## 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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**double**of the same , are equal to one an- other . VII . Things which are halves of the same , are equal to one an- other . VIII . Magnitudes which coincide with one another , that is , which exactly fill the same space , are equal to ... Page 5

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**double**of the same , are equal to one an- other . VII . Things which are halves of the same , are equal to one an- other . VIII . Magnitudes which coincide with one another , that is , which exactly fill the same space , are equal to ... Page 33

... THEOR . If a parallelogram and triangle be upon the same base , and between the same parallels ; the paral- lelogram shall be

... THEOR . If a parallelogram and triangle be upon the same base , and between the same parallels ; the paral- lelogram shall be

**double**of the triangle . BOOK I. Let the parallelogram ABCD and the triangle EBC D 2 OF EUCLID . 35. Page 33

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**double**of the triangle EBC . Join AC ; then the triangle ABC a 37. 1. is equal to the triangle EBC , be- cause they are upon the same base BC , and between the same parallels BC , AE . But the parallelogram b 34. 1. ABCD is doubleb of ... Page 3-6

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**double**of the triangle EBC . Join AC ; then the triangle ABC 37. 1. is equal to the triangle EBC , be- cause they are upon the same base BC , and between the same parallels BC , AE . But the parallelogram 34. 1. ABCD is doubleb of 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.