## 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 120

Book V. the first be greater than that of the second , the multiple of the

greater than that of the fourth . VI . Magnitudes which have the same ratio are

called proportionals . N. B. · When four magnitudes are proportionals , it is usually

...

Book V. the first be greater than that of the second , the multiple of the

**third**is alsogreater than that of the fourth . VI . Magnitudes which have the same ratio are

called proportionals . N. B. · When four magnitudes are proportionals , it is usually

...

Page 121

Permutando , or alternando , by permutation , or alternately ; this word is used

when there are four proportionals , and it is See N. inferred , that the first has the

same ratio to the

the ...

Permutando , or alternando , by permutation , or alternately ; this word is used

when there are four proportionals , and it is See N. inferred , that the first has the

same ratio to the

**third**, which the second has to the fourth ; or that the first is tothe ...

Page 129

1 Book V. PROP . A. THEOR . IF the first of four magnitudes has to the second the

See N. same ratio which the

than the second , the

...

1 Book V. PROP . A. THEOR . IF the first of four magnitudes has to the second the

See N. same ratio which the

**third**has to the fourth ; then , if the first be greaterthan the second , the

**third**is also greater than the fourth ; and , if equal , equal ; if...

Page 212

IF two planes cutting one another be each of them perpendicular to a

their common section shall be perpendicular to the same plane . Let the two

planes AB , BC be each of them perpendicular to a

...

IF two planes cutting one another be each of them perpendicular to a

**third**plane ,their common section shall be perpendicular to the same plane . Let the two

planes AB , BC be each of them perpendicular to a

**third**plane , and let BD be the...

Page 272

But this pyramid is the

and of the same altitude with the cylinder . Therefore this prism is greater than the

cylinder of which the base is the circle ABCD . B But it is also less , for it is ...

But this pyramid is the

**third**part of the prism upon the same E base AEBFCGDH ,and of the same altitude with the cylinder . Therefore this prism is greater than the

cylinder of which the base is the circle ABCD . B But it is also less , for it is ...

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### 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.