Books 10-13 and appendixThe University Press, 1908 |
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Seite 15
... draws no inference from it [ x . 1 ] , not even that which we should more than anything else expect , namely that , if two magnitudes are incom- mensurable , we can always form a magnitude commensurable with the first which shall differ ...
... draws no inference from it [ x . 1 ] , not even that which we should more than anything else expect , namely that , if two magnitudes are incom- mensurable , we can always form a magnitude commensurable with the first which shall differ ...
Seite 19
... Draw FE at right angles to AC meeting BC in E. It is easily proved that BE = EF = FC , CF - AC - AB = d - a CE = CB - CF = a - ( d− a ) . ( 1 ) . = 2a - d ......... ( 2 ) . 2 E ༤ Suppose , if possible , that d , a are commensurable ...
... Draw FE at right angles to AC meeting BC in E. It is easily proved that BE = EF = FC , CF - AC - AB = d - a CE = CB - CF = a - ( d− a ) . ( 1 ) . = 2a - d ......... ( 2 ) . 2 E ༤ Suppose , if possible , that d , a are commensurable ...
Seite 76
Euclid. let EF be drawn at right angles to AB , and let AF , FB be joined . Then , since AB , BC are unequal straight lines , and the square on AB is greater than the square on the square on a straight line incommensurable with A while ...
Euclid. let EF be drawn at right angles to AB , and let AF , FB be joined . Then , since AB , BC are unequal straight lines , and the square on AB is greater than the square on the square on a straight line incommensurable with A while ...
Seite 78
... equal to th on BE and deficient by a square figure , namely the AF , FB ; therefore AF is incommensurable in length with FB . Let FD be drawn from Fat right angles to AB , and let AD , DB be joined . x . 34 ] PROPOSITIONS 33 , 34 79 Since.
... equal to th on BE and deficient by a square figure , namely the AF , FB ; therefore AF is incommensurable in length with FB . Let FD be drawn from Fat right angles to AB , and let AD , DB be joined . x . 34 ] PROPOSITIONS 33 , 34 79 Since.
Seite 117
... drawn from G , E , F either of the straight lines AB , CD ; let the square SN be constructed equal to the pa AH , and the square NQ equal to GK , and let them be placed so that MN is in a straigh NO ; therefore RN is also in a straight ...
... drawn from G , E , F either of the straight lines AB , CD ; let the square SN be constructed equal to the pa AH , and the square NQ equal to GK , and let them be placed so that MN is in a straigh NO ; therefore RN is also in a straight ...
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.