Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes .... the first six books |
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Page 8
... equal to BC . Because the point B is the center of the circle CGH , therefore BC is equal to BG ; ( def . 15. ) and because D is the center of the circle GKL , therefore DL is equal to DG , and DA , DB parts of them are equal ; ( 1. 1 ...
... equal to BC . Because the point B is the center of the circle CGH , therefore BC is equal to BG ; ( def . 15. ) and because D is the center of the circle GKL , therefore DL is equal to DG , and DA , DB parts of them are equal ; ( 1. 1 ...
Page 9
... equal sides are opposite shall be equal , each to each , viz . the angle ABC to the angle DEF , and the angle ACB to the angle DFE . D B C E F For , if the triangle ABC be applied to the triangle DEF , so that the point A may be on D ...
... equal sides are opposite shall be equal , each to each , viz . the angle ABC to the angle DEF , and the angle ACB to the angle DFE . D B C E F For , if the triangle ABC be applied to the triangle DEF , so that the point A may be on D ...
Page 10
... equal to the triangle AGB , also the remaining angles of the one are equal to the remaining angles of the other , each to each , to which the equal sides are opposite ; viz . the angle ACF to the angle ABG , and the angle AFC to the ...
... equal to the triangle AGB , also the remaining angles of the one are equal to the remaining angles of the other , each to each , to which the equal sides are opposite ; viz . the angle ACF to the angle ABG , and the angle AFC to the ...
Page 12
... equal to BD in the triangle BCD , therefore the angle BDC is equal to the angle BCD ( 1. 5. ) but the angle BDC has been proved greater than BCD , wherefore the angle BDC is both equal to , and greater than the angle BCD ; which is ...
... equal to BD in the triangle BCD , therefore the angle BDC is equal to the angle BCD ( 1. 5. ) but the angle BDC has been proved greater than BCD , wherefore the angle BDC is both equal to , and greater than the angle BCD ; which is ...
Page 13
... equal angles . Let BAC be the given rectilineal angle . It is required to bisect it . D A A B F In AB take any point D ; from AC cut off AE equal to AD , ( 1. 3. ) and join DE , on the side of DE remote from A , describe the equilateral ...
... equal angles . Let BAC be the given rectilineal angle . It is required to bisect it . D A A B F In AB take any point D ; from AC cut off AE equal to AD , ( 1. 3. ) and join DE , on the side of DE remote from A , describe the equilateral ...
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Common terms and phrases
A₁ ABCD AC is equal Algebraically angle ABC angle ACB angle BAC Apply Euc axiom base BC chord circle ABC constr describe a circle diagonals diameter divided double draw equal angles equiangular equilateral triangle equimultiples Euclid Euclid's Elements exterior angle Geometrical given circle given line given point given straight line given triangle gnomon greater hypotenuse inscribed intersection isosceles triangle less Let ABC line AC lines be drawn meet the circumference multiple opposite angles parallelogram pentagon perpendicular porism problem produced Prop proportionals proved Q.E.D. PROPOSITION quadrilateral radius ratio rectangle contained rectilineal figure remaining angle right angles right-angled triangle segment semicircle shew shewn similar similar triangles square on AC tangent THEOREM touch the circle trapezium triangle ABC twice the rectangle vertex vertical angle wherefore
Popular passages
Page 54 - If two triangles have two sides of the one equal to two sides of the...
Page 89 - If a straight line be bisected, and produced to any point ; the rectangle contained by the whole line thus produced, and the part of it produced, together with the square...
Page 38 - Let ABCD be the given rectilineal figure, and E the given rectilineal angle. It is required to describe a parallelogram equal to ABCD, and having an angle equal to E.
Page 144 - 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.
Page 18 - Any two angles of a triangle are together less than two right angles.
Page 266 - 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 152 - 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 the square of the line which touches it.
Page 7 - From a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line; it is required to draw from the point A a straight line equal to BC.
Page 3 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Page 96 - 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...