The Elements of Euclid with Many Additional Propositions and Explanatory NotesJ. Weale, 1860 |
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Page 6
... equal angles , as AEC and DEB , are termed vertical angles ; while those contiguous , as AEC and CEB , are termed adjacent angles . When a straight line , as AB , intersects two other straight lines , as CD and EF , the angles CGH and ...
... equal angles , as AEC and DEB , are termed vertical angles ; while those contiguous , as AEC and CEB , are termed adjacent angles . When a straight line , as AB , intersects two other straight lines , as CD and EF , the angles CGH and ...
Page 11
... equal to two sides of the other ( DE and DF to AB and AC ) , and the angles formed by those sides also equal to one another ( D to A ) ; [ 1 ] their bases or third sides ( EF and BC ) will be equal ; [ 2 ] and the angles at the bases ...
... equal to two sides of the other ( DE and DF to AB and AC ) , and the angles formed by those sides also equal to one another ( D to A ) ; [ 1 ] their bases or third sides ( EF and BC ) will be equal ; [ 2 ] and the angles at the bases ...
Page 13
... equal to the side AB ( d ) , the side AF to the side AG ( e ) , and the angle A is common to both ; therefore the base CF is equal to the base BG , the angle ACF ... angles themselves are equal ( c ) ; the greater ELEMENTS OF GEOMETRY . 13.
... equal to the side AB ( d ) , the side AF to the side AG ( e ) , and the angle A is common to both ; therefore the base CF is equal to the base BG , the angle ACF ... angles themselves are equal ( c ) ; the greater ELEMENTS OF GEOMETRY . 13.
Page 14
Eucleides. angles themselves are equal ( c ) ; the greater ABC to the lesser DBC , which is absurd ; therefore neither of the sides AC or AB being greater than the other , they are equal . COROLLARY . Hence every equiangular triangle is ...
Eucleides. angles themselves are equal ( c ) ; the greater ABC to the lesser DBC , which is absurd ; therefore neither of the sides AC or AB being greater than the other , they are equal . COROLLARY . Hence every equiangular triangle is ...
Page 15
... equal ( a ) , therefore the angles BDC and BCD are equal ( b ) . Also , be- cause in the triangle ACD the two sides AC and AD are equal ( a ) , therefore the angles ECD and FDC , on the other side of the base , are equal ( b ) . Now the ...
... equal ( a ) , therefore the angles BDC and BCD are equal ( b ) . Also , be- cause in the triangle ACD the two sides AC and AD are equal ( a ) , therefore the angles ECD and FDC , on the other side of the base , are equal ( b ) . Now the ...
Other editions - View all
The Elements of Euclid: With Many Additional Propositions, & Explanatory ... Euclid No preview available - 2023 |
The Elements of Euclid: With Many Additional Propositions, and Explanatory ... Euclid No preview available - 2013 |
Common terms and phrases
AC is equal altitude angle ABC bisected circle ABCD circumference cone CONSTRUCTION contained COROLLARY cylinder DEMONSTRATION diameter divided double draw duplicate ratio EFGH equal angles equal in area equiangular equilateral equimultiples Euclid external angle fore fourth given line given rectilineal given straight line gnomon greater ratio homologous sides Hypoth HYPOTHESES inscribed join less line AC lines be drawn meet multiple opposite angle parallel parallelogram perpendicular polygon prism proposition pyramid ABCG pyramid DEFH rectangle rectilineal figure remaining angle right angles SCHOLIA SCHOLIUM segment side AC solid angle solid CD solid parallelopipeds sphere square on AB square on AC syllogism THEOREM THEOREM.-If third three plane angles tiple triangle ABC triplicate ratio vertex wherefore
Popular passages
Page 107 - A cone is a solid figure described by the revolution of a right-angled triangle about one of the sides containing the right angle, which side remains fixed.
Page 85 - ... have an angle of the one equal to an angle of the other, and the sides about those angles reciprocally proportional, are equal to une another.
Page 18 - Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding...
Page 82 - From the point A draw a straight line AC, making any angle with AB ; and in AC take any point D, and take AC the same multiple of AD, that AB is of the part which is to be cut off from it : join BC, and draw DE parallel to it : then AE is the part required to be cut off. Because ED is parallel to one of the sides of the triangle ABC, viz. to BC ; as CD is to DA, so is (2.
Page 111 - If two triangles have two angles of the one equal to two angles of the other, each to each, and also one side of the one equal to the corresponding side of the other, the triangles are congruent.
Page 116 - ... plane, from a given point above it. Let A be the given point above the plane BH; it is required to draw from the point A a straight line perpendicular to the plane BH.
Page 115 - For the same reason, CD is likewise at right angles to the plane HGK. Therefore AB, CD are each of them at right angles to the plane HGK.
Page 49 - IF magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately ; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these. Let AB, BE, CD...
Page 32 - 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 34 - Take of B and D any equimultiples whatever E and F; and of A and C any equimultiples whatever G and H. First, let E be greater than G, then G is less than E: and because A is to B, as C is to D, (hyp.) and of A and C...