Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes .... the first six books |
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Page 14
... draw a straight line from the point C at right angles to AB . F " D C E B In AC take any point D , and make CE equal to CD ; ( I. 3. ) upon DE describe the equilateral triangle DEF ( 1. 1 ) , and join CF. Then CF ' drawn from the point ...
... draw a straight line from the point C at right angles to AB . F " D C E B In AC take any point D , and make CE equal to CD ; ( I. 3. ) upon DE describe the equilateral triangle DEF ( 1. 1 ) , and join CF. Then CF ' drawn from the point ...
Page 15
... drawn from the given point C , shall be perpendicular to the given straight line AB . Join FC , and CG . Because FH is equal to HG , ( constr . ) and HC is common to the triangles FHC , GHC ; the two sides FH , HC , are equal to the two ...
... drawn from the given point C , shall be perpendicular to the given straight line AB . Join FC , and CG . Because FH is equal to HG , ( constr . ) and HC is common to the triangles FHC , GHC ; the two sides FH , HC , are equal to the two ...
Page 35
... drawn from A but AD is parallel to BC ; AD is therefore parallel to BC . Wherefore , equal triangles upon , & c . Q. E.D. PROPOSITION XL . THEOREM . Equal triangles upon equal bases in the same straight line , and towards the same parts ...
... drawn from A but AD is parallel to BC ; AD is therefore parallel to BC . Wherefore , equal triangles upon , & c . Q. E.D. PROPOSITION XL . THEOREM . Equal triangles upon equal bases in the same straight line , and towards the same parts ...
Page 43
... drawn on a plane surface , the words toou may mean , that no part of the line which is called a straight line deviates either from one side or the other of the direction which is fixed by the extremities of the line ; and thus it may be ...
... drawn on a plane surface , the words toou may mean , that no part of the line which is called a straight line deviates either from one side or the other of the direction which is fixed by the extremities of the line ; and thus it may be ...
Page 53
... drawn from the point to the line . From this Prop . it follows that only one perpendicular can be drawn from a given point to a given line ; and this perpendicular may be shewn to be less than any other line which can be drawn from the ...
... drawn from the point to the line . From this Prop . it follows that only one perpendicular can be drawn from a given point to a given line ; and this perpendicular may be shewn to be less than any other line which can be drawn from the ...
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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...