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 the theorem is true . Sometimes , however , no construction is required ; and sometimes the construction and demonstration are com- bined . The demonstration is a process of reasoning in ...
... shews that the problem has been solved , or that the theorem is true . Sometimes , however , no construction is required ; and sometimes the construction and demonstration are com- bined . The demonstration is a process of reasoning in ...
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 AC is equal angle ABC angle ACB angle BAC angle EDF angles equal Axiom base BC 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 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 35 - 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 equal to two right angles.
Página 67 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle, is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle, Let ABC be an obtuse-angled triangle, having the obtuse angle...
Página 73 - 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.
Página 284 - 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 50 - In any right-angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle.
Página 57 - 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 on the line between the points of section, is equal to the square on half the line.
Página 10 - 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 227 - If two straight lines be at right angles to the same plane, they shall be parallel to one another. Let the straight lines AB, CD be at right angles to the same plane : AB is parallel to CD.
Página 102 - 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 352 - 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.