## The school edition. Euclid's Elements of geometry, the first six books, by R. Potts. corrected and enlarged. corrected and improved [including portions of book 11,12]. |

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Resultat 6-10 av 22

Side 212

Take any

5th of this book , if the double of the first be greater than the double of the second

, the double of the third is greater than the double of the fourth : but if the first be ...

Take any

**equimultiples**of each of them , as the doubles of each : then , by def .5th of this book , if the double of the first be greater than the double of the second

, the double of the third is greater than the double of the fourth : but if the first be ...

Side 213

Then A shall be to B as Cis to D . AB0 - - D - - - - - - - - HTake of A and Cany

and H . Then , because A is the same multiple of B that C is of D ; ( hyp . ) and that

E is ...

Then A shall be to B as Cis to D . AB0 - - D - - - - - - - - HTake of A and Cany

**equimultiples**whatever E and F ; and of B and D any**equimultiples**whatever Gand H . Then , because A is the same multiple of B that C is of D ; ( hyp . ) and that

E is ...

Side 214

Then A and B shall each of them have the same ratio to C : and C shall have the

same ratio to each of the magnitudes A and B . A B - - - D - EFTake of A and B

any

Then A and B shall each of them have the same ratio to C : and C shall have the

same ratio to each of the magnitudes A and B . A B - - - D - EFTake of A and B

any

**equimultiples**whatever D and E , and of C any multiple whatever F . Then ... Side 215

that is , EG and FG are

FG is not less than K , and , by the construction , EF is greater than D ; therefore

the whole EG is greater than K and D together : but K together with D is equal to ...

that is , EG and FG are

**equimultiples**of AB and CB ; and since it was shewn , thatFG is not less than K , and , by the construction , EF is greater than D ; therefore

the whole EG is greater than K and D together : but K together with D is equal to ...

Side 216

and FG , EG were proved to be

greater ratio than it has to AB . ( v . def . 7 . ) Wherefore , of two unequal

magnitudes , & c . Q . E . D . DPROPOSITION IX . THEOREM . Magnitudes which

have ...

and FG , EG were proved to be

**equimultiples**of CB , AB : therefore D has to CB agreater ratio than it has to AB . ( v . def . 7 . ) Wherefore , of two unequal

magnitudes , & c . Q . E . D . DPROPOSITION IX . THEOREM . Magnitudes which

have ...

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### Vanlige uttrykk og setninger

ABCD Algebraically Apply base bisected Book chord circle circumference common construction contained definition demonstrated described diagonals diameter difference distance divided double draw drawn equal equal angles equiangular equilateral triangle equimultiples Euclid extremities fall figure formed four fourth Geometrical given circle given line given point given straight line greater half Hence inscribed intersection isosceles join length less Let ABC line drawn magnitudes manner mean meet multiple parallel parallelogram pass perpendicular plane problem produced Prop proportionals proved Q.E.D. PROPOSITION radius ratio reason rectangle rectangle contained regular remaining respectively right angles segment semicircle shew shewn sides similar solid square straight line taken tangent THEOREM third touch triangle ABC twice units vertex wherefore whole

### Populære avsnitt

Side 112 - Guido, with a burnt stick in his hand, demonstrating on the smooth paving-stones of the path, that the square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

Side 83 - 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...

Side 48 - If two triangles have two sides of the one equal to two sides of the other, each to each ; and...

Side 238 - 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...

Side 198 - A LESS magnitude is said to be a part of a greater magnitude, when the less measures the greater, that is, ' when the less is contained a certain number of times exactly in the greater.

Side 271 - SIMILAR triangles are to one another in the duplicate ratio of their homologous sides.

Side 81 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Side 115 - angle in a segment' is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment.

Side 341 - On the same base, and on the same side of it, there cannot be two triangles...

Side 24 - ... twice as many right angles as the figure has sides ; therefore all the angles of the figure together with four right angles, are equal to twice as many right angles as the figure has sides.