## 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, The Book of Euclid's Data, in Like Manner Corrected. viz. the first six books, together with the eleventh and twelfthMathew Carey, and sold by J. Conrad & Company, S. F. Bradford, Birch & Small, and Samuel Etheridge. Printed by T. & G. Palmer, 116, High-Street., 1806 - Trigonometry - 518 pages |

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Page 60

Book II . the square of GF ; but the square of EF is equal to the squares w of EG ,

GF ; therefore the square of EF is

equal h to CD ; therefore the square of EF is

...

Book II . the square of GF ; but the square of EF is equal to the squares w of EG ,

GF ; therefore the square of EF is

**double**of the square h 34. 1. GF ; and GF isequal h to CD ; therefore the square of EF is

**double**of the square of CD : but the...

Page 61

1 . fore the square of EA is

equal to FE , the square of GF is equal to the square of FE ; and therefore the

squares of GF , FE are

to ...

1 . fore the square of EA is

**double**of the square of AC : again , because GF isequal to FE , the square of GF is equal to the square of FE ; and therefore the

squares of GF , FE are

**double**of the square of EF : but the square of EG is equal ito ...

Page 85

BAC an angle at the circumference , which have the same circumference BC for

their base ; the angle A BEC is

the circle be within the angle BAC , and join AE , and produce it to P : because ...

BAC an angle at the circumference , which have the same circumference BC for

their base ; the angle A BEC is

**double**of the angle BAC . First , let E the centre ofthe circle be within the angle BAC , and join AE , and produce it to P : because ...

Page 113

FKC ; wherefore the angle BFC is

FKC ; for the same reason , the angle CFD is

...

FKC ; wherefore the angle BFC is

**double**of the angle KFC , and BKC**double**ofFKC ; for the same reason , the angle CFD is

**double**of the angle CFL , and CLD**double**of CLF : and because the circumference BC is equal to the circumference...

Page 129

Take any equimultiples of each of them , as the

5th of this book , if the

, the

...

Take any equimultiples of each of them , as the

**doubles**of each ; then , by def .5th of this book , if the

**double**of the first be greater than the**double**of the second, the

**double**of the third is greater than the**double**of the fourth ; but , if the first be...

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The Elements of Euclid, Viz: The Errors, by Which Theon, Or Others, Have ... Robert Simson,Robert Euclid No preview available - 2018 |

The Elements of Euclid, Viz: The Errors, by Which Theon, Or Others, Have ... Robert Simson,Robert Euclid No preview available - 2015 |

The Elements of Euclid: Viz. The First Six Books, Together With the Eleventh ... Robert Simson No preview available - 2017 |

### Common terms and phrases

ABCD added altitude angle ABC angle BAC base Book centre circle circle ABC circumference common cone cylinder definition demonstrated described diameter divided double draw drawn equal equal angles equiangular equimultiples excess 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 multiple opposite parallel parallelogram pass perpendicular plane prisms produced PROP proportionals proposition pyramid Q. E. D. PROP radius reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid solid angle sphere square square of AC taken THEOR third triangle ABC wherefore whole

### Popular passages

Page 30 - Any two sides of a triangle are together greater than the third side.

Page 64 - 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.

Page 30 - IF, from the ends of the 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. Let...

Page 59 - PROP. VIII. THEOR. IF a straight line be divided into any two parts, tour times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Page 28 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Page 165 - 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 19 - THE angles at the base of an isosceles triangle are equal to one another : and, if the equal sides be produced, the angles upon the other side of the base shall be equal.

Page 191 - In right angled triangles, the rectilineal figure described upon the side opposite to the right angle, is equal to the similar, and similarly described figures upon the sides containing the right angle.

Page 39 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sidef. For any rectilineal figure ABCDE can be divided into as many triangles as the figure has sides, by drawing straight lines from a point F within the figure to each of its angles.

Page 180 - Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.