Books 10-13 and appendixThe University Press, 1908 |
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Seite 1
... common measure . " They called all magnitudes measurable by the same measure commensurable , but those which are not subject to the same measure incommensurable , and again such of these as are measured by some other common measure ...
... common measure . " They called all magnitudes measurable by the same measure commensurable , but those which are not subject to the same measure incommensurable , and again such of these as are measured by some other common measure ...
Seite 10
... common measure . 2. Straight lines are commensurable in square when the squares on them are measured by the same area , and incommensurable in square when the squares on them cannot possibly have any area as a common measure . 3. With ...
... common measure . 2. Straight lines are commensurable in square when the squares on them are measured by the same area , and incommensurable in square when the squares on them cannot possibly have any area as a common measure . 3. With ...
Seite 16
... common extremity and the smaller lies along the other on the same side of the common extremity . If AC be the greater and AB the smaller , we have to prove that there exists an integral number » such that n . AB > AC . Suppose that this ...
... common extremity and the smaller lies along the other on the same side of the common extremity . If AC be the greater and AB the smaller , we have to prove that there exists an integral number » such that n . AB > AC . Suppose that this ...
Seite 18
... common measure . The sign of the incommensurability of two magnitudes is that this operation never comes to an end , while the successive remainders become smaller and smaller until they are less than any assigned magnitude . Observe ...
... common measure . The sign of the incommensurability of two magnitudes is that this operation never comes to an end , while the successive remainders become smaller and smaller until they are less than any assigned magnitude . Observe ...
Seite 19
... common measure . If , therefore , the segments were commensurable , the process would stop . But it clearly does not ; therefore the segments are incommensurable . Allman expresses the opinion that it was rather in connexion with the ...
... common measure . If , therefore , the segments were commensurable , the process would stop . But it clearly does not ; therefore the segments are incommensurable . Allman expresses the opinion that it was rather in connexion with the ...
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.