## 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 30

THEOR , THE straight lines which

straight lines towards the same parts , are also themselves equal and parallel .

Let AB , CD be equal and parallel straight lines , and

same ...

THEOR , THE straight lines which

**join**the extremities of two equal and parallelstraight lines towards the same parts , are also themselves equal and parallel .

Let AB , CD be equal and parallel straight lines , and

**join**- A B ed towards thesame ...

Page 69

If two circles touch each other internally , the straight line which

being produced shall pass through the ... being produced , H / passes through

the point A. For , if not , let it fall otherwise , if D possible , as FGDH , and

If two circles touch each other internally , the straight line which

**joins**their centresbeing produced shall pass through the ... being produced , H / passes through

the point A. For , if not , let it fall otherwise , if D possible , as FGDH , and

**join**AF ... Page 446

E. D it to E , and

and the * 43 Dat , angle AEC is given ; therefore c the triangle ACE is given in

species , and the ratio of EA to AC is therefore given , A and EA is given in ...

E. D it to E , and

**join**EC : The angle ACE В. 631. 3. being a right angle , is given ;and the * 43 Dat , angle AEC is given ; therefore c the triangle ACE is given in

species , and the ratio of EA to AC is therefore given , A and EA is given in ...

Page 468

... given ratio to the third BD : Let AE be that given straight line , therefore the

remainder EB has a given ratio to BD : Let BD be placed at right angles to EB ,

and

angle ...

... given ratio to the third BD : Let AE be that given straight line , therefore the

remainder EB has a given ratio to BD : Let BD be placed at right angles to EB ,

and

**join**DE ; then the triangle EBD • 14 Dat . isb given in species ; wherefore theangle ...

Page 505

AC , AB are quadrants , the angles ADB , ADC are right angles , and AD will be

perpendicular to the plane of BC :. Therefore the angle ADE is a right angle , and

...

**Join**AC , and draw AE , a straight line to any other point E , in BC ;**join**DE : SinceAC , AB are quadrants , the angles ADB , ADC are right angles , and AD will be

perpendicular to the plane of BC :. Therefore the angle ADE is a right angle , and

...

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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.