## The Elements of Euclid, the parts read in the University of Cambridge [book 1-6 and parts of book 11,12] with geometrical problems, by J.W. Colenso |

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Page 6

**Wherefore**CA , AB , BC are equal to one another ; and the triangle ABC is therefore equilateral , and it is described upon the given straight line AB , Q. E , F , PROP . II . PROB . From a given point to draw a straight line equal to a ... Page 7

... 3 ) : But it has been shewn that BC is equal to BG ; therefore AL and BC are each of them equal to BG : And things which are equal to the same are equal to one another ; therefore the straight line AL is equal to BC .

... 3 ) : But it has been shewn that BC is equal to BG ; therefore AL and BC are each of them equal to BG : And things which are equal to the same are equal to one another ; therefore the straight line AL is equal to BC .

**Wherefore**from ... Page 8

whence AE and C are each of them equal to AD :

whence AE and C are each of them equal to AD :

**Wherefore**AE is equal to C , and from AB , the greater of two given straight lines , a part AE has been cut off equal to C the less . Q.E.F. PROP . IV , THEOREM . If two triangles have two ... Page 12

... the angle BCD is equal to the angle BDC : but it has been proved to be greater than it — which is impossible . The case , in which the vertex of one triangle is upon a side of the other , needs no demonstration .

... the angle BCD is equal to the angle BDC : but it has been proved to be greater than it — which is impossible . The case , in which the vertex of one triangle is upon a side of the other , needs no demonstration .

**Wherefore**... Page 15

10 ) ; therefore each of the angles DCF , ECF is a right angle :

10 ) ; therefore each of the angles DCF , ECF is a right angle :

**Wherefore**, from the given point C , in the given straight line AB , CF has been drawn at right angles to AB . Cor . By the help of this problem , it may be demonstrated ...### What people are saying - Write a review

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The Elements of Euclid, the Parts Read in the University of Cambridge [Book ... Euclides No preview available - 2016 |

### Common terms and phrases

ABCD angle ABC angle ACB angle BAC base base BC BC is equal bisected centre chord circle circle ABC circumference common described diameter difference divided double draw drawn equal equal angles equiangular equimultiples extremities fall figure fore four fourth given circle given line given point given straight line greater half inscribed intersection join less Let ABC lines be drawn lines drawn magnitudes manner meet multiple opposite sides parallel parallelogram pass perpendicular plane polygon PROB produced PROP proportionals Q.E.D. PROP rectangle rectangle contained rectilineal figure right angles segment semicircle shew shewn sides similar square square of AC straight lines &c Take taken THEOR third touches the circle triangle ABC Wherefore whole

### Popular passages

Page 42 - 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 4 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 33 - F, which is the common vertex of the triangles: that is », together with four right angles. 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.

Page 62 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

Page 22 - If from the ends of a 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.

Page 58 - If there be two straight lines, one of which is divided into any number of parts, the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line.

Page 146 - ... may be demonstrated from what has been said of the pentagon : and likewise a circle may be inscribed in a given equilateral and equiangular hexagon, and circumscribed about it, by a method like to that used for the pentagon.

Page 194 - 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.

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 69 - 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.