The First Six Books of the Elements of Euclid: With Notes |
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Page 3
... whole is greater than its part . 10. Two right lines cannot inclose a space . 11. All right angles are equal to one another . 12. If a right line meet two right lines , so as to make See N. the two internal angles on the same side of it ...
... whole is greater than its part . 10. Two right lines cannot inclose a space . 11. All right angles are equal to one another . 12. If a right line meet two right lines , so as to make See N. the two internal angles on the same side of it ...
Page 7
... whole , which is absurd : therefore neither of the sides AB or AC is greater than the other ; they are therefore equal to one another . Cor . Hence every equiangular triangle is also equi- lateral ; for whatever side is taken for the ...
... whole , which is absurd : therefore neither of the sides AB or AC is greater than the other ; they are therefore equal to one another . Cor . Hence every equiangular triangle is also equi- lateral ; for whatever side is taken for the ...
Page 11
... whole , which is absurd . There- fore BE is not the continuation of CB ; and in the same manner it can be proved that no other line , ex- cept BD , is the continuation of it : therefore BD forms with BC one continued right line . PROP ...
... whole , which is absurd . There- fore BE is not the continuation of CB ; and in the same manner it can be proved that no other line , ex- cept BD , is the continuation of it : therefore BD forms with BC one continued right line . PROP ...
Page 21
... whole external angle FBC is equal to the two internal angles A and C. ( 1 ) Prop.31 . ( 2 ) Prop.29 . The angle ABC with FBC is equal to two right an- gles ( 3 ) ; but FBC is equal to the two angles A and C ( 3 ) Prop . 13 . E L " ( 4 ) ...
... whole external angle FBC is equal to the two internal angles A and C. ( 1 ) Prop.31 . ( 2 ) Prop.29 . The angle ABC with FBC is equal to two right an- gles ( 3 ) ; but FBC is equal to the two angles A and C ( 3 ) Prop . 13 . E L " ( 4 ) ...
Page 25
... ) ; but BDC is also ( 1 ) Prop.37 . equal to BAC ( 2 ) ; therefore BEC and BDC are equal ( 2 ) Hypoth . ( 3 ) , a part equal to the whole , which is absurd . There- ( 3 ) Ax . 1 . Fig . 59 . See N. f f ( 1 BOOK THE FIRST . 25.
... ) ; but BDC is also ( 1 ) Prop.37 . equal to BAC ( 2 ) ; therefore BEC and BDC are equal ( 2 ) Hypoth . ( 3 ) , a part equal to the whole , which is absurd . There- ( 3 ) Ax . 1 . Fig . 59 . See N. f f ( 1 BOOK THE FIRST . 25.
Common terms and phrases
absurd AC and BD AC and CB AC is equal angle ABC angle ACD angle BAC angle equal base bisected centre circumference Constr constructed contained in CD demonstrated described double the rectangle double the square drawn equal angles equal to AC equal to double equi equi-multiples equi-submultiples equiangular Euclid evident external angle fore form one right four magnitudes proportional four right angles fourth given angle given circle given line given right line given triangle gles half a right Hypoth inscribed less line CD manner multiple oftener opposite parallel parallelogram perpendicular PROB produced Prop proposition radius rectangle under AC right angles right line AB Schol segment side AC sides AB similar squares of AC submultiple taken tangent THEOR third tiple triangle ABC vertex
Popular passages
Page 147 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
Page 3 - Things which are equal to the same thing are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are double of the same are equal to one another.
Page 22 - 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 sides.
Page 16 - If two triangles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles are equal.
Page 148 - A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.
Page 2 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Page 92 - F) a ratio compounded of the ratios which the first has to the second, the second to the third, the third to the fourth (A to D, D to C, and C to F) ; and so on to the last.
Page 93 - AXIOMS. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude.
Page 25 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Page 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.