## The first six books of the Elements of Euclid, with numerous exercises |

### From inside the book

Results 1-5 of 23

Page 89

...

...

**greater**contains the less a certain number of times exactly . III .**Ratio**is a mutual relation of two magnitudes of the same kind to one another , in respect of quantity . IV . Magnitudes are said to have a**ratio**to one another , when ... Page 94

...

...

**greater**than m , 1 is**greater**than n and if equal , equal ; if less , less ( v . def . 5 ) . And k , 1 are any ...**ratio**to the second , which the third has to the fourth , then also any equimultiples whatever of the first and third have the ... Page 95

...

...

**ratio**which the third has to the fourth ; then , if the first be**greater**than the second , the third is also**greater**than the fourth ; and if equal , equal ; if less , less . TAKE any equimultiples of each of them , as the doubles of ... Page 98

...

...

**greater**than fe is**greater**than f ; and if equal , equal ; if less , less . And d , e are any equimultiples of a , b , and f is any multiple of c . Therefore ( v . def . 5 ) , as a is to c , so is b to c . Likewise c has the same**ratio**... Page 99

... greater than fg ; that is , fg is not less than k . And since ef is the same multiple of ac that fg is of cb ; fg is ... ratio than bc has to d . Also d has to be a

... greater than fg ; that is , fg is not less than k . And since ef is the same multiple of ac that fg is of cb ; fg is ... ratio than bc has to d . Also d has to be a

**greater ratio**than it has to ab . For , having made the same ...### Other editions - View all

### Common terms and phrases

a b c abcd adjacent angles angle a cb angle abc angle bac angle edf angle equal base bc bc is equal bisected centre circle abc describe diameter double draw equal angles equal straight lines equal to f equimultiples ex æquali exterior angle fore four magnitudes fourth given circle given straight line gnomon greater ratio greater than f ILLUSTRATED LONDON inscribe less LET abc LET the straight likewise multiple opposite angle parallel parallelogram pentagon perpendicular Q. E. D. PROPOSITION rectangle a d rectangle contained remaining angle right angles segment shewn square of a c square of eg straight line a b straight line ab straight line bc tiple touches the circle triangle abc triangle def twice the rectangle

### Popular passages

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 : XVI.

Page 42 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.

Page 4 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 21 - 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 29 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 38 - If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together with the square of...

Page 15 - If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than, the other two sides of the triangle, but shall contain a greater angle. Let...

Page 13 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Page 4 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...

Page 126 - 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.