## The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh and Twelfth ... Also the Book of Euclid's Data, in Like Manner CorrectedWingrave and Collingwood, 1816 - 528 pages |

### From inside the book

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**join**DC ; therefore , because in the triangles DBC , ACB , DB is equal to AC , and BC common to both the two sides , DB , BC are equal to the two AC , CB each to each ; and the angle DBC is equal to the angle ACB ; therefore the base DC ... Page 13

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**join**DE , and upon it describe b an equilateral triangle DEF ; then**join**AF ; the straight line AF bisects the an- gle BAC . Because AD is equal to AE , and AF is common to the two triangles DAF , EAF ; the two sides DA , AF , are equal ... Page 18

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**join**BE and produce it to F , and make EF equal to BE ;**join**also FC , and produce AC to G. Because AE is equal to EC , and BE to EF ; AE , EB are equal to CE , EF , each to each ; and the angle 15. 1. AEB , is equal to the angle CEF ... Page 22

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**join**D4 E B / a 22.1 . DE ; and make the tri- angle AFG , the sides of which shall be equal to the three straight lines CD , DE , EC , so that CD be equal to AF , CE to AG , and DE to FG ; and because DC , CE are equal to FA , AG , each ... Page 23

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**join**EG , GF . Because AB is equal to DE , and AC to DG , the two sides BA , AC are , equal to the two ED , DG , each to each , and the angle BAC is equal to the angle EDG ; thereforethe base BC is equal to the base EG ; and because DG ...### Common terms and phrases

ABC is given ABCD AC is equal altitude angle ABC angle BAC base BC bisected Book XI centre circle ABC circumference common logarithm cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid excess fore given angle given in magnitude given in position given in species given magnitude given point given ratio given straight line gnomon greater join less Let ABC logarithm multiple opposite parallel parallelogram AC perpendicular point F polygon prism proportionals proposition Q.E.D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopipeds square of BC straight line AB straight line BC tangent THEOR third triangle ABC vertex wherefore

### Popular passages

Page 41 - 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.

Page 180 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical 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 on the straight line which bisects the angle.

Page 166 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. Let AC, CF be equiangular parallelograms having the angle BCD equal to the angle ECG ; the ratio of the parallelogram AC to the parallelogram CF is the same with the ratio which is compounded •f the ratios of their sides. DH Let BC, CG be placed in a straight line ; therefore DC and CE are also in a straight line (14.

Page 2 - A rhomboid, is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.

Page 105 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth...

Page 79 - 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 1 - A straight line is that which lies evenly between its extreme points.

Page 149 - 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 23 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Page 83 - Wherefore from the given circle ABC has been cut off the segment BAC, containing an angle equal to the given angle DQEP PROP. XXXV. THEOR. If two straight lines within a circle cut one another, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. Let the...