The Elements of Euclid with Many Additional Propositions and Explanatory NotesJ. Weale, 1860 |
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Página xx
... [ Hypoth . and I. 34. ] i Therefore ; The lines BC and EH ARE equal to one another . Syllogism 2 . Da ( Straight lines which join the adjacent extremities of two equal and parallel straight lines ) ARE themselves equal and parallel . [ 1 ...
... [ Hypoth . and I. 34. ] i Therefore ; The lines BC and EH ARE equal to one another . Syllogism 2 . Da ( Straight lines which join the adjacent extremities of two equal and parallel straight lines ) ARE themselves equal and parallel . [ 1 ...
Página 13
... Hypoth . ( e ) Constr . ( ƒ ) I. 4 . ( 9 ) Ax . 3 . [ 1. ] Further in the same triangles the remaining angles must be equal , BCF to CBG ( f ) ; and if these equal angles be taken from the equal angles ACF and ABG , the remaining angles ...
... Hypoth . ( e ) Constr . ( ƒ ) I. 4 . ( 9 ) Ax . 3 . [ 1. ] Further in the same triangles the remaining angles must be equal , BCF to CBG ( f ) ; and if these equal angles be taken from the equal angles ACF and ABG , the remaining angles ...
Página 14
... and as BDC is greater than ADC ( c ) , therefore BDC is greater than BCD ; but they have already been proved to be equal , which is absurd . ( a ) Hypoth . 1.5 . Ax . 9 . [ 2. ] Let the vertex of one triangle be 14 ELEMENTS OF GEOMETRY .
... and as BDC is greater than ADC ( c ) , therefore BDC is greater than BCD ; but they have already been proved to be equal , which is absurd . ( a ) Hypoth . 1.5 . Ax . 9 . [ 2. ] Let the vertex of one triangle be 14 ELEMENTS OF GEOMETRY .
Página 15
... Hypoth . ( b ) I. 5 . ( c ) Ax . 9 . SCHOLIA . 1. The only use made of this proposition is to prove that which follows it , which can , however , be demonstrated without it , as is done in the scholium attached to that proposition . 2 ...
... Hypoth . ( b ) I. 5 . ( c ) Ax . 9 . SCHOLIA . 1. The only use made of this proposition is to prove that which follows it , which can , however , be demonstrated without it , as is done in the scholium attached to that proposition . 2 ...
Página 16
... Hypoth . I. 5 . SCHOLIA . 1. This proposition may be demonstrated in the following manner , without any reference to the seventh . Let the triangle ABC be applied to the tri- angle DEF , so that their bases may coincide , and that the ...
... Hypoth . I. 5 . SCHOLIA . 1. This proposition may be demonstrated in the following manner , without any reference to the seventh . Let the triangle ABC be applied to the tri- angle DEF , so that their bases may coincide , and that the ...
Otras ediciones - Ver todas
The Elements of Euclid: With Many Additional Propositions, & Explanatory ... Euclid Sin vista previa disponible - 2023 |
The Elements of Euclid: With Many Additional Propositions, and Explanatory ... Euclid Sin vista previa disponible - 2013 |
Términos y frases comunes
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
Pasajes populares
Página 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.
Página 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.
Página 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...
Página 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.
Página 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.
Página 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.
Página 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.
Página 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...
Página 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.
Página 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...