## Euclid's Elements of Geometry: Chiefly from the Text of Dr. Simson, with Explanatory Notes .... the first six books |

### From inside the book

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**one equal to two angles of the other , each to each , and**one side equal to one side , viz . either the sides adja- cent to the equal angles in each , or the sides opposite to them ; then shall the other sides be equal , each to each ... Page 178

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**one equal to two angles of the other , each to each ; and**the side BD , which is opposite to one of the equal angles in each , is common to both ; therefore their other sides are equal ; ( 1. 26. ) wherefore DE is equal to DF : for the ... Page 185

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**one equal to two angles of the other , each to each ; and**the side FC which is adjacent to the equal angles in each , is com- mon to both ; therefore the other sides are equal to the other sides , and the third angle to the third angle ... Page 187

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**one equal to two angles of the other , each to each ; and**the side FC , which is opposite to one of the equal angles in each , is common to both ; therefore the other sides are equal , each to each ; ( 1. 26. ) wherefore the ...### Other editions - View all

### Common terms and phrases

A₁ ABCD Algebraically angle BAC Apply base base BC bisected Book chord circle circumference common construction definition demonstrated described diagonals diameter difference divided double draw drawn equal equal angles equiangular equilateral triangle equimultiples Euclid exterior angle extremities fall figure four fourth Geometrical given circle given line given point given straight line greater half Hence inscribed intersection join less Let ABC line drawn magnitudes manner mean meet multiple opposite sides parallel parallelogram pass perpendicular problem produced Prop proportionals PROPOSITION proved radius ratio reason rectangle rectangle contained remaining respectively right angles segment shew shewn sides similar square straight line taken tangent term THEOREM third triangle ABC twice units wherefore whole

### 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...