## The First Six Books with Notes |

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Page 7

6 . and BD are equal ( 4 ) , therefore the angles BDC and BCD are equal ( 5 ) but the angle BDC has been 18

6 . and BD are equal ( 4 ) , therefore the angles BDC and BCD are equal ( 5 ) but the angle BDC has been 18

**proved**to be greater than BCD , which is absurd . Therefore the triangles constructed upon the same E } 19 o F 110 right line ... Page 8

... but the angle BDC has been

... but the angle BDC has been

**proved**to be greater than BCD , which is absurd . Therefore triangles constructed upon the same right line cannot have their conterminous sides equal , if the vertex of one of them is within the other . Page 11

Therefore BE is not the continuation of CB ; and in the same manner it can be

Therefore BE is not the continuation of CB ; and in the same manner it can be

**proved**that no other line , except BD , is the continuation of it ; therefore BD forms with BC one continued right line . Page 46

( 6 ) , in the same manner it is

( 6 ) , in the same manner it is

**proved**that CEB is half a right angle , therefore AEB is a right angle , and because DG and EC are parallel ( 5 ) , the alternate angles GDB , ECB are equal , therefore GDB is a right angle ... Page 53

11 . and in the same manner it can be

11 . and in the same manner it can be

**proved**that no other B. 1 . point without the line CE is the centre ; therefore the ( 7 ) Cons . centre is in the line CE , and therefore is the point F. ( 8 ) Ax . 11 . Schol .### 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.