The Elements of Euclid, with many additional propositions, and explanatory notes, by H. Law. Pt. 2, containing the 4th, 5th, 6th, 11th, & 12th books1855 |
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Side 24
... proportional magnitudes , but it should be understood that any similar magnitudes might have been employed , such as plane figures , solid bodies , or angles . ROPOSITION I. THEOREM . - If any number of magnitudes 24 ELEMENTS OF GEOMETRY .
... proportional magnitudes , but it should be understood that any similar magnitudes might have been employed , such as plane figures , solid bodies , or angles . ROPOSITION I. THEOREM . - If any number of magnitudes 24 ELEMENTS OF GEOMETRY .
Side 109
... planes , is perpendicular to the other plane . SCHOLIUM . The common section of two planes is the line in which they mutually cut or intersect each other . 4. The inclination of a straight line AC to a plane is the acute angle O formed ...
... planes , is perpendicular to the other plane . SCHOLIUM . The common section of two planes is the line in which they mutually cut or intersect each other . 4. The inclination of a straight line AC to a plane is the acute angle O formed ...
Side 110
... plane , and the other BC upon the other plane . 6. PARALLEL PLANES are such as do not meet one another , though produced ever so far in every direction . 7. A SOLID ANGLE is that which is made by the meeting in one point of more than two ...
... plane , and the other BC upon the other plane . 6. PARALLEL PLANES are such as do not meet one another , though produced ever so far in every direction . 7. A SOLID ANGLE is that which is made by the meeting in one point of more than two ...
Side 112
... plane figures . SCHOLIUM . When all the plane figures are equal and similar , the poly- hedron is said to be regular . 25. A CUBE , or HEXAHEDRON , is a solid figure contained by six equal squares . 26. A TETRAHEDRON is a solid figure ...
... plane figures . SCHOLIUM . When all the plane figures are equal and similar , the poly- hedron is said to be regular . 25. A CUBE , or HEXAHEDRON , is a solid figure contained by six equal squares . 26. A TETRAHEDRON is a solid figure ...
Side 113
... plane , if another part is above it . DEMONSTRATION . If it be possible , let AB , part of the straight line ABC , be in the plane , and the part BC above it : and since the straight line AB is in the plane , it can be produced in that ...
... plane , if another part is above it . DEMONSTRATION . If it be possible , let AB , part of the straight line ABC , be in the plane , and the part BC above it : and since the straight line AB is in the plane , it can be produced in that ...
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Vanlige uttrykk og setninger
algebraically expressed altitude angle ABC angle BAC axis base ABC base DEF base EH circle ABCD circle EFGH circumference common section cone contained COROLLARY cylinder DEMONSTRATION diameter divided duplicate ratio equal and similar equal angles equi equiangular equimultiples Euclid ex æquali fore four magnitudes fourth given circle given straight line gnomon greater ratio homologous sides Hypoth inscribed join less meet multiple opposite planes paral parallel parallelogram pentagon perpendicular polygon prism PROPOSITION pyramid ABCG pyramid DEFH reciprocally proportional rectangle rectilineal figure remaining angle right angles SCHOLIUM segments solid angle solid CD solid parallelopipeds solid polyhedron square on BD THEOREM THEOREM.-If third three plane angles tiple triangle ABC triplicate ratio vertex vertex the point wherefore
Populære avsnitt
Side 198 - ... 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.
Side 75 - ... if the segments of the base have the same ratio which the other sides of the triangle have to one another...
Side 115 - 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.
Side 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.
Side 198 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.
Side 53 - Convertendo, by conversion ; when there are four proportionals, and it is inferred, that the first is to its excess above the second, as the third to its excess above the fourth.
Side 40 - A and B are not unequal ; that is, they are equal. Next, let C have the same ratio to each of the magnitudes A and B ; then A shall be equal to B.
Side 119 - 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.
Side 115 - FB ; (i. 4.) for the same reason, CF is equal to FD : and because AD is equal to BC, and AF to FB, the two sides FA, AD are equal to the two FB, BC, each to each ; and the base DF was proved equal to the base FC ; therefore the angle FAD is equal to the angle FBC: (i. 8.) again, it was proved that GA is equal to BH, and also AF to FB; therefore FA and AG are equal...
Side 94 - C, they are equiangular, and also have their sides about the equal angles proportionals (def. 1. 6.). Again, because B is similar to C, they are equiangular, and have their sides about the equal angles proportionals (def. 1. 6.) : therefore the figures A, B, are each of them equiangular to C, and have the sides about the equal angles of each of them, and of C, proportionals. Wherefore the rectilineal figures A and B are equiangular (1. Ax. 1.), and have their sides about the equal angles proportionals...