## 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]. |

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Page 31

The opposite sides and angles of parallelograms are equal to one another , and

the

parallelogram is a four - sided figure , of which the opposite sides are parallel ;

and ...

The opposite sides and angles of parallelograms are equal to one another , and

the

**diameter**bisects them , that is , divides them into two equal parts . N. B. Aparallelogram is a four - sided figure , of which the opposite sides are parallel ;

and ...

Page 65

Ir any point be taken in the

straight lines which can be drawn from it to the circunference , the greatest is that

in which the centre is , and the other part of that

...

Ir any point be taken in the

**diameter**of a circle which is not the centre , of all thestraight lines which can be drawn from it to the circunference , the greatest is that

in which the centre is , and the other part of that

**diameter**is the least ; and , of any...

Page 95

In a given circle to place a straight line equal to a given straight line not greater

than the

straight line , not greater than the

the ...

In a given circle to place a straight line equal to a given straight line not greater

than the

**diameter**of the circle . Let ABC be the given circle , and D the givenstraight line , not greater than the

**diameter**of the circle . Draw BC the**diameter**ofthe ...

Page 278

Let the spheres be cut by a plane passing through the centre ; the common

sections of it with the spheres shall be circles ; because the sphere is described

by the revolution of a semicircle about the

that in ...

Let the spheres be cut by a plane passing through the centre ; the common

sections of it with the spheres shall be circles ; because the sphere is described

by the revolution of a semicircle about the

**diameter**remaining unmoveable ; sothat in ...

Page 451

IF from the given point in the

and from that point a straight line be drawn at right angles to the first , and from

the point ...

IF from the given point in the

**diameter**of a circle given in position , or in the**diameter**produced , a straight line be drawn to any point in the circumference ,and from that point a straight line be drawn at right angles to the first , and from

the point ...

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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.