF , that is , it is equal to it ; and AB is equal to DE ; therefore the two AB , BC , are equal to the two DE , EF , each to each ; and ...
Page 35
... interior and opposite upon the same side of the line ; or makes the interior an- gles upon the same side together equal to two right an- gles ; the two straight lines are parallel to one another . E A G Let the straight line EF , which ...
... interior and opposite upon the same side of the line ; or makes the interior an- gles upon the same side together equal to two right an- gles ; the two straight lines are parallel to one another . E A G Let the straight line EF , which ...
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Common terms and phrases
ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC angle EDF arch AC base BC bisected centre circle ABC circumference cosine cylinder demonstrated diameter draw equal and similar equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given straight line greater hypotenuse inscribed join less Let ABC Let the straight line BC meet multiple opposite angle parallel parallelogram perpendicular polygon prism PROB produced proportionals proposition Q. E. D. COR Q. E. D. PROP radius ratio rectangle contained rectilineal figure remaining angle segment semicircle side BC sine solid angle solid parallelopipeds spherical angle spherical triangle straight line AC THEOR third touches the circle triangle ABC triangle DEF wherefore
Popular passages
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 39 - THE straight lines which join the extremities of two equal and parallel straight lines, towards the same parts, are also themselves equal and parallel. Let AB, CD be equal and parallel straight lines, and joined towards the same parts by the straight lines AC, BD ; AC, BD are also equal and parallel.
Page 41 - Parallelograms upon the same base and between the same parallels, are equal to one another.
Page 19 - BG; and things that are equal to the same are equal to one another; therefore the straight line AL is equal to BC. Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC.
Page 145 - If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another ; the remaining sides shall be in a straight line. Let ABC, DCE be two triangles which have the two sides BA, AC proportional to the two CD, DE, viz.
Page 30 - If, from the ends of the 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 136 - FGL, have an angle in one equal to an angle in the other, and their sides about these equal angles proportionals ; the triangle ABE is equiangular (6.
Page 51 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.
Page 20 - DEF, and be equal to it ; and the other angles of the one shall coincide with the remaining angles of the other and be equal to them, viz. the angle ABC to the angle DEF, and the angle ACB to DFE.
Page 55 - If a straight line be divided into two equal, and also into two unequal parts ; the squares on the two unequal parts are together double of the square on half the line, and of the square on the line between the points of section.