## 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 Corrected |

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Now this Proposition is a Theorem , not a Definition ; because the equality of figures of any kind must be

Now this Proposition is a Theorem , not a Definition ; because the equality of figures of any kind must be

**demonstrated**, and not assumed , and l ; therefore , though this were a true proposition , it ought to have been**demonstrated**. Page 14

Cor : By help of this problem , it may be

Cor : By help of this problem , it may be

**demonstrated**, that two straight lines cannot have a common segment . If it be possible , let the two straight lines ABC , ABD have the segment AB common to both of them . Page 16

... EBA , add to these equals the angle ABC , therefore the angles DBA , ABC are equal to the three angles DBE , EBA , ABC : but the angles CBE , EBD have been

... EBA , add to these equals the angle ABC , therefore the angles DBA , ABC are equal to the three angles DBE , EBA , ABC : but the angles CBE , EBD have been

**demonstrated**to be equal to the same three angles ; and d 1 Ax ... Page 17

And , in like manner , it may be

And , in like manner , it may be

**demonstrated**, that no other can be in the same straight line with it but BD , which therefore is in the same straight line with CB . Wherefore , if at a point , & c . Q. E. D. PROP . XV . THEOR . Page 18

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**demonstrated**that the angle BCG , that isą , the angle ACD , is greater than the angle ABC . Therefore , if one side , & c . Q.E.D. GU PROP . XVII . THEOR . Any two angles of a triangle are together less than two right angles .### What people are saying - Write a review

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### Common terms and phrases

ABCD added altitude angle ABC angle BAC base Book centre circle circle ABC circumference common cone contained cylinder definition demonstrated described diameter difference divided double draw drawn Edition equal equiangular equimultiples excess fore four fourth given angle given in magnitude given in position given in species given magnitude given ratio given straight line greater Greek half join less likewise logarithm magnitude manner meet multiple opposite parallel parallelogram pass perpendicular plane Price produced PROP proportionals proposition proved pyramid radius reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid solid angle space sphere square square of BC Take taken THEOR third triangle ABC wherefore whole

### 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...