## The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected and Some of Euclid's Demonstrations are Restored. Also, to this Second Edition is Added the Book of Euclid's Data. In Like Manner Corrected. viz. the first six books, together with the eleventh and twelfth |

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Page 51

Let the straight line AB be divided at the point C into two equal , and at D into two

unequal parts . the squares of AD , DB are toģether double of the

CD . From the point C draw a CE at right angles to AB , and make it a . is equal ...

Let the straight line AB be divided at the point C into two equal , and at D into two

unequal parts . the squares of AD , DB are toģether double of the

**squares of AC**,CD . From the point C draw a CE at right angles to AB , and make it a . is equal ...

Page 52

1. square of EF is double of the square of CD , but the square of AE is likewise

double of the

1. square of EF is double of the square of CD , but the square of AE is likewise

double of the

**square of AC**; therefore the squares of G AE , EF are double of the**squares of AC**, CD . and the square of g . 47.1 . AF is equal to the squares of А ... Page 53

47. s to the squares of EC , CA ; therefore the square of EA is double of the

the square of FE ; and therefore the squares of GF , FE are double of the square

of EF ...

47. s to the squares of EC , CA ; therefore the square of EA is double of the

**square of AC**. again , because GF is equal to FE , the square of GF is equal tothe square of FE ; and therefore the squares of GF , FE are double of the square

of EF ...

Page 56

1. dicular · AD from the opposite angle . the

, is less than the squares of CB , BA by twice the rectangle CB , BD . First , Let AD

fall within the triangle ABC ; and because the straight line CB is divided into two ...

1. dicular · AD from the opposite angle . the

**square of AC**opposite to the angle B, is less than the squares of CB , BA by twice the rectangle CB , BD . First , Let AD

fall within the triangle ABC ; and because the straight line CB is divided into two ...

Page 440

than GH , GL , that is than the

AF is less than the

the

A ...

than GH , GL , that is than the

**square of AC**; wherefore twice the rectangle CA ,AF is less than the

**square of AC**, and the rectangle CA , AF itself less than halfthe

**square of AC**, that is than the rectangle contained by the diameter AC and itsA ...

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### Common terms and phrases

added alſo altitude angle ABC angle BAC baſe BC is given becauſe Book Book XI caſe circle circle ABCD circumference common cone cylinder Definition demonſtrated deſcribed diameter divided double draw drawn equal equal angles equiangular equimultiples exceſs fame fides firſt folid fore four fourth given angle given in poſition given in ſpecies given magnitude given ratio given ſtraight line greater Greek half join leſs likewiſe magnitude manner meet muſt oppoſite parallel parallelogram perpendicular plane priſm produced PROP proportionals Propoſition pyramid reaſon rectangle remaining right angles ſame ſecond ſegment ſhall ſhewn ſides ſimilar ſolid ſquare ſquare of AC taken THEOR theſe third thro triangle ABC wherefore whole

### Popular passages

Page 5 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 163 - 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 48 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.

Page 73 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity between that straight line and the circumference, so as not to cut the circle...

Page 105 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...

Page 3 - 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.

Page 167 - Similar triangles are to one another in the duplicate ratio of their homologous sides.

Page 54 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.

Page 47 - 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 37 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.