The Contents of the Fifth and Sixth Books of EuclidThe University Press, 1900 - 143 σελίδες |
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Σελίδα v
... Definition , which is retained not for the purpose of proving all the properties of ratio required in the Sixth Book ... definition , which is the test for the sameness of two ratios . ( II ) The seventh definition , which is the test ...
... Definition , which is retained not for the purpose of proving all the properties of ratio required in the Sixth Book ... definition , which is the test for the sameness of two ratios . ( II ) The seventh definition , which is the test ...
Σελίδα vi
Euclid, Micaiah John Muller Hill. * ( III ) The tenth definition , which defines " Duplicate Ratio . " * ( IV ) The definition marked A by Simson , which defines the process for compounding ratios . In order to make things clear , it is ...
Euclid, Micaiah John Muller Hill. * ( III ) The tenth definition , which defines " Duplicate Ratio . " * ( IV ) The definition marked A by Simson , which defines the process for compounding ratios . In order to make things clear , it is ...
Σελίδα vii
... definitions . To write down a definition , and then draw conclusions from it , is a process which is useful in Advanced Mathematics ; but it is wholly unsuitable for elementary teaching . It seems not unlikely that Euclid reached his ...
... definitions . To write down a definition , and then draw conclusions from it , is a process which is useful in Advanced Mathematics ; but it is wholly unsuitable for elementary teaching . It seems not unlikely that Euclid reached his ...
Σελίδα viii
... definition of Proportion given in treatises on Algebra ; but even with this assistance it remains difficult for him to remember its details . He may and frequently does learn to apply it correctly in demonstrating the 1st and 33rd ...
... definition of Proportion given in treatises on Algebra ; but even with this assistance it remains difficult for him to remember its details . He may and frequently does learn to apply it correctly in demonstrating the 1st and 33rd ...
Σελίδα ix
... Definition alone ; and all , with a single exception , fall under one or other of two well recognised types . These types correspond to the two forms of the conditions for the sameness of two relative multiple scales ( or two ratios ) ...
... Definition alone ; and all , with a single exception , fall under one or other of two well recognised types . These types correspond to the two forms of the conditions for the sameness of two relative multiple scales ( or two ratios ) ...
Άλλες εκδόσεις - Προβολή όλων
The Contents of the Fifth and Sixth Books of Euclid Euclid,Micaiah John Muller Hill Πλήρης προβολή - 1900 |
Συχνά εμφανιζόμενοι όροι και φράσεις
ABCD ABEF angle equal angles reciprocally proportional BCGE BCHD BEFG BEHC bisected BLNO centre circle corresponding sides cross-ratio cutting AC DÊE definition divided drawn perpendicular duplicate ratio ENUNCIATION EQPS equal angles equimultiples Euclid's EXAMPLE exhibited in Fig expressed by Fig fact exhibited four harmonic points four straight lines greater Hence by Prop Hence the triangles hypotenuse integer inversion kind locus mean proportional middle point parallel to BC parallelogram PQRST PROPOSITION Proposition 48 rA rB rA sC radical axis ratio compounded ratio of equality rect rectangle contained relative multiple scale required to prove respectively equal right angle segments shows the fact side BC side corresponding similar figures similar triangles similarly described square on AB square on AC supplementary angles tangents three magnitudes triangle ABC triangle DEF triangles are similar vertex
Δημοφιλή αποσπάσματα
Σελίδα 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.
Σελίδα 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.
Σελίδα 99 - Prove that similar triangles are to one another in the duplicate ratio of their homologous sides.
Σελίδα 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...
Σελίδα 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...
Σελίδα xvi - In a right triangle, if a perpendicular is drawn from the vertex of the right angle to the hypotenuse : I.
Σελίδα 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...
Σελίδα 84 - If they do not intersect, show that the radical axis is perpendicular to the line joining the centres of the circles...
Σελίδα 100 - If an 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 of the straight line bisecting the angle.
Σελίδα 80 - If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means.