Elements of Geometry, Containing the First Six Books of EuclidBaldwin, Cradock, and Joy, 1826 - 180 Seiten |
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Seite 9
... Whence the base Dc is equal to 4.1 . to the base AB , and the triangle DBC equal to the triangle ACB , the less to the greater , which is absurd . Therefore the sides AB , AC , are not unequal . Whence they are equal . Wherefore if two ...
... Whence the base Dc is equal to 4.1 . to the base AB , and the triangle DBC equal to the triangle ACB , the less to the greater , which is absurd . Therefore the sides AB , AC , are not unequal . Whence they are equal . Wherefore if two ...
Seite 10
... Whence the angle ADC is greater than the angle Wherefore the angle BDC will be much greater than the angle BCD . Again , because CB is equal to DB , the angle BDC will be equal to the angle BCD . BCD . But it C D A B has been shown to ...
... Whence the angle ADC is greater than the angle Wherefore the angle BDC will be much greater than the angle BCD . Again , because CB is equal to DB , the angle BDC will be equal to the angle BCD . BCD . But it C D A B has been shown to ...
Seite 12
... Whence it was only necessary to demonstrate this , and assume the rest as consequents . PROPOSITION IX . PROBLEM . To bisect a given rectilineal angle , that is , to divide it into two equal parts . D E Let BAC be the given rectilineal ...
... Whence it was only necessary to demonstrate this , and assume the rest as consequents . PROPOSITION IX . PROBLEM . To bisect a given rectilineal angle , that is , to divide it into two equal parts . D E Let BAC be the given rectilineal ...
Seite 18
... whence ABC , BCA , are less than two right angles . In like manner , we may demonstrate also that the angles BAC , ACB , and also CAB , ABC , are less than two right angles . Therefore two angles of every triangle , & c . Q. E. D. ...
... whence ABC , BCA , are less than two right angles . In like manner , we may demonstrate also that the angles BAC , ACB , and also CAB , ABC , are less than two right angles . Therefore two angles of every triangle , & c . Q. E. D. ...
Seite 19
... Whence AC is greater than AB . There- fore the greater angle , & c . Q. E. D. Deduction . If from a given point there be drawn any number of right lines to another right line given in position , the perpendicular is the shortest right ...
... Whence AC is greater than AB . There- fore the greater angle , & c . Q. E. D. Deduction . If from a given point there be drawn any number of right lines to another right line given in position , the perpendicular is the shortest right ...
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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.