## Euclid's Elements: Or, Second Lessons in Geometry,in the Order of Simson's and Playfair's Editions ... |

### From inside the book

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Page 17

... angle ABC : therefore the re- mainders ABE , ABD , are equal ( b ) ; the less to the greater , which is impossible ...

... angle ABC : therefore the re- mainders ABE , ABD , are equal ( b ) ; the less to the greater , which is impossible ...

**exterior angle**( ACD ) is greater than either of the interior opposite angles ( at A , or B ) . Constr . Bisect ... Page 18

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**angles**. Argument . Produce the side BC to D ( a ) . Then since the interior**angle**B is less than the**exterior**and opposite**angle**ACD ( b ) , to each add ACB ; then ACB and B are less than ACB and ACD ( c ) : but these latter two are ... Page 19

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**angle**( b ) ; therefore BC is less than BD , which is the sum of BA and AC . B C In this manner it may be proved ...**exterior**of the triangle BAE , and interior of the triangle CED ; it is therefore less than the**exterior**BDC , and ... Page 21

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**angles**B , E equal by hypoth . and the sides BA , BH equal to he sides ED , EF ( b ) : therefore the**angle**BHA equals the**angle**F , or its equal C ; that is , the**exterior**equals the interior on the same side ( ^ ) , which is impossible ... Page 22

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**exterior angle**( EGB ) to the interior opposite angle ( GHD on the same side of the line ; or make the interior angles ( BGH , GHD ) equal to two right angles , the two lines shall be parallel . Argument 1. Since , by hyp . EGB is equal ...### Other editions - View all

Euclid's Elements, Or Second Lessons in Geometry, in the Order of Simson's ... D. M'Curdy No preview available - 2017 |

Euclid's Elements, Or Second Lessons in Geometry, in the Order of Simson's ... D. M'Curdy No preview available - 2017 |

### Common terms and phrases

ABCD alternate angles angle ACD angles ABC angles equal antecedents Argument base BC bisected centre Chart chord circle ABC circumference Constr Denison Olmsted diameter draw drawn equal angles equal arcs equal radii equal sides equals the squares equi equiangular equilateral equilateral polygon equimultiples exterior angle fore Geometry given circle given rectilineal given straight line gles gnomon greater half inscribed isosceles isosceles triangle join less meet multiple opposite angles parallelogram parallelopipeds pentagon perimeter perpendicular plane polygon produced Q. E. D. Recite radius ratio rectangle rectangle contained rectilineal figure School segment semicircle similar similar triangles sine square of AC tangent touches the circle triangle ABC unequal Wherefore

### Popular passages

Page 90 - If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.

Page 117 - In the same way it may be proved that a : b : : sin. A : sin. B, and these two proportions may be written a : 6 : c : : sin. A : sin. B : sin. C. THEOREM III. t8. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference. By Theorem II. we have a : b : : sin. A : sin. B.

Page 92 - IN a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.

Page 79 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.

Page 87 - If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or those sides produced, proportionally...

Page 26 - Triangles upon equal bases, and between the same parallels, are equal to one another.

Page 133 - If a straight line stand at right angles to each of two straight lines at the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.

Page 13 - AB be the greater, and from it cut (3. 1.) off DB equal to AC the less, and join DC ; therefore, because A in the triangles DBC, ACB, DB is equal to AC, and BC common to both, the two sides DB, BC are equal to the two AC, CB. each to each ; and the angle DBC is equal to the angle ACB; therefore the base DC is equal to the base AB, and the triangle DBC is< equal to the triangle (4. 1.) ACB, the less to 'the greater; which is absurd.

Page 71 - If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth ; then shall...

Page 83 - IF there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last. NB This is usually cited by the words