Elements of Geometry, Containing the First Six Books of EuclidBaldwin, Cradock, and Joy, 1826 - 180 Seiten |
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Seite 120
Euclid. PROPOSITION I. THEOREM . If there be any number of magnitudes equimultiples of as many other magnitudes , each of each ; whatsoever mul tiple one magnitude is of one , the same multiple shall all be of all . Let AB , CD , be any ...
Euclid. PROPOSITION I. THEOREM . If there be any number of magnitudes equimultiples of as many other magnitudes , each of each ; whatsoever mul tiple one magnitude is of one , the same multiple shall all be of all . Let AB , CD , be any ...
Seite 121
... magnitudes as are in the whole AG equal to C , so many will there be in B the whole DH equal to F. Wherefore a ... magnitudes m , n ; also b m , bn , equimultiples of the same magni- tudes m , n ; then shall a m + bm be the same multiple ...
... magnitudes as are in the whole AG equal to C , so many will there be in B the whole DH equal to F. Wherefore a ... magnitudes m , n ; also b m , bn , equimultiples of the same magni- tudes m , n ; then shall a m + bm be the same multiple ...
Seite 122
... magnitudes as are in EF equal to A , so many will there be in GH equal C. Divide EF into magnitudes E A B H -A D EK , KF , equal to A ; also divide F CH into magnitudes equal to c ; viz . GL , LH therefore the mul- K titude of EK , KF ...
... magnitudes as are in EF equal to A , so many will there be in GH equal C. Divide EF into magnitudes E A B H -A D EK , KF , equal to A ; also divide F CH into magnitudes equal to c ; viz . GL , LH therefore the mul- K titude of EK , KF ...
Seite 123
... magnitudes have the same ratio to the second , which the third has to the fourth , then shall any equimultiples of the first and second have the same ratio which the third has to the fourth . a * Ax . 1. 5 . a 1.5 . PROPOSITION V ...
... magnitudes have the same ratio to the second , which the third has to the fourth , then shall any equimultiples of the first and second have the same ratio which the third has to the fourth . a * Ax . 1. 5 . a 1.5 . PROPOSITION V ...
Seite 124
... magnitude , as a part taken away from the first is to a part taken away from the other , the remainder shall be the same multiple of the remainder as the whole is of the whole ... magnitudes be equimultiples 124 [ Book V. EUCLID'S ELEMENTS .
... magnitude , as a part taken away from the first is to a part taken away from the other , the remainder shall be the same multiple of the remainder as the whole is of the whole ... magnitudes be equimultiples 124 [ Book V. EUCLID'S ELEMENTS .
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ABC is equal adjacent angles Algebra angle ABC angle ACB angle BAC angles equal base BC bisected centre circle ABC circum circumference BC diameter double draw equal angles equal circles equal right lines equal to F equi equiangular equimultiples Euclid EUCLID'S ELEMENTS exceed exterior angle fore four magnitudes fourth Geometry given circle given point given right line gnomon greater ratio hence inscribed join less Let ABC multiple parallel parallelogram perpendicular polygon proportional Q. E. D. Deduction Q. E. D. PROPOSITION rectangle contained remaining angle right angles right line AB right line AC sector HEF segment side BC similar and similarly square of AC subtending THEOREM tiple touches the circle triangle ABC triangle DEF whence whole
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Seite xxvi - A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. XVIII. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter. XIX. "A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.
Seite 74 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle...
Seite 33 - The complements of the parallelograms, which are about the diameter of any parallelogram, are equal to one another.
Seite 148 - 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 27 - And because the angle ABC is equal to the angle BCD, and the angle CBD to the angle ACB, therefore the whole angle ABD is equal to the whole angle ACD • (ax.
Seite 8 - To bisect a given finite straight line, that is, to divide it into two equal parts. Let AB be the given straight line : it is required to divide it intotwo equal parts.
Seite 73 - DH; (I. def. 15.) therefore DH is greater than DG, the less than the greater, which is impossible : therefore no straight line can be drawn from the point A, between AE and the circumference, which does not cut the circle : or, which amounts to the same thing, however great an acute angle a straight line makes with the diameter at the point A, or however small an angle it makes with AE, the circumference must pass between that straight line and the perpendicular AE.
Seite 99 - To describe a square about a given circle. Let ABCD be the given circle ; it is required to describe a square about it. . Draw two diameters AC, BD of the circle ABCD, at right angles to one another, and through the points A, B, • 17.3. C, D, draw...
Seite 7 - ... equal to them, of the other. Let ABC, DEF be two triangles which have the two sides AB, AC equal to the two sides DE, DF, each to each, viz. AB equal to DE, and AC to DF ; but the base CB greater than the base EF ; the angle BAC is likewise greater than the angle EDF.
Seite 88 - From a given circle to cut off a segment, which shall contain an angle equal to a given rectilineal angle.