## Elements of Geometry, Geometrical Analysis, and Plane Trigonometry: With an Appendix, Notes and IllustrationsJames Ballantyne and Company, 1809 - 493 pages |

### From inside the book

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Page 13

... angle equal to the given

... angle equal to the given

**angle BAC**. In the sides AB and AC of the given angle , assume the points G and H , join GH , from DE cut off DI equal to AG , and on DI constitute ( I. 1. ) a triangle DKI , having the sides DK and IK equal G ... Page 17

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**angle BAC**is equal to DCA ; the two tri- angles ( I. 3. ) are , therefore , equal . But this conclusion is manifestly absurd . To suppose then the in- equality of AB and BC , involves a contradiction ; and con- sequently those sides ... Page 18

... angle BCF ( Def . 10. ) , and hence BCF must be greater than either the angle CBA or CAB , PROP . XI . THEOR . Any two angles of a triangle are together less than two right angles . The two

... angle BCF ( Def . 10. ) , and hence BCF must be greater than either the angle CBA or CAB , PROP . XI . THEOR . Any two angles of a triangle are together less than two right angles . The two

**angles BAC**and BCA of the triangle ABC are ... Page 19

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**angle**. Consequently the**angles**CAB and CBA are both of them acute . Lastly , let the triangle have the**angle**at C acute . If one of the remaining**angles**, such as**BAC**, be likewise acute , the two**angles**ACB and**BAC**are both of them ... Page 26

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**angle**CEG is obtuse . Where fore CEG is much greater than CGE , and the opposite side CG greater ( I. 15. ) than CE ...**BAC**equal to EDF , the**angle**BCA to EFD , and a side of the one equal to a side of the other , whether it be ...### Other editions - View all

### Common terms and phrases

ABCD ANALYSIS angle ABC angle ACB angle BAC bisect centre chord circumference COMPOSITION conse consequently the angle decagon describe a circle diameter distance diverging lines drawn equal to BC exterior angle fall the perpendicular given angle given circle given in position given point given ratio given space given straight line greater hence hypotenuse inflected inscribed intercepted intersection isosceles triangle join let fall mean proportional parallel perpendicular point F polygon porism PROB PROP quently radius rectangle rectangle contained regular polygon rhomboid right angles right-angled triangle Scholium segments semicircle semiperimeter sequently side AC similar sine square of AB square of AC tangent THEOR triangle ABC twice the square vertex vertical angle whence wherefore

### Popular passages

Page 28 - ... if a straight line, &c. QED PROPOSITION 29. — Theorem. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another ; and the exterior angle equal to the interior and opposite upon the same side ; and likewise the two interior angles upon the same side together equal to two right angles.

Page 147 - The first and last terms of a proportion are called the extremes, and the two middle terms are called the means.

Page 92 - THE angle at the centre of a circle is double of the angle at the circumference, upon the same base, that is, upon the same

Page 458 - 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...

Page 99 - ... a circle. 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. The opposite angles of any quadrilateral inscribed in a circle are supplementary; and the converse.

Page 155 - Componendo, by composition ; when there are four proportionals, and it is inferred that the first together with the second, is to the second, as the third together with the fourth, is to the fourth.

Page 408 - C' (89) (90) (91) (92) (93) 112. In any plane triangle, the sum of any two sides is to their difference as the tangent of half the sum of the opposite angles is to the tangent of half their difference.

Page 16 - PROP. V. THEOR. The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced, the angles -upon the other side of the base shall be equal. Let ABC be an isosceles triangle, of which the side AB is equal to AC, and let the straight lines AB, AC, be produced to D and E: the angle ABC shall be equal to the angle ACB, and the angle...

Page 36 - 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.

Page 60 - Prove, geometrically, that the rectangle of the sum and the difference of two straight lines is equivalent to the difference of the squares of those lines.