The Elements of Euclid: The First Six Books and the Eleventh and Twelfth from the Text of Robert Simson

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Longmans, Green, Reader, and Dyer, 1866 - 388 pages
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Page 30 - [QED] PROP. XXVI. THEOR. 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 angles, or the sides opposite to equal
Page 7 - side of it taken together less than two right " angles, these straight lines, being continually " produced, shall at length meet upon that side " on which are the angles which are less than " two right angles." PROP. I. PROBLEM. To describe an equilateral triangle upon a given finite straight
Page 63 - [QED] PROP. VIII. THEOR. If a straight line be divided into any two parts, four times the rectangle contained by the whole line and one of the parts, together with the square of the other part, is equal to the square of the straight line, which is made up of the whole and that part.
Page 60 - THEOR. 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. Let | AB be
Page 70 - 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
Page 58 - a straight line, &íc. [QED] PROP. IV. THEOR. 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. Let | AB be
Page 116 - [QED]] PROP. XXXVI. THEOR. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to
Page 245 - ED] PROP. B. THEOR. If an 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 of the straight line which, bisects the angle. Let
Page 65 - [QE.D.] PROP. IX. THEOR. If a straight line be divided into two equal, and also into two unequal parts ; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section. Let | AB be
Page 199 - PROP. III. THEOR. If the angle of a triangle be divided into two equal angles, by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle hare to one another: and if the segments of the base have, the

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