## The Elements of Euclid; viz. the first six books, together with the eleventh and twelfth. Also the book of Euclid's Data. By R. Simson. To which is added, A treatise on the construction of the trigonometrical canon [by J. Christison] and A concise account of logarithms [by A. Robertson].1814 |

### From inside the book

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Page 474

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**radius**of the circle of which the measure of a given angle is an arch , that arch will con- tain 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 475

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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 ; therefore , if the**radius**be supposed to be divided into any ... Page 476

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**radius**, the sides become the sines of the angles opposite to them ; and if either side be made**radius**, the remaining side is the tangent of the angle opposite to it , and the hypothenuse the secant of the same angle . Let ABC be a ... Page 477

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**radius**, CE is the sine of the angle CBA , and BD the sine of the angle ACB ; but the two triangles CAE , DAB have each a right angle at D and E ; and likewise the common angle CAB ; therefore they are similar , and consequently , CA is ... Page 478

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**radius**is to the tangent of an angle ; and the**radius**is to the tangent of the excess of this angle above half a right angle , as the tangent of half the sum of the angles B and C at the base , is to the tangent of half their difference ...### Other editions - View all

### Common terms and phrases

ABC is given AC is equal altitude angle ABC angle BAC base BC bisected BOOK XI centre circle ABCD circumference common logarithm cone cylinder demonstrated described diameter drawn equal angles equiangular equimultiples Euclid 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 logarithm meet multiple opposite parallel parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius rectangle CB rectangle contained rectilineal figure remaining angle right angles segment side BC 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 3-7 - IF a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts.

Page 16 - Any two sides of a triangle are together greater than the third side.

Page 26 - Therefore all the angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.

Page 16 - If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than, the other two sides of the triangle, but shall contain a greater angle. Let...

Page 304 - Again ; the mathematical postulate, that " things which are equal to the same are equal to one another," is similar to the form of the syllogism in logic, which unites things agreeing in the middle term.

Page 4 - DL is equal to DG, and DA, DB, parts of them, are equal ; therefore the remainder AL is equal to the remainder (3. Ax.) BG : But it has been shewn that BC is equal to BG ; wherefore AL and BC are each of them equal to BG ; and things that are equal to the same are equal to one another ; therefore the straight line AL is equal to BC.

Page 147 - 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 3-16 - To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, may be equal to the square of the other part.

Page 159 - SIMILAR triangles are to one another in the duplicate ratio of their homologous sides.