The Elements of Euclid for the Use of Schools and Colleges: Comprising the First Six Books and Portions of the Eleventh and Twelfth Books : with Notes, an Appendix, and ExercisesMacmillan, 1867 - 400 páginas |
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Página xv
... shews that the problem has been solved , or that tho theorem is true . Sometimes , however , no construction is required ; and sometimes the construction and demonstration are com- bined . 1 The demonstration is a process of reasoning ...
... shews that the problem has been solved , or that tho theorem is true . Sometimes , however , no construction is required ; and sometimes the construction and demonstration are com- bined . 1 The demonstration is a process of reasoning ...
Página 255
... shew that a theorem is true rather than to shew why it is true . Euclid uses the reductio ad absur- dum chiefly when he is demonstrating the converse of some former theorem ; see I. 14 , 19 , 25 , 40 . Some remarks on indirect ...
... shew that a theorem is true rather than to shew why it is true . Euclid uses the reductio ad absur- dum chiefly when he is demonstrating the converse of some former theorem ; see I. 14 , 19 , 25 , 40 . Some remarks on indirect ...
Página 260
... shew it in the following manner . Let H denote the point of intersection of DF and EG . Then , the angle DHG is greater than the angle DEG , by I. 16 ; the angle DEG is not less than the angle DGE , by I. 19 ; therefore the angle DHG is ...
... shew it in the following manner . Let H denote the point of intersection of DF and EG . Then , the angle DHG is greater than the angle DEG , by I. 16 ; the angle DEG is not less than the angle DGE , by I. 19 ; therefore the angle DHG is ...
Página 263
... shew that if AB and EF are each of them parallel to CD , they are parallel to each other . It has been said that the case considered in the text is so obvious as to need no demonstration ; for if AB and CD can never meet EF , which lies ...
... shew that if AB and EF are each of them parallel to CD , they are parallel to each other . It has been said that the case considered in the text is so obvious as to need no demonstration ; for if AB and CD can never meet EF , which lies ...
Página 265
... prevent misconception . Cresswell , in his Treatise of Geometry , has given a demon . stration of I. 35 which shews that the parallelograms may be divided into pairs of pieces admitting of superposition and coin- EUCLID'S ELEMENTS . 265.
... prevent misconception . Cresswell , in his Treatise of Geometry , has given a demon . stration of I. 35 which shews that the parallelograms may be divided into pairs of pieces admitting of superposition and coin- EUCLID'S ELEMENTS . 265.
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Términos y frases comunes
ABCD angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC BC is equal bisects the angle centre chord circle ABC circle described circumference Construction Corollary describe a circle diameter double draw a straight equal angles equal to F equiangular equilateral equimultiples Euclid Euclid's Elements exterior angle given circle given point given straight line gnomon Hypothesis inscribed intersect isosceles triangle less Let ABC magnitudes middle point multiple opposite angles opposite sides parallelogram perpendicular plane polygon produced proportionals PROPOSITION 13 Q.E.D. PROPOSITION quadrilateral radius rectangle contained rectilineal figure remaining angle rhombus right angles right-angled triangle segment shew shewn side BC square on AC straight line &c straight line drawn tangent THEOREM touches the circle triangle ABC triangle DEF twice the rectangle vertex Wherefore
Pasajes populares
Página 286 - 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.
Página 30 - 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...
Página 34 - IF a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite upon the same side; and likewise the two interior angles upon the same side together equal to two right angles...
Página 37 - If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles of every triangle are together equal to two right angles.
Página 12 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.
Página 302 - Describe a circle which shall pass through a given point and touch a given straight line and a given circle.
Página 220 - If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Página 354 - Prove that the square on any straight line drawn from the vertex of an isosceles triangle to the base, is less than the square on a side of the triangle by the rectangle contained by the segments of the base : and conversely.
Página 104 - 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.
Página 96 - The opposite angles of any quadrilateral figure inscribed in a circle, are together equal to two right angles.