## 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

Results 6-10 of 58

Page 133

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**proportionals**, See N. they shall also be**proportionals**when taken jointly , that is , if the first be to the second , as the third to the fourth , the first and second together shall be to the second , as the third and fourth together ... Page 141

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**proportion- als**, they shall likewise be**proportionals**F H CDF a 22. 5 . when taken jointly b ; as therefore AG is to GB , so is DH 18. 5 . to HE : but as GB to C , so is HE to F. Therefore ex æqualia , as AG is to C , so is DH to F ... Page 142

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**proportion- als**, viz . AB to CD , as E to F ; and let AB be the greatest A. & 14.5 . of them , and consequently F the least . AB , together with F , are greater than CD , together with E. Take AG equal to E , and CH equal to F : Then ... Page 147

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**proportionals**. II . Book VI . " Reciprocal figures , viz . triangles and parallelograms , are See N. " such as have their sides about two of their angles pro- " portionals in such manner , that a side of the first figure " is to a side ... Page 153

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**proportionals**; and those which are op- posite to the equal angles are homologous sides , that is , are the antecedents or consequents of the ratios . Let ABC , DCE be equiangular triangles , having the angle ABC equal to the angle DCE ...### 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.