## 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 16

Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC . Which was to be done . PROP . III . PROB . FROM the greater of two given straight lines to cut off a part equal to the

Wherefore from the given point A a straight line AL has been drawn equal to the given straight line BC . Which was to be done . PROP . III . PROB . FROM the greater of two given straight lines to cut off a part equal to the

**less**. Page 19

1 . equal to AC , the

1 . equal to AC , the

**less**, and join DC ; thereA fore , because in the triangles DBC , ACB , DB is equal to AC , and BC common to both , the two sides DB , BC are equal to D to the two AC , CB , each to each ; and the angle DBC is ... Page 27

In like manner , it may be demonstrated , that BAC , ACB , as also CAB , ABC , are

In like manner , it may be demonstrated , that BAC , ACB , as also CAB , ABC , are

**less**than two right angles . Therefore , any two angles , & c . Q. E. D. PROP . XVIII . THEOR . THE greater side of every triangle is opposite to the ... Page 28

Book I. but it is not ; therefore the side AC is not

Book I. but it is not ; therefore the side AC is not

**less**than AB ; and it has been shown that it is not equal to AB ; therefore AC is greater than AB . Wherefore , the greater angle , & c . Q. E. D. PROP . XX . THEOR . Page 32

Book I. For , if it be not greater , it must either be equal to it , or

Book I. For , if it be not greater , it must either be equal to it , or

**less**; but the angle BAC is not equal to the angle EDF , because then the base BC would 24. 1 . be equala to EF ; but A D it is not ; therefore the angle BAC is not ...### What people are saying - Write a review

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

The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh ... Robert Simson,Robert Euclid No preview available - 2016 |

### Common terms and phrases

added altitude angle ABC angle BAC base Book centre circle circle ABCD 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 26 - Any two sides of a triangle are together greater than the third side.

Page 60 - 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 26 - 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 55 - 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 24 - If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Page 161 - 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 15 - 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 187 - 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 35 - 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 176 - Therefore, universally, similar rectilineal figures are to one another in the duplicate ratio of their homologous sides.