## 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 1-5 of 87

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**base BC**shall coincide with the base EF , because the point B coinciding with E , and C with F , if the**base BC**does not coincide with the base EF , two straight lines would inclose a space , which is impossible . Therefore the**base BC**... Page 10

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**base BC**is common to the two triangles BFC , CGB ; wherefore the triangles are equal , and their remaining angles , each to each , to which the equal sides are opposite ; therefore the angle FBC is equal to the angle GCB , and the angle ... Page 11

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**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 is equal to the**base**AB , and the triangle DBC is equal to the tri- angle ACB ... Page 12

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**base**CD are 5. 1. equal a to one another : but the an- gle ECD is greater ...**base**, and on the same side of it , there cannot be two triangles that have their sides which ...**BC**is equal to EF ; therefore**BC**coinciding with 12 THE ELEMENTS. Page 13

Euclides Robert Simson. BC is equal to EF ; therefore BC coinciding with EF ; Book I. BA and AC shall coincide with ED and DF ; for , if the

Euclides Robert Simson. BC is equal to EF ; therefore BC coinciding with EF ; Book I. BA and AC shall coincide with ED and DF ; for , if the

**base BC**coincides with the base EF ; but the sides BA , CA do not coincide with the sides ED ...### 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.