The Contents of the Fifth and Sixth Books of EuclidThe University Press, 1900 - 143 Seiten |
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Seite v
... proving all the properties of ratio required in the Sixth Book , but only for demonstrating two important propositions , viz . , the 1st and 33rd . The other properties of ratio required in the Sixth Book are usually assumed , or so ...
... proving all the properties of ratio required in the Sixth Book , but only for demonstrating two important propositions , viz . , the 1st and 33rd . The other properties of ratio required in the Sixth Book are usually assumed , or so ...
Seite vii
... proved that A , B , E , F also satisfy the conditions of that definition . Remarks of a somewhat similar nature apply to the 13th Proposition . In this book it is shewn that two commensurable magnitudes determine a real number ; and ...
... proved that A , B , E , F also satisfy the conditions of that definition . Remarks of a somewhat similar nature apply to the 13th Proposition . In this book it is shewn that two commensurable magnitudes determine a real number ; and ...
Seite viii
... prove any of the properties of equal ratios . Its use only renders the proofs of the propositions indirect and artificial and consequently difficult . Not only does no inconvenience result from avoiding its use , but it is possible to ...
... prove any of the properties of equal ratios . Its use only renders the proofs of the propositions indirect and artificial and consequently difficult . Not only does no inconvenience result from avoiding its use , but it is possible to ...
Seite ix
... in striking contrast to the directness of the proofs of the other propositions . The plan followed in this work is to explain at an early stage what the relative multiple scale of two magnitudes is , then to prove PREFACE . ix.
... in striking contrast to the directness of the proofs of the other propositions . The plan followed in this work is to explain at an early stage what the relative multiple scale of two magnitudes is , then to prove PREFACE . ix.
Seite x
... prove a few of the simpler properties of relative multiple scales , then to point out that these propositions accord with the ideas of ratio , formed by learners long before they commenced to read Geometry , in so far as those ideas ...
... prove a few of the simpler properties of relative multiple scales , then to point out that these propositions accord with the ideas of ratio , formed by learners long before they commenced to read Geometry , in so far as those ideas ...
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Häufige Begriffe und Wortgruppen
arc CD BEFG BEHC bisected BLNO CÂD centre circle Commutative Law congruent corresponding sides cross-ratio DÊE draw duplicate ratio EFGH ENUNCIATION equal angles Equal Ratios equimultiples EXAMPLE expressed by Fig Fifth Book four harmonic points greater Hence the scale hypotenuse integers kind mean proportional middle point parallel to BC parallelogram point at infinity point of division PQRST Prop Proportion 66 PROPOSITION rA rB rA sB rA sC radical axis ratio compounded ratio of equality rect rectangle contained relative multiple scale required to prove respectively equal right angle second column segments side AC side corresponding similar figures similar triangles similarly described Sixth Book square ABLK supplementary angles Theory of Relative three magnitudes triangle ABC triangle are respectively triangle DEF triangles are similar unequal ratios vertex whole numbers
Beliebte Passagen
Seite 99 - IF two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals : the triangles shall be equiangular, and shall have those angles equal which are opposite to the homologous sides.
Seite xviii - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.
Seite 102 - If two similar parallelograms have a common angle, and be similarly situated ; they are about the same diameter.
Seite 99 - Prove that similar triangles are to one another in the duplicate ratio of their homologous sides.
Seite xvi - If the vertical angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another...
Seite 80 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.
Seite 99 - ABC, DEF have one angle in the one equal to one angle in the other, viz. the angle BAC to the angle EDF, and the sides about...
Seite 73 - P moves in a plane so that the ratio of its distances from two fixed points A, B in that plane is always the same.
Seite 35 - The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth ; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth: or, if the 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...
Seite 84 - If they do not intersect, show that the radical axis is perpendicular to the line joining the centres of the circles...