## 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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Side 20

13, 1. angle ABC would be less than the angle ACB; but it is not; therefore the

side AC is not less than AB; and it has been shown that it is not equal to AB;

therefore AC is

xx.

13, 1. angle ABC would be less than the angle ACB; but it is not; therefore the

side AC is not less than AB; and it has been shown that it is not equal to AB;

therefore AC is

**greater**than A.B. Wherefore the**greater**angle, &c., Q.E.D. PROP.xx.

Side 127

THAT magnitude which has a

same magnitude, is the

has a

THAT magnitude which has a

**greater**ratio than see N. another has unto thesame magnitude, is the

**greater**of the two: And that magnitude to which the samehas a

**greater**ratio than it has unto another magnitude, is the lesser of the two. Side 130

See N. B: Then, because A is to B, as C to D, and of A and C, M and G are

equimultiples: And of B and D, N and Kare equimultiples; if M be

G is

See N. B: Then, because A is to B, as C to D, and of A and C, M and G are

equimultiples: And of B and D, N and Kare equimultiples; if M be

**greater**than N,G is

**greater**than K; and if equal, equal; and if less, less"; but G is**greater**than K; ... Side 285

is

GHK is

because it is contained within it, which is impossible: Therefore the sphere ABC

has ...

is

**greater**than the solid polyhedron in it: thereforee also Book XII. the sphereGHK is

**greater**than the solid polyhedron in the . sphere DEF: But it is also less,because it is contained within it, which is impossible: Therefore the sphere ABC

has ...

Side 326

... because, as was said, the conclusion would have been immediately deduced

without this superfluous step by permutation: This we have shown at the

length, both because it affords a certain proof of the vitiation of the text of Euclid; ...

... because, as was said, the conclusion would have been immediately deduced

without this superfluous step by permutation: This we have shown at the

**greater**length, both because it affords a certain proof of the vitiation of the text of Euclid; ...

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ABC is given ABCD AC is equal altitude angle ABC angle BAC arch base BC bisected Book XI centre circle ABC circumference common logarithm cone cosine cylinder demonstration described diameter 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 Greek text hypothenuse join less Let ABC logarithm multiple opposite parallel parallelogram AC perpendicular polygon prisms proportionals proposition Q.E.D. PROP radius rectangle CB rectangle contained rectilineal figure right angles segment side BC similar sine solid angle solid parallelopipeds spherical angle spherical triangle square of BC straight line BC tangent THEOR third triangle ABC vertex wherefore

### Populære avsnitt

Side 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.

Side 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.

Side 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.

Side 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.

Side 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...

Side 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.

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

Side 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.

Side 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.

Side 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...