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 17
... coincide in part , without coinciding altogether . " IV . A superficies is that which has only length and breadth . " COR . The extremities of a superficies are lines ; and the intersec- " tions of one superficies with another are also ...
... coincide in part , without coinciding altogether . " IV . A superficies is that which has only length and breadth . " COR . The extremities of a superficies are lines ; and the intersec- " tions of one superficies with another are also ...
Page 21
... coincide with one another , that is , which exactly fill the same space , are equal to one another . IX . The whole is greater than its part . X. All right angles are equal to one another . XI . " Two straight lines which intersect one ...
... coincide with one another , that is , which exactly fill the same space , are equal to one another . IX . The whole is greater than its part . X. All right angles are equal to one another . XI . " Two straight lines which intersect one ...
Page 24
... coincide with the point E , because AB is equal to DE ; and AB coinciding with DE , AC shall coincide with DF , because the angle BAC is equal to the angle EDF ; wherefore also the point C shall coincide with the point F , because AC is ...
... coincide with the point E , because AB is equal to DE ; and AB coinciding with DE , AC shall coincide with DF , because the angle BAC is equal to the angle EDF ; wherefore also the point C shall coincide with the point F , because AC is ...
Page 27
... coincide with the point F , because BC is equal to EF : therefore BC coinciding with EF , BA and AC shall coincide with ED , and DF ; for , if BA , and CA do not coincide with ED , and FD , but have a different situation as EG and FG ...
... coincide with the point F , because BC is equal to EF : therefore BC coinciding with EF , BA and AC shall coincide with ED , and DF ; for , if BA , and CA do not coincide with ED , and FD , but have a different situation as EG and FG ...
Page 40
... coinciding with one an- other , which is impossible C- ( 11. Ax . ) . The angles AGH , GHD therefore are not un- equal , that is , they are equal to one another . Now , the angle EGB is equal to AGH ( 15. 1. ) ; and AGH is proved to be ...
... coinciding with one an- other , which is impossible C- ( 11. Ax . ) . The angles AGH , GHD therefore are not un- equal , that is , they are equal to one another . Now , the angle EGB is equal to AGH ( 15. 1. ) ; and AGH is proved to be ...
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Common terms and phrases
ABC is equal ABCD 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 magnitudes meet 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 shewn side BC sine solid angle solid parallelepipeds spherical angle spherical triangle square straight line AC THEOR third touches the circle triangle ABC triangle DEF wherefore
Popular passages
Page 56 - 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 19 - 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.
Page 33 - THE greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it.
Page 62 - In every triangle, the square on the side subtending either of the acute angles, is less than the squares on the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the...
Page 62 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle, is greater than the squares of 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...
Page 130 - 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 76 - THE diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote ; and the greater is nearer to the centre than the less.* Let ABCD be a circle, of which...
Page 36 - IF two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle shall be greater than the base of the other.
Page 18 - 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.
Page 55 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.