The Elements of Euclid: Viz. the First Six Books, Together with the Eleventh and Twelfth ... Also the Book of Euclid's Data, in Like Manner CorrectedWingrave and Collingwood, 1816 - 528 Seiten |
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Seite 364
... ratio of the given magni- tudes , DE , EF : And because ... the given magnitude AB has to A BC the given ratio of DE to EF , if unto DE , EF , AB , a D fourth proportional can be C B FE Dat . found , this which is BC is given ; and ...
... ratio of the given magni- tudes , DE , EF : And because ... the given magnitude AB has to A BC the given ratio of DE to EF , if unto DE , EF , AB , a D fourth proportional can be C B FE Dat . found , this which is BC is given ; and ...
Seite 365
... given ratios , though they be not the same , to some other magnitudes D , E , F : The magnitudes D , E , F , have given ratios to one another . Because the ratio of A to B is given , and likewise the ratio of A to D ; there- Α . fore the ...
... given ratios , though they be not the same , to some other magnitudes D , E , F : The magnitudes D , E , F , have given ratios to one another . Because the ratio of A to B is given , and likewise the ratio of A to D ; there- Α . fore the ...
Seite 366
... given ratio , and the parts have to the parts given , but not the same , ratios every one of them , whole or part , shall have to every one a given ratio . Let the whole AB have a given ratio to the whole CD , and the parts AE , EB , have ...
... given ratio , and the parts have to the parts given , but not the same , ratios every one of them , whole or part , shall have to every one a given ratio . Let the whole AB have a given ratio to the whole CD , and the parts AE , EB , have ...
Seite 367
... given , because its sides D , E , are given ; wherefore the square of F , and the straight line F is given : And ... ratio of A to B is given , because the ratio of the given straight lines D , F , which is the same with it , has been ...
... given , because its sides D , E , are given ; wherefore the square of F , and the straight line F is given : And ... ratio of A to B is given , because the ratio of the given straight lines D , F , which is the same with it , has been ...
Seite 368
... given magnitude has a given ratio to AB . Because the ratio of AE to CD is given ; as AE to CD so make BE to FD ; therefore the ra- tio of BE to FD is given ... given ratio . Let the excess of the magnitude AB above a given 368 EUCLID'S.
... given magnitude has a given ratio to AB . Because the ratio of AE to CD is given ; as AE to CD so make BE to FD ; therefore the ra- tio of BE to FD is given ... given ratio . Let the excess of the magnitude AB above a given 368 EUCLID'S.
Häufige Begriffe und Wortgruppen
ABC is given ABCD AC is equal altitude angle ABC angle BAC base BC bisected Book XI centre circle ABC circumference common logarithm cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid excess fore given angle given in magnitude given in position given in species given magnitude given point given ratio given straight line gnomon greater join less Let ABC logarithm multiple opposite parallel parallelogram AC perpendicular point F polygon prism proportionals proposition Q.E.D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopipeds square of BC straight line AB straight line BC tangent THEOR third triangle ABC vertex wherefore
Beliebte Passagen
Seite 41 - If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.
Seite 180 - 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 166 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides. Let AC, CF be equiangular parallelograms having the angle BCD equal to the angle ECG ; the ratio of the parallelogram AC to the parallelogram CF is the same with the ratio which is compounded •f the ratios of their sides. DH Let BC, CG be placed in a straight line ; therefore DC and CE are also in a straight line (14.
Seite 2 - A rhomboid, is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles.
Seite 105 - 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...
Seite 79 - The angle in a semicircle is a right angle; the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.
Seite 1 - A straight line is that which lies evenly between its extreme points.
Seite 149 - If two triangles have one angle of the one equal to one angle of the other and the sides about these equal angles proportional, the triangles are similar.
Seite 23 - That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
Seite 83 - Wherefore from the given circle ABC has been cut off the segment BAC, containing an angle equal to the given angle DQEP PROP. XXXV. THEOR. If two straight lines within a circle cut one another, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other. Let the...