## The Elements of Euclid: The Errors, by which Theon, Or Others, Have Long Ago Vitiated These Books, are Corrected and Some of Euclid's Demonstrations are Restored. Also, to this Second Edition is Added the Book of Euclid's Data. In Like Manner Corrected. viz. the first six books, together with the eleventh and twelfth |

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Page 104

3. fore the angle BAE is equal é to the angle AED . for the same

the angles ABC , BCD , CDE is equal to the angle BAE or AED . therefore the pen

. tagon ABCDE is equiangular ; and it has been shewn that it is equilateral .

3. fore the angle BAE is equal é to the angle AED . for the same

**reason**, each ofthe angles ABC , BCD , CDE is equal to the angle BAE or AED . therefore the pen

. tagon ABCDE is equiangular ; and it has been shewn that it is equilateral .

Page 169

5 . bout the equal angles EBC , LGH are proportionals ; therefore the triangle

EBC is equiangular to the triangle LGH , and fimilar to it c . for the same А. M

the ...

5 . bout the equal angles EBC , LGH are proportionals ; therefore the triangle

EBC is equiangular to the triangle LGH , and fimilar to it c . for the same А. M

**reason**the triF angle ECDE B likewise is fiL G milar to the triangle LHK . thereforethe ...

Page 182

B. 5. lyd , as DB to BC , so is the figure upon B A to that upon B D с BC . for the

same

as BD and DC together to t . 24. 5. BC , so are the figures upon BA , AC to that ...

B. 5. lyd , as DB to BC , so is the figure upon B A to that upon B D с BC . for the

same

**reason**, as DC to CB , so is the figure upon CA to that upon CB . Whereforeas BD and DC together to t . 24. 5. BC , so are the figures upon BA , AC to that ...

Page 219

Defil therefore each of the angles FGD , FGE is a right angle . for the same

each to each , and contain equal angles , therefore the base BK is equal to the

base ...

Defil therefore each of the angles FGD , FGE is a right angle . for the same

**reason**, HKA , HKB are right angles . and because KA , AB are equal to GD , DE ,each to each , and contain equal angles , therefore the base BK is equal to the

base ...

Page 250

Book XII.equal to the base KD , and the triangle AEH equal d and similar to the

triangle HKD . for the same

to the triangle HLD . and because the two straight lines EH , HG which meet one

...

Book XII.equal to the base KD , and the triangle AEH equal d and similar to the

triangle HKD . for the same

**reason**, the triangle AGH is equal and d . 4. 1. fimilarto the triangle HLD . and because the two straight lines EH , HG which meet one

...

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### Common terms and phrases

added alſo altitude angle ABC angle BAC baſe BC is given becauſe Book Book XI caſe circle circle ABCD circumference common cone cylinder Definition demonſtrated deſcribed diameter divided double draw drawn equal equal angles equiangular equimultiples exceſs fame fides firſt folid fore four fourth given angle given in poſition given in ſpecies given magnitude given ratio given ſtraight line greater Greek half join leſs likewiſe magnitude manner meet muſt oppoſite parallel parallelogram perpendicular plane priſm produced PROP proportionals Propoſition pyramid reaſon rectangle remaining right angles ſame ſecond ſegment ſhall ſhewn ſides ſimilar ſolid ſquare ſquare of AC taken THEOR theſe third thro triangle ABC wherefore whole

### Popular passages

Page 5 - Let it be granted that a straight line may be drawn from any one point to any other point.

Page 163 - 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 48 - If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. Let the straight line AB be divided into any two parts in the point C; the squares of AB, BC are equal to twice the rectangle AB, BC, together with the square of AC.

Page 73 - The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn from the extremity between that straight line and the circumference, so as not to cut the circle...

Page 105 - If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, viz. either the sides adjacent to the equal...

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.

Page 54 - AB be the given straight line ; it is required to divide it into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.

Page 47 - If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line.

Page 37 - To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.