## 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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Side 17

and be equal to it ; and the other angles of the one thall coincide with the

remaining angles of the other , and be equal to them , viz . the angle iBC to the

angle DÉF ...

**Wherefore**the whole triangle ABC shall coincide with the whole triangle DEF ,and be equal to it ; and the other angles of the one thall coincide with the

remaining angles of the other , and be equal to them , viz . the angle iBC to the

angle DÉF ...

Side 25

two right angles ; and CEA , AED have been demonstrated to be equal to two

right angles ;

.

**Wherefore**, if at a point , & c . ... E B AED , DEB , these also are together equal totwo right angles ; and CEA , AED have been demonstrated to be equal to two

right angles ;

**wherefore**the angles CEA , AED are equal to the angles AED , DEB.

Side 33

... therefore the angle GCB is equal to the angle DFE ; but DFE is , by the

hypothesis , equal to the angle BCA ;

the angle BCA , the less to the greater , which is impossible ; therefore AB is not

unequal ...

... therefore the angle GCB is equal to the angle DFE ; but DFE is , by the

hypothesis , equal to the angle BCA ;

**wherefore**also the angle BCG is equal tothe angle BCA , the less to the greater , which is impossible ; therefore AB is not

unequal ...

Side 42

31.1 . RIANGLEs upon equal bases , and between the same parallels , are equal

to one another . Let the triangles ABC , DEF be upon equal bases BC , EF ...

**Wherefore**triangles , & c . Q. E. D. C 34. I. 7 . Ax . PRO P. XXXVIII . THE O R. TE a31.1 . RIANGLEs upon equal bases , and between the same parallels , are equal

to one another . Let the triangles ABC , DEF be upon equal bases BC , EF ...

Side 43

triangles upon equal bases , in the fame straight line , and towards the fame parts

, are bea tween the same parallels . Let the equal triangles ABC , DEF be upon ...

**Wherefore**equal triangles upon , & c . Q. E. D. PRO P. XL . THEOR . E QUALtriangles upon equal bases , in the fame straight line , and towards the fame parts

, are bea tween the same parallels . Let the equal triangles ABC , DEF be upon ...

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The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago ... Robert Simson Uten tilgangsbegrensning - 1762 |

The Elements of Euclid: The Errors by which Theon, Or Others, Have Long ... Robert Simson Uten tilgangsbegrensning - 1827 |

The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago ... Robert Simson Uten tilgangsbegrensning - 1781 |

### Vanlige uttrykk og setninger

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

### Populære avsnitt

Side 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...

Side 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.

Side 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.

Side 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...

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.

Side 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.