Books 10-13 and appendixThe University Press, 1908 |
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Seite 1
... roots ( dvváμes ) , proving that the square roots of 1 I have already noted ( Vol . I. p . 351 ) that G. Junge ( Wann haben die Griechen das Irrationale entdeckt ? ) disputes this , maintaining that it was the Pythagoreans , but not ...
... roots ( dvváμes ) , proving that the square roots of 1 I have already noted ( Vol . I. p . 351 ) that G. Junge ( Wann haben die Griechen das Irrationale entdeckt ? ) disputes this , maintaining that it was the Pythagoreans , but not ...
Seite 2
... root up to that of 17 square feet , at which for some reason he stopped . No mention is here made of √2 , doubtless for the reason that its incommensurability had been proved before , i.e. by Pythagoras . We know that Pythagoras ...
... root up to that of 17 square feet , at which for some reason he stopped . No mention is here made of √2 , doubtless for the reason that its incommensurability had been proved before , i.e. by Pythagoras . We know that Pythagoras ...
Seite 3
... roots ( dvvápμes ) appeared to be unlimited in multitude , to try to arrive at one collective term by which we could designate all these square roots .... I divided number in general into two classes . The number which can be expressed ...
... roots ( dvvápμes ) appeared to be unlimited in multitude , to try to arrive at one collective term by which we could designate all these square roots .... I divided number in general into two classes . The number which can be expressed ...
Seite 4
... roots of equations of the second degree as are incommensurable with the given magnitudes cannot be expressed by means of the latter and of numbers , it is conceivable that the Greeks , in exact investigations , introduced no approximate ...
... roots of equations of the second degree as are incommensurable with the given magnitudes cannot be expressed by means of the latter and of numbers , it is conceivable that the Greeks , in exact investigations , introduced no approximate ...
Seite 5
... roots do not come in , since x must be a straight line . The omission however to bring in negative roots constitutes no loss of generality , since the Greeks would write the equation leading to negative roots in another form so as to ...
... roots do not come in , since x must be a straight line . The omission however to bring in negative roots constitutes no loss of generality , since the Greeks would write the equation leading to negative roots in another form so as to ...
Häufige Begriffe und Wortgruppen
area a medial base bimedial binomial straight line bisected circle ABCD commensurable in length commensurable in square cone cut in extreme cylinder decagon diameter dihedral angle dodecahedron equal equilateral Euclid extreme and mean greater segment height icosahedron inscribed irrational straight line kp² Lemma let the square magnitudes mean ratio medial area medial straight line medial whole parallel parallelepipedal solids parallelogram pentagon perpendicular plane of reference polygon prism Proclus PROPOSITION proved rational and incommensurable rational area rational straight line rectangle AC rectangle contained right angles second apotome side Similarly solid angle sphere square number square on AB squares on AC straight lines commensurable surable triangle twice the rectangle vertex whence
Beliebte Passagen
Seite 310 - 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.
Seite 372 - Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out?
Seite 260 - The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the...
Seite 295 - BAE; and they are in one plane, which is impossible. Also, from a point above a plane, there can be but one perpendicular to that plane ; for, if there could be two, they would be parallel (6. PI.) to one another, which is absurd. Therefore, from the same point, &c.
Seite 279 - AB, CD. In like manner, it may be proved, that FE makes right angles with every straight line which meets it in that plane. But a straight line is at right angles to a plane when it makes right angles with every straight line which meets it in that plane : (xi. def. 3.) therefore EF is at right angles to the plane in which are AB, CD. Wherefore, if a straight line, &c.
Seite 389 - The upper end of the frustum of a pyramid or cone is called the upper base...
Seite 324 - AE is a parallelogram : join AH, DF ; and because AB is parallel to DC, and BH to CF ; the two straight lines AB, BH, which meet one another, are parallel to DC and CF, which meet one another...
Seite 294 - To erect a straight line at right angles to a given plane, from a point given in the plane. Let A be the point given in the plane.
Seite 304 - And because the plane AB is perpendicular to the third plane, and DE is drawn in the plane AB at right angles to AD their common section...
Seite 345 - N. equiangular to one another, each to each, that is, of which the folid angles are equal, each to each ; have to one another the ratio which is the fame with the ratio compounded of the ratios of their fides.