The Elements of Euclid with Many Additional Propositions and Explanatory NotesJ. Weale, 1860 |
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Side 16
... multiple the greater was of the less ; this latter method is the one employed by Euclid in the fifth book , and by it we are enabled to ascertain their relative value . 2. A greater magnitude is said to be a multiple of a less , when ...
... multiple the greater was of the less ; this latter method is the one employed by Euclid in the fifth book , and by it we are enabled to ascertain their relative value . 2. A greater magnitude is said to be a multiple of a less , when ...
Side 17
... multiple one is of the other . In the following definition given by Wood in his Algebra , this objection is removed : - " Ratio is the relation which one quantity bears to another in respect of magnitude , the compari- son being made by ...
... multiple one is of the other . In the following definition given by Wood in his Algebra , this objection is removed : - " Ratio is the relation which one quantity bears to another in respect of magnitude , the compari- son being made by ...
Side 18
... multiple of the first be equal to that of the second , the multiple of the third is also equal to that of the fourth : or if the multiple of the first be greater than that of the second , the multiple of the third is also greater than ...
... multiple of the first be equal to that of the second , the multiple of the third is also equal to that of the fourth : or if the multiple of the first be greater than that of the second , the multiple of the third is also greater than ...
Side 20
... multiple of the first is greater than that of the second , but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the ...
... multiple of the first is greater than that of the second , but the multiple of the third is not greater than the multiple of the fourth ; then the first is said to have to the second a greater ratio than the third magnitude has to the ...
Side 24
... multiple of a greater magnitude is greater than the same multiple of a less . 4. Of two magnitudes that one of which a multiple is greater than the same multiple of the other , is the greater . In the following propositions lines are ...
... multiple of a greater magnitude is greater than the same multiple of a less . 4. Of two magnitudes that one of which a multiple is greater than the same multiple of the other , is the greater . In the following propositions lines are ...
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Vanlige uttrykk og setninger
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
Populære avsnitt
Side 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.
Side 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.
Side 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...
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. Because ED is parallel to one of the sides of the triangle ABC, viz. to BC ; as CD is to DA, so is (2.
Side 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.
Side 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.
Side 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.
Side 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...
Side 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.
Side 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...