## 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 30

Cor . point F which is the common Vertex of the triangles ; that is , together with

angles , are equal to twice as many right angles as the figure has fides . COR . 2.

Cor . point 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**rightangles , are equal to twice as many right angles as the figure has fides . COR . 2.

Page 50

1. rallelogram CO ; therefore also BN is equal to GR . and the

CK , GR , RN , are therefore equal to one another , and so are quadruple of one

of them CK . again , because CB is equal d . Cor.4.2 . to BD , and that BD is ...

1. rallelogram CO ; therefore also BN is equal to GR . and the

**four**rectangles BN ,CK , GR , RN , are therefore equal to one another , and so are quadruple of one

of them CK . again , because CB is equal d . Cor.4.2 . to BD , and that BD is ...

Page 113

See it word is used when there are

first has the same ratio to the third , which the fecond has to the fourth ; or that the

first is to the third , as the fecond to the fourth . as is shewn in the 16th Prop . of ...

See it word is used when there are

**four**proportionals , and it is infer red , that thefirst has the same ratio to the third , which the fecond has to the fourth ; or that the

first is to the third , as the fecond to the fourth . as is shewn in the 16th Prop . of ...

Page 338

... D , E are not greater than the angles A , B , E. but these last three are less than

are together less than

... D , E are not greater than the angles A , B , E. but these last three are less than

**four**right angles , as has been demonstrated , wherefore also the angles C , D , Eare together less than

**four**right angles , and every two of them are greater than ... Page 463

... the ratio of which spaces is the next thing said to be given , and from this it

follows that

rectangle BC , CD is to the square of BD ; and , by Composition ,

square ...

... the ratio of which spaces is the next thing said to be given , and from this it

follows that

**four**times the square of FE is to the square of GL , as**four**times therectangle BC , CD is to the square of BD ; and , by Composition ,

**four**times thesquare ...

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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.