## The First Six Books with Notes |

### From inside the book

Results 1-5 of 37

Page 7

... therefore the triangles themselves DBC and ACB , are equal , a part ( 2 ) hypoth . equal to the whole , which is

... therefore the triangles themselves DBC and ACB , are equal , a part ( 2 ) hypoth . equal to the whole , which is

**absurd**: therefore neither of the sides AB or AC is greater than the other ; they are therefore equal to one another . Page 8

... but the angle BDC has been proved to be greater than BCD , which is

... 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

... to the whole , which is

... to the whole , which is

**absurd**. 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 12

16. greater than the same angle BCE ( 2 ) which is

16. greater than the same angle BCE ( 2 ) which is

**absurd**: BA , therefore , cannot fall at the side of the obtuse angle , and therefore falls at the side of the acute angle . Fig . 33 . Cor . 2. Two perpendiculars cannot be drawn from ... Page 17

4 . also equal to DFE ( 4 ) , therefore the angle BCG is ( 4 ) hypoth . equal to BCA ( 5 ) , which is

4 . also equal to DFE ( 4 ) , therefore the angle BCG is ( 4 ) hypoth . equal to BCA ( 5 ) , which is

**absurd**: neither of the ( 5 ) ax , 1 . sides BA and DE , therefore , is greater than the other , therefore they are equal ; and also ...### 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.