## 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].1834 |

### From inside the book

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**base BC**shall coincide with the base EF ; because , the point B coin- ciding 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 9

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**base BC**is com- mon to the two triangles BFC , CGB ; wherefore these triangles are equal , and their remaining angles , each to each , to which 4. 1 . the equal sides are opposite : therefore the angle FBC is equal to the angle GCB ... Page 10

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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 angie 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 triangle ACB ... Page 11

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**base BC**equal to the base EF : the angle BAC shall be equal to the angle EDF . A Ꭰ Ꮐ B CE For if the triangle ABC be applied to DEF , so that the point B may be on E , and the straight line BC upon EF , the point C shall also coin ... Page 12

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**BC**, CD , each to each ; and the angle ACD is equal + to the angle BCD ; therefore the**base**AD is equal to the**base*** DB , and the straight line AB is divided into two equal parts in the point D. Which was to be done . PROPOSITION XI ...### Other editions - View all

### Common terms and phrases

ABC is given AC is equal altitude angle ABC angle BAC base BC bisected centre circle ABCD circle EFGH circumference common logarithm cone Constr cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid ex æquali 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 multiple parallel parallelogram perpendicular point F polygon prism Prop proportionals Q. E. D. PROPOSITION radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment shewn sides BA similar sine solid angle solid parallelopiped square of AC straight line AB straight line BC tangent THEOR.-If tiple triangle ABC vertex wherefore

### Popular passages

Page 32 - To a given straight line, to apply a parallelogram which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle...

Page 138 - 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 39 - If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the aforesaid part.

Page 22 - If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another...

Page 41 - If a straight line be bisected and produced to any point, the rectangle contained by the whole line thus produced and the part of it produced, together •with the square on half the line bisected, is equal to the square on the straight line which is made up of the half and the part produced.

Page 5 - If two triangles have two sides of the one equal to two sides of the other, each to each, but the...

Page 38 - IF a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line. Let the straight line AB be divided...

Page 262 - 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 89 - PBOR. —To describe an isosceles triangle, having each of the angles at the base, double of the third angle. Take any straight...

Page 165 - Wherefore, in equal circles &c. QED PROPOSITION B. THEOREM If the vertical angle of a triangle be bisected by a straight line which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.