## 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 44

Let A and BC be two straight lines ; and let BC be

points D , E ; the rectangle contained by the straight lines A , BC is equal to the

rectB D E C angle contained by A , BD ; and to that contained by A , DE ; and also

to ...

Let A and BC be two straight lines ; and let BC be

**divided**into any parts in thepoints D , E ; the rectangle contained by the straight lines A , BC is equal to the

rectB D E C angle contained by A , BD ; and to that contained by A , DE ; and also

to ...

Page 45

IF F a straight line be

whole and one of the parts , is equal to the rectangle contained by the two parts ,

together with the square of the foresaid part . . a . 46. s . b . 31. I. Let the straight ...

IF F a straight line be

**divided**into any two parts , the rectangle contained by thewhole and one of the parts , is equal to the rectangle contained by the two parts ,

together with the square of the foresaid part . . a . 46. s . b . 31. I. Let the straight ...

Page 159

TO divide a given straight line fimilarly to a given

into parts that shall have the same ratios to one another which the parts of the

and ...

TO divide a given straight line fimilarly to a given

**divided**straight line , that is ,into parts that shall have the same ratios to one another which the parts of the

**divided**given straight line have . Let AB be the straight line given to be**divided**,and ...

Page 252

See N. IF F there be two pyramids of the same altitude , upon tri : angular bases ,

and each of them be

pyramid , and also into two equal prisms ; and if each of these pyramids be

See N. IF F there be two pyramids of the same altitude , upon tri : angular bases ,

and each of them be

**divided**into two equal pyramids similar to the wholepyramid , and also into two equal prisms ; and if each of these pyramids be

**divided**in ... Page 397

... of AB to D C b.9 . Dat . BC is given ; and therefore b the ratio of AD to BC is

given . 2. 43. Dat . PRO P. LI . 47 RECTILINE.al figures given in species , are

ABCDE be ...

... of AB to D C b.9 . Dat . BC is given ; and therefore b the ratio of AD to BC is

given . 2. 43. Dat . PRO P. LI . 47 RECTILINE.al figures given in species , are

**divided**into triangles which are given in species . 1 Let the rectilineal figureABCDE be ...

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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.