The First Six Books of the Elements of Euclid: With NotesPrinted at the University Press, for R. Milliken and son, 1833 - 183 Seiten |
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Seite 91
... submultiple whatsoever of the first is contained in the second , as often as an equi - submultiple of the third is contained in the fourth . 6. Magnitudes which have the same ratio are called proportionals . 7. If a submultiple of the ...
... submultiple whatsoever of the first is contained in the second , as often as an equi - submultiple of the third is contained in the fourth . 6. Magnitudes which have the same ratio are called proportionals . 7. If a submultiple of the ...
Seite 93
... equi - submultiples of the same See N. or of equal magnitudes are equal . 2. Those magnitudes , of which the same or equal magnitudes are equi - multiples or equi - submultiples , are equal . 3. A multiple or submultiple of a greater ...
... equi - submultiples of the same See N. or of equal magnitudes are equal . 2. Those magnitudes , of which the same or equal magnitudes are equi - multiples or equi - submultiples , are equal . 3. A multiple or submultiple of a greater ...
Seite 95
... submultiple of the first be always contained in the third as often as an equi- submultiple of the second is contained in the fourth , the third is equal to the fourth . PROP . III . THEOR . If there be four magnitudes ( A , B , CD and ...
... submultiple of the first be always contained in the third as often as an equi- submultiple of the second is contained in the fourth , the third is equal to the fourth . PROP . III . THEOR . If there be four magnitudes ( A , B , CD and ...
Seite 96
... equi - multiples of A and B ( 1 ) , and A is ( 2 ) Hypoth . greater than B ( 2 ) , Cm is greater than EF ( 3 ) ; but EF ( 3 ) Ax . 3 . is equal to CD ( 2 ) , therefore Cm is greater than CD , which is absurd . 2. Let B be not a submultiple ...
... equi - multiples of A and B ( 1 ) , and A is ( 2 ) Hypoth . greater than B ( 2 ) , Cm is greater than EF ( 3 ) ; but EF ( 3 ) Ax . 3 . is equal to CD ( 2 ) , therefore Cm is greater than CD , which is absurd . 2. Let B be not a submultiple ...
Seite 97
... equi - submultiples Fig . 7 . of two other magnitudes ( AC and BD ) , and there be taken two others ( x and 2 ) which are equi - submultiples of the former , they are also equi - submultiples of the latter . Divide AC into parts Ao , oC ...
... equi - submultiples Fig . 7 . of two other magnitudes ( AC and BD ) , and there be taken two others ( x and 2 ) which are equi - submultiples of the former , they are also equi - submultiples of the latter . Divide AC into parts Ao , oC ...
Häufige Begriffe und Wortgruppen
absurd AC and BD AC and CB AC is equal angle ABC angle ACD angle BAC angle equal base bisected centre circumference Constr constructed contained in CD demonstrated described double the rectangle double the square drawn equal angles equal to AC equal to double equi equi-multiples equi-submultiples equiangular Euclid evident external angle fore form one right four magnitudes proportional four right angles fourth given angle given circle given line given right line given triangle gles half a right Hypoth inscribed less line CD manner multiple oftener opposite parallel parallelogram perpendicular PROB produced Prop proposition radius rectangle under AC right angles right line AB Schol segment side AC sides AB similar squares of AC submultiple taken tangent THEOR third tiple triangle ABC vertex
Beliebte Passagen
Seite 147 - A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.
Seite 3 - Things which are equal to the same thing are equal to one another. 2. If equals be added to equals, the wholes are equal. 3. If equals be taken from equals, the remainders are equal. 4. If equals be added to unequals, the wholes are unequal. 5. If equals be taken from unequals, the remainders are unequal. 6. Things which are double of the same are equal to one another.
Seite 22 - 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 16 - If two triangles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles are equal.
Seite 148 - A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction.
Seite 2 - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.
Seite 92 - F) a ratio compounded of the ratios which the first has to the second, the second to the third, the third to the fourth (A to D, D to C, and C to F) ; and so on to the last.
Seite 93 - AXIOMS. 1. Equimultiples of the same, or of equal magnitudes, are equal to one another. 2. Those magnitudes, of which the same or equal magnitudes are equimultiples, are equal to one another. 3. A multiple of a greater magnitude is greater than the same multiple of a less. 4. That magnitude, of which a multiple is greater than the same multiple of another, is greater than that other magnitude.
Seite 25 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Seite 2 - A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.