The Elements of Euclid with Many Additional Propositions and Explanatory NotesJ. Weale, 1860 |
Dentro del libro
Resultados 1-5 de 45
Página 7
... double of the same are equal to one another . 7. Magnitudes which are halves of the same are equal to one another . SCHOLIUM . A similar extension may be given to the 6th and 7th axioms as that given above to the 1st by substituting ...
... double of the same are equal to one another . 7. Magnitudes which are halves of the same are equal to one another . SCHOLIUM . A similar extension may be given to the 6th and 7th axioms as that given above to the 1st by substituting ...
Página 36
... double of A and B ; but the four angles together are equal to four right angles ( c ) , therefore A and B are equal to two right angles , and the line AC is parallel to BD ( d ) ; and in a similar manner it may be proved that AB is ...
... double of A and B ; but the four angles together are equal to four right angles ( c ) , therefore A and B are equal to two right angles , and the line AC is parallel to BD ( d ) ; and in a similar manner it may be proved that AB is ...
Página 41
... double of the triangle ABC ( c ) , therefore ABCD is also double of EBC . B C ( a ) Hypoth . ( b ) I. 37 . ( c ) I. 34 . E COROLLARY . Hence , if a parallelogram and a triangle have equal bases and are between the same parallels , the ...
... double of the triangle ABC ( c ) , therefore ABCD is also double of EBC . B C ( a ) Hypoth . ( b ) I. 37 . ( c ) I. 34 . E COROLLARY . Hence , if a parallelogram and a triangle have equal bases and are between the same parallels , the ...
Página 42
... double of the triangle ABE ; but , because the paral- lelogram FECG and the triangle AEC are upon the same base EC and between the same parallels BC and AG ( d ) , therefore the parallelogram is double of the triangle ( f ) ; and the ...
... double of the triangle ABE ; but , because the paral- lelogram FECG and the triangle AEC are upon the same base EC and between the same parallels BC and AG ( d ) , therefore the parallelogram is double of the triangle ( f ) ; and the ...
Página 45
... doubles of the triangles EAB and GCD ( c ) , therefore they are equal ( d ) . COROLLARY 2. If two squares AF and CH are equal . DEMONSTRATION . For , if it be possible , let one of them AF be the greater ; take AK equal to CD , and AI ...
... doubles of the triangles EAB and GCD ( c ) , therefore they are equal ( d ) . COROLLARY 2. If two squares AF and CH are equal . DEMONSTRATION . For , if it be possible , let one of them AF be the greater ; take AK equal to CD , and AI ...
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...