## 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 33

... . 1 . to DBCF , because they are upon the same base BC , and between the same parallels BC , EF ; and the triangle ABC is the

... . 1 . to DBCF , because they are upon the same base BC , and between the same parallels BC , EF ; and the triangle ABC is the

**half**of the parallelogram , EBCA , because the D € 34. 1 . с Book I diameter AB bisects OF EUCLID . 33. Page 33

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**half**of the parallelogram GBCA , because the diameter AB bisects it ; and the triangle DEF is the**half**of the parallelogram DEFH , because the dia- meter DF bisects it : But the halves of equal things are 7 Ax . equald ; therefore the ... Page 3-17

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**half**the line . . Let the straight line AB be divided into two equal parts in the point C , and into two unequal parts at the point D ; the rectangle AD , DB , together with the square of CD , is equal to the square of CB . b 46. 1 ... Page 3-18

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**half**the line bisected , is equal to the square of the straight line which is made up of the**half**and the part produced . Let the straight line AB be bisected in C , and produced to the point D ; the rectangle AD , DB , together with ... Page 3-21

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**half**the line , and of the square of the line be- tween the points of section . Let the straight line AB be divided ...**half**of a right angle . For the same reason each of the angles CEB , EBC is**half**a right angle ; and therefore the ...### 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.