## The Elements of Euclid: Viz, the First Six Books, Together with the Eleventh and Twelfth ; the Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected, and Some of Euclid's Demonstrations are Restored ; Also the Book of Euclid's Data, in Like Manner Corrected |

### From inside the book

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**radius**of the circle of which the measure of a given angle is an arch , that arch will contain the same number of degrees , minutes , seconds , & c . as is manifest from Lemma 2 . III . Let AB be produced till it meet the circle again ... Page 481

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**radius**CB to the**radius**NB , and AE to MP as AB to BM , and BC or BA to BD as BN or BM to BO ; and , by conversion , DA to MO as AB to MB . Hence the corollary is manifest ; there- fore , if the**radius**be supposed to be divided into any ... Page 482

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**radius**, the sides become the sines of the angles op- posite to them and if either side be made**radius**, the re- maining side is the tangent of the angle opposite to it , and the hypothenuse the secant of the same angle . Let ABC be a ... Page 483

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**radius**, BE , BG , are the tangents of the angles EFB , BFG ; but it is manifest that EC is the sum of the sides BA ...**radius**is to the tangent of an angle ; and the**radius**is to the tangent of the excess of this angle above half a ... Page 484

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**radius**is to the tangent of the angle ABD ; and because BGD is a right angle , BG is to GD or GF as the**radius**is to the tangent of GBD , which is the excess of the angle ABD above ABF half a right angle . But because EB is parallel to ...### Other editions - View all

The Elements of Euclid: Viz, the First Six Books, Together with the Eleventh ... Robert Simson,Euclid Euclid No preview available - 2018 |

### Common terms and phrases

altitude angle ABC angle BAC base BC BC is equal BC is given bisected centre circle ABCD circumference co-sine cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid ex æquali excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gnomon greater join less Let ABC multiple parallel parallelogram parallelogram AC perpendicular point F polygon prism proportionals proposition Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar sine solid angle solid parallelopipeds square of AC straight line AB straight line BC tangent THEOR third triangle ABC triplicate ratio vertex wherefore

### Popular passages

Page 9 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 81 - The angles in the same segment of a circle are equal to one another.

Page 315 - Equiangular parallelograms have to one another the ratio which is compounded of the ratios of their sides.

Page 33 - 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 49 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Page 96 - If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square of the line which touches it.

Page 155 - 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.

Page 22 - ANY two angles of a triangle are together less than two right angles.

Page 25 - Let A, B, C be the three given straight lines, of which any two whatever are greater than the third, viz.

Page 24 - Any two sides of a triangle are together greater than the third side.