## 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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Page 111

V. The first of four magnitudes is said to have the same ratio to the second , which the third has to the fourth , when any

V. 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 ... Page 112

When of the

When of the

**equimultiples**of four magnitudes ( taken as in the fifth definition ) , the multiple of the first is greater than that of the second , but the multiple of the third is not greater than the multiple of the fourth ; then the ... Page 114

I.

I.

**EQUIMULTIPLES**of the same , or of equal magnitudes , are equal to one another . Book V. II . Those magnitudes of which the same. * 4 Prop . lib . 2. Archimedis de sphæra et cylindro . 114 THE ELEMENTS. Page 115

Those magnitudes of which the same , or equal magnitudes , are

Those magnitudes of which the same , or equal magnitudes , are

**equimultiples**, are equal to one another . Iti . A multiple of a greater magnitude is greater than the same multiple of a less . IV . That magnitude of which a multiple is ... Page 117

If the first be the same multiple of the second , which the third is of the fourth ; and if of the first and third there be taken

If the first be the same multiple of the second , which the third is of the fourth ; and if of the first and third there be taken

**equimultiples**, these shall be**equimultiples**, the one of the second , and the other of the fourth .### 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...