## 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 |

### From inside the book

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**angles**contained by those sides equal to one another , their bases ...**angle**BAC be also equal to the**angle**EDF : then shall he base BC be equal to ...**ACB**to DFE . For , if the triangle ABC be applied to the triangle DEF , so ... Page 14

... angles of the one shall coincide with the remaining angles of the other , and be equal to them , viz . the angle ABC to the angle DEF , and the

... angles of the one shall coincide with the remaining angles of the other , and be equal to them , viz . the angle ABC to the angle DEF , and the

**angle ACB**to the angle DFE . Therefore , if two triangles have two sides of the one equal to ... Page 15

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**angle**FBC is equal to the**angle**GCB , which are the**angles**upon the other ...**ACB**; the side AB is also equal to the side AC . For , if AB be not equal to ...**ACB**, DB is equal to AC , and BC common to both , the two sides DB ... Page 16

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**ACB**; produce AC , AD to E , F ; therefore , because AC is equal to AD in the triangle ACD , the**angles**ECD , FDC upon the other side of the base CD are equal ( 5. 1. ) to one another , but the**angle**ECD is greater than the**angle**BCD ... Page 17

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**angle**BAC coincides with the**angle**EDF , and is equal ( 8. Ax . ) to it . PROP . IX . PROB . To bisect a given ...**ACB**by the straight line CD . AB is cut into two equal parts in the point D. Because AC is equal to CB , and CD common to the ...### Other editions - View all

### Common terms and phrases

ABC is equal ABCD adjacent angles altitude angle ABC angle ACB angle BAC angle BCD base BC bisected centre chord circle ABC circumference cosine cylinder demonstrated described diameter divided draw equal angles equiangular equilateral equilateral polygon equimultiples Euclid exterior angle fore four right angles given rectilineal given straight line greater Hence hypotenuse inscribed join less Let ABC Let the straight line BC magnitudes meet opposite angle parallel parallelogram parallelopiped perpendicular polygon prism PROB produced PROP proportional proposition radius ratio rectangle contained rectilineal figure remaining angle right angled triangle SCHOLIUM segment semicircle shewn side BC sine solid angle solid parallelopiped spherical angle spherical triangle square straight line AC tangent THEOR touches the circle triangle ABC triangle DEF wherefore

### Popular passages

Page 51 - 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 81 - IF a straight line touch a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.

Page 14 - The angles at the base of an Isosceles triangle are equal to one another ; and if the equal sides be produced, the angles upon the other side of the base shall be equal.

Page 19 - The angles which one straight line makes with another upon one side uf it, are, either two right angles, or are together equal to two right angles. Let the straight line AB make with CD, upon one side of it the angles CBA, ABD ; these are either two right angles, or are together equal to two right angles. For, if the angle CBA be equal to ABD, each of them is a right angle (Def.

Page 52 - If a straight line be divided into any two parts, the squares on the whole line, and on one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square on the other part. Let the straight line AB be divided into any two parts in the point C. Then the squares on AB, BC shall be equal to twice the rectangle AB, BC, together with the square on A C.

Page 147 - If the vertical angle of a triangle be bisected by a straight line which also cute 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 on the straight line which bisects the angle.

Page 242 - 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.

Page 9 - 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 119 - Reciprocal figures, viz. triangles and parallelograms, " are such as have their sides about two of their " angles proportionals in such a manner, that a side

Page 72 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...