The Contents of the Fifth and Sixth Books of EuclidThe University Press, 1900 - 143 Seiten |
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Seite v
... whole , though they readily admit the truth of the propositions singly as consequences of the fundamental definitions , which are ( I ) The fifth definition , which is the test for the sameness of two ratios . ( II ) The seventh ...
... whole , though they readily admit the truth of the propositions singly as consequences of the fundamental definitions , which are ( I ) The fifth definition , which is the test for the sameness of two ratios . ( II ) The seventh ...
Seite viii
... no inconvenience result from avoiding its use , but it is possible to get rid of the latter part of the 8th Proposition , and of the whole of the 10th and 13th Propositions , which deal with unequal ratios , and of viii PREFACE .
... no inconvenience result from avoiding its use , but it is possible to get rid of the latter part of the 8th Proposition , and of the whole of the 10th and 13th Propositions , which deal with unequal ratios , and of viii PREFACE .
Seite xii
... whole numbers . Prop . 12 relates to differing relative multiple scales ; and with Props . 9 and 11 is very useful in developing the idea of ratio . Props . 24 , 59 , 60 and 63 appear here chiefly but not entirely on account of their ...
... whole numbers . Prop . 12 relates to differing relative multiple scales ; and with Props . 9 and 11 is very useful in developing the idea of ratio . Props . 24 , 59 , 60 and 63 appear here chiefly but not entirely on account of their ...
Seite xvi
... whole triangle ; the perpendicular is a mean proportional between the segments of the hypotenuse ; and each side is a mean proportional between the adjacent segment of the hypotenuse and the hypotenuse . PROP . 33 . 34 . SECTION VI ...
... whole triangle ; the perpendicular is a mean proportional between the segments of the hypotenuse ; and each side is a mean proportional between the adjacent segment of the hypotenuse and the hypotenuse . PROP . 33 . 34 . SECTION VI ...
Seite xviii
... whole and to one another . If two similar parallelograms have a common angle and be similarly situated they are about the same diagonal . 55. If OAB be a given triangle it is required to find a point P on AB or AB produced so that if PQ ...
... whole and to one another . If two similar parallelograms have a common angle and be similarly situated they are about the same diagonal . 55. If OAB be a given triangle it is required to find a point P on AB or AB produced so that if PQ ...
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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...