## 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 495

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**logarithm**of N , by article 8. This series , however , if x be a whole number , does not converge . 1 Let M be a ...**LOGARITHMS**. 495. Page 496

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**logarithm**of M. 5 11. As , by the last article , the hyperbolic**logarithm**of N x2 204 x5 .6 or 1 + x is x + 2 3 4 5 ...**logarithms**may be calculated , 496 OF**LOGARITHMS**. Page 497

Euclides Robert Simson. From the preceding articles hyperbolic

Euclides Robert Simson. From the preceding articles hyperbolic

**logarithms**may be calculated , as in the following examples . Example 1. Required the hyperbolic**logarithm**of 2. Put N + 1 = 2 , and then n = l , n 2n + 2 4 2n + 13 ' and 2n ... Page 498

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**logarithm**of the number 2 is 0.57536414488 + 0.1177830356 = 0.69514718054 . The hyperbolic**logarithm**of 2 being thus found , that of 4 , 8 , 16 , and all the other powers of 2 , may be obtained by multiplying the**logarithm**of 2 , by 2 ... Page 499

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**logarithm**may converge quickly . For as = ac + 1 ac -X ac + acx = ac + 1 - acx - x , and therefore 2 acx + x = 1 , and x = 1 2ac + · Example 2. Required the hyperbolic**logarithm**of 5 . 1 1 Here a = 4 , c = 6 , and x = ; = 2ac + 1 49 Sum ...### 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.