## 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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**join**FC , GB . Because AF is equal to AG , and AB to AC , the two sides FA , AC are equal to the two GA , AB , each to each ; and they contain the augle FAG com- mon to the two triangles AFC , AGB ; therefore the base FC is 4. 1. equal ... Page 11

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**join**DC ; therefore , because in the triangles DBC , ACB , DB is equal to AC , and BC common to both the two sides , DB , BC are equal to the two AC , CB each to each ; and the angle DBC is equal to the angle ACB ; therefore the base DC ... Page 13

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**join**DE , and upon it describeb an equilateral triangle DEF ; then**join**AF ; the straight line AF bisects the an- gle BAC . D Because AD is equal to AE , and AF is common to the two triangles DAF , EAF ; the two sides DA , AF , are ... Page 18

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**join**BE and produce it to F , and make EF equal to BE ;**join**also FC , and produce AC to G. 4 Because AE is equal to EC , and BE to EF ; AE , EB are equal to CE , EF , each to each ; and the angle B 15. 1. AEB , is equal to the angle ... Page 20

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**join**DC . Because DA is equal to AC , the angle ADC is likewise 5. 1. equal to ACD ; but the angle BCD is greater than the angle ACD ; therefore the angle BCD is greater than the angle ADC ; and because the angle BCD of the triangle DCB ...### 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.