The Elements of Euclid: With Many Additional Propositions, & Explanatory Notes, Etc, Part 1 |
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Page 10
... DEMONSTRATION . For the lines AD and AE being both radii of the same circle DEF are equal ( c ) . But AD and C are equal ( d ) ; therefore , because AE and C are both equal to the same line AD , they are equal to each other ( e ) ; and ...
... DEMONSTRATION . For the lines AD and AE being both radii of the same circle DEF are equal ( c ) . But AD and C are equal ( d ) ; therefore , because AE and C are both equal to the same line AD , they are equal to each other ( e ) ; and ...
Page 11
... DEMONSTRATION . For , if the triangle ABC be applied to DEF , so that the point A may be on the point D , the point B on the straight line DE , and that AC and DF may lie on the same side ; then AB must lie wholly on DE , for otherwise ...
... DEMONSTRATION . For , if the triangle ABC be applied to DEF , so that the point A may be on the point D , the point B on the straight line DE , and that AC and DF may lie on the same side ; then AB must lie wholly on DE , for otherwise ...
Page 12
... demonstration of the first of these is an example of the negative or in- direct proof termed " Reductio ad Absurdum , " which consists in proving a proposition by showing that if it is denied an obvious absurdity follows . Concerning ...
... demonstration of the first of these is an example of the negative or in- direct proof termed " Reductio ad Absurdum , " which consists in proving a proposition by showing that if it is denied an obvious absurdity follows . Concerning ...
Page 13
... DEMONSTRATION . For if AB be not equal to AC , one of them is greater than the other ; let AB be the greater , from it cut off DB equal to AC the less ( a ) , and draw the line DC . Then in the triangles ABC and DBC , because the side ...
... DEMONSTRATION . For if AB be not equal to AC , one of them is greater than the other ; let AB be the greater , from it cut off DB equal to AC the less ( a ) , and draw the line DC . Then in the triangles ABC and DBC , because the side ...
Page 15
... demonstration of this proposition is that termed a dilemma , and this is the only instance through- out the Elements in which it is employed . The argument will also be seen to be by the " reductio ad absurdum . ' PROPOSITION VIII ...
... demonstration of this proposition is that termed a dilemma , and this is the only instance through- out the Elements in which it is employed . The argument will also be seen to be by the " reductio ad absurdum . ' PROPOSITION VIII ...
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Common terms and phrases
AC and CB AC is equal angle ABC angle BCD angle equal area to double area to twice bisect chord circle ABC circumference Constr CONSTRUCTION COROLLARY DB is equal DEMONSTRATION diagonal divided double the rectangle draw equal angles equal in area equal to AC equilateral Euclid external angle Find the center finite straight line Geometry given angle given line greater Hypoth HYPOTHESES intersect join less line BC lines be drawn magnitude major premiss opposite sides parallel parallelogram perpendicular predicate premises produced proposition quadratic equation reductio ad absurdum right angles SCHOLIA SCHOLIUM second power segment sides AC square on AC square on half squares on AB syllogism termed THEOREM THEOREM.-If triangle ABC twice the rectangle twice the square vertex whole line
Popular passages
Page 24 - 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 114 - In obtuse-angled triangles, if a perpendicular be drawn from either of the acute angles to the opposite side produced, the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle, by twice the rectangle contained by the side...
Page xiv - 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 : 16. And this point is called the centre of the circle.
Page 13 - The difference between any two sides of a triangle is less than the third side.
Page 111 - AFC. (in. 21.) Hence in the triangles ADE, AFC, there are two angles in the one respectively equal to two angles in the other, consequently, the third angle CAF is equal to the third angle DAB ; therefore the arc DB is equal to the arc CF, (in.
Page 89 - ... the centre of the circle shall be in that line. Let the straight line DE touch the circle ABC in C, and from C let CA be drawn at right angles to DE ; the centre of the circle is in CA.
Page 70 - EQUAL circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal. ' This is not a definition, but a theorem, the truth of ' which is evident; for, if the circles be applied to one ' another, so that their centres coincide, the circles ' must likewise coincide, since the straight lines from
Page 34 - To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle.
Page 22 - If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles...