## 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 329

and of the

judges that the definition of compound

same manner in which the definitions of duplicate and triplicate

viz .

and of the

**ratio**of L to M. ” This definition therefore of Book VI . ... upon it , rightlyjudges that the definition of compound

**ratio**might have been made after thesame manner in which the definitions of duplicate and triplicate

**ratio**are**given**,viz .

Page 374

See N. IF the whole have to the whole a

parts given , but not the same , ratios , every cne of them , whole or part , shall

have to every cnc a

CD ...

See N. IF the whole have to the whole a

**given ratio**, and the parts have to theparts given , but not the same , ratios , every cne of them , whole or part , shall

have to every cnc a

**given ratio**. Let the whole AB have a**given ratio**to the wholeCD ...

Page 377

Let the excess of the magnitude AB above a given magnitude , have a

to the magnitude BC ; the excess of AC , both of them together , above the given

magnitude , has a

Let the excess of the magnitude AB above a given magnitude , have a

**given ratio**to the magnitude BC ; the excess of AC , both of them together , above the given

magnitude , has a

**given ratio**to BC . Let AD be the given magnitude , the ... Page 378

Let AD be the given magnitude ; and because DB , the excess of AB above AD ,

has a

of AD to DE the same with this ratio ; therefore the ratio of A E D B C AD to DE is ...

Let AD be the given magnitude ; and because DB , the excess of AB above AD ,

has a

**given ratio**to BC ; the ratio of DC a 7. dat . to DB is givena ; make the ratioof AD to DE the same with this ratio ; therefore the ratio of A E D B C AD to DE is ...

Page 382

and let CD be taken from the given magnitude CF : the remainder EB is given ,

together with the magnitude to which the other remainder DF has a

Because the ratio of AB to CD is given , make as AB to CD , so AG to CF : the

ratio ...

and let CD be taken from the given magnitude CF : the remainder EB is given ,

together with the magnitude to which the other remainder DF has a

**given ratio**.Because the ratio of AB to CD is given , make as AB to CD , so AG to CF : the

ratio ...

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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.