## 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]. |

### From inside the book

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Page 24

Book I. For , if it be not greater , it must either be equal to it , or less ; but the

equala A D to EF : but it is not ; therefore the

...

Book I. For , if it be not greater , it must either be equal to it , or less ; but the

**angle****BAC**is not equal to the angle EDF , because then the base * 4. 1. BC would beequala A D to EF : but it is not ; therefore the

**angle BAC**is not equal to the angle...

Page 85

Let ABCD be a circle , of which the diameter is BC , and centre E ; and draw CA ,

dividing the circle into the segments ABC , ADC , and join BA , AD , DC ; the

Let ABCD be a circle , of which the diameter is BC , and centre E ; and draw CA ,

dividing the circle into the segments ABC , ADC , and join BA , AD , DC ; the

**angle**in the semicircle**BAC**is a right**angle**; and the**angle**in the segment ABC ... Page 205

Ir a solid angle be contained by three plane angles , See N. any two of them are

greater than the third . Let the solid angle at A he contained by the three plane

angles ...

Ir a solid angle be contained by three plane angles , See N. any two of them are

greater than the third . Let the solid angle at A he contained by the three plane

**angles**,**BAC**, CAD , DAB . Any two of them are greater than the third . If theangles ...

Page 425

milar to OPQ : Because , as has been shown , the ratio of AD to BC is the same

with the ratio of ( HF to FL , that is , by the construction , with the ratio of ) OR to

PQ ; and the

...

milar to OPQ : Because , as has been shown , the ratio of AD to BC is the same

with the ratio of ( HF to FL , that is , by the construction , with the ratio of ) OR to

PQ ; and the

**angle BAC**is equal to the angle POQ . Therefore the triangle ABC is...

Page 426

Let the triangle ABC have the given

drawn to the base BC , making the given angle ADB , divide CB into the

segments BD , DC , which have a given ratio to one another ; the triangle ABC is

given in ...

Let the triangle ABC have the given

**angle BAC**, and let the straight line AD ,drawn to the base BC , making the given angle ADB , divide CB into the

segments BD , DC , which have a given ratio to one another ; the triangle ABC is

given in ...

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### Common terms and phrases

added altitude angle ABC angle BAC arch base Book centre circle circle ABCD circumference common cone contained cylinder definition demonstrated described diameter difference divided double draw drawn equal equal angles equiangular equimultiples excess fore four fourth given angle given in position given in species given magnitude given ratio given straight line greater Greek half join less likewise logarithm magnitude manner meet multiple opposite parallel parallelogram pass perpendicular plane prism produced PROP proportionals proposition pyramid radius reason rectangle rectangle contained remaining right angles segment shown sides similar sine solid solid angle sphere square square of AC Take taken THEOR third triangle ABC wherefore whole

### Popular passages

Page 45 - 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 20 - Any two sides of a triangle are together greater than the third side.

Page 30 - 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 20 - 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 312 - 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 8 - 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 155 - 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 54 - 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 3 - A circle is a plane figure contained by one line, which is called the circumference, and is such, that all straight lines drawn from a certain point within the figure to the circumference are equal to one another.

Page 167 - SIMILAR triangles are to one another in the duplicate ratio of their homologous sides.