## 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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**angles ACD**. ADC , are also equal ( a ) . But the angle BCD is less than the**angle ACD**, therefore less than ADC , and still less than BDC . Again , because CB is equal to DB , the angles BCD , BDC , are equal : but BCD has been proved ... Page 15

... angle ( BAC ) ; that is , to divide it into two equal angles . Constr . In AB take any point D ; make AE equal to AD ... ACD to BCD , and CD is common to the two tri-

... angle ( BAC ) ; that is , to divide it into two equal angles . Constr . In AB take any point D ; make AE equal to AD ... ACD to BCD , and CD is common to the two tri-

**angles ACD**, BCD : therefore , the bases AD and BD are equal ... Page 17

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**angle**ABC : therefore the re- mainders ABE , ABD , are equal ( b ) ; the ... (**ACD**) is greater than either of the interior opposite**angles**( at A , or B ) ...**angle**ECD is greater than ECF , or A. B In like manner , if BC be ... Page 18

... angles . Argument . Produce the side BC to D ( a ) . Then since the interior angle B is less than the exterior and opposite

... 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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**angles ACD**, ADC are equal , being opposite to equal sides ( a ) ; but either of them is less than BCD ; and the less side subtends the less angle ( b ) ; therefore BC is less than BD , which is the sum of BA and B AC . In this manner ...### 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 propositions Q. E. D. Recite radius ratio rectangle 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 94 - Equal parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional ; and parallelograms that have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

Page 12 - 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 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.