## 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 169

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**ratio**of two**given**homogeneous quantities . Let A and B be two quantities of the same kind , whose numerical**ratio**it is required to discover . Find , by the last Proposition , the greatest common mea- sure E of the two quantities ; and ... Page 180

With an Appendix, Notes and Illustrations Sir John Leslie. lines AB and BC

With an Appendix, Notes and Illustrations Sir John Leslie. lines AB and BC

**proportion**...**ratio**as AB has to AD . From what has been already demonstrated , it appears ...**given**straight lines . Let A , B , and C be three straight 1 180 ELEMENTS ... Page 184

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**given ratio**. Let AB be a straight line , which it is required to cut harmonically , in the ratio of K to L. Through A draw the diverging line AC , and produce it both ways till AC and AD be each equal to K , make AE equal to L , join ... Page 191

... ratio of the adjacent sides , it will bisect the vertical angle ; for it is evident from VI . 6. cor . , that a straight line is only capable of a single section , whether internal or external , in a

... ratio of the adjacent sides , it will bisect the vertical angle ; for it is evident from VI . 6. cor . , that a straight line is only capable of a single section , whether internal or external , in a

**given proportion**. Scholium . The ... Page 198

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**proportion**, the hypotenuse will be divided into extreme and mean**ratio**, by the perpendi- cular let fall upon it ...**given**straight lines . Let it be required to find the mean A proportional between the straight lines A and B. Find ...### 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.