## 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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**diameter**of a circle is a straight line drawn through the centre , and terminated both ways by the circumference . XVIII . A semicircle is the figure contained by a**diameter**and the part of the circumference it cut off . xix . Page 34

The opposite sides and angles of parallelograms are equal to one another , and the

The opposite sides and angles of parallelograms are equal to one another , and the

**diameter**bisects them , that is , divides them into two equal parts . N.B. - A parallelogram is a four - sided figure , of which the opposite sides are ... Page 35

A B V C D Let ACDB be a parallelogram , of which BC is a

A B V C D Let ACDB be a parallelogram , of which BC is a

**diameter**: the opposite sides and angles of the figure are equal to one another , and the**diameter**BC bisects it . Because AB is parallel to CD , and BC meets them , the alternate ... Page 38

ABC is half of the parallelogram EBCA , because the

ABC is half of the parallelogram EBCA , because the

**diameter**AB bisects it ( 1. 34 ) ; and the triangle DBC is half of the parallelogram DBCF , because the**diameter**DC bisects it : But the halves of equal things are equal ( Ax . 7 ) ... Page 40

Join AC : Then the triangle ABC is equal to the triangle EBC , because they are upon the same base BC , and between the same parallels BC , AE ( 1. 37 ) : But the parallelogram ABCD is double of the triangle B ABC , because the

Join AC : Then the triangle ABC is equal to the triangle EBC , because they are upon the same base BC , and between the same parallels BC , AE ( 1. 37 ) : But the parallelogram ABCD is double of the triangle B ABC , because the

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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.