## The First Six Books with Notes |

### From inside the book

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Page 90

If a submultiple of the first be contained in the second

If a submultiple of the first be contained in the second

**oftener**than an equi - submultiple of the third is contained in the fourth ; the first is said to have a less ratio to the second than the third has to the fourth ... Page 93

First , let one of the given magnitudes BC be a multiple of A , and A is not

First , let one of the given magnitudes BC be a multiple of A , and A is not

**oftener**contained in one of them than in the other . For , if it be possible , let A be**oftener**contained in ( 1 ) Az . 1 . DE than in BC , and as often as A ... Page 94

For , if it be possible , let one of them BC be greater than the other , and let its excess be nC , take a submultiple a of A less than nC , a is

For , if it be possible , let one of them BC be greater than the other , and let its excess be nC , take a submultiple a of A less than nC , a is

**oftener**contained in BC than in Bn , but Bn is equal to DE , therefore a is con( 1 ) Prop ... Page 95

Let B be not a submultiple of EF , and , if it be possible , let A be contained in CD

Let B be not a submultiple of EF , and , if it be possible , let A be contained in CD

**oftener**than B is contained in EF ; take away B as often as possible from EF , and there shall remain nF less than B , take away A as often from CD ... Page 96

If there be given two magnitudes ( AB and CD ) , and any third can be found ( e ) which is contained in one ( AB )

If there be given two magnitudes ( AB and CD ) , and any third can be found ( e ) which is contained in one ( AB )

**oftener**than in the other ( CD ) , the latter ( CD ) is less than the former . Fig . 4 . If e be a submultiple of CD ...### What people are saying - Write a review

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### Common terms and phrases

absurd added alternate angles angle ABC applied arches base bisected centre circle circumference common Constr constructed contained contained in CD continued definition demonstrated described difference divided draw drawn equal equal angles equi-multiples equi-submultiples equiangular equilateral Euclid evident external extremities fall figure fore four magnitudes fourth given line given right line greater half Hence Hypoth inscribed internal join less line AC manner meet multiple oftener parallel parallelogram pass perpendicular placed possible PROB produced Prop proportional proposition proved radius ratio rectangle rectilineal figure remaining right angles right line ruler Schol segment side AC similar squares of AC submultiple taken tangent THEOR third triangle ABC vertex whole

### Popular passages

Page 145 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

Page 2 - 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 : 16.

Page 28 - To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 22 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 118 - A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment as the greater segment is to the less.

Page 146 - A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.

Page 168 - 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 3 - Things which are equal to the same thing are also equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be subtracted from equals, the remainders are equal. 4. Things which coincide with one another are equal to one another.

Page 25 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.

Page 3 - A rhombus is that which has all its sides equal, but its angles are not right angles.