## The Elements of Euclid, Viz: The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected; and Some of Euclid's Demonstrations are Restored. Also the Book of Euclid's Data, in Like Manner Corrected. the first six books, together with the eleventh and twelfth |

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Page 13

produced ever so far both ways , do not meet . POST U L A T E S. 1 L I. ET it be

granted that a straight line may be drawn from any one point to any other point . II

.

**Parallel**straight lines , are such as are in the same plane , and which , beingproduced ever so far both ways , do not meet . POST U L A T E S. 1 L I. ET it be

granted that a straight line may be drawn from any one point to any other point . II

.

Page 34

For , if it be not

D or towards A , C ; let them be produced and meet towards B , D in the point G ;

therefore GEF is a triangle , and its exterior angle AEF is greater than the interior

...

For , if it be not

**parallel**, AB and CD being produced fhall meet either towards B ,D or towards A , C ; let them be produced and meet towards B , D in the point G ;

therefore GEF is a triangle , and its exterior angle AEF is greater than the interior

...

Page 35

angle EGB equal to the angle AGH , the angle AGH is equal Book T. to the angle

GHD ; and they are the alternate angles ; therefore AB is

because the angles BGH , GHD a 15. 1 . are equal to two right angles , and that ...

angle EGB equal to the angle AGH , the angle AGH is equal Book T. to the angle

GHD ; and they are the alternate angles ; therefore AB is

**parallel**to CD . Again ,because the angles BGH , GHD a 15. 1 . are equal to two right angles , and that ...

Page 36

Traight lines which are

another . Let AB , CD be each of them

Let the straight line GHK cut AB , EF , CD ; and because GHK cuts the

straight ...

Traight lines which are

**parallel**to the same straight line are**parallel**to oneanother . Let AB , CD be each of them

**parallel**to EF ; AB is also**parallel**to CD .Let the straight line GHK cut AB , EF , CD ; and because GHK cuts the

**parallel**straight ...

Page 37

EAF is drawn through the given point A

BC . Which was to be done . PRO P : XXXII . THE O R. IF a side of any triangle be

produced , the exterior angle is equal to the two interior and opposite angles ...

EAF is drawn through the given point A

**parallel**to the given Book I. straight lineBC . Which was to be done . PRO P : XXXII . THE O R. IF a side of any triangle be

produced , the exterior angle is equal to the two interior and opposite angles ...

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### Common terms and phrases

added alſo altitude angle ABC angle BAC baſe 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 magnitude given ratio given ſtraight line greater Greek half join leſs likewiſe magnitude manner meet muſt oppoſite parallel parallelogram perpendicular plane produced prop proportionals propoſition pyramid radius rectangle rectangle contained remaining right angles ſame ſecond ſegment ſhall ſides ſimilar ſolid ſphere ſquare ſquare of AC Take taken theſe third triangle ABC wherefore whole

### Popular passages

Page 32 - 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 165 - D ; wherefore the remaining angle at C is equal to the remaining angle at F ; Therefore the triangle ABC is equiangular to the triangle DEF.

Page 170 - 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 10 - When several angles are at one point B, any ' one of them is expressed by three letters, of which ' the letter that is at the vertex of the angle, that is, at ' the point in which the straight lines that contain the ' angle meet one another, is put between the other two ' letters, and one of these two is...

Page 55 - 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 32 - ... then shall the other sides be equal, each to each; and also the third angle of the one to the third angle of the other. Let ABC, DEF be two triangles which have the angles ABC, BCA equal to the angles DEF, EFD, viz.

Page 45 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Page 211 - AB shall be at right angles to the plane CK. Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK ; take any point F in CE, from which draw FG in the plane DE at right D angles to CE ; and because AB is , perpendicular to the plane CK, therefore it is also perpendicular to every straight line in that plane meeting it (3.

Page 38 - 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 304 - Thus, if B be the extremity of the line AB, or the common extremity of the two lines AB, KB, this extremity is called a point, and has no length : For if it have any, this length must either be part of the length of the line AB, or of the line KB.